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Statistics - A Full Lecture to learn Data Science (2025 Version)

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hi everyone Welcome to our full and free

tutorial about statistics before we dive

in we want to say a huge thank you last

year we launched our first ever full

course on statistics and the response

was absolutely amazing we are on track

to reach 1 million views within just one

year wow this inspired us to create a

revised version for

2025 it was a big project but there it

is thank you for being such an amazing

community so let's start this video is

designed to guide you through the

fundamental concepts and most powerful

statistical tests used in research today

from the basics of descriptive

statistics to the complexities of

regression and Beyond we'll explore how

each method fits into the bigger picture

of data analysis and don't worry if you

have no no clue about statistics will go

through everything step by step if you

like you can find all topics in our book

as well the link is in the video

description so what is the outline of

the video our video has three major

parts in the first part we discuss what

statistics is and what the differences

between descriptive and inferential

statistics are in the second part we go

through the most common hypothesis test

like T Test and an NOA and discuss the

difference between parametric and

non-parametric tests in the third part

we take a look at correlation analysis

and regression analysis and finally we

talk about clust analysis we have

prepared detailed videos for each

section so let's start with the video

that explains what statistics is after

this video you will know what statistics

is what descriptive statistics is and

what inferential statistics is so let's

start with the first question what is

statistics statistics deals with the

collection analysis and presentation of

data an example we would like to

investigate whether gender has an

influence on the preferred newspaper

then gender and newspaper are our

so-called variables that we want to

analyze in order to analyze whether

gender has an influence on the preferred

newspaper we first need to collect data

to do this we create a questionnaire

that asks about gender and preferred

newspaper we will then send out the

survey and wait 2 weeks afterwards we

can display the received answers in a

table in this table we have one column

for each variable one for gender and one

for newspaper on the other hand each row

is the response of one served person the

first respondent is mail and stated New

York Post the second is female and

stated USA Today and so on and so forth

of course the data does not have to be

from a survey the data can also come

from an experiment in which you for

example want to study the effect of Two

drugs on blood pressure now the first

step is done we have collected data and

we can start analyzing the data but what

do we actually want to analyze we did

not survey the entire population but we

took a sample now the big question is do

we just want to describe the sample data

or do we want to make a statement about

the whole population if our aim is

limited to the sample itself I.E we only

want to describe the collected data we

will use descriptive statistics

descriptive statistics will provide a

detailed summary of the sample

however if we want to draw conclusions

about the population as a whole

inferential statistics are used this

approach allows us to make educated

guesses about the population based on

the sample data let us take a closer

look at both methods starting with

descriptive statistics why is

descriptive statistics so important

let's say a company wants to know how

its employees travel to work so the

company creates a survey to answer this

question once enough data has been

collected this data can be analyzed

using descriptive statistics but what is

descriptive statistics descriptive

statistics aims to describe and

summarize a data set in a meaningful way

but it is important to note that

descriptive statistics only describe the

collected data without drawing

conclusions about a larger population

put simply just because we know how some

people from one company get to work we

cannot say how all working people of the

company get to work this is the task of

inferential statistics which we will

discuss later to describe data

descriptively we now look at the four

key components measures of central

tendency measures of dispersion

frequency tables and charts let's start

with the first one measures of central

tendency measures of central tendency

are for example the mean the median and

the mode Let's first have a look at the

mean the arithmetic mean is the sum of

all observations divided by the number

of observations an example imagine we

have the test scores of five students to

find the mean score we sum up all the

scores and divide by the number of

scores the mean test score of these five

students is therefore

86.6 what about the median when the

values in a data set are are arranged in

ascending order the median is the middle

value if there is an odd number of data

points the median is simply the middle

value if there is an even number of data

points the median is the average of the

two middle values it is important to

note that the median is resistant to

extreme values or outliers let's look at

this example no matter how tall the last

person is the person in the middle

Remains the person in the middle so the

median does not change but if we look at

the mean it does have an effect on how

tall the last person is the mean is

therefore not robust to outliers let's

continue with the mode the mode refers

to the value or values that appear most

frequently in a set of data for example

if 14 people travel to work by car six

by bike five walk and five take public

transport then car occurs most often and

is therefore the mode great let's

continue with the measures of dispersion

measures of dispersion describe how

spread out the values in a data set are

measures of dispersion are for example

the variance and standard deviation the

range and the interquartile range let's

start with the standard deviation the

standard deviation indicates the average

distance between each data point and the

mean but what does that mean each person

has some deviation from the mean now we

want to know how much the person's

deviate from the mean value on average

in this example the average deviation

from the mean value is 11.5 cm to

calculate the standard deviation we can

use this equation Sigma is the standard

deviation n is the number of persons it

x i is the size of each person and xbar

is the mean value of all persons but

attention there are two slightly

different equations for the standard

deviation the difference is that we have

once 1 / n and once's 1 /

nus1 to keep it simple if our servy

doesn't cover the whole population we

always use this equation to estimate the

standard deviation likewise if we have

conducted a clinical study then we also

use this equation to estimate the

standard deviation but what is the

difference between the standard

deviation and the variance as we now

know the standard deviation is the

quadratic mean of the distance from the

mean the variance now is the squared

standard deviation if you want to know

more details about the standard

deviation and the variance please watch

our video let's move on to range and

interquartile range it is easy to

understand the range is simply the

difference between the maximum and

minimum value inter quartile range

represents the middle 50% of the data it

is the difference between the first

quartile q1 and the third quartile Q3

therefore 25% of the values are smaller

than the inter quartile range and 25% of

the values are larger the inter quartal

range contains exactly the middle 50% of

the values before we get to the last two

points let's briefly compare measures of

central tendency and measures of

dispersion let's say we measured the

blood pressure of patients measures of

central tendency provide a single value

that represents the entire data set

helping to identify a central value

around which data points tend to Cluster

measures of dispersion like the standard Devi

Devi

the range and the inter quarle range

indicate how spread out the data points

are whether they are closely packed

around the center or spread far from it

in summary while measures of central

tendency provide a central point of the

data set measures of dispersion describe

how the data is spread around the center

let's move on to tables here we will

have a look at the most important ones

frequency tables and contingency tables

a frequency table displays how often

each distinct value appears in a data

set let's have a closer look at the

example from the beginning a company

surveyed its employees to find out how

they get to work the options given were

car bicycle walk and public transport

here are the results from 30 employees

the first answer car the next walk and

so on and so forth now we can create a

fre table to summarize this data to do

this we simply enter the four possible

options car bicycle walk and public

transport in the First Column and then

count how often they occurred from the

table it is evident that the most common

mode of Transport among the employees is

by car with 14 employees preferring it

the frequency table thus provides a

clear and concise summary of the data

but what if we have not only one but two

categorical variables this is where the

contingency table also called cross tab

comes in Imagine the company doesn't

have one Factory but two one in Detroit

and one in Cleveland so we also ask the

employees at which location they work if

we want to display both variables we can

use a contingency table a contingency

table provides a way to analyze and

compare the relationship between two

categorical variables the rows of a

contingency table represent the

categories of one variable while the

columns represent the categories of

another variable each cell in the table

shows the number of observations that

fall into the corresponding category

combination for example the first cell

shows that car and Detroit were answered

six times and what about the charts

let's take a look at the most important

one to do this let's simply use data.net

if you like you can load this sample

data set with the link in the video

description or you just copy your own

data into this table here below you can

see the variables distance to work mode

of transport and site data daab gives

you a hint about the level of

measurement but you can also change it

here now if we only click on mode of

Transport we get a frequency table and

we can also display the percentage

values if we scroll down we get a bar

chart and a pie chart here on the left

we can adjust further settings for

example we can specify whether we want

to display the frequences or the

percentage values or whether the bars

should be vertical or

horizontal if you also select site we

get a cross table here and a grouped bar

chart for for the diagrams here we can

specify whether we want the chart to be

grouped or stacked if we click on

distance to work and mode of Transport

we get a bar chart where the height of

the bar shows the mean value of the

individual groups here we can also

display the

dispersion we also get a histogram a box

plot a violin plot and a rainbow plot if

you would like to know more about what a

Bo Bo plot a violin plot and a rainbow

plot are take a look at my videos let's

continue with inferential statistics at

the beginning we briefly go through what

inferential statistics is and then I'll

explain the six key components to you so

what is inferential statistics

inferential statistics allows us to make

a conclusion or inference about a

population based on data from a sample

what is the population and what is the

sample the population is the whole group

we're interested in if you want to study

the average height of all adults in the

United States then the population would

be all adults in the United States the

sample is a smaller group we actually

study chosen from the population for

example 150 adults were selected from

the United States and now we want to use

the sample to make a statement about the

population and here are the six steps

how to do that number one hypothesis

first we need a statement a hypothesis

that we want to test for example you

want to know whether a drug will have a

positive effect on blood pressure in

people with high blood pressure but

what's next in our hypothesis we stated

that we would like to study people with

high blood pressure so our population is

all people with high blood pressure in

for example the US obious obviously we

cannot collect data from the whole

population so we take a sample from the

population now we use this sample to

make a statement about the population

but how do we do that for this we need a

hypothesis test hypothesis testing is a

method for testing a claim about a

parameter in a population using data

measured in a sample great that's

exactly what we need there are many

different hypothesis tests and at the

end of this video I will give you a

guide on how to find the right test and

of course you can find videos about many

more hypothesis tests on our Channel but

how does a hypothesis test work when we

conduct a hypothesis test we start with

a research hypothesis also called

alternative hypothesis this is the

hypothesis we are trying to find

evidence for in our case the research

hypothesis is the drug has an effect on

blood pressure but we cannot test this

hypothesis directly with a classical

hypothesis test so we test the opposite

hypothesis that the drug has no effect

on blood pressure but what does that

mean first we assume that the drug has

no effect in the population we therefore

assume that in general people who take

the drug and people who don't take the

drug have the same blood pressure on

average if we now take a random sample

and it turns out that the drug has a

large effect in a sample then we can ask

How likely it is to draw such a sample

or one that deviates even more if the

drug actually has no effect so in

reality on average there is no

difference in the population if this

probability is very low we can ask

ourselves maybe the drug has an effect

in the population and we may have enough

evidence to reject the Nile hypothesis

that the drug has no effect

and it is this probability that is

called the P value let's summarize this

in three simple steps number one the

null hypothesis states that there is no

difference in the population number two

the hypothesis test calculates how much

the sample deviates from the null

hypothesis number three the P value

indicates the probability of getting a

sample that deviates as much as our

sample or one that even deviates more

than our sample assuming the null

hypothesis is true but at what point is

the P value small enough for us to

reject the Nile hypothesis this brings

us to the next Point statistical

significance if the P value is less than

a predetermined threshold the result is

considered statistically significant

this means that the result is unlikely

to have occurred by chance alone and

that we have enough evidence to re dect

anal hypothesis this threshold is often

0.05 therefore a small P value suggests

that the observed data or sample is

inconsistent with the null hypothesis

this leads us to reject the null

hypothesis in favor of the alternative

hypothesis a large P value suggests that

the observed data is consistent with the

null hypothesis and we will not reject

it but note there is always a risk of

making an error a small P value does not

prove that the alternative hypothesis is

true it is only saying that it is

unlikely to get such result or a more

extreme when the Nal hypothesis is true

and again if the Nal hypothesis is true

there is no difference in the population

and the other way around a large P value

does not prove that the Nal hypothesis

is true it is only saying that it is

likely to get such a result or a more extreme

extreme

when the null hypothesis is true so

there are two types of Errors which are

called type one and type two error let's

start with the type one error in

hypothesis testing a type one error

occurs when a true null hypothesis is

rejected so in reality the null

hypothesis is true but we make the

decision to reject the null hypothesis

in our example it means that the drug

actually had no effect so in reality

there there is no difference in blood

pressure whether the drug is taken or

not the blood pressure Remains the Same

in both cases but our sample happened to

be so far off the True Value that we

mistakenly thought the drag was working

and a type two error occurs when a false

null hypothesis is not rejected so in

reality the null hypothesis is false but

we make the decision not to reject an Al

hypothesis in our example this means the

drug actually did work there is a

difference between those who have taken

the drug and those who have not but it

was just a coincidence that the sample

taken did not show much difference and

we mistakenly thought the drug was not

working and now I'll show you how data

helps you to find a suitable hypothesis

test and of course calculates it and

interprets the results for you let's go

to data.net and copy your own data in

here we will just use this example data

set after copying your data into the

table the variables appear down here

data tab automatically tries to

determine the correct level of

measurement but you can also change it

up here now we just click on hypothesis

testing and select the varibles we want

to use for the calculation of a

hypothesis test data tab will then

suggest a suitable test for example in

this case a Ki Square test or in that

case an analysis of

variance then you will see the

hypothesis and the results if you're not

sure how to interpret the results click

on summary inverts further you can check

the assumptions and decide whether you

want to calculate a parametric or a

nonparametric test now we know the

differences between descriptive and

inferential statistics our next step is

to take a closer look at inferential

statistics and a choosing the

appropriate hypothesis test which

hypothesis test you can use depends on

the level of measurement of your data

there are four types of levels of

measurement nominal ordinal interval and

ratio and here is an easy explanation

for you in this video we we are going to

explore the four levels of measurement

nominal ordinal interval and ratio each

level gives us important information

about the variable and supports

different types of statistical analysis

by the end of this video you will know

what the levels of measurement are and

especially you will understand why you

need these levels so whether you are

analyzing survey data optimizing

business operations or studying for a

statistics Exam

stay tuned what are levels of

measurement levels of measurement refer

to different ways that variables can be

Quantified or categorized if you have a

data set then every variable in the data

set corresponds to one of the four

primary levels of measurement these

levels are nominal ordinal interval and

Rao in practice interval and ratio data

are often used to perform the same

analysis therefore the term matric level

is used to combine these two levels why

do you need levels of measurement the

level of measurement is crucial in

statistics for several key reasons it

tells us how our data can be collected

analyzed and interpreted here's why

understanding these levels is so

important different levels of

measurement support different

statistical analysis for instance mean

and standard deviation are suitable for

matric data in some cases it may be

suitable for ordinal data but only if

you know how to interpret the results

correctly and it definitely makes no

sense to calculated for nominal data the

level of measurement also tells us which

hypothesis tests are possible and

determines the most effective type of

data visualization for example bar

charts are great for nominal data while

histograms are better suited for metric

data so each level provides different

information and supports various types

of statistical analysis but attention

the level of measurement is mainly

relevant at the end of the research

process however the types of data to be

collected and are formed are determined

at the beginning therefore it is crucial

to consider the level of measurement of

the data from the start part to ensure

that the desired tests can be conducted

at the end so let's take a closer look

at each level of measurement what

characterizes nominal variables this is

the most basic level of measurement

nominal data can be categorized but it

is not possible to rank the categories

in a meaningful way examples of nominal

variables are gender with the categories

male and female types of animals with

for example the categories dog cat bird

or preferred newspaper in all these

cases you can tell whether one value

corresponds to the other so you can

distinguish the values but it is not

possible to put the categories in a

meaningful order an example we would

like to investigate whether chender has

an influence on the preferred newspaper

both variables are nominal so when we

create a questionnaire we simply list

the possible answers for both variables

since there is no meaningful order pH

nominal variables it usually does not

matter in which order the categories are

listed in the questionnaire then we can

display the collected data in a table

where each row is a person with the

respective answer we can now use our

data to create frequency tables or bar

charts but what about the ordinal level

of measurement ordinal data can be

categorized and in comparison with ninal

data it is possible to have a meaningful

ranking of the categories but

differences between ranks do not have a

mathematical meaning this means the

intervals between the data points are

not necessarily equal examples of

ordinant variables are all kinds of

rankings like first second third

satisfaction ratings very unsatisfied

unsatisfied neutral satisfied very

satisfied and levels of Education High

School Bachelors Masters in a

questionnaire you could ask how

satisfied are you with your current job

in this case we have these five possible

options the answers can be categorized

and there is a logic order that's why

the variable satisfaction with the job

is an ordinal variable

what about matric variables matric

variables are the highest level of

measurement matric data is like ordinal

but the intervals between values are

equally spaced this means the

differences and sums can be formed

meaningfully examples of mic variables

are Income weight age and electricity

consumption if you ask for a matric

variable in a questionnaire there's

usually just an input field in which the

person directly enters the value for

example age or body weight let's look at

what we've learned so far using an

example imagine you're conducting a

survey in a school to understand how

pupils get to school here are questions

you might ask each corresponding to a

different level of measurement the first

question could be what mode of

transportation do you use to get to

school bus car bicycle walk this is of

course a nominal variable the answers

can be categorized but there is no

meaningful order this means that bus is

not higher than bicycle walk is not

higher than car and so on and so forth

if you want to analyze the results of

this question you can count how many

students use each mode of transportation

and present it in a bar chart further

you can ask ask how satisfied are you

with your current mode of transportation

choices might include very unsatisfied

unsatisfied neutral satisfied very

satisfied this is of course an ordinal

variable you can rank the responses to

see which mode of transportation ranks

higher in satisfaction but the exact

difference between satisfied and very

satisfied for example or other options

isn't quantifiable and the last question

how many minutes does it take you to get

to school minutes to get to school is a

metric variable here you can calculate

the average time to get the school and

use all standard statistical measures we

can visualize this data with a histogram

showing the distribution of times you

get to school and compare the different

Transportation modes so using nominal

data we can categorize and count

responses but cannot inere any order

ordinal data allows us to rank responses

but not to measure precise differences

between ranks matric data enables us to

measure exact differences between data

points as already mentioned matric level

of measurement can be further subdivided

into interval scale and ratio scale but

what is the difference between interval

and ratio level let's look at an example

in a marathon of course the time of the

marathon runners is measured let's say

the first one took 2 hours and the last

one finished the marathon in 6 hours

here we can say that the fastest runner

was three times as fast as the slowest

or to put it the other way around the

slowest one took three times as long as

the fastest one this is possible because

there is a true zero point at the

beginning of the marathon where all

Runners start from zero in this case we

have ratio level of measurement if

however the stopwatch is forgotten to

start at the beginning of the race and

only the differences are measured

starting from the fastest runner we

don't have this true zero now the

runners cannot be put in proportion in

this case we can say how big the

interval between the runners is for

example the fastest runner is 4 hours

was faster than the slowest runner but

we cannot say that the fastest runner

was three times as fast as the slowest

this is because we don't know the

absolute values for both Runners we

still have equal intervals we can say

things like Runner B finished one hour

after the fastest runner and Runner C

finished 1 hour and 45 minutes after the

fastest runner the time differences are

measurable and meaningful but since

there is no true zero point we can say

the fastest runner was x times as fast

as the slowest runner we only know how

much lat that the other Runners finished

relative to the fastest runner but not

that total running times and in this

case we have an interval level of

measurement in summary while both

interval and ratio scales have equal

intervals and support similar operations

like addition and subtraction ratio

scales have a true zero point zero

represents the absence of the quantity

being measured this allows meaningful

multiplication and division and now a

little exercise to check whether

everything is clear to you first we have

state of the US which is a nominal level

of measurement this means the data is

used for labeling or naming categories

without without any quantitative value

in this case the states are names with

no inherent order or ranking next we

have product ratings on a scale from 1

to five this is an example of ordinal

data here the numbers do have an order

or rank five is better than one but the

intervals between the ratings are not

necessarily equal moving on to religious

confession like the state stes this is

also nominal the categories here such as

different religions are for

categorization and do not imply any

order next we have CO2 emissions in the

year which is measured on a metric ratio

scale this level allows for the full

range of mathematical operations

including meaningful ratios zero

emissions mean no emissions at all then

we have telephone numbers although

telephone numbers are numeric they are

categorized as nominal they are just

identifiers with no numerical value for

analysis care level of patience is

another ordinal example this might

include levels such as low medium and

high care which indicate an order but

not the exact difference between these

levels living space in square meters is

measured on a ratio scale like CO2

emissions Co sare meters mean there is

no living space and comparisons like

double or half are meaningful lastly we

have chop satisfaction on a scale from 1

to four this is ordinal data it ranks

satisfaction levels but the difference

between each level isn't

Quantified now we know what the level of

measurement is and we can go through the

hypothesis tests that are most popular

and discuss when to use which tests

let's start with the video on the most

common hypothesis test the tea test this

video is about everything you need to

know about the Tea test after this video

you know what a te test is and when you

use it what types of te tests there are

what the hypotheses and the assumptions

are how a te test is calculate it and

how you interpret the results let's

start with the first question what is a

T Test the T Test is a statistical test

procedure hm and what does the T Test do

the T Test analyzes whether there is a

significant difference between the means

of two groups for example the two groups

may be patients who received once drag a

and once drag B we would now like to

know if there is a difference in blood

pressure between these two groups now

there are three different types of T

tests the one sample T Test the

independent samples T Test and the pair

samples T Test when do we use a one

sample T Test we use the one sample T

Test when we want to compare the mean of

a sample with a known reference mean

example a chocolate bar manufacturer

claims that its chocolate bars weigh an

average of 50 g to check this we take a

sample of 30 bars and weigh them the

mean value of this sample is 48 G now we

can use a one sample T test to check if

the mean of 48 G is significantly

different from the claimed 50 g when do

we use the independent samples T Test we

use the T test for independent samples

when we want to compare the means of two

independent groups or samples we want to

know if there is a significant

difference between these means example

we would like to compare the

effectiveness of two

painkillers we randomly divide 60 people

into two groups the first group receives

track a and the second second group

receives Str B using an independent T

Test we can now test whether there is a

significant difference in pain relief

between the two drugs when we use the

pair samples T Test we use the paired

samples T Test to compare the means of

two dependent groups example we want to

know how effective a diet is to do this

we weigh 30 people before the diet and

then weigh exactly the same people after

the diet now we can look at the

difference in weight between before and

after for each subject we can now use a

paired samples T test to test whether

there is a significant difference in a

paired sample the measurements are

available in pairs the pairs result for

example from repeated measurements with

the same people independ dep samples are

made up of people and measurements that

are independent of each other here's an

interesting note the paired samples T

Test is very similar to the one sample T

Test we can also think of the paired

samples T Test as having one sample that

was measured at two different times we

then calculate the difference between

the paired values giving us a value for

one sample the differ difference is 1's

- 5 1's + 2 1's -1 and so on and so

forth now we want to test whether the

mean value of the difference just

calculated deviates from a reference

value in this case zero this is exactly

what the one sample T Test does what are

the assumptions for a t test of course

we first need a suitable sample in the

one sample T Test we need a sample and

the reference value in the independent T

Test we need two independent samples and

in the case of a pair T Test a paired

sample the variable for which we want to

test whether there is a difference

between the means must be metric

examples of matric variables are age

body weight and income for example a

person's level of education is not a

metric variable in addition the matric

variable must be normally distributed in

all three test variants to learn how to

test if your data is normally

distributed watch my video test for

normal distribution in case of an

independent T Test the variances in the

two groups must be approximately equal

you can check whether the variances are

equal using lavine's test for more

information watch my video on lavine's

test so what are the hypotheses of the T

Test let's start with the one sample T

Test in the one sample T Test the null

hypothesis is the sample mean is equal

to the given reference value so there's

no difference and the alternative

hypothesis is the sample mean is not

equal to the given reference value what

about the independent samples T Test in

the independent T Test the null

hypothesis is the mean values in both

groups are the same so there is no

difference between the two groups and

the alternative hypothesis is the mean

values in both groups are not equal so

there is a difference between the two

groups and finally the pair samples T

Test in a pair T Test the Nile

hypothesis is the mean of the difference

between the pairs is zero and the

alternative hypothesis is the mean of

the difference between the pairs is not

zero so now we know what the hypotheses

are before we look at how the te test is

calculated let us look at an example of

why we actually need a te test let's say

there is a difference in the length of

study for a bachelor's degree between

men and women in Germany our population

is therefore made up of all graduates of

a bachelor who have studied in Germany

however as we cannot survey all Bachelor

graduates we draw a sample that is as

representative as possible we now use

the test to test the null hypothesis

that there is no difference in the

population if there is no difference in

the population we will certainly still

see a difference in study duration in

the sample it would be very unlikely

that we drew a sample where the

difference would be exactly zero in

simple terms we now want to know at what

difference measured in a sample we can

say that the duration of study of men

and women is significantly different and

this is exactly what the T Test answers

but how do we calculate a T Test to do

this we first calculate the T value to

calculate the T value we need two values

first we need the difference between the

means and then we need the standard

deviation from the mean this is also Al

known as the standard error in the one

sample T Test we calculate the

difference between the sample mean and

the known reference mean s is the

standard deviation of the collected data

and N is the number of cases s / the

square root of n is then the standard

deviation from the mean which is the

standard error in the dependent samples

T Test we simply calculate the

difference between the the two sample

means to calculate the standard error we

need the standard deviation and the

number of cases from the first and

second sample depending on whether we

can assume equal or unequal variance for

our data there are different formulas

for the standard error read more about

this in our tutorial on data.net in a

paired sample T Test we only need to

calculate the difference between the

paired values and calculate to mean from

that the standard error is then the same

as for a one sample T Test so what have

we learned so far about the T value no

matter which T Test we calculate the T

value will be greater if we have a

greater difference between the means and

the T value will be smaller if the

difference between the means is smaller

further the T value becomes smaller when

we have a larger disperson of the mean

so the more scattered the data the less

meaningful a given mean difference is

now we want to use the T test to see if

we can reject the null hypothesis or not

to do this we can now use the T value in

two ways either we read the critical T

value from a table or we simply

calculate the P value from the T value

we'll go through both in a moment but

what is the P value a T Test over always

test the null hypothesis that there is

no difference so first we assume that

there is no difference in the population

when we draw a sample this sample

deviates from the null hypothesis by a

certain amount the P value tells us How

likely it is that we would draw a sample

that deviates from the population by the

same amount or more than a sample we

drew thus the more the sample deviates

from the null hypothesis the smaller the

P value becomes if this probability is

very very small we can of course ask

whether the null hypothesis holds for

the population perhaps there is a

difference but at what point can we

reject an Al hypothesis this border is

called the significance level which is

usually set at 5% so if there is only a

5% chance that we draw such a sample or

one that is more different then we have

enough evidence to assume that we reject

the null hypothesis and to put it simple

we assume that there is a difference

that the alternative hypothesis is true

now that we know what the P value is we

can finally look at how the T value is

used to determine whether or not the

null hypothesis is rejected let's start

with the path through the critical T

value which you can read from from a

table to do this we first need a table

of critical T values which we can find

on data.net under tutorials and T

distribution let's start with the

two-tailed case we'll briefly look at

the one tail case at the end of this

video here below we see the table first

we need to decide what level of

significance we want to use let's choose

a significance level of

0.05 or 5% then we look in this column

at 1 -

0.05 which is

0.95 now we need the degrees of freedom

in the one sample T Test and the paired

samples T Test the degrees of freedom

are simply the number of cases minus one

so if we have a sample of 10 people

there are 9 degrees of freedom in the

independent sample T Test we add the

number of people from both samples and

calculate that minus two because we have

two samples note that the degrees of

freedom can be determined in a different

way depending on whether we assume equal

or unequal variance so if we have a 5%

significance level and 9 degrees of

freedom we get a critical T value of

2.262 now on the one hand we've

calculated a t value with the T Test and

we have the critical T value if our

calculated T value is greater than the

critical T value we reject the null

hypothesis for example suppose we

calculate a t value of 2.5 this value is

greater than

2.262 and therefore the two means are so

different that we can reject the N

hypothesis on the other hand we can also

calculate the P value for the T value

we've calculated if we enter 2.5 for the

T value and 9 for the degrees of freedom

we get a P value of

0.034 the P value is less than

0.05 and we therefore reject the null

hypothesis as a control we copy the T

value of

2.262 here we get exactly a P value of

0.05 which is exactly the limit if you

want to calculate a t test with data tab

you just need to copy your own data in

this table click on hypothesis test and

then select the variables of interest

for example if you want to test whether

gender has an effect on income you

simply click on the two variables and

automatically get a T Test calculated

for independent samples here below you

can read the P value if you're still

unsure about the interpretation of the

results you can simply click on

interpretation in words a two-tail t

test for independent samples equal

variances assumed showed that the

difference between female and male with

respect to the dependent variable salary

was not statistically significant thus

the null hypothesis is retained the

final question now is what is the

difference between directed hypothesis

and undirected hypothesis in the

undirected case the alternative

hypothesis is that there is a difference

for example there is a difference

between the salary of men and women in

Germany we don't care who earns more we

just want to know if there is a

difference or not in a directed

hypothesis we are also interested in the

direction of the difference for example

example the alternative hypothesis might

be that men earn more than women or

women earn more than men if we look at

the T distribution graphically we can

see that in the two-sided case we have a

range on the left and the range on the

right we want to reject the null

hypothesis if we are either here or

there with a 5% significance level both

ranges have a probability of two .5%

together just 5% if we do a one tail T

Test the null hypothesis is rejected

only if we are in this range or

depending on the direction which we want

to test in that range with a 5%

significance level all 5% fall within

this range we've seen how the test is a

powerful tool for comparing the means of

two groups to determine why they differ sign

sign

significantly but what if we want to

extend our analysis to more than two

groups this is where analysis of

variance or Anova comes into play so

let's get started with Anova this video

is about analysis of variance or Anova

for short we discuss what an anova is

and why you need it we look at the

hypotheses and the assumptions for an

anova and I will show you how to

calculate an anova and what the

equations behind an anova are finally we

will discuss how to interpret the

results and take a look at the posst hog

test let's get started first of all

there are different types of analysis of

variant the simplest and most common one

is the oneway Anova and that's exactly

what this video is about what is a

oneway Anova the oneway Anova is a

hypothesis test okay and what does the

One Way Anova test and an NOA tests

whether there are statistically

significant differences between three or

more groups more precisely it is tested

whether there is a significant

difference between the mean values of

the groups an example you want to

investigate whether there is a

difference between fertilizers a b and c

in terms of plant growth the analysis of

variance now helps you to find out

whether there is a significant

difference in plant growth between the

three groups but doesn't it T Test do

something similar that's true the T test

tests whether there is a difference

between two groups the analysis of

variance is now an extension of the T

test and it tests whether there is a

difference between more than two groups

so if you only had two fertilizers A and

B that you wanted to compare you would

use a t test if you have three or more

different fertilizers you would use an

anova okay that makes sense what are the

hypothesis test it in an anova in an

anova the null hypothesis is that the

mean values of the groups are equal

mathematically this can be expressed as

mu1 is equal to mu2 is equal to Mu K

where mu1 to Mu K represents the mean

values of the different groups and the

alternative hypothesis is that the mean

values of the groups are not equal this

can be expressed mathematically as

follows at least one mu is different

where e stands for one of the groups

under consideration what are the

assumptions of an anova and how is it

calculated let's look at this with the

help of an example let's say you are a

researcher and you want to find out

whether three different drugs have a

different effect on blood pressure so

your research question is do the three

drugs have a different effect on blood

pressure the null hypothesis is there is

no different between the three drugs in

terms of blood pressure and the

alternative hypothesis is there is a

difference between the three drugs in

terms of blood pressure in order to

calculate an anova we of course need

data how do we get the data we obtain

data by taking a sample from the

population we need to analyze for

example all people suffering from high

blood pressure let's say we took a

sample of 24 tests subjects we can then

divide these 24 test subjects into three

groups with each group receiving a

different rack okay but now we still

don't have any data we only have groups

for now that's right in order to be able

to calculate an anova we have to measure

something for each subject in our case

the blood pressure we measure the blood

pressure for each test person once at

the beginning and once at the end of the

treatment the difference between the two

values tells us how the blood pressure

has changed while taking the medication

we have now collected data but what

about the assumptions we still have to

check them don't we exactly so that you

can calculate an anova these assumptions

must be met number one level of

measurement first the levels of

measurement for your variable must be

appropriate for an anova the independent

variable should be nominal such as the

type of medic ation or the type of

fertilizer applied the dependent

variable on the other hand must be

metric like blood pressure or plant

growth to summarize we need a variable

that defines the different groups such

as the different medications and the

variable that reflects what is measured

in a different groups for example blood

pressure but now to the second

assumption Independence the measurements

should be independent I.E the measured

value of one group should not be

influenced by the measured value of

another group the third assumption is

that of normal distribution the data

within the groups should be normally

distributed this assumption becomes less

important as the sample size increases

but should still be evaluated especially

for small samples if this assumption is

violated the cross called Wallace test

can be used as an alternative if you

would like to know how to test data for

normal distribution please watch my

video test for normal distribution and

thus to the fourth assumption the

variances in each group should be

roughly the same in other words it

should not be the case that we have a

very large variance or a very small

variance in one group compared to the

other group this assumption can be

tested with the Levine test you can find

more information in my video on the

Levine test if this assumption is not

met the Welch Anova can be used as

alternative okay but how does an anova

work we want to know whether there is a

difference between the groups to test

whether there is a difference the Anova

uses the ratio of the variance between

the groups and the variance within the

groups what is the between group

variance and what is the within group

variance the variance between the groups

measures how much the mean values of the

groups differ from each other in our

case the deviation of the mean values of

groups a b and c in this case we would

have a big variance in that case a small

one what is the variance within the

groups the within group variance

measures how much the individual data

points in each group fluctuate in this

case we would have a large variance

within the groups in that case a small

variance the ratio between the variance

between the groups and the variance with

within the groups is now the so-called F

value or F statistic if the variance

between the groups is small and the

variance within the groups is large then

we get a small F value in this case it

is likely that these deviations or even

larger ones occur purely by chance even

if in reality there is no difference

between the groups however if we have a

small spread within the groups and a

large spread between the groups then the

F value is large it is then very

unlikely that such a difference or an

even more extreme difference would arise

purely by chance or to be more precise

it is very unlikely that we will obtain

such an F value or an even larger one

therefore to assess whether the

differences between the groups are

statistically significant we use the F

value however this F value does not give

us a DE definite answer to assess the

significance of the results more

precisely we need the P value so let's

calculate the P value let's say this is

our data we have the values from the

first second and third group so now we

start and calculate the variance between

the groups and the variance within the

groups first of all we need the total

mean and the mean values of the

individual groups we obtain the total

mean by adding up all the values and

dividing by the number of values we have

a total of 24 values this gives us a total mean of

total mean of 5.3 we get the mean values of the groups

5.3 we get the mean values of the groups by simply adding up all the values of

by simply adding up all the values of the respective group and dividing by the

the respective group and dividing by the number of values per group now we've

number of values per group now we've calculated all the required mean values

calculated all the required mean values and we can calculate the so-called sum

and we can calculate the so-called sum of squares between the groups to do this

of squares between the groups to do this we calculate the difference between each

we calculate the difference between each group mean and the total mean Square

group mean and the total mean Square this and add up the results n is the

this and add up the results n is the number of values in the E Group so we

number of values in the E Group so we get 2.5 - 5.3 sared + 8 - 5.3 sared +

get 2.5 - 5.3 sared + 8 - 5.3 sared + 5.5 - 5.3 sared the sum of the squares

5.5 - 5.3 sared the sum of the squares between the groups is therefore

calculate the sum of squares within the groups to calculate this we subtract the

groups to calculate this we subtract the mean of each group from each individual

mean of each group from each individual value within that group so we subtract

value within that group so we subtract the mean of the first group from each of

the mean of the first group from each of its values the mean of the second group

its values the mean of the second group from each of its values and the mean of

from each of its values and the mean of the third group from each of its values

the third group from each of its values we Square the differences obtained and

we Square the differences obtained and add everything up for example in group

add everything up for example in group one we substract the the group average

one we substract the the group average of 2.5 from the first value from the

of 2.5 from the first value from the second value and so on in total we get a

second value and so on in total we get a sum of squares within the groups of 36

sum of squares within the groups of 36 so now we've almost made it we've

so now we've almost made it we've calculated the total mean and the group

calculated the total mean and the group mean values we have calculated the sum

mean values we have calculated the sum of squares between the groups and within

of squares between the groups and within the groups and now we can calculate the

the groups and now we can calculate the variances within and between the groups

variances within and between the groups in most cases however the term variance

in most cases however the term variance is not used but mean squares which we

is not used but mean squares which we will now also adopt we obtain the mean

will now also adopt we obtain the mean squares between the groups by dividing

squares between the groups by dividing the sum of squares between the groups by

the sum of squares between the groups by the number of degrees of freedom between

the number of degrees of freedom between the groups we've just calculated the sum

the groups we've just calculated the sum of squares between the groups and the

of squares between the groups and the degrees of freedom result from the

degrees of freedom result from the number of groups minus one so in our

number of groups minus one so in our case 3 - 1 which is equal to 2 we get a

case 3 - 1 which is equal to 2 we get a mean square of 60.6 7 so now we need the

mean square of 60.6 7 so now we need the mean squares within the groups these are

mean squares within the groups these are obtained by dividing the calculated sum

obtained by dividing the calculated sum of squares with in the groups by the

of squares with in the groups by the degrees of freedom within the groups the

degrees of freedom within the groups the degrees of freedom result from the total

degrees of freedom result from the total number of values in our case 24 minus

number of values in our case 24 minus the number of groups in our case three

the number of groups in our case three so we get

so we get 21 now we can calculate the F value to

21 now we can calculate the F value to do this we divide the mean Square

do this we divide the mean Square between the groups by the mean Square

between the groups by the mean Square within the groups or if we take the

within the groups or if we take the expression from the beginning the

expression from the beginning the variance between the groups by the

variance between the groups by the variance within the groups we get an F

variance within the groups we get an F value of

value of 3539 but what about the P value we can

3539 but what about the P value we can calculate the P value with an F

calculate the P value with an F distribution to do this we simply go to

distribution to do this we simply go to data.net we need the degrees of freedom

data.net we need the degrees of freedom between the groups in our case two the

between the groups in our case two the degrees of freedom within the groups in

degrees of freedom within the groups in our case 21 and we need the F value

our case 21 and we need the F value which is calculated I.E

which is calculated I.E 3539 we get a P value that is smaller

3539 we get a P value that is smaller than

than 0.01 so if we use the usual significance

0.01 so if we use the usual significance level of 5% we get a significant

level of 5% we get a significant difference and we reject the null

difference and we reject the null hypothesis that there is no difference

hypothesis that there is no difference between the three groups of course you

between the three groups of course you can simply calculate an anova online

can simply calculate an anova online with data tab just copy your data into

with data tab just copy your data into this table and click on Hy hypothesis

this table and click on Hy hypothesis test then simply select your desired

test then simply select your desired variables for example blood pressure and

variables for example blood pressure and medication and an NOA will now be

medication and an NOA will now be calculated automatically if you don't

calculated automatically if you don't know exactly how to interpret the

know exactly how to interpret the results simply click on summary inverts

results simply click on summary inverts or you can simply click on AI

or you can simply click on AI interpretation for most tables in this

interpretation for most tables in this table we now get exactly the values that

table we now get exactly the values that we just calculated by hand and we can

we just calculated by hand and we can also see here that we get a significant

also see here that we get a significant result as I said you can simply click on

result as I said you can simply click on AI interpretation to help you understand

AI interpretation to help you understand the results now there's one more topic

the results now there's one more topic to cover the post talk test a post talk

to cover the post talk test a post talk test becomes necessary when an anova

test becomes necessary when an anova indicates a significant difference

indicates a significant difference between groups while a NOA can confirm

between groups while a NOA can confirm that differences exist it doesn't

that differences exist it doesn't specify which groups are different from

specify which groups are different from each other a post talk test helps to

each other a post talk test helps to pinpoint exactly which groups differ

pinpoint exactly which groups differ from each other it performs a pairwise

from each other it performs a pairwise comparison between the groups indicating

comparison between the groups indicating for example whether group a differs from

for example whether group a differs from group b or group C in our case the post

group b or group C in our case the post talk test reveals that all groups differ

talk test reveals that all groups differ significantly from one another however

significantly from one another however this isn't always the outcome it's

this isn't always the outcome it's possible that only group one differs

possible that only group one differs from group two or that no individual

from group two or that no individual groups shows a significant difference

groups shows a significant difference even if the unov result is significant

even if the unov result is significant overall but what if you have not just

overall but what if you have not just one factor but two in this case you will

one factor but two in this case you will need to perform a two-way

need to perform a two-way Anova let's explore the two-way Anova

Anova let's explore the two-way Anova what is a two-way Anova a two-way Anova

what is a two-way Anova a two-way Anova is a statistical method used to test the

is a statistical method used to test the effect of two categorical variables on a

effect of two categorical variables on a continuous variable the categorical

continuous variable the categorical variables are the independent variables

variables are the independent variables for example the variable drag type with

for example the variable drag type with drug A and B and gender with female and

drug A and B and gender with female and male and the continuous variable is the

male and the continuous variable is the dependent variable for example the

dependent variable for example the reduction in blood pressure so the

reduction in blood pressure so the two-way Anova is the extension of the

two-way Anova is the extension of the one-way Anova while a oneway Anova tests

one-way Anova while a oneway Anova tests the effect of a single independent

the effect of a single independent variable on a dependent variable a twv

variable on a dependent variable a twv Anova tests the effects of two

Anova tests the effects of two independent variables the independent

independent variables the independent variables are called factors but what is

variables are called factors but what is a factor a factor is for example gender

a factor a factor is for example gender of a person with the levels male and

of a person with the levels male and female type of therapy with therapy a b

female type of therapy with therapy a b and c or the field of study with

and c or the field of study with medicine business administration

medicine business administration psychology and Mathematics in an

psychology and Mathematics in an analysis of variance a factor is

analysis of variance a factor is therefore a nominal variable we use an

therefore a nominal variable we use an anova whenever we want to test whether

anova whenever we want to test whether these levels have an influence on the

these levels have an influence on the so-call dependent variable you might

so-call dependent variable you might want to test whether gender has an

want to test whether gender has an effect on salary whether therapy has an

effect on salary whether therapy has an effect on blood pressure or whether

effect on blood pressure or whether field of study has an effect on length

field of study has an effect on length of study celer blood pressure and length

of study celer blood pressure and length of study will then be the dep dependent

of study will then be the dep dependent variables in each of these cases you

variables in each of these cases you test whether the factor has an effect on

test whether the factor has an effect on the dependent variable since you only

the dependent variable since you only have one factor in these cases you would

have one factor in these cases you would use a oneway Anova okay you're right in

use a oneway Anova okay you're right in the first case we have a variable with

the first case we have a variable with only two categories so of course we

only two categories so of course we would use the independent samples T Test

would use the independent samples T Test but when do we use a two-way Anova we

but when do we use a two-way Anova we use a two factor analysis of variance

use a two factor analysis of variance when we have a second factor and we want

when we have a second factor and we want to know whether this Factor also has an

to know whether this Factor also has an effect on the dependent variable we

effect on the dependent variable we would also like to know whether in

would also like to know whether in addition to gender the highest level of

addition to gender the highest level of education has an impact on salary or we

education has an impact on salary or we would like to include gender in addition

would like to include gender in addition to type of therapy or in the third case

to type of therapy or in the third case we would also like to know whether the

we would also like to know whether the university attended in addition to the

university attended in addition to the field of study has an in fluence on the

field of study has an in fluence on the length of study now we don't have one

length of study now we don't have one factor in all three cases but two

factor in all three cases but two factors in each case and since we now

factors in each case and since we now have two factors we use a two-way

have two factors we use a two-way analysis of variance so in a one-way

analysis of variance so in a one-way Anova we have one factor from which we

Anova we have one factor from which we create the groups if the factor we're

create the groups if the factor we're looking at has three levels for example

looking at has three levels for example three different types of drug we will

three different types of drug we will have three groups to compare in the the

have three groups to compare in the the case of a two-way analysis of variance

case of a two-way analysis of variance the group results from the combination

the group results from the combination of the levels of the two factors if we

of the levels of the two factors if we have one factor with three levels and

have one factor with three levels and one with two levels we have a total of

one with two levels we have a total of six groups to compare but what kind of

six groups to compare but what kind of statements can we make with a two-way

statements can we make with a two-way Anova with the help of a two-way Anova

Anova with the help of a two-way Anova we can answer three things whether the

we can answer three things whether the first Factor has an effect on the

first Factor has an effect on the dependent variable whether the second

dependent variable whether the second Factor has an effect on the dependent

Factor has an effect on the dependent variable and whether there is an

variable and whether there is an interaction effect between the two

interaction effect between the two factors but what about the hypotheses in

factors but what about the hypotheses in a two-way Anova there are three null

a two-way Anova there are three null hypotheses and therefore also three

hypotheses and therefore also three alternative hypothesis the first null

alternative hypothesis the first null hypothesis is there is no significant

hypothesis is there is no significant difference between the groups of the

difference between the groups of the first factor and the alternative

first factor and the alternative hypothesis there is a significant

hypothesis there is a significant difference between the groups of the

difference between the groups of the first Factor the second Nile hypothesis

first Factor the second Nile hypothesis is there is no significant difference

is there is no significant difference between the groups of the second factor

between the groups of the second factor and the alternative hypothesis there is

and the alternative hypothesis there is a significant difference between the

a significant difference between the groups of the second factor and the

groups of the second factor and the third null hypothesis reflects the

third null hypothesis reflects the interaction Effect one factor has no

interaction Effect one factor has no effect on the fact of the other factor

effect on the fact of the other factor and the alternative hypothesis at least

and the alternative hypothesis at least one fact Factor has an influence on the

one fact Factor has an influence on the effect of the other factor and what

effect of the other factor and what about the assumptions for the test

about the assumptions for the test results to be valid several assumptions

results to be valid several assumptions must be met number one normality the

must be met number one normality the data within the groups should be

data within the groups should be normally distributed or alternatively

normally distributed or alternatively the residual should be normally

the residual should be normally distributed this can be checked with a

distributed this can be checked with a quantile quantile plot Number Two

quantile quantile plot Number Two homogeneity of variances the variance of

homogeneity of variances the variance of data in groups should be equal this can

data in groups should be equal this can be checked with the Lin test number

be checked with the Lin test number three Independence the measurements

three Independence the measurements should be independent I.E the measured

should be independent I.E the measured value of one group should not be

value of one group should not be influenced by the measured value of

influenced by the measured value of another group number four measurement

another group number four measurement level the dependent variable should have

level the dependent variable should have a metric scale level but how to

a metric scale level but how to calculate a two-way Anova let's look at

calculate a two-way Anova let's look at the example from the beginning we would

the example from the beginning we would like to know if Dr type and gender have

like to know if Dr type and gender have an influence on the reduction in blood

an influence on the reduction in blood pressure Dr type has the two levels drug

pressure Dr type has the two levels drug A and B and gender has the two levels

A and B and gender has the two levels male and female to answer the question

male and female to answer the question we collect data we randomly assigned

we collect data we randomly assigned patients to the treatment combinations

patients to the treatment combinations and measured the reduction in blood

and measured the reduction in blood pressure after a month for example the

pressure after a month for example the first patient receives track a is male

first patient receives track a is male and after 1 month a reduction in blood

and after 1 month a reduction in blood pressure of six was measured now let us

pressure of six was measured now let us answer the questions is there a main

answer the questions is there a main effect of drag type on the reduction in

effect of drag type on the reduction in blood pressure is there a main effect of

blood pressure is there a main effect of gender on the reduction in blood

gender on the reduction in blood pressure and is there an interaction

pressure and is there an interaction effect between drag type and gender on

effect between drag type and gender on the reduction in blood pressure for the

the reduction in blood pressure for the calculation we can use either a

calculation we can use either a statistical software like data tab or do

statistical software like data tab or do it by hand first I will show you how to

it by hand first I will show you how to calculate it with data Tab and how to

calculate it with data Tab and how to interpret the results at the end I will

interpret the results at the end I will show you how to calculate the Anova by

show you how to calculate the Anova by hand and go through all the equations to

hand and go through all the equations to calculate a two-way Anova online simply

calculate a two-way Anova online simply visit data.net and copy your data into

visit data.net and copy your data into this table then click on hypothesis test

this table then click on hypothesis test under this tab you will find a lot of

under this tab you will find a lot of hypothesis tests and depending on which

hypothesis tests and depending on which variable able you select you will get an

variable able you select you will get an appropriate hypothesis test suggested we

appropriate hypothesis test suggested we want to know if drag type and gender

want to know if drag type and gender have an influence on the reduction in

have an influence on the reduction in blood pressure so let's just click on

blood pressure so let's just click on all three variables data now

all three variables data now automatically gives us a two-way Anova

automatically gives us a two-way Anova we can read the three null and the three

we can read the three null and the three alternative hypotheses here afterwards

alternative hypotheses here afterwards we get the descriptive statistics and

we get the descriptive statistics and the LaVine test for equal variance with

the LaVine test for equal variance with the LaVine test we can check if the

the LaVine test we can check if the variances within the groups are equal

variances within the groups are equal the P value is greater than

the P value is greater than 0.05 so we assume equality of variance

0.05 so we assume equality of variance in the groups for this data and here we

in the groups for this data and here we see the results of the analysis of

see the results of the analysis of variance we'll look at these in more

variance we'll look at these in more detail in a moment but if you don't know

detail in a moment but if you don't know exactly how to interpret the results you

exactly how to interpret the results you can also just click on summary in words

can also just click on summary in words in addition you can check here if the

in addition you can check here if the requirements for the analysis of

requirements for the analysis of variance are met at all but now back to

variance are met at all but now back to the results let's take a closer look at

the results let's take a closer look at this table the first row tests the N

this table the first row tests the N hypothesis where drag type has an effect

hypothesis where drag type has an effect on the reduction in blood pressure the

on the reduction in blood pressure the second row tests whether gender has an

second row tests whether gender has an effect on the reduction in blood

effect on the reduction in blood pressure and the third row tests if the

pressure and the third row tests if the interaction has an effect you you can

interaction has an effect you you can read the P value in each case right at

read the P value in each case right at the back here let's say we set the

the back here let's say we set the significance level at 5% if our

significance level at 5% if our calculated P value is less than

calculated P value is less than 0.05 the null hypothesis is rejected and

0.05 the null hypothesis is rejected and if the calculated P value is greater

if the calculated P value is greater than

than 0.05 the N hypothesis is not rejected

0.05 the N hypothesis is not rejected thus we see that all three p values are

thus we see that all three p values are greater than

greater than 0.05 and therefore we cannot not reject

0.05 and therefore we cannot not reject any of the three null hypotheses

any of the three null hypotheses therefore neither the drug type nor

therefore neither the drug type nor gender have a significant effect on the

gender have a significant effect on the reduction in blood pressure and there's

reduction in blood pressure and there's also no significant interaction effect

also no significant interaction effect but what does an analysis of variance

but what does an analysis of variance actually do and why is the word variance

actually do and why is the word variance in analysis of variance in a two-way

in analysis of variance in a two-way analysis of variance the total variance

analysis of variance the total variance of the dependent variable is divided

of the dependent variable is divided into the variance that can be explained

into the variance that can be explained by factor a the variance that can be

by factor a the variance that can be explained by Factor B the variance of

explained by Factor B the variance of the interaction and the arrow variance

the interaction and the arrow variance actually SS is not the variance but the

actually SS is not the variance but the sum of squares we will discuss how to

sum of squares we will discuss how to calculate the variance in this case in a

calculate the variance in this case in a moment but how can I imagine that the

moment but how can I imagine that the dependent variable has some variance in

dependent variable has some variance in our example not every everyone will have

our example not every everyone will have the same reduction in blood pressure we

the same reduction in blood pressure we now want to know if we can explain some

now want to know if we can explain some of this variance by the variabl Str type

of this variance by the variabl Str type gender and their interaction the part

gender and their interaction the part that we cannot explain by these three

that we cannot explain by these three terms accumulates in the eror if the

terms accumulates in the eror if the result looked like this we would be able

result looked like this we would be able to explain almost all the variance by

to explain almost all the variance by factors A and B and their interaction

factors A and B and their interaction and we would only have a very small

and we would only have a very small proportion that could not be explained

proportion that could not be explained this means that we can make a very good

this means that we can make a very good statement about the reduction in blood

statement about the reduction in blood pressure by the variable drag type sex

pressure by the variable drag type sex and interaction in this case it would be

and interaction in this case it would be the other way around drug type gender

the other way around drug type gender and the interaction almost have no

and the interaction almost have no effect on the reduction in blood

effect on the reduction in blood pressure and it all adds up in the arrow

pressure and it all adds up in the arrow but how do we calculate the sum of

but how do we calculate the sum of squares the F value and the P value

squares the F value and the P value here we have our data one's drug type

here we have our data one's drug type with drug A and B and one's gender with

with drug A and B and one's gender with male and female so these individuals are

male and female so these individuals are for example all male and have been given

for example all male and have been given drug a first we calculate the mean

drug a first we calculate the mean values we need we calculate the mean

values we need we calculate the mean value of each group so male and drag a

value of each group so male and drag a that is 5.8 then male and Drug B that is

that is 5.8 then male and Drug B that is 5.4 and we do the same for female then

5.4 and we do the same for female then we calculate the mean value of all males

we calculate the mean value of all males and females and the mean value of drug A

and females and the mean value of drug A and B finally we need the total mean we

and B finally we need the total mean we can now start to calculate the sum of

can now start to calculate the sum of squares let's start with the total sum

squares let's start with the total sum of squares we do this by subtracting the

of squares we do this by subtracting the total mean from each individual value

total mean from each individual value squaring the result and adding up all

squaring the result and adding up all the values the total mean is 5. 4 so we

the values the total mean is 5. 4 so we calculate 6 - 5.4 2ar + 4 - 5.4 squared

calculate 6 - 5.4 2ar + 4 - 5.4 squared to finally 3 - 5.4 squared so we get a

to finally 3 - 5.4 squared so we get a sum of squares of

sum of squares of 84.8 the degrees of freedom are given by

84.8 the degrees of freedom are given by n * P * Q - 1 n is the number of people

n * P * Q - 1 n is the number of people in a group in our case five and P and Q

in a group in our case five and P and Q are the number of categories in each of

are the number of categories in each of the factors in both cases we have two

the factors in both cases we have two groups the total variance is calculated

groups the total variance is calculated by dividing the sum of squares by the

by dividing the sum of squares by the degrees of freedom so we get

degrees of freedom so we get 4.46 now we can calculate the sum of

4.46 now we can calculate the sum of squares between the groups for this we

squares between the groups for this we calculate the group mean minus the total

calculate the group mean minus the total mean so 5.8 - 5.4 sared + 5. 4 - 5.4

mean so 5.8 - 5.4 sared + 5. 4 - 5.4 sared and the same for these two values

sared and the same for these two values we get

we get 7.6 in this case the degrees of freedom

7.6 in this case the degrees of freedom are three which gives us a variance of

are three which gives us a variance of 2.53 now we can calculate the sum of

2.53 now we can calculate the sum of squares of factor a a dash is the mean

squares of factor a a dash is the mean value of the categories of factor a so

value of the categories of factor a so we calculate

we calculate 5.9 minus the total mean value and and

5.9 minus the total mean value and and 4.9 minus the total mean value this

4.9 minus the total mean value this results in five together with the

results in five together with the degrees of freedom we can now calculate

degrees of freedom we can now calculate the variance for factor a which is five

the variance for factor a which is five we do the same for Factor B in this case

we do the same for Factor B in this case we use the mean values of male and

we use the mean values of male and female and we get the variance of

female and we get the variance of 0.8 now we can calculate the sum of

0.8 now we can calculate the sum of squares for the interaction we obtain

squares for the interaction we obtain this by calculating the sum of squares

this by calculating the sum of squares minus the sum of squares of A and B the

minus the sum of squares of A and B the degrees of freedom result to one for the

degrees of freedom result to one for the interaction we get a variance of

interaction we get a variance of 1.8 finally we can calculate the sum of

1.8 finally we can calculate the sum of squares of the error we substract the

squares of the error we substract the mean value of each group from the

mean value of each group from the respective group values so in this group

respective group values so in this group we subtract 5.8 from each individual

we subtract 5.8 from each individual value in this group we subtract

value in this group we subtract 5.4 here we substract six and then we

5.4 here we substract six and then we substract 4.4 this gives us a sum of

substract 4.4 this gives us a sum of squares of

squares of 77.2 the degrees of freedom are 16 and

77.2 the degrees of freedom are 16 and we get a variance of

we get a variance of 4.83 and now we calculate the F values

4.83 and now we calculate the F values these are obtained by dividing the

these are obtained by dividing the variance of factor a b or the

variance of factor a b or the interaction by the arrow variance so we

interaction by the arrow variance so we get the F value for factor a by dividing

get the F value for factor a by dividing in the variance of factor a by the eror

in the variance of factor a by the eror variance which is equal to

variance which is equal to 1.04 we can now do exactly the same for

1.04 we can now do exactly the same for FB and faab to verify we get exactly the

FB and faab to verify we get exactly the same values with data tab

same values with data tab 1.04

1.04 0.17 and

0.17 and 0.37 for the calculation of the P value

0.37 for the calculation of the P value you need the degrees of freedom and the

you need the degrees of freedom and the F distribution so with these three

F distribution so with these three values you can either read the critical

values you can either read the critical P value in a table or as usual you just

P value in a table or as usual you just use a software to calculate the P values

use a software to calculate the P values you can find a table of critical F

you can find a table of critical F values on data tab a g for a

values on data tab a g for a significance level of 5% you can use

significance level of 5% you can use this table if the red value is greater

this table if the red value is greater than the calculated F value the null

than the calculated F value the null hypothesis is rejected otherwise not

hypothesis is rejected otherwise not we've seen how an NOA allows us to

we've seen how an NOA allows us to compare PA means across different groups

compare PA means across different groups to determine if there are significant

to determine if there are significant differences but what if our research

differences but what if our research design involves measurements taken from

design involves measurements taken from the same subjects at different time

the same subjects at different time points this is where the pendency among

points this is where the pendency among the observations comes into play Let's

the observations comes into play Let's dive into how repeated measures and NOA

dive into how repeated measures and NOA adjusts our approach to interconnected

adjusts our approach to interconnected data points this video is about the

data points this video is about the repeated measures and NOA we will go

repeated measures and NOA we will go through the following questions what is

through the following questions what is a repeated measures analysis of variance

a repeated measures analysis of variance what are the hypotheses and the

what are the hypotheses and the assumptions how is a repeated measures

assumptions how is a repeated measures and over calculated how are the results

and over calculated how are the results interpreted and what is a post talk test

interpreted and what is a post talk test and how do you interpret it we'll go

and how do you interpret it we'll go through all points using a simple

through all points using a simple example let's start with the first

example let's start with the first question what is a repeated measures

question what is a repeated measures Anova a repeated measures an is of

Anova a repeated measures an is of variance tests whether there is a

variance tests whether there is a statistically significant difference

statistically significant difference between three or more dependent samples

between three or more dependent samples what are dependent samples in a

what are dependent samples in a dependent sample the same participants

dependent sample the same participants are measured multiple times under

are measured multiple times under different conditions or at different

different conditions or at different time points we therefore have several

time points we therefore have several measurements from each person involved

measurements from each person involved let's take a look at an example let's

let's take a look at an example let's say we want to investigate the effect

say we want to investigate the effect effectiveness of a training program for

effectiveness of a training program for this we've started looking for

this we've started looking for volunteers to participate in order to

volunteers to participate in order to investigate the effectiveness of the

investigate the effectiveness of the program we measure the physical fitness

program we measure the physical fitness of the participants at several points in

of the participants at several points in time before the training program

time before the training program immediately after completion and two

immediately after completion and two months later so for each participant we

months later so for each participant we have a value for physical fitness before

have a value for physical fitness before the program a value immediately after

the program a value immediately after after completion and a value 2 months

after completion and a value 2 months later and since we are measuring the

later and since we are measuring the same participants at different points in

same participants at different points in time we are dealing with dependent

time we are dealing with dependent samples now of course it doesn't have to

samples now of course it doesn't have to be about people or points in time in a

be about people or points in time in a generalized way we can say in a

generalized way we can say in a dependent sample the same test units are

dependent sample the same test units are measured several times under different

measured several times under different conditions the test units can be people

conditions the test units can be people animals or cells for example

animals or cells for example and the conditions can be time points or

and the conditions can be time points or treatments for example but what is the

treatments for example but what is the purpose of repeated measures and over we

purpose of repeated measures and over we want to know whether the fitness program

want to know whether the fitness program has an influence on physical fitness and

has an influence on physical fitness and it is precisely this question that we

it is precisely this question that we can answer with the help of an anova

can answer with the help of an anova with repeated measures physical fitness

with repeated measures physical fitness is therefore our dependent variable and

is therefore our dependent variable and time is our independent variable with

time is our independent variable with time points as levels so the analysis of

time points as levels so the analysis of variance with repeated measures checks

variance with repeated measures checks whether there is a significant

whether there is a significant difference between the different time

difference between the different time points but isn't that what the paired

points but isn't that what the paired samples T Test does doesn't it also test

samples T Test does doesn't it also test whether there is a difference between

whether there is a difference between dependent samples that's correct the

dependent samples that's correct the paired samples T Test evaluates whether

paired samples T Test evaluates whether there is a difference between two

there is a difference between two dependent groups the repeated measures a

dependent groups the repeated measures a Nova extends this concept allow allowing

Nova extends this concept allow allowing you to examine differences among three

you to examine differences among three or more dependent groups what are the

or more dependent groups what are the hypotheses for repeated measures and NOA

hypotheses for repeated measures and NOA the null hypothesis for a repeated

the null hypothesis for a repeated measures a Nova is that there are no

measures a Nova is that there are no differences between the means of the

differences between the means of the different conditions or time points in

different conditions or time points in other words the null hypothesis assumes

other words the null hypothesis assumes that each person has the same value at

that each person has the same value at all times the values of the individual

all times the values of the individual persons themselves May differ but one

persons themselves May differ but one the same person always has the same

the same person always has the same value the alternative hypothesis on the

value the alternative hypothesis on the other hand is that there is a difference

other hand is that there is a difference between the dependent groups in our

between the dependent groups in our example the null hypothesis states that

example the null hypothesis states that the training program has no influence on

the training program has no influence on physical fitness I.E that physical

physical fitness I.E that physical fitness does not change over time and

fitness does not change over time and the alternative hypothesis assumes that

the alternative hypothesis assumes that the training program does have an

the training program does have an influence I.E that physical fitness

influence I.E that physical fitness changes over time to correctly apply

changes over time to correctly apply repeated measures and over certain

repeated measures and over certain assumptions about the data must be

assumptions about the data must be fulfilled number one normality the

fulfilled number one normality the dependent variable should be

dependent variable should be approximately normally distributed this

approximately normally distributed this can be tested using the QQ plot or the

can be tested using the QQ plot or the karov smof test for more information

karov smof test for more information please watch my video test for normal

please watch my video test for normal distribution you can find the link in

distribution you can find the link in the video description

the video description number two sphericity the variances of

number two sphericity the variances of the differences between all combinations

the differences between all combinations of factor levels or time points should

of factor levels or time points should be the same this can be tested with the

be the same this can be tested with the help of Marley's test for sphericity if

help of Marley's test for sphericity if the resulting P value is greater than

the resulting P value is greater than 0.05 we can assume that the variances

0.05 we can assume that the variances are equal and the assumption is not

are equal and the assumption is not violated in this case the P value is

violated in this case the P value is greater than

greater than 0.05 then therefore this assumption is

0.05 then therefore this assumption is fulfilled if the assumption is violated

fulfilled if the assumption is violated adjustments such as Greenhouse Geer or

adjustments such as Greenhouse Geer or hind F can be made now I'll show you how

hind F can be made now I'll show you how to calculate and interpret an analysis

to calculate and interpret an analysis of variance online with data Tab and

of variance online with data Tab and then we'll go through the formulas to

then we'll go through the formulas to explain how to calculate the analysis of

explain how to calculate the analysis of variance with repeated measures by hand

variance with repeated measures by hand to calculate an anova online you simply

to calculate an anova online you simply go to data.net and copy your own data

go to data.net and copy your own data into this table I use this example data

into this table I use this example data set you can find a link to load this

set you can find a link to load this example data set in the video

example data set in the video description make sure that your data is

description make sure that your data is structured correctly I.E one row per

structured correctly I.E one row per participant and one column per condition

participant and one column per condition or time now we click on the hypothesis

or time now we click on the hypothesis test tab at the bottom we see the three

test tab at the bottom we see the three variables before in the middle and end

variables before in the middle and end from the data set if we now click on all

from the data set if we now click on all of them a repeated measures an NOA is

of them a repeated measures an NOA is automatically calculated firstly we can

automatically calculated firstly we can check the assumptions here we see that

check the assumptions here we see that the Mist test for sphericity results in

the Mist test for sphericity results in a P value of

a P value of 0.357 this value is greater than

0.357 this value is greater than 0.05 and thus the assumption is

0.05 and thus the assumption is fulfilled if this is not the case you

fulfilled if this is not the case you can take sphericity

correction I will explain how to test the normal distribution in a separate

the normal distribution in a separate video in our example now we will assume

video in our example now we will assume normal distribution with a lot of coning

normal distribution with a lot of coning of teeth if the assumption is not

of teeth if the assumption is not fulfilled you can simply calculate the

fulfilled you can simply calculate the non-parametric counterpart to the

non-parametric counterpart to the repeated measures and NOA the fredman

repeated measures and NOA the fredman test this does not require your data to

test this does not require your data to be normally

be normally distributed first of all if you do not

distributed first of all if you do not know exactly how to interpret the

know exactly how to interpret the individual tables in your analysis you

individual tables in your analysis you you can simply click on summary in words

you can simply click on summary in words or on AI interpretation for the tables

or on AI interpretation for the tables but now back to the results first we see

but now back to the results first we see the null and the alternative hypothesis

the null and the alternative hypothesis the null hypothesis is that there is no

the null hypothesis is that there is no difference between the dependent

difference between the dependent variables before in the middle and end

variables before in the middle and end and the alternative hypothesis is that

and the alternative hypothesis is that there is a difference at the end of the

there is a difference at the end of the test we can say whether we reject this

test we can say whether we reject this null hypothesis or not

null hypothesis or not now we see the descriptive statistics

now we see the descriptive statistics and a box plot we then get the results

and a box plot we then get the results of the Anova with repeated measures in

of the Anova with repeated measures in this table the P value is the most

this table the P value is the most important value it is

important value it is 0.01 and indicates the probability that

0.01 and indicates the probability that a sample deviates as much or even more

a sample deviates as much or even more from the null hypothesis as our sample

from the null hypothesis as our sample with a P value of

with a P value of 0.01 the results are statistically

0.01 the results are statistically significant at the conventional

significant at the conventional significance level of

significance level of 0.05 which means that there are

0.05 which means that there are significant differences between the mean

significant differences between the mean values of the three levels before in the

values of the three levels before in the middle and end this rejects the null

middle and end this rejects the null hypothesis and we assume that there is a

hypothesis and we assume that there is a difference between the groups and that

difference between the groups and that the training program or therapy has a

the training program or therapy has a significant effect if you want an

significant effect if you want an interpretation of the other values in

interpretation of the other values in this table simply click on AI

this table simply click on AI interpretation finally here is the table

interpretation finally here is the table for the bonferoni posst Haw test since

for the bonferoni posst Haw test since the p value of the analysis of variance

the p value of the analysis of variance is smaller than

is smaller than 0.05 we know that there is a difference

0.05 we know that there is a difference between one or more groups with the POs

between one or more groups with the POs hog test we can now determine between

hog test we can now determine between which groups this different exists we

which groups this different exists we see that there is a significant

see that there is a significant difference between before and end and in

difference between before and end and in the midle middle and end both have a P

the midle middle and end both have a P value of less than

value of less than 0.05 how do you calculate an analysis of

0.05 how do you calculate an analysis of variance with repeated measures by hand

variance with repeated measures by hand let's say this is our data we have five

let's say this is our data we have five people each of whom we measured at three

people each of whom we measured at three different points in time now we can

different points in time now we can calculate the necessary mean values

calculate the necessary mean values first we calculate the mean value of all

first we calculate the mean value of all the data which is

the data which is 5.4 then we calculate the mean value Val

5.4 then we calculate the mean value Val of the three groups for the first groups

of the three groups for the first groups we get a mean value of five for the

we get a mean value of five for the second a value of 6.1 and for the third

second a value of 6.1 and for the third a value of

a value of 5.1 and finally we can calculate the

5.1 and finally we can calculate the mean value of the three measurements for

mean value of the three measurements for each person so for the first person for

each person so for the first person for example we have an average value of

example we have an average value of eight over the three measurements and

eight over the three measurements and for the last person we have an average

for the last person we have an average value of five now that we have all all

value of five now that we have all all the mean values we need to calculate the

the mean values we need to calculate the required sums of squares but note our

required sums of squares but note our goal is the so-called F value and

goal is the so-called F value and subsequently calculate a P value from it

subsequently calculate a P value from it there are different ways for getting

there are different ways for getting this F value I will demonstrate one

this F value I will demonstrate one common way how to do this depending on

common way how to do this depending on which statistics textbook you use you

which statistics textbook you use you may come across a different formula but

may come across a different formula but back to the calculation let's start with

back to the calculation let's start with the sum of squares within the subject we

the sum of squares within the subject we obtain this by calculating each

obtain this by calculating each individual value

individual value xmi minus the mean value of the

xmi minus the mean value of the respective subject squaring this and

respective subject squaring this and adding it up so we start with 7 - 8 2ar

adding it up so we start with 7 - 8 2ar + 9 - 8 2ar until finally 3 - 5 and 7 -

+ 9 - 8 2ar until finally 3 - 5 and 7 - 5 we can then calculate the sum of

5 we can then calculate the sum of squares of the treatment I.E the sum of

squares of the treatment I.E the sum of squares of the three points in time we

squares of the three points in time we obtain this by subtracting the total

obtain this by subtracting the total mean value from each group mean value

mean value from each group mean value squaring it and adding it Up N is the

squaring it and adding it Up N is the number of people in a group so we get 5

number of people in a group so we get 5 - 5.4 sared + 6.1 - 5.4 sared + 5.1 -

- 5.4 sared + 6.1 - 5.4 sared + 5.1 - 5.4 squar now we can calculate the sum

5.4 squar now we can calculate the sum of squares of the residual we get this

of squares of the residual we get this by simply calculating the sum of squares

by simply calculating the sum of squares within the subjects minus the sum of

within the subjects minus the sum of squares of the treatment alternatively

squares of the treatment alternatively we can also use this formula here xmi is

we can also use this formula here xmi is again the value of each individual

again the value of each individual person AI is the mean value of the

person AI is the mean value of the respective group PM is the mean value of

respective group PM is the mean value of the respective person of the three

the respective person of the three points in time and G is the total mean

points in time and G is the total mean value we can then calculate the mean

value we can then calculate the mean squares to do this we divide the

squares to do this we divide the respective sum of squares by the degrees

respective sum of squares by the degrees of freedom the mean square of the

of freedom the mean square of the treatment is therefore calculated by

treatment is therefore calculated by dividing the sum of squares of the

dividing the sum of squares of the treatment by the degrees of freedom of

treatment by the degrees of freedom of the treatment the degrees of freedom of

the treatment the degrees of freedom of the treatment are the number of factor

the treatment are the number of factor levels minus one so we have three time

levels minus one so we have three time points minus one which is two the mean

points minus one which is two the mean Square of the residual is obtained in

Square of the residual is obtained in the same way here the degrees of freedom

the same way here the degrees of freedom are the number of factor levels minus 1

are the number of factor levels minus 1 times the number of subjects minus 1 we

times the number of subjects minus 1 we get 2 * 7 which is equal to 14 now we

get 2 * 7 which is equal to 14 now we calculate the F value which is done by

calculate the F value which is done by dividing the mean square of the

dividing the mean square of the treatment by the mean square of the

treatment by the mean square of the residual or error finally we calculate

residual or error finally we calculate the P value using the F value and the

the P value using the F value and the degrees of freedom from the treatment

degrees of freedom from the treatment and residual to calculate the P value

and residual to calculate the P value you can simply go to this page on data

you can simply go to this page on data tab the link can be found in the video

tab the link can be found in the video description here you can enter your

description here you can enter your values our F value is

values our F value is 1.69 the numerator degree of Freedom I.E

1.69 the numerator degree of Freedom I.E that of the treatments is two and the

that of the treatments is two and the denominator degree of Freedom I.E that

denominator degree of Freedom I.E that of the error is four 14 we get a P value

of the error is four 14 we get a P value of

of 0.22 the P value is greater than

0.22 the P value is greater than 0.05 and therefore we do not have enough

0.05 and therefore we do not have enough evidence to reject the null hypothesis

evidence to reject the null hypothesis of course we can then compare the

of course we can then compare the results with data tab to do this we copy

results with data tab to do this we copy the data back into this table and click

the data back into this table and click on the

on the variables we can see that we also get a

variables we can see that we also get a P value of 0.22 here after after

P value of 0.22 here after after exploring how repeated measures Anova

exploring how repeated measures Anova can be used to analyze data we might

can be used to analyze data we might wonder how to handle even more complex

wonder how to handle even more complex designs this is where mixed model Anova

designs this is where mixed model Anova comes in let's find out how this

comes in let's find out how this powerful tool can help us what is a

powerful tool can help us what is a mixed model Anova what are the

mixed model Anova what are the hypotheses and assumptions and how to

hypotheses and assumptions and how to interpret the results of a mixed model

interpret the results of a mixed model Anova this is what we discussed in this

Anova this is what we discussed in this video Let's let's start with the first

video Let's let's start with the first question what is a mixed model Anova a

question what is a mixed model Anova a mixed model Anova is a statistical

mixed model Anova is a statistical method used to analyze data that

method used to analyze data that involves both between subject factors

involves both between subject factors and within subject factors but what are

and within subject factors but what are between subjects factors and within

between subjects factors and within subjects factors let's look at an

subjects factors let's look at an example let's say we want to test

example let's say we want to test whether different diets have an effect

whether different diets have an effect on cholesterol levels we would like to

on cholesterol levels we would like to compare the three diets a b and c so the

compare the three diets a b and c so the factor diet has the three levels a b and

factor diet has the three levels a b and c to test whether there is a difference

c to test whether there is a difference between the diets we are conducting a

between the diets we are conducting a study with 18 participants the

study with 18 participants the individual participants are called

individual participants are called subjects now we randomly assign six

subjects now we randomly assign six participants to each of the three groups

participants to each of the three groups each participant or subject is assigned

each participant or subject is assigned to only one group in this case we have a

to only one group in this case we have a between subjects Factor difference

between subjects Factor difference subjects are exposed to different levels

subjects are exposed to different levels of a factor in this analysis our

of a factor in this analysis our objective is to determine whether

objective is to determine whether significant differences exist in the

significant differences exist in the mean cholesterol levels among the

mean cholesterol levels among the various groups on the study and this is

various groups on the study and this is exactly what a One Way Anova does now of

exactly what a One Way Anova does now of course we could also examine the impact

course we could also examine the impact of one diet across multiple time points

of one diet across multiple time points we could measure the cholesterol levels

we could measure the cholesterol levels at each participant at the start of the

at each participant at the start of the diet after 2 weeks and after 4 weeks so

diet after 2 weeks and after 4 weeks so the factor time has the three level

the factor time has the three level start two weeks and four weeks and in

start two weeks and four weeks and in this case the same subjects are being

this case the same subjects are being exposed to all levels of the factor and

exposed to all levels of the factor and this is called a within subjects Factor

this is called a within subjects Factor the same subjects are exposed to all

the same subjects are exposed to all levels or conditions in this case we

levels or conditions in this case we want to know if there is a difference in

want to know if there is a difference in the mean value of the cholesterol levels

the mean value of the cholesterol levels between the different points in time and

between the different points in time and this is exactly what a repeated measures

this is exactly what a repeated measures Anova does therefore in a repeated

Anova does therefore in a repeated measures Anova we have within subject

measures Anova we have within subject factors so in a between subjects design

factors so in a between subjects design each subject or participant is only

each subject or participant is only assigned to one factor level so that the

assigned to one factor level so that the different subjects only have the

different subjects only have the influence of the respective group in

influence of the respective group in contrast in the within subject design

contrast in the within subject design the same subjects op participants are

the same subjects op participants are exposed to all Factor levels which

exposed to all Factor levels which enables a direct comparison of the

enables a direct comparison of the reactions to each factor level but what

reactions to each factor level but what if we want to test if there is a

if we want to test if there is a difference between Diet a b and c over

difference between Diet a b and c over the different points in time so we want

the different points in time so we want to test if there is a difference between

to test if there is a difference between the diets and if there is a difference

the diets and if there is a difference between the different time points then

between the different time points then we need a mixed model and NOA because we

we need a mixed model and NOA because we have both one between subject factor and

have both one between subject factor and one within subject Factor so in a mixed

one within subject Factor so in a mixed model and NOA we have at least one

model and NOA we have at least one between subjects factor and at least one

between subjects factor and at least one within subjects factor in the same

within subjects factor in the same analysis note a mixed modan NOA is also

analysis note a mixed modan NOA is also called a two Way Anova with repeated

called a two Way Anova with repeated measures because there are two factors

measures because there are two factors and one of them results from repeated

and one of them results from repeated measures therefore the mixed model and

measures therefore the mixed model and Nova tests whether there is a difference

Nova tests whether there is a difference between more than two samples which are

between more than two samples which are divided at least between two factors one

divided at least between two factors one of the factors is a result of

of the factors is a result of measurement repetition with the help of

measurement repetition with the help of a mixed model and NOA you can now answer

a mixed model and NOA you can now answer three things first whether the within

three things first whether the within subject Factor has an effect on the

subject Factor has an effect on the dependent variable second whether the

dependent variable second whether the between subject Factor has an effect on

between subject Factor has an effect on a dependent variable and third whether

a dependent variable and third whether there is a so-called interaction effect

there is a so-called interaction effect between the two factors this gives us a

between the two factors this gives us a good transition to the hypothesis the

good transition to the hypothesis the first null hypothesis is the mean values

first null hypothesis is the mean values of the different measurement times do

of the different measurement times do not differ there are no significant

not differ there are no significant differences between the groups of the

differences between the groups of the within subject Factor then of course

within subject Factor then of course there is the second the means of the

there is the second the means of the different groups of the between subject

different groups of the between subject Factor do not differ and the third Nile

Factor do not differ and the third Nile hypothesis reflects the interaction

hypothesis reflects the interaction effect one factor has no effect on the

effect one factor has no effect on the effect of the other Factor what are the

effect of the other Factor what are the assumptions of a mixed model Nova

assumptions of a mixed model Nova normality the dependent variable should

normality the dependent variable should be approximately normally distributed

be approximately normally distributed within each group of the dependent

within each group of the dependent variables this assumption is especially

variables this assumption is especially important when a sample size is small

important when a sample size is small when a sample size is large and NOA is

when a sample size is large and NOA is somewhat robust to violations of

somewhat robust to violations of normality homogeneity of variances the

normality homogeneity of variances the variances in each group should be equal

variances in each group should be equal in mixed modela NOA this needs to be

in mixed modela NOA this needs to be true for both the within subjects and

true for both the within subjects and between subjects factors the leine stask

between subjects factors the leine stask can be used to check this assumption

can be used to check this assumption homogeneity of co-variances

homogeneity of co-variances sphericity this applies to the within

sphericity this applies to the within subjects factors and assumes that the

subjects factors and assumes that the variance of the differences between all

variance of the differences between all combinations of the different groups are

combinations of the different groups are equal what does that mean let's start

equal what does that mean let's start with the differences between all

with the differences between all combinations to do this we simply need

combinations to do this we simply need to calculate the difference of the first

to calculate the difference of the first group minus the second the difference of

group minus the second the difference of the first group and the third group and

the first group and the third group and the difference of the second group and

the difference of the second group and the third group these calculated

the third group these calculated differences should now have the same

differences should now have the same variance this assumption can be tested

variance this assumption can be tested using marless test of stcity when this

using marless test of stcity when this assumption is violated adjustments to

assumption is violated adjustments to the degrees of freedom such as

the degrees of freedom such as Greenhouse Geer or hind Feld can be used

Greenhouse Geer or hind Feld can be used independence of observations this

independence of observations this assumes that the observations are

assumes that the observations are independent of each other this is a

independent of each other this is a fundamental assumption in an NOA and is

fundamental assumption in an NOA and is usually assured by the study design no

usually assured by the study design no significant outliers outliers can have a

significant outliers outliers can have a disproportionate effect on Anova

disproportionate effect on Anova potentially causing misleading results

potentially causing misleading results it's important to identify and address

it's important to identify and address outliers let let's calculate an example

outliers let let's calculate an example and I'll show you how to interpret the

and I'll show you how to interpret the results let's say this is our data that

results let's say this is our data that we want to analyze we want to know

we want to analyze we want to know whether therapy a b and c and three

whether therapy a b and c and three different time points have an effect on

different time points have an effect on cholesterol levels each row is one

cholesterol levels each row is one person the therapy is the between

person the therapy is the between subject factor and the time with the

subject factor and the time with the levels before the therapy in the middle

levels before the therapy in the middle and at the end of the therapy is the

and at the end of the therapy is the within subject Factor

within subject Factor so the first patient on therapy a had a

so the first patient on therapy a had a cholesterol level of 165 before therapy

cholesterol level of 165 before therapy a cholesterol level of 145 in the middle

a cholesterol level of 145 in the middle and a cholesterol level of 140 at the

and a cholesterol level of 140 at the end let's first calculate the example

end let's first calculate the example online with data Tab and then discuss

online with data Tab and then discuss how to interpret the results to

how to interpret the results to calculate an anova online simply go to

calculate an anova online simply go to data.net and copy your data into this

data.net and copy your data into this table table you can also load this

table table you can also load this example data set using the link in the

example data set using the link in the video description then click on

video description then click on hypothesis testing under this tab you

hypothesis testing under this tab you will find a lot of hypothesis tests and

will find a lot of hypothesis tests and depending on which variable you click on

depending on which variable you click on you will get an appropriate hypothesis

you will get an appropriate hypothesis test suggested if you copy your data up

test suggested if you copy your data up here the variables will appear down here

here the variables will appear down here if the correct scale level is not

if the correct scale level is not automatically detected you can simply

automatically detected you can simply simply change it on variables view for

simply change it on variables view for example if we click on before middle and

example if we click on before middle and end a repeated measur Anova is

end a repeated measur Anova is automatically

automatically calculated but we also want to include

calculated but we also want to include the therapy so we just click on therapy

the therapy so we just click on therapy now we get a mixed model Anova we can

now we get a mixed model Anova we can read the three null and the three

read the three null and the three alternative hypotheses here then we get

alternative hypotheses here then we get the descriptive statistics output and

the descriptive statistics output and here we see the results of the analysis

here we see the results of the analysis of variance and also the post talk test

of variance and also the post talk test we'll look at these again in detail in a

we'll look at these again in detail in a moment but if you don't know exactly how

moment but if you don't know exactly how to interpret the results you can also

to interpret the results you can also click on summary in

click on summary in wordss but now back to the results most

wordss but now back to the results most important in this table are these three

important in this table are these three rows with these three rows you can check

rows with these three rows you can check if the three n hypotheses we stated

if the three n hypotheses we stated before are rejected or not the first row

before are rejected or not the first row test

test hypothesis whether cholesterol level

hypothesis whether cholesterol level changes over time so whether the

changes over time so whether the therapies have an effect on cholesterol

therapies have an effect on cholesterol level the second row tests whether there

level the second row tests whether there is a difference between the respective

is a difference between the respective therapy forms with respect to

therapy forms with respect to cholesterol level and the last row

cholesterol level and the last row checks if there is an interaction

checks if there is an interaction between the two factors you can read the

between the two factors you can read the P value in each case right at the back

P value in each case right at the back here let's say we set the significant

here let's say we set the significant level at 5% if our calculated P value is

level at 5% if our calculated P value is less than

less than 0.05 then the respective null hypothesis

0.05 then the respective null hypothesis is rejected and if the calculated P

is rejected and if the calculated P value is greater than

value is greater than 0.05 then the null hypothesis is not

0.05 then the null hypothesis is not rejected thus we see here that the P

rejected thus we see here that the P value of before middle and end is less

value of before middle and end is less than

than 0.05 and therefore the values at before

0.05 and therefore the values at before middle and end are significantly

middle and end are significantly different in terms of cholesterol levels

different in terms of cholesterol levels the P value in the second row is greater

the P value in the second row is greater than

than 0.05 therefore the types of therapy have

0.05 therefore the types of therapy have no significant influence on the

no significant influence on the cholesterol level it is important to

cholesterol level it is important to note that the mean value over the three

note that the mean value over the three time points is considered here it could

time points is considered here it could also be that in one therapy the blood

also be that in one therapy the blood pressure increases and in the other

pressure increases and in the other therapy the blood pressure pressure

therapy the blood pressure pressure decreases but on average over the time

decreases but on average over the time points the blood pressure is the same if

points the blood pressure is the same if this was the case however we would have

this was the case however we would have an interaction between the therapies and

an interaction between the therapies and the time we test this with the last

the time we test this with the last hypothesis in this case there is no

hypothesis in this case there is no significant interaction between therapy

significant interaction between therapy and time so there is an influence over

and time so there is an influence over time but it does not matter which

time but it does not matter which therapy is used the therapy has has no

therapy is used the therapy has has no significant influence if one of the two

significant influence if one of the two factors has a significant influence the

factors has a significant influence the following two tables show which of the

following two tables show which of the combinations differ significantly so far

combinations differ significantly so far we've explored various types of Anova

we've explored various types of Anova and test now these are all so-called

and test now these are all so-called parametric tests and require certain

parametric tests and require certain assumptions about the data like

assumptions about the data like normality but what happens if our data

normality but what happens if our data doesn't meet these assumptions

doesn't meet these assumptions this is where nonparametric tests come

this is where nonparametric tests come into play let's compare these two

into play let's compare these two families of tests to understand their

families of tests to understand their differences hi in this video I explain

differences hi in this video I explain the difference between parametric and

the difference between parametric and non-parametric hypothesis testing why

non-parametric hypothesis testing why are you interested in this topic you

are you interested in this topic you want to calculate a hypothesis test but

want to calculate a hypothesis test but you don't know exactly what the

you don't know exactly what the difference is between a parametric and a

difference is between a parametric and a non-parametric metric test and you're

non-parametric metric test and you're wondering when to use which test if you

wondering when to use which test if you want to calculate a hypothesis test you

want to calculate a hypothesis test you must first check the assumptions for the

must first check the assumptions for the test one of the most common assumptions

test one of the most common assumptions is that your data is normally

is that your data is normally distributed in simple terms if your data

distributed in simple terms if your data is normally distributed parametric tests

is normally distributed parametric tests are used such as the T Test analysis of

are used such as the T Test analysis of variance or peeron correlation if your

variance or peeron correlation if your data is not not normally distributed

data is not not normally distributed nonparametric tests are used such as men

nonparametric tests are used such as men with u test or spearman's correlation

with u test or spearman's correlation what about the other assumptions of

what about the other assumptions of course you still need to check whether

course you still need to check whether there are other assumptions for the test

there are other assumptions for the test in general however nonparametric tests

in general however nonparametric tests make fewer assumptions than parametric

make fewer assumptions than parametric tests so why use parametric tests at all

tests so why use parametric tests at all parametric tests are generally more

parametric tests are generally more powerful than non-parametric tests what

powerful than non-parametric tests what does that mean here's an example you

does that mean here's an example you have formulated your null hypothesis man

have formulated your null hypothesis man and women are paid equally whether this

and women are paid equally whether this null hypothesis is rejected depends on

null hypothesis is rejected depends on the difference in salary the dispersion

the difference in salary the dispersion of the data and the sample size in a

of the data and the sample size in a parametric test a smaller difference in

parametric test a smaller difference in salary or a smaller sample is usually

salary or a smaller sample is usually sufficient to reject Al hypothesis if

sufficient to reject Al hypothesis if possible always use parametric tests

possible always use parametric tests what is the structural difference

what is the structural difference between parametric and non-parametric

between parametric and non-parametric tests let's take a look at pearon

tests let's take a look at pearon correlation and spearman's rank

correlation and spearman's rank correlation as well as a t test for

correlation as well as a t test for independent samples and the Man withney

independent samples and the Man withney U test let's start with the Pearson and

U test let's start with the Pearson and Spearman correlation the span rank

Spearman correlation the span rank correlation is the nonparametric

correlation is the nonparametric counterpart to the the Pearson

counterpart to the the Pearson correlation what is the difference

correlation what is the difference between the two correlation coefficients

between the two correlation coefficients spean correlation does not use raw data

spean correlation does not use raw data but the ranks of the data let's look at

but the ranks of the data let's look at an example we measure the reaction time

an example we measure the reaction time of eight computer players and ask their

of eight computer players and ask their age when we calculate a Pearson

age when we calculate a Pearson correlation we simply take the two

correlation we simply take the two varibles reaction time and age and

varibles reaction time and age and calculate the PE and correlation

calculate the PE and correlation coefficient

coefficient however we now want to calculate span's

however we now want to calculate span's rank correlation so first we assign a

rank correlation so first we assign a rank to each person for reaction time

rank to each person for reaction time and age the reaction time is already

and age the reaction time is already sorted by size 12 is the smallest value

sorted by size 12 is the smallest value so gets rank one 15 the second smallest

so gets rank one 15 the second smallest so gets rank two and so on and so forth

so gets rank two and so on and so forth we are now doing the same with ag here

we are now doing the same with ag here we have the smallest value there the

we have the smallest value there the second small

second small here the third smallest fourth smallest

here the third smallest fourth smallest and so on and so forth let's take a look

and so on and so forth let's take a look at this in a scatter plot here we see

at this in a scatter plot here we see the raw data of age and reaction time

the raw data of age and reaction time but now we would like to use the

but now we would like to use the rankings so we form ranks from the

rankings so we form ranks from the variables age and reaction time through

variables age and reaction time through this transformation we have now

this transformation we have now distributed our data more evenly to get

distributed our data more evenly to get span's correlation we we simply

span's correlation we we simply calculate peon correlation from the

calculate peon correlation from the ranks so Spearman correlation is equal

ranks so Spearman correlation is equal to peon correlation only that the ranks

to peon correlation only that the ranks are used instead of raw values what

are used instead of raw values what about a test for independent samples and

about a test for independent samples and the man with u test the T test for

the man with u test the T test for independent samples and the man with u

independent samples and the man with u test check whether there is a difference

test check whether there is a difference between two groups an example is the

between two groups an example is the there a difference between the reaction

there a difference between the reaction time of man and women the man with u

time of man and women the man with u test is the nonparametric counterpart to

test is the nonparametric counterpart to the T test for independent samples but

the T test for independent samples but there is an important difference between

there is an important difference between the two tests the T test for independent

the two tests the T test for independent samples tests whether there is a mean

samples tests whether there is a mean difference for both samples the mean

difference for both samples the mean value is calculated and it is tested

value is calculated and it is tested whether these mean values differ

whether these mean values differ significantly the man with u test on the

significantly the man with u test on the other hand checks whether there is a

other hand checks whether there is a rank sum difference how do we calculate

rank sum difference how do we calculate the rank sums for this purpose we sort

the rank sums for this purpose we sort all persons from the smallest to the

all persons from the smallest to the largest value this person has the

largest value this person has the smallest value so gets rank one that

smallest value so gets rank one that person has the second smallest value so

person has the second smallest value so gets rank two and this person has the

gets rank two and this person has the third smallest value and so on and so

third smallest value and so on and so forth now we have assigned a rank rank

forth now we have assigned a rank rank to each person then we can simply add up

to each person then we can simply add up the ranks of the first group and the

the ranks of the first group and the second group in the first group we get a

second group in the first group we get a rank sum of 42 and in the second group a

rank sum of 42 and in the second group a rank sum of 36 now we can investigate

rank sum of 36 now we can investigate whether there is a significant

whether there is a significant difference between these rank sums if

difference between these rank sums if you want to know more about the man with

you want to know more about the man with u test check out my related video so we

u test check out my related video so we can summarize the raw data I used to

can summarize the raw data I used to parametric test tests and the ranks of

parametric test tests and the ranks of the raw data are used for nonparametric

the raw data are used for nonparametric tests the hypothesis test you use

tests the hypothesis test you use usually depends on how many variables

usually depends on how many variables you have and whether it is an

you have and whether it is an independent or dependent sample in most

independent or dependent sample in most cases there is always a nonparametric

cases there is always a nonparametric counterpart to parametric tests so if

counterpart to parametric tests so if you do not meet the assumptions for the

you do not meet the assumptions for the parametric test you can use the

parametric test you can use the non-parametric counterpart but don't

non-parametric counterpart but don't worry data tab will do its best to help

worry data tab will do its best to help you choose the right hypothesis test of

you choose the right hypothesis test of course you can calculate the most common

course you can calculate the most common parametric and nonparametric test with

parametric and nonparametric test with data tab online simply copy your own

data tab online simply copy your own data into the table and your variables

data into the table and your variables will appear here below now click on the

will appear here below now click on the variables you want to calculate a

variables you want to calculate a hypothesis test for for example if you

hypothesis test for for example if you choose salary and gender a t test will

choose salary and gender a t test will be calculated

here you can check the assumptions if the assumptions are not

assumptions if the assumptions are not met you can simply click on

met you can simply click on nonparametric and a man with the U test

nonparametric and a man with the U test will be

will be calculated if you click on salary and

calculated if you click on salary and Company an analysis of variance is

Company an analysis of variance is calculated or in the nonparametric case

calculated or in the nonparametric case the cross Vol

the cross Vol test as we've seen parametric tests rely

test as we've seen parametric tests rely heavily on the assumption that data are

heavily on the assumption that data are normally

normally distributed this leads us to an

distributed this leads us to an essential step in data analysis checking

essential step in data analysis checking our data for

our data for normality before applying parametric

normality before applying parametric tests it's crucial to check this

tests it's crucial to check this assumption otherwise we would get

assumption otherwise we would get inaccurate results let's now look into

inaccurate results let's now look into different methods and statistical tests

different methods and statistical tests to check out data for normal

to check out data for normal distribution in this video I show you

distribution in this video I show you how to test your data for normal

how to test your data for normal distribution first of all why do you

distribution first of all why do you need normal

need normal distribution let's say you've collected

distribution let's say you've collected data and you want to analyze this data

data and you want to analyze this data with an appropriate hypothesis test for

with an appropriate hypothesis test for example a t test or an analysis of

example a t test or an analysis of variant one of the most common

variant one of the most common requirements for hypothesis testing is

requirements for hypothesis testing is that the data used must be normally

that the data used must be normally distributed data are normally

distributed data are normally distributed if the frequency

distributed if the frequency distribution of the data has this bell

distribution of the data has this bell curve now of course the big question is

curve now of course the big question is how do you know if your data is normally

how do you know if your data is normally distributed or not or how can you test

distributed or not or how can you test that there are two ways either you can

that there are two ways either you can check the normal distribution

check the normal distribution analytically or graphically we now look

analytically or graphically we now look at both in

at both in detail let's start with the analytical

detail let's start with the analytical test for normal distribution in order to

test for normal distribution in order to test your data analytically for normal

test your data analytically for normal distribution there are several test

distribution there are several test procedures the best known are the colog

procedures the best known are the colog of smov test the Shapiro wil test and

of smov test the Shapiro wil test and the Anderson darling test with all these

the Anderson darling test with all these tests you test the null hypothesis that

tests you test the null hypothesis that the data are normally distributed so the

the data are normally distributed so the null hypothesis is that the frequency

null hypothesis is that the frequency distribution of your data fits the

distribution of your data fits the normal distribution in in order to

normal distribution in in order to reject or not reject the null hypothesis

reject or not reject the null hypothesis you get a P value out of all these tests

you get a P value out of all these tests now the big question is whether this P

now the big question is whether this P value is greater or less than

value is greater or less than 0.05 if the P value is less than

0.05 if the P value is less than 0.05 this is interpreted as a

0.05 this is interpreted as a significant deviation from the normal

significant deviation from the normal distribution and you can assume that

distribution and you can assume that your data are not normally distributed

your data are not normally distributed if the P value is greater than

if the P value is greater than 0.05 and you want to be statistically

0.05 and you want to be statistically clean you cannot necessarily say that

clean you cannot necessarily say that the frequency distribution corresponds

the frequency distribution corresponds to the normal distribution you just

to the normal distribution you just cannot disprove the null hypothesis in

cannot disprove the null hypothesis in practice however values greater than

practice however values greater than 0.05 are assumed to be normally

0.05 are assumed to be normally distributed to be on a safe side you

distributed to be on a safe side you should always take a look at the graphic

should always take a look at the graphic solution which we will talk about in a

solution which we will talk about in a moment so in summary all these tests

moment so in summary all these tests give you a P value if this P value is

give you a P value if this P value is less than

less than 0.05 you assume no normal distribution

0.05 you assume no normal distribution if it is greater than 0.05 you assume

if it is greater than 0.05 you assume normal distribution for your information

normal distribution for your information with the Koger of smof test and with the

with the Koger of smof test and with the Anderson darling test you can also test

Anderson darling test you can also test distributions other than the normal

distributions other than the normal distribution now unfortunately there is

distribution now unfortunately there is a big disadvantage of the analytical

a big disadvantage of the analytical methods which is why more and more

methods which is why more and more people are switching to using the

people are switching to using the graphical methods the problem is that

graphical methods the problem is that the calculated P value is influenced by

the calculated P value is influenced by the size of the sample therefore if you

the size of the sample therefore if you have a very small sample your P value

have a very small sample your P value may be much larger than

may be much larger than 0.05 but if you if you have a very large

0.05 but if you if you have a very large sample your P value may be smaller than

sample your P value may be smaller than 0.05 let's assume the distribution in

0.05 let's assume the distribution in your population deviates very slightly

your population deviates very slightly from the normal distribution then if you

from the normal distribution then if you take a very small sample you will get a

take a very small sample you will get a very large P value and thus you will

very large P value and thus you will assume that it is normally distributed

assume that it is normally distributed data however if you take a larger sample

data however if you take a larger sample then a P value Bec becomes smaller and

then a P value Bec becomes smaller and smaller even though the samples come

smaller even though the samples come from the same population with the same

from the same population with the same distribution therefore if you have a

distribution therefore if you have a minimal deviation from the normal

minimal deviation from the normal distribution which isn't actually

distribution which isn't actually relevant the larger your sample the

relevant the larger your sample the smaller the P value becomes with a very

smaller the P value becomes with a very large sample you may even get a P value

large sample you may even get a P value smaller than

smaller than 0.05 and thus reject the hypothesis that

0.05 and thus reject the hypothesis that it is a normal distribution to get

it is a normal distribution to get around this problem graphical methods

around this problem graphical methods are being used more and more we'll come

are being used more and more we'll come to that now if the normal distribution

to that now if the normal distribution is checked graphically you either look

is checked graphically you either look at the histogram or even better at the

at the histogram or even better at the QQ plot if you use the histogram you

QQ plot if you use the histogram you plot the normal distribution in the

plot the normal distribution in the histogram of your data and then you can

histogram of your data and then you can see whether the curve of the normal

see whether the curve of the normal distribution ution roughly corresponds

distribution ution roughly corresponds to that of the normal distribution curve

to that of the normal distribution curve however it is better if you use the

however it is better if you use the so-called quantile quantile plot or QQ

so-called quantile quantile plot or QQ plot for short here the theoretical

plot for short here the theoretical quantiles that the data should have if

quantiles that the data should have if they are perfectly normally distributed

they are perfectly normally distributed and the quantiles of the measured values

and the quantiles of the measured values are compared if the data is perfectly

are compared if the data is perfectly normally distributed all points would

normally distributed all points would lie on the line the more the data

lie on the line the more the data deviates from the line the less it is

deviates from the line the less it is normally distributed in addition data de

normally distributed in addition data de plots the 95% confidence interval if all

plots the 95% confidence interval if all or almost all of your data lies within

or almost all of your data lies within this interval it is a very strong

this interval it is a very strong indication that your data is normally

indication that your data is normally distributed your data would not be

distributed your data would not be normally distributed if for example they

normally distributed if for example they form an arc and lie far away from the

form an arc and lie far away from the line in some areas if you use data tap

line in some areas if you use data tap in order to test for normal distribution

in order to test for normal distribution you get the following evaluation first

you get the following evaluation first you get the analytical test procedures

you get the analytical test procedures clearly arranged in a table then come

clearly arranged in a table then come the graphical test procedures how you

the graphical test procedures how you can test your data with data tab for

can test your data with data tab for normal distribution I will show you now

normal distribution I will show you now just copy your data into to this table

just copy your data into to this table click on descriptive statistics and then

click on descriptive statistics and then select the variable you want to test for

select the variable you want to test for normal distribution for example age

normal distribution for example age after that you can simply click on test

after that you can simply click on test for normal distribution here and you

for normal distribution here and you will get the results down here I know

will get the results down here I know the test procedures are not actually

the test procedures are not actually descriptive methods but if you want to

descriptive methods but if you want to get an overview of your data it's

get an overview of your data it's usually also relevant to look at the

usually also relevant to look at the distribution of your data furthermore if

distribution of your data furthermore if you calculate a hypothesis test for

you calculate a hypothesis test for example whether gender has an influence

example whether gender has an influence on the salary of a person then you can

on the salary of a person then you can check the precond conditions for each

check the precond conditions for each hypothesis test and you will also get

hypothesis test and you will also get the test for normal distribution if the

the test for normal distribution if the pr condition is not met you would click

pr condition is not met you would click on this and a non-parametric test the

on this and a non-parametric test the man Whitney U test would be calculated

man Whitney U test would be calculated the man with u test does not need

the man with u test does not need normally distributed data another

normally distributed data another important assumption is the equality of

important assumption is the equality of variance you can check whether two or

variance you can check whether two or more groups have the same variance using

more groups have the same variance using the LaVine test let's take a look at it

the LaVine test let's take a look at it what is a lavine's test lavine's test

what is a lavine's test lavine's test tests the hypothesis that the variances

tests the hypothesis that the variances are equal in different groups the aim is

are equal in different groups the aim is to determine whether the variances in

to determine whether the variances in different groups are significantly

different groups are significantly different from each other the hypotheses

different from each other the hypotheses for Lavin's test are as follows the null

for Lavin's test are as follows the null hypothesis is the variances of the

hypothesis is the variances of the groups are equal and the alternative

groups are equal and the alternative hypothesis is at least one of the groups

hypothesis is at least one of the groups has a different variance when is

has a different variance when is lavine's test most commonly used

lavine's test most commonly used lavine's test is most often used to test

lavine's test is most often used to test a assumptions for another hypothesis

a assumptions for another hypothesis test what does that mean let's say your

test what does that mean let's say your hypothesis is there is a difference

hypothesis is there is a difference between two medications in terms of

between two medications in terms of perceived pain relief to test this

perceived pain relief to test this hypothesis you've collected data now to

hypothesis you've collected data now to test the hypothesis based on your data

test the hypothesis based on your data you use a hypothesis test such as a T

you use a hypothesis test such as a T Test many hypothesis tests have the

Test many hypothesis tests have the assumption that the variances in each

assumption that the variances in each group are equal and this is where thein

group are equal and this is where thein test comes in it tells us whether this

test comes in it tells us whether this assumption is fulfilled or not how is a

assumption is fulfilled or not how is a lavine's test calculated here's an

lavine's test calculated here's an example we want to know if there is a

example we want to know if there is a significant difference in variance

significant difference in variance between these groups first we simply

between these groups first we simply calculate the mean of each group then we

calculate the mean of each group then we subtract the respective group mean from

subtract the respective group mean from each person

each person the amount of each value is now formed

the amount of each value is now formed so that the negative values become

so that the negative values become positive from these new values the group

positive from these new values the group mean can now be calculated again the

mean can now be calculated again the larger the group mean the greater the

larger the group mean the greater the variance within each group thus there is

variance within each group thus there is a smaller variance in this group than in

a smaller variance in this group than in that group in addition we can calculate

that group in addition we can calculate the total mean value now we can

the total mean value now we can calculate the Square deviations of the

calculate the Square deviations of the group means from the overall mean and

group means from the overall mean and sum them up and then we can calculate

sum them up and then we can calculate the square deviation of the individual

the square deviation of the individual values from the respective group mean

values from the respective group mean and add them up we can now compare the

and add them up we can now compare the two calculated sums and that is exactly

two calculated sums and that is exactly what lavine's test does the test

what lavine's test does the test statistic of lavine's test is obtained

statistic of lavine's test is obtained with this equation n is the number of

with this equation n is the number of cases and I the number of cases in the E

cases and I the number of cases in the E Group set I is the mean value of the E

Group set I is the mean value of the E Group set is the overall average set i j

Group set is the overall average set i j is the respective value in the groups

is the respective value in the groups and K is the number of groups the

and K is the number of groups the calculated test statistic L is equal to

calculated test statistic L is equal to the F statistic therefore with the F

the F statistic therefore with the F value and the degrees of freedom the p

value and the degrees of freedom the p value can be calculated the degrees of

value can be calculated the degrees of freedom result with number of groups

freedom result with number of groups minus one and number of cases minus

minus one and number of cases minus number of groups if the P value is

number of groups if the P value is greater than

greater than 0.05 the null hypothesis that the

0.05 the null hypothesis that the variances are equal is not rejected thus

variances are equal is not rejected thus equality of variance can be assumed if

equality of variance can be assumed if you use data Tab and calculate an

you use data Tab and calculate an analysis of variance you can find

analysis of variance you can find lavine's test under test

assumptions in an independent T Test you will find lavine's test at the bottom of

will find lavine's test at the bottom of the results if equality of variance is

the results if equality of variance is not given you can use the T test for

not given you can use the T test for unequal

unequal variance now that we understand the

variance now that we understand the importance of testing for normal

importance of testing for normal distribution we might find ourselves in

distribution we might find ourselves in situations where these assumption are

situations where these assumption are not met in this case we turn to

not met in this case we turn to nonparametric methods that are less

nonparametric methods that are less sensitive to the distribution of data we

sensitive to the distribution of data we will discuss various kinds of tests for

will discuss various kinds of tests for this purpose like man with u test cross

this purpose like man with u test cross called Wallace test will coxen signed

called Wallace test will coxen signed rank test and fredman test let's start

rank test and fredman test let's start with the non-parametric counterpart to

with the non-parametric counterpart to the T test for independent samples The

the T test for independent samples The Man withney U test what is a man Whitney

Man withney U test what is a man Whitney U test and how is it calculated that's

U test and how is it calculated that's what we will discuss in this video let's

what we will discuss in this video let's start with the first question what is a

start with the first question what is a man Whitney UT test a man Whitney UT

man Whitney UT test a man Whitney UT test tests whether there is a difference

test tests whether there is a difference between two independent samples an

between two independent samples an example is there a difference between

example is there a difference between the reaction time of women and men but

the reaction time of women and men but the T test for independent samples does

the T test for independent samples does the same it also tests whether there is

the same it also tests whether there is the difference between two independent

the difference between two independent samples that's right the manw U test is

samples that's right the manw U test is the nonparametric counter part to the T

the nonparametric counter part to the T test for independent samples but there

test for independent samples but there is an important difference between the

is an important difference between the two tests the T test for independent

two tests the T test for independent samples tests whether there is a mean

samples tests whether there is a mean difference for both samples the mean

difference for both samples the mean value is calculated and it is tested

value is calculated and it is tested whether these mean values differ

whether these mean values differ significantly the man would need you

significantly the man would need you test on the other hand checks whether

test on the other hand checks whether there is a rank sum difference how do we

there is a rank sum difference how do we calculate the rank sum for this purpose

calculate the rank sum for this purpose we sort all persons from the smallest to

we sort all persons from the smallest to the largest value this person has the

the largest value this person has the smallest value so gets rank one this

smallest value so gets rank one this person has the second smallest value so

person has the second smallest value so gets rank two and this person has the

gets rank two and this person has the third smallest value and so on and so

third smallest value and so on and so forth now we have assigned a rank to to

forth now we have assigned a rank to to each person then we can simply add up

each person then we can simply add up the ranks of the first group and the

the ranks of the first group and the second group in the first group we get a

second group in the first group we get a rank of 42 and in the second group a

rank of 42 and in the second group a rank of 36 now we can investigate

rank of 36 now we can investigate whether there is a significant

whether there is a significant difference between these rank sums but

difference between these rank sums but more on that later the advantage of

more on that later the advantage of taking the rank sums rather than the

taking the rank sums rather than the difference in means is that the data

difference in means is that the data need not to be normally distributed so

need not to be normally distributed so in contrast to the T Test the data in

in contrast to the T Test the data in the man with u test do not have to be

the man with u test do not have to be normally distributed what are the

normally distributed what are the hypotheses of the Man witne U test the

hypotheses of the Man witne U test the null hypothesis is in the two samples

null hypothesis is in the two samples the rank sums do not differ

the rank sums do not differ significantly the alternative hypothesis

significantly the alternative hypothesis is in the two samples the rank sums do

is in the two samples the rank sums do differ significantly now let's go

differ significantly now let's go through everything with an example first

through everything with an example first we calculate the example with data Tab

we calculate the example with data Tab and then we see if we can get the same

and then we see if we can get the same results by hand if you like you can load

results by hand if you like you can load the sample data set to follow the

the sample data set to follow the example you can find the link in the

example you can find the link in the video description we simply click on

video description we simply click on data.net and open the statistics

data.net and open the statistics calculator I've already loaded the data

calculator I've already loaded the data from the link here you can also copy

from the link here you can also copy your own data into this table then all

your own data into this table then all you have to do is click on the

you have to do is click on the hypothesis test Tab and then simply

hypothesis test Tab and then simply select the desired variables we measure

select the desired variables we measure the reaction time of a group of men and

the reaction time of a group of men and women and want to know if there is a

women and want to know if there is a difference in reaction time so we click

difference in reaction time so we click on response time and gender we don't

on response time and gender we don't want to calculate a t test for

want to calculate a t test for independent samples but a man Whitney U

independent samples but a man Whitney U test so let's just click on

test so let's just click on non-parametric test here we see the

non-parametric test here we see the results of the Man whitne U test if

results of the Man whitne U test if you're not sure how to interpret the

you're not sure how to interpret the results just click on summary inverts

results just click on summary inverts for the given data a Man withney U test

for the given data a Man withney U test showed that the difference between

showed that the difference between female and male with respect to the

female and male with respect to the dependent variable response time was not

dependent variable response time was not statistically significant thus the null

statistically significant thus the null hypothesis is not

hypothesis is not rejected so we now calculate the man

rejected so we now calculate the man with u test by hand for this we have

with u test by hand for this we have plotted the values in a table on one

plotted the values in a table on one side we have gender with female and male

side we have gender with female and male and on the other side the values for

and on the other side the values for reaction time unfortunately the data is

reaction time unfortunately the data is not normally distributed so we cannot

not normally distributed so we cannot use a t test and we calculate the Man

use a t test and we calculate the Man withney U test instead first we assign a

withney U test instead first we assign a rank to each value we pick the smallest

rank to each value we pick the smallest value which is 33 which gets the rank

value which is 33 which gets the rank one the second smallest value is 34

one the second smallest value is 34 which gets the rank two the third

which gets the rank two the third smallest value is 35 five which gets the

smallest value is 35 five which gets the rank three now we do the same for all

rank three now we do the same for all other values so now we have all ranks

other values so now we have all ranks assigned and we can just add up all the

assigned and we can just add up all the ranks from women and all the ranks from

ranks from women and all the ranks from Men the rank sum is abbreviated with t

Men the rank sum is abbreviated with t and we get T1 for female with 2 + 4 + 7

and we get T1 for female with 2 + 4 + 7 + 9 + 10 + 5 which is 37 now we do the

+ 9 + 10 + 5 which is 37 now we do the same for male here we get get 11 + 1 + 3

same for male here we get get 11 + 1 + 3 + 6 + 8 which is 29 again our Nal

+ 6 + 8 which is 29 again our Nal habesis is that both rank sums are equal

habesis is that both rank sums are equal now we want to calculate the P value for

now we want to calculate the P value for this we have once calculated the rank

this we have once calculated the rank sum for the female participants and we

sum for the female participants and we have the number of cases of six

have the number of cases of six therefore we have six female subjects we

therefore we have six female subjects we can now calculate the U1 that that is

can now calculate the U1 that that is the U for the female participants using

the U for the female participants using this formula here we have N1 and N2 that

this formula here we have N1 and N2 that is the number of cases of female and

is the number of cases of female and male minus the rank sum of the female

male minus the rank sum of the female participants if we insert our values we

participants if we insert our values we get a U1 of 14 we now do exactly the

get a U1 of 14 we now do exactly the same for the male participants and we

same for the male participants and we get a U2 of 16 so now we have calculated

get a U2 of 16 so now we have calculated U1 and U2 the U for the man with u test

U1 and U2 the U for the man with u test is now given by the smaller value of the

is now given by the smaller value of the two so in our case we take the minimum

two so in our case we take the minimum of 14 and 16 this is of course 14 next

of 14 and 16 this is of course 14 next we need to calculate the expected value

we need to calculate the expected value of U which we get by N1 * N2 / 2 in our

of U which we get by N1 * N2 / 2 in our case it is 6 * 5 / 2 and that is equal

case it is 6 * 5 / 2 and that is equal to 15 last but not least we need the

to 15 last but not least we need the standard error of U the standard error

standard error of U the standard error can be calculated with this formula and

can be calculated with this formula and in our case it is equal to

54772 with all these values we can now calculate Z the Z value results with u

calculate Z the Z value results with u minus mu U divided by the standard error

minus mu U divided by the standard error in our case we get 14 - 15 / 5

in our case we get 14 - 15 / 5 54772 which is equal to -

01825 so now we have the set value and with the set value we can calculate the

with the set value we can calculate the P value however it should be noted

P value however it should be noted depending on how large the sample is the

depending on how large the sample is the P value for the man with u test is

P value for the man with u test is calculated in different ways for up to

calculated in different ways for up to 25 cases the exact values are used which

25 cases the exact values are used which can be read from a table table for large

can be read from a table table for large samples the normal distribution of the U

samples the normal distribution of the U value can be used as an approximation in

value can be used as an approximation in our example we would actually use the

our example we would actually use the exact values nevertheless we assume a

exact values nevertheless we assume a normal distribution for this we can

normal distribution for this we can simply go to data and calculate the P

simply go to data and calculate the P value for a given set value the P value

value for a given set value the P value of

of 0.855 is significantly greater than the

0.855 is significantly greater than the significance level of 0.05

significance level of 0.05 and thus the null hypothesis cannot be

and thus the null hypothesis cannot be rejected based on this sample how to

rejected based on this sample how to calculate the man with you test on tide

calculate the man with you test on tide ranks you can learn in our tutorial on

ranks you can learn in our tutorial on data.net you find the link in the video

data.net you find the link in the video description but what if we want to

description but what if we want to compare to dependent samples and need a

compare to dependent samples and need a non-parametric test let's take a look at

non-parametric test let's take a look at the Willcox signed rank test in this

the Willcox signed rank test in this video I will explain the will coxen test

video I will explain the will coxen test to you with will go through what will

to you with will go through what will coxen test is what the assumptions are

coxen test is what the assumptions are and how it is calculated and at the end

and how it is calculated and at the end I will show you how you can easily

I will show you how you can easily calculate the will coxen test online

calculate the will coxen test online with data Tab and we get started right

with data Tab and we get started right now the will coxen test analyzes whether

now the will coxen test analyzes whether there's a difference between two

there's a difference between two dependent samples or not therefore if

dependent samples or not therefore if you have two dependent groups you want

you have two dependent groups you want to test whether there is a difference

to test whether there is a difference between these two groups then and you

between these two groups then and you can use the will coxen test now you

can use the will coxen test now you rightly say hey the T test for dependent

rightly say hey the T test for dependent samples does the same thing it also

samples does the same thing it also tests whether there's a difference

tests whether there's a difference between two dependent groups that's

between two dependent groups that's correct of course the will coxen test is

correct of course the will coxen test is the nonparametric counterpart of the T

the nonparametric counterpart of the T test for dependent samples the special

test for dependent samples the special thing about will coxen test is that your

thing about will coxen test is that your data do not have to be normally

data do not have to be normally distributed to put it simple if your

distributed to put it simple if your data are normally distributed you use a

data are normally distributed you use a parametric test in the case of two

parametric test in the case of two dependent samples this is the T test for

dependent samples this is the T test for dependent

dependent samples if your data is not normally

samples if your data is not normally distributed you use a nonparametric test

distributed you use a nonparametric test in the case of two dependent samples

in the case of two dependent samples this would be the wil coxen

this would be the wil coxen test now of course you could say hm well

test now of course you could say hm well then I'll just always use Willl coxen

then I'll just always use Willl coxen test and I don't even have to check the

test and I don't even have to check the normal distribution at the end of this

normal distribution at the end of this video I will show you why you should

video I will show you why you should always use the T Test if it's possible

always use the T Test if it's possible to do

to do so first I have a little reminder for

so first I have a little reminder for you what dependent samples are in

you what dependent samples are in dependent samples the measured values

dependent samples the measured values are always available in pairs the pairs

are always available in pairs the pairs result from for example repeated

result from for example repeated measures of the same same person but

measures of the same same person but what is the difference now between the T

what is the difference now between the T test for dependent samples and the will

test for dependent samples and the will coxen test the T test for dependent

coxen test the T test for dependent samples tests whether there's a

samples tests whether there's a difference in

difference in means if we have a dependent sample say

means if we have a dependent sample say we took a value from each person once in

we took a value from each person once in the morning and once in the evening then

the morning and once in the evening then we can calculate the difference from

we can calculate the difference from each pair so for example we would have

each pair so for example we would have 45 5 - 34 which equals

45 5 - 34 which equals 11 the T test for dependent sample now

11 the T test for dependent sample now tests whether these differences differ

tests whether these differences differ from zero or not in The Wil coxon test

from zero or not in The Wil coxon test we don't use the differences of means

we don't use the differences of means but we form ranks and then we compare

but we form ranks and then we compare these ranks with each other three is the

these ranks with each other three is the smallest value in terms of amount it

smallest value in terms of amount it gets rank one four four is the second

gets rank one four four is the second smallest value and it gets rank two six

smallest value and it gets rank two six gets rank three and 11 gets rank four we

gets rank three and 11 gets rank four we assign a plus to all positive values and

assign a plus to all positive values and a minus to all negative values but don't

a minus to all negative values but don't worry we will go through this slowly and

worry we will go through this slowly and we'll also look at an example now we go

we'll also look at an example now we go to the assumptions and the hypotheses in

to the assumptions and the hypotheses in The Wil coxon test only two dependent r

The Wil coxon test only two dependent r random samples with at least ordinary

random samples with at least ordinary scaled characteristics need to be

scaled characteristics need to be present the variables do not have to

present the variables do not have to satisfy a distribution curve what should

satisfy a distribution curve what should be mentioned now however is that the

be mentioned now however is that the distribution shape of the differences of

distribution shape of the differences of the two dependent samples should be

the two dependent samples should be approximately symmetric the null

approximately symmetric the null hypothesis in the will coxen test is

hypothesis in the will coxen test is that there is no difference in the

that there is no difference in the so-called central tendency of the two

so-called central tendency of the two samples in the population that is that

samples in the population that is that there is no difference between the

there is no difference between the dependent groups the alternative

dependent groups the alternative hypothesis is that there is a difference

hypothesis is that there is a difference in the central tendency in the

in the central tendency in the population so we expect that the two

population so we expect that the two dependent groups are different so now we

dependent groups are different so now we finally look at a quite simple example

finally look at a quite simple example let's say you have measured the reaction

let's say you have measured the reaction time of a small group group of people

time of a small group group of people once in the morning and once in the

once in the morning and once in the evening and you want to know if there is

evening and you want to know if there is a difference between morning and evening

a difference between morning and evening in order to do this you measure the

in order to do this you measure the reaction time of seven people in the

reaction time of seven people in the morning and in the evening the measured

morning and in the evening the measured values are therefore available now in

values are therefore available now in pairs and now you want to calculate the

pairs and now you want to calculate the difference between morning and evening

difference between morning and evening if the difference would be normally

if the difference would be normally distributed you would use a t test for

distributed you would use a t test for dependent samples if it is not we would

dependent samples if it is not we would use a wil coxen test let's just assume

use a wil coxen test let's just assume that there is no normal distribution and

that there is no normal distribution and we need to calculate a will coxen test

we need to calculate a will coxen test in order to do this the first thing we

in order to do this the first thing we do is we form ranks we look for the

do is we form ranks we look for the smallest value in terms of amount that

smallest value in terms of amount that is minus 2 which gets rank one what is

is minus 2 which gets rank one what is the second smallest value that is three

the second smallest value that is three which gets ranked two and so on and so

which gets ranked two and so on and so forth until we have ranked all the

forth until we have ranked all the values the next thing is that we look at

values the next thing is that we look at the differences and we try to figure out

the differences and we try to figure out which difference is positive and which

which difference is positive and which is a negative one for the negative

is a negative one for the negative differences we simply add a minus then

differences we simply add a minus then we can add up the positive ranks and the

we can add up the positive ranks and the negative ranks for the positive r ranks

negative ranks for the positive r ranks we get 7 + 2 + 3 + 4 + 6 which is equal

we get 7 + 2 + 3 + 4 + 6 which is equal to

to 22 for the negative ranks we get 5 + 1

22 for the negative ranks we get 5 + 1 which is equal to 6 if there is no

which is equal to 6 if there is no difference between morning and evening

difference between morning and evening the positive and negative ranks should

the positive and negative ranks should be approximately equal therefore the

be approximately equal therefore the null hypothesis is that both rank sums

null hypothesis is that both rank sums are equal but how can we test test this

are equal but how can we test test this now we have the sum of ranks and we use

now we have the sum of ranks and we use it to calculate the test statistic W

it to calculate the test statistic W this is simply the minimum value of t+

this is simply the minimum value of t+ and T minus in our case it is the

and T minus in our case it is the minimum value of 22 and six and

minimum value of 22 and six and therefore the test statistic W in our

therefore the test statistic W in our case is six now we can further calculate

case is six now we can further calculate the value for t or W that we would

the value for t or W that we would expect if there was no difference

expect if there was no difference between morning and evening in this case

between morning and evening in this case we would get a value of

we would get a value of 14 therefore if there is no difference

14 therefore if there is no difference between morning and evening we would

between morning and evening we would actually expect a value for t plus and T

actually expect a value for t plus and T minus of 14 and thus W would also be 14

minus of 14 and thus W would also be 14 further we can calculate the standard

further we can calculate the standard deviation this is given by this to be

deviation this is given by this to be fair a little bit complicated formula

fair a little bit complicated formula once we finished with that we can now

once we finished with that we can now calculate the set value the set value is

calculate the set value the set value is obtained by calculating W minus mu and

obtained by calculating W minus mu and dividing that by the standard deviation

dividing that by the standard deviation so we compare the value that would be

so we compare the value that would be expected if there was no difference with

expected if there was no difference with the value that actually occurred note

the value that actually occurred note that if there are more than 25 cases

that if there are more than 25 cases normal distribution is assumed in which

normal distribution is assumed in which case we can calculate the set value

case we can calculate the set value using this formula if there are less

using this formula if there are less than 25 values the critical T value is

than 25 values the critical T value is read from a table of critical T values

read from a table of critical T values therefore in our case we would actually

therefore in our case we would actually use the table now I will show you how

use the table now I will show you how you can easily calculate the will coxen

you can easily calculate the will coxen test online and then I will go into why

test online and then I will go into why I should always prefer the dependent T

I should always prefer the dependent T Test to the will coxen test if it's

Test to the will coxen test if it's possible in order to calculate the will

possible in order to calculate the will coxen test simply go to data.net you

coxen test simply go to data.net you will also find the link in the video

will also find the link in the video description and you copy your own data

description and you copy your own data into this

table then you click on this tab and you will see the names of all your variables

will see the names of all your variables that you copied into the table above

that you copied into the table above underneath this tab many hypothesis

underneath this tab many hypothesis tests are summarized and data tab

tests are summarized and data tab automatically suggests the appropriate

automatically suggests the appropriate hypothesis test for your data if you now

hypothesis test for your data if you now select morning and evening data daab

select morning and evening data daab automatically recognizes that it is a

automatically recognizes that it is a dependent sample and calculates the

dependent sample and calculates the dependent T Test but we don't want to

dependent T Test but we don't want to calculate a T Test we want to calculate

calculate a T Test we want to calculate the Willl coxen test so we just click

the Willl coxen test so we just click here now data automatically calculates

here now data automatically calculates The Wil coxen test here we can read the

The Wil coxen test here we can read the negative and positive ranks and here we

negative and positive ranks and here we see the Z value and the P value if you

see the Z value and the P value if you don't know exactly how this is

don't know exactly how this is interpreted just look at the summary in

interpreted just look at the summary in words it says that the morning group had

words it says that the morning group had lower values than the evening group and

lower values than the evening group and a wil coxen test showed that this

a wil coxen test showed that this difference was not statistically

difference was not statistically significant P equals 312 and now we come

significant P equals 312 and now we come as promised to the point why you should

as promised to the point why you should always prefer a parametric test for

always prefer a parametric test for example the T Test to a non-parametric

example the T Test to a non-parametric test we already discussed that The Wil

test we already discussed that The Wil coxen test has fewer requirements than a

coxen test has fewer requirements than a t test now of course the question may be

t test now of course the question may be why do I use parametric tests like the T

why do I use parametric tests like the T Test at all parametric tests usually

Test at all parametric tests usually have a greater test strength than

have a greater test strength than nonparametric tests what does that mean

nonparametric tests what does that mean say you have formulated your null

say you have formulated your null hypothesis for example reactivity is the

hypothesis for example reactivity is the same in the morning and in the evening

same in the morning and in the evening whether the null hypothesis is rejected

whether the null hypothesis is rejected depends among other things on the

depends among other things on the difference in

difference in responsiveness and also on the sample

responsiveness and also on the sample size in a parametric test usually a

size in a parametric test usually a smaller difference or a smaller sample

smaller difference or a smaller sample is sufficient to reject the null

is sufficient to reject the null hypothesis so if possible always use a

hypothesis so if possible always use a parametric test and finally we can take

parametric test and finally we can take a look at the non-parametric

a look at the non-parametric counterparts of the Anova let's start

counterparts of the Anova let's start with the cral Wallace test this tutorial

with the cral Wallace test this tutorial is about the cral Wallace test if you

is about the cral Wallace test if you want to know what the cral Wallace test

want to know what the cral Wallace test is and how it can be calculated and

is and how it can be calculated and interpreted you're at the right place

interpreted you're at the right place and at the end of this video I will show

and at the end of this video I will show you how you can easily calculate the

you how you can easily calculate the Cross C Wallace test online and we get

Cross C Wallace test online and we get started right now the cral Wallis test

started right now the cral Wallis test is a hypothesis test that is used when

is a hypothesis test that is used when you want to test whether there is a

you want to test whether there is a difference between several independent

difference between several independent groups now you may wand a little bit and

groups now you may wand a little bit and say hey if there are several independent

say hey if there are several independent groups I use an analysis of variant

groups I use an analysis of variant that's right but if your data are not

that's right but if your data are not normally dist distributed and the

normally dist distributed and the assumptions for the analysis of variance

assumptions for the analysis of variance are not met the cral Wallace test is

are not met the cral Wallace test is used the cral Wallis test is the

used the cral Wallis test is the non-parametric counterpart of the single

non-parametric counterpart of the single factor analysis of variance I will now

factor analysis of variance I will now show you what that means there is an

show you what that means there is an important difference between the two

important difference between the two tests the analysis of variance tests if

tests the analysis of variance tests if there is a difference in means so when

there is a difference in means so when we have our groups we calculate the mean

we have our groups we calculate the mean of the group and check if all the means

of the group and check if all the means are equal when we look at a cross Cod

are equal when we look at a cross Cod wace test on the other hand we don't

wace test on the other hand we don't check if the means are equal we check if

check if the means are equal we check if the rank sums of all the groups are

the rank sums of all the groups are equal what does that mean now what is a

equal what does that mean now what is a rank and what is a rank sum in the cral

rank and what is a rank sum in the cral volis test we do not use the actual

volis test we do not use the actual measured values but we sort all people

measured values but we sort all people by size and then the person with the

by size and then the person with the small smallest value gets the new value

small smallest value gets the new value or rank one the person with the second

or rank one the person with the second smallest value gets rank two the person

smallest value gets rank two the person with the third smallest value gets rank

with the third smallest value gets rank three and so on and so forth until each

three and so on and so forth until each person has been assigned a rank now we

person has been assigned a rank now we have assigned a rank to each person and

have assigned a rank to each person and then we can simply add up the ranks from

then we can simply add up the ranks from the first group add up the ranks from

the first group add up the ranks from the second group and add up the ranks

the second group and add up the ranks from the Third group in this case we get

from the Third group in this case we get a rank sum of 42 for the first group 70

a rank sum of 42 for the first group 70 for the second group and 47 for the

for the second group and 47 for the third group the big Advantage is that if

third group the big Advantage is that if we do not look at the main difference

we do not look at the main difference but at the rank sum the data does not

but at the rank sum the data does not have to be normally distributed when

have to be normally distributed when using the cral wace test our data does

using the cral wace test our data does not have to satisfy any distributional

not have to satisfy any distributional form and therefore we also don't need

form and therefore we also don't need needed to be normally distributed before

needed to be normally distributed before we discuss how the crustal wace test is

we discuss how the crustal wace test is calculated and don't worry it's really

calculated and don't worry it's really not complicated we first take a look at

not complicated we first take a look at the assumptions when do we use the cral

the assumptions when do we use the cral Wallace test we use the cral volis test

Wallace test we use the cral volis test if we have a nominal or ordinal variable

if we have a nominal or ordinal variable with more than two values and a matric

with more than two values and a matric variable a nominal or ordinal variable

variable a nominal or ordinal variable with more than two values is for example

with more than two values is for example the variable preferred newspaper with

the variable preferred newspaper with the values Washington Post New York

the values Washington Post New York Times USA Today it could also be

Times USA Today it could also be frequency of Television viewing with

frequency of Television viewing with daily several times a week rarely never

daily several times a week rarely never a matric variable is for example salary

a matric variable is for example salary well-being or weight of people what are

well-being or weight of people what are the assumptions now only several

the assumptions now only several independent random samples with at least

independent random samples with at least ordinary scaled characteristics must be

ordinary scaled characteristics must be available the variables do not have to

available the variables do not have to satisfy a distribution curve so the null

satisfy a distribution curve so the null hypothesis is the independent samples

hypothesis is the independent samples all have the same central tendency and

all have the same central tendency and therefore come from the same population

therefore come from the same population or in other words there's no difference

or in other words there's no difference in the rank sums and the alternative

in the rank sums and the alternative hypothesis could be at least one of the

hypothesis could be at least one of the independent samples does not have the

independent samples does not have the same central tendency as as the other

same central tendency as as the other samples and therefore comes from a

samples and therefore comes from a different population or to say it in

different population or to say it in other words again at least one group

other words again at least one group differs in rank sums so the next

differs in rank sums so the next question is how do we calculate a ccal

question is how do we calculate a ccal Wallace test it's not

Wallace test it's not difficult let's say you have measured

difficult let's say you have measured the reaction time of three groups group

the reaction time of three groups group a group b and Group C and now you want

a group b and Group C and now you want to know if there's a difference between

to know if there's a difference between the groups in terms of reaction time

the groups in terms of reaction time time let's say you have written down the

time let's say you have written down the measured reaction time in a table let's

measured reaction time in a table let's just assume that the data is not

just assume that the data is not normally distributed and therefore you

normally distributed and therefore you have to use the cral volis test so then

have to use the cral volis test so then our null hypothesis is that there is no

our null hypothesis is that there is no difference between the

difference between the groups and we're going to test that

groups and we're going to test that right

right now first we assign a rank to each

now first we assign a rank to each person this is the smallest value so

person this is the smallest value so this person gets rank one this is the

this person gets rank one this is the second smallest value so this person

second smallest value so this person gets rank two and we do this now for all

gets rank two and we do this now for all people if the groups have no influence

people if the groups have no influence on reaction time the ranks should

on reaction time the ranks should actually be distributed purely randomly

actually be distributed purely randomly in the second step we now calculate the

in the second step we now calculate the rank sum and the mean rank sum for the

rank sum and the mean rank sum for the first group the rank sum is 2 + 4 + 7 +

first group the rank sum is 2 + 4 + 7 + 9 which is equal to 22 and we have four

9 which is equal to 22 and we have four people in the group so the mean rank sum

people in the group so the mean rank sum is 22 / 4 which equals

is 22 / 4 which equals 5.5 now we do the same for the second

5.5 now we do the same for the second group here we get a rank sum of 27 and

group here we get a rank sum of 27 and the mean rank sum of

the mean rank sum of 6.75 and for the third group we get a

6.75 and for the third group we get a rank sum of 29 and a mean rank sum of

rank sum of 29 and a mean rank sum of 7.25 now we can calculate the expected

7.25 now we can calculate the expected value of the rank sums the expected

value of the rank sums the expected value if there is no difference in the

value if there is no difference in the groups would be that each group would

groups would be that each group would have a rank sum of

have a rank sum of 6.5 we've now almost got everything we

6.5 we've now almost got everything we need we interviewed 12 people so the

need we interviewed 12 people so the number of cases is 12 the expected value

number of cases is 12 the expected value of the ranks is

of the ranks is 6.5 we've also calculated the mean rank

6.5 we've also calculated the mean rank sums of the individual groups the

sums of the individual groups the degrees of freedom in our case are two

degrees of freedom in our case are two and these are simply given by the number

and these are simply given by the number of groups minus one which makes 3 - 1

of groups minus one which makes 3 - 1 last we need the variance the variance

last we need the variance the variance of ranks is given by n to the^ of 2 -1

of ranks is given by n to the^ of 2 -1 divided by 12 n is again the number of

divided by 12 n is again the number of people so 12 so we get a variance of

people so 12 so we get a variance of 11.9 to now we've got everything we need

11.9 to now we've got everything we need with these values we can now calculate

with these values we can now calculate our test value H the test statistic H

our test value H the test statistic H corresponds to the G Square value and is

corresponds to the G Square value and is given by this formula n is the number of

given by this formula n is the number of cases R Bar is the mean rank sum of the

cases R Bar is the mean rank sum of the individual groups and E is the expected

individual groups and E is the expected value of the ranks Sigma squared is the

value of the ranks Sigma squared is the variance of the ranks in our case the

variance of the ranks in our case the number of cases is 12 we always have

number of cases is 12 we always have four people per group so we can pull out

four people per group so we can pull out the ne 5.5 is the mean rank of group a

the ne 5.5 is the mean rank of group a 6.75 is the mean rank of Group B and

6.75 is the mean rank of Group B and 7.25 is the mean rank of Group C this

7.25 is the mean rank of Group C this gives us a rounded AG value of

gives us a rounded AG value of 0.5 as we just said this value

0.5 as we just said this value corresponds to the G Square value so now

corresponds to the G Square value so now we can easily read the critical gsquare

we can easily read the critical gsquare value in the table of critical G Square

value in the table of critical G Square values you find this table on data.net

values you find this table on data.net we have two degrees of freedom and if we

we have two degrees of freedom and if we assume that we have a significance level

assume that we have a significance level of

of 0.05 we get a critical G squ value of

0.05 we get a critical G squ value of 5991 so of course our value is smaller

5991 so of course our value is smaller than the critical G squ value and So

than the critical G squ value and So based on our example data the Nile

based on our example data the Nile hypothesis is retained and now I will

hypothesis is retained and now I will show you how I can easily calculate the

show you how I can easily calculate the cral volis test online with data tab in

cral volis test online with data tab in order to do this you simply visit

order to do this you simply visit data.net net you will find a link in the

data.net net you will find a link in the video description and then you click on

video description and then you click on the statistics calculator and insert

the statistics calculator and insert your own data into this

table further you click on this tab and under this tab you will find many

under this tab you will find many hypothesis tests and when you select the

hypothesis tests and when you select the variables you want to test data tab will

variables you want to test data tab will suggest the appropriate test after

suggest the appropriate test after you've copied your data into the table

you've copied your data into the table you will see the reaction time and group

you will see the reaction time and group right here at the bottom now we simply

right here at the bottom now we simply click on reaction time and group and

click on reaction time and group and data tab automatically calculates an

data tab automatically calculates an analysis of variance for us but we don't

analysis of variance for us but we don't want an analysis of variance we want the

want an analysis of variance we want the non-parametric test so we just click

non-parametric test so we just click here now data automatically calculates

here now data automatically calculates the cral volis test we also get a CH

the cral volis test we also get a CH Square value of

Square value of 0.5 the degrees of freedom are two and

0.5 the degrees of freedom are two and the calculated P value is

the calculated P value is 0.779 and here below you can read the

0.779 and here below you can read the interpretation in words across calist

interpretation in words across calist has showed that there is no significant

has showed that there is no significant difference between the categories P

difference between the categories P makes

makes 0.779 therefore with the data used the

0.779 therefore with the data used the null hypothesis is not rejected if we

null hypothesis is not rejected if we have three or more dependent samples we

have three or more dependent samples we can use the fredman test as a

can use the fredman test as a nonparametric alternative to the

nonparametric alternative to the repeated measures Anover this video is

repeated measures Anover this video is about the Freedman test and we start

about the Freedman test and we start right away with the first question what

right away with the first question what is the fredman

is the fredman test the fredman test analyzes whether

test the fredman test analyzes whether there are statistically significant

there are statistically significant differences between three or more

differences between three or more dependent

dependent samples what is a dependent sample again

samples what is a dependent sample again in a dependent sample the measured

in a dependent sample the measured values are linked for example if a

values are linked for example if a sample is drawn of people who have knee

sample is drawn of people who have knee surgery and these people are interviewed

surgery and these people are interviewed before the surgery and one and two weeks

before the surgery and one and two weeks after the surgery it is a dependent

after the surgery it is a dependent sample this is the case because the same

sample this is the case because the same person was interviewed at multiple time

person was interviewed at multiple time points now you might rightly say that

points now you might rightly say that the analysis of variance with repeated

the analysis of variance with repeated measures tests exactly the same thing

measures tests exactly the same thing since it also tests whether there is a

since it also tests whether there is a difference between three or more

difference between three or more dependent samples that is correct the

dependent samples that is correct the Freedman test is the nonparametric

Freedman test is the nonparametric counter part of the analysis of variance

counter part of the analysis of variance with repeated measures but what is the

with repeated measures but what is the difference between the two tests the

difference between the two tests the analysis of variance tests the extent to

analysis of variance tests the extent to which the measured values of the

which the measured values of the dependent sample differ in the Freedman

dependent sample differ in the Freedman test on the other hand it is not the

test on the other hand it is not the actual measured values that are used but

actual measured values that are used but the ranks the time where a person has

the ranks the time where a person has the highest value gets rank one the time

the highest value gets rank one the time with the second highest value gets rank

with the second highest value gets rank two and the time with the smallest value

two and the time with the smallest value gets rank three this is now done for all

gets rank three this is now done for all people or for all rows afterwards the

people or for all rows afterwards the ranks of the single points of time are

ranks of the single points of time are added up at the first time point we get

added up at the first time point we get a sum of seven at the second time point

a sum of seven at the second time point we get a sum of eight and at the third

we get a sum of eight and at the third time point we get a sum of nine now we

time point we get a sum of nine now we can check how much these rank sums

can check how much these rank sums differ from each

differ from each other but why are rank sums used the big

other but why are rank sums used the big Advantage is that if you don't look at

Advantage is that if you don't look at the mean differences but at the rank sum

the mean differences but at the rank sum the data doesn't have to be normal

the data doesn't have to be normal distributed to put it simple if your

distributed to put it simple if your data are normally distributed parametric

data are normally distributed parametric tests are used for more than two

tests are used for more than two dependent samples this is the analysis

dependent samples this is the analysis of variance with repeated measures if

of variance with repeated measures if your data are not normally distributed

your data are not normally distributed nonparametric tests are used for more

nonparametric tests are used for more than two dependent samples this is the

than two dependent samples this is the fredman test this leads us to the

fredman test this leads us to the research question that you can answer

research question that you can answer with with the freedan test is there a

with with the freedan test is there a significant difference between more than

significant difference between more than two dependent groups let's have a look

two dependent groups let's have a look at that with an example you might be

at that with an example you might be interested to know whether therapy after

interested to know whether therapy after a slipped disc has an influence on the

a slipped disc has an influence on the patient's perception of pain for this

patient's perception of pain for this purpose you measure the pain perception

purpose you measure the pain perception before the therapy in the middle of the

before the therapy in the middle of the therapy and at the end of the therapy

therapy and at the end of the therapy now you want to know if there is a

now you want to know if there is a difference between the different times

difference between the different times so your independent variable is time or

so your independent variable is time or therapy progressing over time your

therapy progressing over time your dependent variable is the pain

dependent variable is the pain perception you now have a history of the

perception you now have a history of the pain perception of each person over time

pain perception of each person over time and now you want to know whether the

and now you want to know whether the therapy has an influence on the pain

therapy has an influence on the pain perception simplified in this case the

perception simplified in this case the therapy has an influence and in that

therapy has an influence and in that that case the therapy has no influence

that case the therapy has no influence on the pain perception in the course of

on the pain perception in the course of time the pain perception does not change

time the pain perception does not change here in this case it

here in this case it does now we also have a good transition

does now we also have a good transition to the

to the hypotheses in the Freedman test the null

hypotheses in the Freedman test the null hypothesis is there are no significant

hypothesis is there are no significant differences between the dependent groups

differences between the dependent groups and the alternative hypothesis is there

and the alternative hypothesis is there is a significant difference between the

is a significant difference between the dependent groups of course as already

dependent groups of course as already mentioned the Freedman test does not use

mentioned the Freedman test does not use the True Values but the ranks we will go

the True Values but the ranks we will go through the formula behind the Freedman

through the formula behind the Freedman test in a moment this brings us to the

test in a moment this brings us to the point of how to calculate the fredman

point of how to calculate the fredman test for the calculation of the Freedman

test for the calculation of the Freedman test you can of course simply use data

test you can of course simply use data or calculate it by hand to be honest

or calculate it by hand to be honest hardly anyone will calculate the fredman

hardly anyone will calculate the fredman test by hand but it will help you to

test by hand but it will help you to understand how the fredman test works

understand how the fredman test works and don't worry it's not that

and don't worry it's not that complicated first I will show you how to

complicated first I will show you how to calculate the fredman test with data Tab

calculate the fredman test with data Tab and then I will show you how to do it by

and then I will show you how to do it by hand in order to do this simply go to

hand in order to do this simply go to data.net and copy your own data into

data.net and copy your own data into this table let's say you want to

this table let's say you want to investigate whether there is a

investigate whether there is a difference in the response time of

difference in the response time of people in the morning at noon and in the

people in the morning at noon and in the evening we simply click on this tab

evening we simply click on this tab under this tab you will find many

under this tab you will find many hypothesis tests and data tab will

hypothesis tests and data tab will automatically suggest an appropriate

automatically suggest an appropriate test if we click on all three variables

test if we click on all three variables morning noon and evening data tab will

morning noon and evening data tab will automatically calculate an analysis of

automatically calculate an analysis of variance with repeated measures but in

variance with repeated measures but in our case we want to calculate the

our case we want to calculate the nonparametric test so we click here now

nonparametric test so we click here now now we get the results for the Freedman

now we get the results for the Freedman test up here you can read the

test up here you can read the descriptive

descriptive statistics and down here you can find

statistics and down here you can find the P value if you don't know exactly

the P value if you don't know exactly how to interpret the P value you can

how to interpret the P value you can just read the interpretation in words

just read the interpretation in words down here a fre has showed that there is

down here a fre has showed that there is no significant difference between the

no significant difference between the variables Kai Square =

variables Kai Square = 2.57 p = 0

2.57 p = 0 .276 if your P value is greater than

.276 if your P value is greater than your set significance level then your

your set significance level then your null hypothesis is retained the null

null hypothesis is retained the null hypothesis is that there is no

hypothesis is that there is no difference between the groups usually a

difference between the groups usually a significance level of 0.05 is used so

significance level of 0.05 is used so this P value is greater than

this P value is greater than 0.05 additionally data gives you the

0.05 additionally data gives you the post Hawk test if your p value is

post Hawk test if your p value is smaller than

smaller than 0.05 the post talk test helps you to

0.05 the post talk test helps you to examine which of the groups really

examine which of the groups really differ so now let's look at what the

differ so now let's look at what the equation behind the Freedman test are

equation behind the Freedman test are and recalculate this example by hand

and recalculate this example by hand here we have the measured values of the

here we have the measured values of the seven people in the first step we have

seven people in the first step we have to assign ranks to the values in order

to assign ranks to the values in order to do this we look at each rows

to do this we look at each rows separately in the first row which is the

separately in the first row which is the first person 45 is the largest value

first person 45 is the largest value this gets rank one then comes 36 with

this gets rank one then comes 36 with rank two and 34 with rank three we do

rank two and 34 with rank three we do the same for the second row here 36 is

the same for the second row here 36 is the largest value and gets rank one then

the largest value and gets rank one then comes 33 with rank two and 31 with rank

comes 33 with rank two and 31 with rank three

three now we do this for each row so for all

now we do this for each row so for all people afterwards we can calculate the

people afterwards we can calculate the rank sum for each point in time for

rank sum for each point in time for example we just sum up all ranks at one

example we just sum up all ranks at one point in time in the morning we get 17

point in time in the morning we get 17 at noon 11 and in the evening

at noon 11 and in the evening 14 if there were no differences between

14 if there were no differences between the different time points in terms of

the different time points in terms of reaction time we would expect the

reaction time we would expect the expected value at all time points the

expected value at all time points the expected value is obtained with this

expected value is obtained with this formula and in this case it is 14 so if

formula and in this case it is 14 so if there is no difference between morning

there is no difference between morning noon and evening we would actually

noon and evening we would actually expect a ranksum of 14 at all three time

expect a ranksum of 14 at all three time points next we can calculate the kai

points next we can calculate the kai Square value we get it with this formula

Square value we get it with this formula n is the number of people which is which

n is the number of people which is which is seven K is the number of time points

is seven K is the number of time points so three and the sum of R SAR is 17

so three and the sum of R SAR is 17 squared + 11 SAR + 14 squar so we get a

squared + 11 SAR + 14 squar so we get a Kai Square value of

Kai Square value of 2.57 now we need the number of degrees

2.57 now we need the number of degrees of freedom this is given by the number

of freedom this is given by the number of time points minus one so in our case

of time points minus one so in our case two finally we can read the critical Kai

two finally we can read the critical Kai Square value in the table of critical

Square value in the table of critical Kai Square values for this we take the

Kai Square values for this we take the predefined significance level let's say

predefined significance level let's say it is

it is 0.05 and the number of degrees of

0.05 and the number of degrees of freedom here we can read that the

freedom here we can read that the critical Kai Square value is

critical Kai Square value is 5.99 this is greater than our calculated

5.99 this is greater than our calculated value therefore the null hypothesis is

value therefore the null hypothesis is not rejected and based on these data

not rejected and based on these data there is no difference between the

there is no difference between the responsiveness at a different time

responsiveness at a different time points therefore the null hypothesis is

points therefore the null hypothesis is not rejected and based on these data

not rejected and based on these data there's no difference between the

there's no difference between the responsiveness at a different time

responsiveness at a different time points if the calculated Kai Square

points if the calculated Kai Square value was greater than the critical one

value was greater than the critical one we would reject the null

we would reject the null hypothesis having explored various

hypothesis having explored various nonparametric tests for ordinal or

nonparametric tests for ordinal or nonnormal distributed matric data now

nonnormal distributed matric data now let's shift our Focus to nominal data

let's shift our Focus to nominal data when our variables are nominal such as

when our variables are nominal such as gender or color preferences we require a

gender or color preferences we require a different kind of statistical test the

different kind of statistical test the kai Square test the kai Square test is a

kai Square test the kai Square test is a powerful tool for analyzing nominal data

powerful tool for analyzing nominal data let's get started what is a Kai Square

let's get started what is a Kai Square test and how is the kai Square test

test and how is the kai Square test calculat

calculat that's what we will discuss in this

that's what we will discuss in this video let's start with the first

video let's start with the first question what is a Kai Square test the

question what is a Kai Square test the kai Square test is a hypothesis test

kai Square test is a hypothesis test that is used when you want to determine

that is used when you want to determine if there is a relationship between two

if there is a relationship between two categorical variables what are

categorical variables what are categorical variables again categorical

categorical variables again categorical variables are for example gender with

variables are for example gender with the categories male and female the

the categories male and female the preferred newspaper with the CATE cies

preferred newspaper with the CATE cies USA Today The Wall Street Journal the

USA Today The Wall Street Journal the New York Times and so on or the highest

New York Times and so on or the highest educational level with the categories

educational level with the categories without graduation College bachelor's

without graduation College bachelor's degree master's degree so gender

degree master's degree so gender preferred newspaper and highest

preferred newspaper and highest educational level are all categorical

educational level are all categorical variables for example no categorical

variables for example no categorical variables are the weight of a person the

variables are the weight of a person the salary of a person or the power consum

salary of a person or the power consum assumption if we now have two

assumption if we now have two categorical variables and we want to

categorical variables and we want to test whether there is a relationship we

test whether there is a relationship we use a Kai Square test for example is

use a Kai Square test for example is there a relationship between gender and

there a relationship between gender and the preferred newspaper we have two

the preferred newspaper we have two categorical variables so we use a Kai

categorical variables so we use a Kai Square test another example is there a

Square test another example is there a relationship between perferred newspaper

relationship between perferred newspaper and highest educational level here again

and highest educational level here again we have two categorical variables so we

we have two categorical variables so we use a Kai Square test however there are

use a Kai Square test however there are two things to note first the Assumption

two things to note first the Assumption for the kai Square test is that the

for the kai Square test is that the expected frequencies per cell are

expected frequencies per cell are greater than five we'll go over what

greater than five we'll go over what that means in a moment second the Ki

that means in a moment second the Ki Square test uses only the categories and

Square test uses only the categories and not the rankings however in the case of

not the rankings however in the case of the highest educational level there's a

the highest educational level there's a ranking of categories if you want to

ranking of categories if you want to account for rankings check out our

account for rankings check out our tutorials on Spearman correlation manw U

tutorials on Spearman correlation manw U test or cross Wallace test but how do we

test or cross Wallace test but how do we calculate the kai Square test let's go

calculate the kai Square test let's go through that with an example we would

through that with an example we would like to investigate whether gender has

like to investigate whether gender has an influence on the preferred newspaper

an influence on the preferred newspaper so our question is is there a

so our question is is there a relationship between gender and the

relationship between gender and the preferred newspaper our null hypothesis

preferred newspaper our null hypothesis is there is no relationship between

is there is no relationship between gender and the preferred newspaper and

gender and the preferred newspaper and our alternative hypothesis is there is a

our alternative hypothesis is there is a relationship between gender and the

relationship between gender and the preferred newspaper so first we create a

preferred newspaper so first we create a questionnaire that asks about gender and

questionnaire that asks about gender and the preferred newspaper we will then

the preferred newspaper we will then send out the questionnaire the results

send out the questionnaire the results of the survey are displayed in a table

of the survey are displayed in a table in this table we see one respondent in

in this table we see one respondent in each row the first respondent is male

each row the first respondent is male and stated New York Post the second

and stated New York Post the second respondent is female and stated USA

respondent is female and stated USA Today We can now copy this table into a

Today We can now copy this table into a statistic software like data deab data

statistic software like data deab data daab then gives us the so-called

daab then gives us the so-called contingency table in this table you can

contingency table in this table you can see the variable newspaper and the

see the variable newspaper and the variable gender

variable gender the number of times each combination

the number of times each combination occurs is plotted in the cells for

occurs is plotted in the cells for example in this survey There are 16

example in this survey There are 16 people who stated New York Post and male

people who stated New York Post and male or 13 people who stated female and New

or 13 people who stated female and New York Post now we want to know if gender

York Post now we want to know if gender has an influence on the preferred

has an influence on the preferred newspaper or put another way is there a

newspaper or put another way is there a relationship between gender and the

relationship between gender and the preferred newspaper now to answer this

preferred newspaper now to answer this question question we use the kai Square

question question we use the kai Square test there are two ways we can calculate

test there are two ways we can calculate the kai Square test either we use a

the kai Square test either we use a statistical software like data tab or we

statistical software like data tab or we calculate the kai Square test by hand we

calculate the kai Square test by hand we start with the uncomplicated variant and

start with the uncomplicated variant and use data tab if you like you can load

use data tab if you like you can load the sample data set for calculation you

the sample data set for calculation you can find the link in the video

can find the link in the video description to calculate a Kai Square

description to calculate a Kai Square test online simply copy your own data

test online simply copy your own data into this t table or use the link to

into this t table or use the link to load this data set then the variables

load this data set then the variables gender and newspaper appear here below

gender and newspaper appear here below now we click on hypothesis test here you

now we click on hypothesis test here you will find a variety of tests and data

will find a variety of tests and data tab will help you to choose the right

tab will help you to choose the right one for example if we click on chender

one for example if we click on chender and newspaper the kai Square test will

and newspaper the kai Square test will be automatically

be automatically calculated now we get the results for

calculated now we get the results for the kai Square test above we see the

the kai Square test above we see the contingency table for the variables

contingency table for the variables gender and newspaper the contingency

gender and newspaper the contingency table shows us how often the respective

table shows us how often the respective combinations occur in our survey female

combinations occur in our survey female and USA Today for example occurs six

and USA Today for example occurs six times in the second table we can see

times in the second table we can see what the contingency table should

what the contingency table should actually look like if the two variables

actually look like if the two variables were perfectly independent that is if

were perfectly independent that is if gender had no influence on the preferred

gender had no influence on the preferred newspaper here it is important to note

newspaper here it is important to note that all of the frequencies should be

that all of the frequencies should be larger than five so that the assumptions

larger than five so that the assumptions of the kai Square test are fulfilled but

of the kai Square test are fulfilled but this is the case here the kai Square

this is the case here the kai Square test now Compares this table with that

test now Compares this table with that table and here we see the results the P

table and here we see the results the P value is

value is 0.91 which is much higher than our

0.91 which is much higher than our significance level of

significance level of 0.05 and therefore we keep the null

0.05 and therefore we keep the null hypothesis if you don't know exactly how

hypothesis if you don't know exactly how to interpret the results just click on

to interpret the results just click on summary in words a Kai Square test was

summary in words a Kai Square test was performed between gender and newspaper

performed between gender and newspaper all expected cell frequences were

all expected cell frequences were greater than five thus the assumptions

greater than five thus the assumptions for the kai Square test were met there

for the kai Square test were met there was no statistically significant

was no statistically significant relationship between gender and

relationship between gender and newspaper this results in a P value of

newspaper this results in a P value of 0.918 which is above the defined

0.918 which is above the defined significance level of 5% the kai Square

significance level of 5% the kai Square test is therefore not significant and

test is therefore not significant and the null hypothesis is not rejected if

the null hypothesis is not rejected if you're unsure what exactly the P value

you're unsure what exactly the P value means just watch our video about the P

means just watch our video about the P value and now we come to the question

value and now we come to the question how to calculate the kai Square test by

how to calculate the kai Square test by hand and we go through the formulas

hand and we go through the formulas needed don't worry it's not difficult we

needed don't worry it's not difficult we need the contingency table with the

need the contingency table with the observed frequencies the contingency

observed frequencies the contingency table with the expected frequencies that

table with the expected frequencies that is those frequencies that would occur

is those frequencies that would occur with perfectly independent variables you

with perfectly independent variables you can find how to calculate the expected

can find how to calculate the expected frequencies on data tab in the tutorial

frequencies on data tab in the tutorial on the kai Square test we can now

on the kai Square test we can now calculate the kai square with this

calculate the kai square with this formula the index k stands for the

formula the index k stands for the respective cell o is the observed

respective cell o is the observed frequency e k is the expected frequency

frequency e k is the expected frequency so we get 6 -

so we get 6 - 6.08 squared divided by

6.08 squared divided by 6.08 plus the next cell 7 -

6.08 plus the next cell 7 - 6.92 SAR divided by

6.92 SAR divided by 6.92 if we do this for all cell

6.92 if we do this for all cell and sum them up we get a Kai square of

0.504 so this results in a Kai Square value of

value of 0.504 now we would like to calculate the

0.504 now we would like to calculate the critical Kai Square value what do we

critical Kai Square value what do we need it for if we use a statistical

need it for if we use a statistical software we simply get a P value

software we simply get a P value displayed if the value is smaller than

displayed if the value is smaller than the significance level for example

the significance level for example 0.05 the the null hypothesis is rejected

0.05 the the null hypothesis is rejected otherwise not in our example case the

otherwise not in our example case the null hypothesis is not rejected by hand

null hypothesis is not rejected by hand however you can't really calculate the P

however you can't really calculate the P value therefore you read off in a table

value therefore you read off in a table which Kai Square value you would get

which Kai Square value you would get with a P value of

with a P value of 0.05 this Kai Square value is called the

0.05 this Kai Square value is called the critical Kai Square value in order to

critical Kai Square value in order to calculate the critical Kai Square value

calculate the critical Kai Square value we need the degrees of freedom these are

we need the degrees of freedom these are obtained by taking the number of rows

obtained by taking the number of rows minus one times the number of columns

minus one times the number of columns minus one we have four rows and two

minus one we have four rows and two columns therefore we get 3 * 1 and thus

columns therefore we get 3 * 1 and thus three degrees of freedom now let's take

three degrees of freedom now let's take a look at the table of critical Kai

a look at the table of critical Kai Square values you can find this table on

Square values you can find this table on data tab the link is in the video

data tab the link is in the video description we select a significance

description we select a significance level of

level of 0.05 and have three degrees of freedom

0.05 and have three degrees of freedom therefore our critical Kai Square value

therefore our critical Kai Square value is

is 7.81 15 the critical Kai Square value of

7.81 15 the critical Kai Square value of 7.81 five is larger than our calculated

7.81 five is larger than our calculated Kai Square value of

Kai Square value of 0.504 thus the null hypothesis is

0.504 thus the null hypothesis is retained up until now we focused on

retained up until now we focused on statistical tests designed to compare

statistical tests designed to compare groups or categories another fundamental

groups or categories another fundamental aspect of data analysis is understanding

aspect of data analysis is understanding the relationship between

the relationship between variables this is where correlation

variables this is where correlation analysis comes into play Let's

analysis comes into play Let's transition from tests of differences to

transition from tests of differences to measures of Association in the following

measures of Association in the following video this video is about correlation

video this video is about correlation analysis

analysis we start by asking what a correlation

we start by asking what a correlation analysis is we will then look at the

analysis is we will then look at the most important correlation analysis

most important correlation analysis Pearson correlation Spearman correlation

Pearson correlation Spearman correlation candles Tow and Point by zerial

candles Tow and Point by zerial correlation and finally we will discuss

correlation and finally we will discuss the difference between correlation and

the difference between correlation and cation let's start with the first

cation let's start with the first question what is a correlation analysis

question what is a correlation analysis correlation analysis is a statistical

correlation analysis is a statistical method used to measure meure the

method used to measure meure the relationship between two variables for

relationship between two variables for example is there a relationship between

example is there a relationship between a person's salary and age in this CER

a person's salary and age in this CER plot every single point is a person in

plot every single point is a person in correlation analysis we usually want to

correlation analysis we usually want to know two things number one how strong

know two things number one how strong the correlation is and number two in

the correlation is and number two in which direction the correlation goes we

which direction the correlation goes we can read both in the correlation

can read both in the correlation coefficient which is is between min-1

coefficient which is is between min-1 and 1 the strength of the correlation

and 1 the strength of the correlation can be read in a table if R is between 0

can be read in a table if R is between 0 and 0.1 we speak of no correlation if R

and 0.1 we speak of no correlation if R is between 0.7 and one we speak of a

is between 0.7 and one we speak of a very strong correlation a positive

very strong correlation a positive correlation exists when high values of

correlation exists when high values of one variable go along with high values

one variable go along with high values of the other variable or when small

of the other variable or when small values of one variable go along with

values of one variable go along with small values of the other variable a

small values of the other variable a positive correlation is found for

positive correlation is found for example for body size and shoe size the

example for body size and shoe size the result is a positive correlation

result is a positive correlation coefficient a negative correlation

coefficient a negative correlation exists when high values of one variable

exists when high values of one variable go along with low values of the other

go along with low values of the other variable and vice versa a negative

variable and vice versa a negative correlation usually exists between

correlation usually exists between product price price and sales volume the

product price price and sales volume the result is a negative correlation

result is a negative correlation coefficient now we have different

coefficient now we have different correlation coefficients the most

correlation coefficients the most popular are peon correlation coefficient

popular are peon correlation coefficient R Spearman correlation coefficient RS

R Spearman correlation coefficient RS canel Tow and Point by zal correlation

canel Tow and Point by zal correlation coefficient

coefficient rpb let's start with the first the peon

rpb let's start with the first the peon correlation coefficient what is a peon

correlation coefficient what is a peon correlation as all all correlation

correlation as all all correlation coefficients the pon correlation R is a

coefficients the pon correlation R is a statistical measure that quantifies the

statistical measure that quantifies the relationship between two variables in

relationship between two variables in the case of pearon correlation the

the case of pearon correlation the linear relationship of matric variables

linear relationship of matric variables is measured more about metric variables

is measured more about metric variables later so with the help of peon

later so with the help of peon correlation we can measure the linear

correlation we can measure the linear relationship between two variables and

relationship between two variables and of course the peon correlation

of course the peon correlation coefficient R tells tells us how strong

coefficient R tells tells us how strong the correlation is and in which

the correlation is and in which direction the correlation goes how is p

direction the correlation goes how is p and correlation calculated the p and

and correlation calculated the p and correlation coefficient is obtained via

correlation coefficient is obtained via this equation where R is the p and

this equation where R is the p and correlation coefficient x i are the

correlation coefficient x i are the individual values of one variable for

individual values of one variable for example h y i are the individual values

example h y i are the individual values of the other variable for example cell

of the other variable for example cell X Das and Y dash are respectively the

X Das and Y dash are respectively the mean values of the two variables in the

mean values of the two variables in the equation we can see that the respective

equation we can see that the respective mean value is first substracted from

mean value is first substracted from both values so in our example we

both values so in our example we calculate the mean values of age and

calculate the mean values of age and salary we then subtract the mean values

salary we then subtract the mean values from each person's age and

from each person's age and salary then we multiply both Val values

salary then we multiply both Val values and we sum up the individual results of

and we sum up the individual results of the multiplication the expression in the

the multiplication the expression in the denominator ensures that the correlation

denominator ensures that the correlation coefficient is scaled between minus1 and

coefficient is scaled between minus1 and 1 if we now multiply two positive values

1 if we now multiply two positive values we get a positive value so all values

we get a positive value so all values that lie in this area have a positive

that lie in this area have a positive influence on the correlation coefficient

influence on the correlation coefficient if we multiply two negative values we

if we multiply two negative values we also get a positive value minus * minus

also get a positive value minus * minus is plus so all values that lie in this

is plus so all values that lie in this area also have a positive influence on

area also have a positive influence on the correlation coefficient if we

the correlation coefficient if we multiply a positive value and a negative

multiply a positive value and a negative value we get a negative value minus *

value we get a negative value minus * plus is minus so all values that lie in

plus is minus so all values that lie in these ranges have a negative influence

these ranges have a negative influence on the correlation coefficient therefore

on the correlation coefficient therefore if our values are predominantly in these

if our values are predominantly in these two areas we get a positive correlation

two areas we get a positive correlation coefficient and thus a positive

coefficient and thus a positive relationship if our values are

relationship if our values are predominantly in these two areas we get

predominantly in these two areas we get a negative correlation coefficient and

a negative correlation coefficient and thus a negative relationship if the

thus a negative relationship if the points are distributed over all four

points are distributed over all four areas the positive terms and the

areas the positive terms and the negative terms cancel each other out and

negative terms cancel each other out and we get a very small or no correlation

we get a very small or no correlation but now there's one more thing to

but now there's one more thing to consider the correlation coefficient is

consider the correlation coefficient is usually calculated with data taken from

usually calculated with data taken from a sample however we often want to test a

a sample however we often want to test a hypothesis about a population in the

hypothesis about a population in the case of correlation analysis we then

case of correlation analysis we then want to know if there is a correlation

want to know if there is a correlation in the population for this we check

in the population for this we check whether the correlation coefficient in

whether the correlation coefficient in the sample is the statistically

the sample is the statistically significantly different from zero the

significantly different from zero the null hypothesis in the Pearson

null hypothesis in the Pearson correlation is the correlation

correlation is the correlation coefficient does not differ

coefficient does not differ significantly from zero there is no

significantly from zero there is no linear relationship and the alternative

linear relationship and the alternative hypothesis is the correlation

hypothesis is the correlation coefficient differs significantly from

coefficient differs significantly from zero there is a linear relationship

zero there is a linear relationship attention it is always tested whether

attention it is always tested whether the null hypothesis is rejected or not

the null hypothesis is rejected or not in our example the research question is

in our example the research question is is there a correlation between age and

is there a correlation between age and salary in the British population to find

salary in the British population to find out we draw a sample and test whether in

out we draw a sample and test whether in this sample the correlation coefficient

this sample the correlation coefficient is significantly different from zero the

is significantly different from zero the null hypothesis then is there is no

null hypothesis then is there is no correlation between salary and age in

correlation between salary and age in the British population and the

the British population and the alternative hypothesis there is a

alternative hypothesis there is a correlation between salary and age in

correlation between salary and age in the British population whether the

the British population whether the correlation coefficient is significantly

correlation coefficient is significantly different from zero based on the sample

different from zero based on the sample collected can be checked using a T Test

collected can be checked using a T Test where R is the correlation coefficient

where R is the correlation coefficient and N is the sample size A P value can

and N is the sample size A P value can then be calculated from the test

then be calculated from the test statistic T if the P value is smaller

statistic T if the P value is smaller than the specified significance level

than the specified significance level which is usually 5% % then the null

which is usually 5% % then the null hypothesis is rejected otherwise it is

hypothesis is rejected otherwise it is not but what about the assumptions for a

not but what about the assumptions for a Pon correlation here we must distinguish

Pon correlation here we must distinguish whether we just want to calculate the

whether we just want to calculate the peon correlation or whether we want to

peon correlation or whether we want to test a hypothesis to calculate the peon

test a hypothesis to calculate the peon correlation coefficient only two matric

correlation coefficient only two matric variables need to be present matric

variables need to be present matric variables are for example a person's

variables are for example a person's weight a person's salary or electric it

weight a person's salary or electric it consumption the peon correlation

consumption the peon correlation coefficient then tells us how large the

coefficient then tells us how large the linear relationship is if there is a

linear relationship is if there is a nonlinear relationship we cannot tell

nonlinear relationship we cannot tell from the peon correlation coefficient

from the peon correlation coefficient however if we want to test whether the

however if we want to test whether the peon correlation coefficient is

peon correlation coefficient is significantly different from zero the

significantly different from zero the two variables must always be normally

two variables must always be normally distributed if this is not given the

distributed if this is not given the calculated test statistic t or the P

calculated test statistic t or the P value cannot be interpreted reliably

value cannot be interpreted reliably let's continue with the Spearman

let's continue with the Spearman correlation the Spearman rank

correlation the Spearman rank correlation is the nonparametric

correlation is the nonparametric counterpart of the pearon correlation

counterpart of the pearon correlation but there is an important difference

but there is an important difference between both correlation coefficients

between both correlation coefficients Spearman correlation does not use the

Spearman correlation does not use the raw data but the ranks of the data let's

raw data but the ranks of the data let's look at this with an example we measure

look at this with an example we measure the reaction time of eight computer

the reaction time of eight computer players and ask their age when we

players and ask their age when we calculate a peon correlation we simply

calculate a peon correlation we simply take the two variables reaction time and

take the two variables reaction time and age and calculate the peering

age and calculate the peering correlation coefficient however we now

correlation coefficient however we now want to calculate the spe and rank

want to calculate the spe and rank correlation so first we assign a rank to

correlation so first we assign a rank to each person for reaction time and age

each person for reaction time and age the reaction time is already sorted by

the reaction time is already sorted by size 12 is the smallest value so gets

size 12 is the smallest value so gets rank one 15 the second smallest value so

rank one 15 the second smallest value so gets rank two and so on and so forth we

gets rank two and so on and so forth we are now doing the same with h here we

are now doing the same with h here we have the smallest value there the second

have the smallest value there the second smallest there the third smallest fourth

smallest there the third smallest fourth smallest and so on and so forth let's

smallest and so on and so forth let's take a look at this in the skatter plot

take a look at this in the skatter plot here we see the raw data of age and

here we see the raw data of age and reaction time but but now we would like

reaction time but but now we would like to use the rankings so we form ranks

to use the rankings so we form ranks from the variables age and reaction

from the variables age and reaction time through this transformation we have

time through this transformation we have now distributed the data more evenly to

now distributed the data more evenly to calculate the pearon correlation we

calculate the pearon correlation we simply calculate the peon correlation

simply calculate the peon correlation from the ranks so the Spearman

from the ranks so the Spearman correlation is equal to the pon

correlation is equal to the pon correlation only that the ranks are used

correlation only that the ranks are used instead of the raw values let's have a

instead of the raw values let's have a quick look at that in data tab here we

quick look at that in data tab here we have the reaction time and age and there

have the reaction time and age and there we have the just created ranks of

we have the just created ranks of reaction time and age now we can either

reaction time and age now we can either calculate spean correlation of reaction

calculate spean correlation of reaction time and AG where we get a correlation

time and AG where we get a correlation of

of 0.9 or we can calculate pieron

0.9 or we can calculate pieron correlation from the ranks where we also

correlation from the ranks where we also get at 0.9 so exactly the same as before

get at 0.9 so exactly the same as before if you like you can download the data

if you like you can download the data set you can find the link in the video

set you can find the link in the video description if there are no rank ties we

description if there are no rank ties we can also use this equation to calculate

can also use this equation to calculate the pon correlation RS is the spean

the pon correlation RS is the spean correlation n is the number of cases and

correlation n is the number of cases and D is the difference in ranks between the

D is the difference in ranks between the two variables referring to our example

two variables referring to our example we get a different D's with this

we get a different D's with this 1 - 1 is = to 0 2 - 3 is -1 3 - 2 is 1

1 - 1 is = to 0 2 - 3 is -1 3 - 2 is 1 and so on now we Square the individual

and so on now we Square the individual D's and add them all up so the sum of d

D's and add them all up so the sum of d i squ is 8 n which is the number of

i squ is 8 n which is the number of people is eight if you put everything in

people is eight if you put everything in we get a correlation coefficient of

we get a correlation coefficient of 0.9 just like the pieron correlation

0.9 just like the pieron correlation coefficient R spe correlation

coefficient R spe correlation coefficient RS also varies between minus

coefficient RS also varies between minus one and one let's continue with candle's

one and one let's continue with candle's tow candle's tow is a correlation

tow candle's tow is a correlation coefficient and is thus a measure of the

coefficient and is thus a measure of the relationship between two variables but

relationship between two variables but what is the difference between peeron

what is the difference between peeron correlation and candles rank correlation

correlation and candles rank correlation in contrast to pearon correlation kless

in contrast to pearon correlation kless rank correlation is a nonparametric test

rank correlation is a nonparametric test procedure thus for the calculation of

procedure thus for the calculation of candle St the data need not be normally

candle St the data need not be normally distributed and the variables need only

distributed and the variables need only have ordinal scale levels exactly the

have ordinal scale levels exactly the same is true for the spe and rank

same is true for the spe and rank correlation right that's right candle

correlation right that's right candle tow is very similar to spearman's rank

tow is very similar to spearman's rank correlation coefficient however candle's

correlation coefficient however candle's tow should be preferred over spean

tow should be preferred over spean correlation if very few data with many

correlation if very few data with many rank ties are available

rank ties are available but how is candle toar calculated we can

but how is candle toar calculated we can calculate candle Tower with this formula

calculate candle Tower with this formula where C is the number of concordant

where C is the number of concordant pairs and D is the number of discordant

pairs and D is the number of discordant pairs what are concordant and discordant

pairs what are concordant and discordant pairs we will now go through this with

pairs we will now go through this with an example suppose two doctors are asked

an example suppose two doctors are asked to rank six patients according to the

to rank six patients according to the physical health one of the two doctors

physical health one of the two doctors is now defined as a reference and the

is now defined as a reference and the patients are sorted from 1 to six now

patients are sorted from 1 to six now the sorted ranks are matched with the

the sorted ranks are matched with the ranks of the other doctor EG the patient

ranks of the other doctor EG the patient who is in third place with the reference

who is in third place with the reference Doctor Is In fourth place with the other

Doctor Is In fourth place with the other doctor now using candle's tow we want to

doctor now using candle's tow we want to know if there is a correlation between

know if there is a correlation between the two rankings for the calculation of

the two rankings for the calculation of candle Tower we only need these ranks we

candle Tower we only need these ranks we now look at each individual Rank and

now look at each individual Rank and note whether the values below are

note whether the values below are smaller or greater than itself so we

smaller or greater than itself so we start at the first rank three one is

start at the first rank three one is smaller than three so gets a minus four

smaller than three so gets a minus four is greater so it gets a plus two is

is greater so it gets a plus two is smaller so it gets a minus six is

smaller so it gets a minus six is greater so it gets a plus and five is

greater so it gets a plus and five is also greater so it also gets a plus now

also greater so it also gets a plus now we do the same for one here of course

we do the same for one here of course each subsequent rank is greater than one

each subsequent rank is greater than one so we have a plus everywhere at rank

so we have a plus everywhere at rank four two is smaller and six and five are

four two is smaller and six and five are greater now we do this for rank two and

greater now we do this for rank two and rank six then we can easily calculate

rank six then we can easily calculate the number of concordant and discordant

the number of concordant and discordant pairs we get the number of concordant

pairs we get the number of concordant pairs by counting all the Plus in our

pairs by counting all the Plus in our example we have 11 plus in total we get

example we have 11 plus in total we get a number of this coordinate pairs by

a number of this coordinate pairs by counting through all the minus in our

counting through all the minus in our example we have a total of 4 minus C is

example we have a total of 4 minus C is thus 11 and D is four candle tow now is

thus 11 and D is four candle tow now is 11 - 4 ided by 11 + 4 and we get a

11 - 4 ided by 11 + 4 and we get a candle tow of

candle tow of 0.47 we get an alternative formula for

0.47 we get an alternative formula for kandle tow here with s is C minus D

kandle tow here with s is C minus D therefore 7 n is the number of cases I.E

therefore 7 n is the number of cases I.E 6 if we insert everything we also get 7

6 if we insert everything we also get 7 ided by 15 just like the peeron

ided by 15 just like the peeron correlation coefficient R candle tow

correlation coefficient R candle tow also varies between minus1 and + one we

also varies between minus1 and + one we have again calculated the correlation

have again calculated the correlation coefficient using data from a sample now

coefficient using data from a sample now we can test if the correlation

we can test if the correlation coefficient is significantly different

coefficient is significantly different from zero thus the null hypothesis is

from zero thus the null hypothesis is the correlation coefficient too is equal

the correlation coefficient too is equal to zero there is no relationship and the

to zero there is no relationship and the alternative hypothesis is the

alternative hypothesis is the correlation coefficient to is unequal to

correlation coefficient to is unequal to zero there is a relationship therefore

zero there is a relationship therefore we want to know if the correlation

we want to know if the correlation coefficient is significantly different

coefficient is significantly different from zero you can analyze this either by

from zero you can analyze this either by hand or with a software like data tab

hand or with a software like data tab for the calculation by hand we can use

for the calculation by hand we can use use the set distribution as an

use the set distribution as an approximation however for this we should

approximation however for this we should at least have 40 cases so the six cases

at least have 40 cases so the six cases from our example are actually too few we

from our example are actually too few we get the set value with this formula here

get the set value with this formula here we have Tow and N is the number of cases

we have Tow and N is the number of cases this brings us to the last correlation

this brings us to the last correlation analysis the point by zerial correlation

analysis the point by zerial correlation Point by zal correlation is a special

Point by zal correlation is a special case of Pon correlation and examines the

case of Pon correlation and examines the relationship between a dichotomous

relationship between a dichotomous variable and the metric variable what is

variable and the metric variable what is a dichotomus variable and what is a

a dichotomus variable and what is a matric variable a damous variable is a

matric variable a damous variable is a variable with two values for example

variable with two values for example gender with male and female or smoking

gender with male and female or smoking status with smoker and non-smoker a

status with smoker and non-smoker a matric variable is for example the

matric variable is for example the weight of a person the salary of a

weight of a person the salary of a person or the electricity consumption

person or the electricity consumption so if we have a dichotomus variable and

so if we have a dichotomus variable and a matric variable and we want to know if

a matric variable and we want to know if there is a relationship we can use a

there is a relationship we can use a Point by Zer correlation of course we

Point by Zer correlation of course we need to check the assumptions beforehand

need to check the assumptions beforehand but more about that later how is the

but more about that later how is the point by zerial correlation calculated

point by zerial correlation calculated as stated at the beginning the point by

as stated at the beginning the point by zerial correlation is a special case of

zerial correlation is a special case of the peon correlation but how can we

the peon correlation but how can we calculate the p correlation when a

calculate the p correlation when a variable is nominal let's look at this

variable is nominal let's look at this with an example let's say we are

with an example let's say we are interested in investigating the

interested in investigating the relationship between the number of ours

relationship between the number of ours studied for a test and the test result

studied for a test and the test result pass failed we've calculated data from a

pass failed we've calculated data from a sample of 20 students where 12 students

sample of 20 students where 12 students passed the test and eight students

passed the test and eight students failed we have recorded the number of

failed we have recorded the number of hours each student studed for the test

hours each student studed for the test to calculate the point by Zer

to calculate the point by Zer correlation we first need to convert the

correlation we first need to convert the test result into numbers we can assign a

test result into numbers we can assign a score of one to students who passed the

score of one to students who passed the test and a score of zero to students who

test and a score of zero to students who failed the test now we can either

failed the test now we can either calculate the pon correlation of time

calculate the pon correlation of time and test result or we use the equation

and test result or we use the equation for the point by zero correlation X1

for the point by zero correlation X1 Dash is the mean value of the people who

Dash is the mean value of the people who have passed pass and X2 Dash is the mean

have passed pass and X2 Dash is the mean value of the people who failed N1 is the

value of the people who failed N1 is the number of people who passed and N2 the

number of people who passed and N2 the number of people who failed and N is the

number of people who failed and N is the total number but whether we calculate

total number but whether we calculate the peon correlation or we use the

the peon correlation or we use the equation for the point by zerial

equation for the point by zerial correlation we get the same result both

correlation we get the same result both times let's take a quick look at this in

times let's take a quick look at this in data tab here we have the learning hours

data tab here we have the learning hours the test result was passed and fail fail

the test result was passed and fail fail and there the test result with 0er and

and there the test result with 0er and one we Define the test result with Zer

one we Define the test result with Zer and one as

and one as metric if we now go to correlation and

metric if we now go to correlation and calculate the peeron correlation for

calculate the peeron correlation for these two variables we get a correlation

these two variables we get a correlation coefficient of

coefficient of 0.31 if we calculate the point by zal

0.31 if we calculate the point by zal correlation for learning hours and the

correlation for learning hours and the exam result was passed and failed we

exam result was passed and failed we also get a correlation of

also get a correlation of 0.31 just like the pon correlation

0.31 just like the pon correlation coefficient R the point by zal

coefficient R the point by zal correlation coefficient rpb also varies

correlation coefficient rpb also varies between minus1 and 1 if we have a

between minus1 and 1 if we have a coefficient between minus1 and less than

coefficient between minus1 and less than one there is a negative correlation thus

one there is a negative correlation thus a negative relationship between the

a negative relationship between the variables if we have a coefficient

variables if we have a coefficient between greater than zero and one there

between greater than zero and one there is a positive correlation that is is a

is a positive correlation that is is a positive relationship between the two

positive relationship between the two variables if the result is zero we have

variables if the result is zero we have no correlation as always with the point

no correlation as always with the point by Zer correlation we can also check

by Zer correlation we can also check whether the correlation coefficient is

whether the correlation coefficient is significantly different from zero thus

significantly different from zero thus the null hypothesis is the correlation

the null hypothesis is the correlation coefficient R is equal to zero there is

coefficient R is equal to zero there is no relationship and the alternative

no relationship and the alternative hypothesis is the correlation

hypothesis is the correlation coefficient R is unequal to Z there is a

coefficient R is unequal to Z there is a relationship before we get to the

relationship before we get to the assumptions here's an interesting note

assumptions here's an interesting note when we compute a point by zerial

when we compute a point by zerial correlation we get the same P value as

correlation we get the same P value as when we compute a t test for independent

when we compute a t test for independent samples for the same data so whether we

samples for the same data so whether we test a correlation hypothesis with the

test a correlation hypothesis with the point by zero correlation or a

point by zero correlation or a difference hypothesis with the T Test we

difference hypothesis with the T Test we get the same P value now if we compute a

get the same P value now if we compute a test in data tab with these data and we

test in data tab with these data and we have the null hypothesis there is no

have the null hypothesis there is no difference between the groups failed and

difference between the groups failed and passed in terms of the variable learning

passed in terms of the variable learning hours we get a P value of

hours we get a P value of 0.179 and also if we calculate a point

0.179 and also if we calculate a point by zero correlation and have the null

by zero correlation and have the null hypothesis there is no correlation

hypothesis there is no correlation between learning hours and test results

between learning hours and test results we get a P value of 0.1

we get a P value of 0.1 179 in our example the P value is

179 in our example the P value is greater than

greater than 0.05 which is most often used as a

0.05 which is most often used as a significance level and thus the null

significance level and thus the null hypothesis is not rejected but what

hypothesis is not rejected but what about the assumptions for a point by zal

about the assumptions for a point by zal correlation here we must distinguish

correlation here we must distinguish whether we just want to calculate the

whether we just want to calculate the correlation coefficient or whether we

correlation coefficient or whether we want to test a hypothesis to calculate a

want to test a hypothesis to calculate a correlation coefficient only one matric

correlation coefficient only one matric variable and one damus variable must be

variable and one damus variable must be present however if we want to test

present however if we want to test whether the correlation coefficient is

whether the correlation coefficient is significantly different from zero the

significantly different from zero the one matric variable must also be

one matric variable must also be normally distributed if this is not

normally distributed if this is not given the calculated test statistic t or

given the calculated test statistic t or the P value cannot be interpreted

the P value cannot be interpreted reliably this brings us to the last

reliably this brings us to the last question what is causality and what is

question what is causality and what is the difference between causality and

the difference between causality and correlation causality is the

correlation causality is the relationship between a cause and an

relationship between a cause and an effect in a causal relationship we have

effect in a causal relationship we have a cause and a resultant effect an

a cause and a resultant effect an example coffee contains caffeine a

example coffee contains caffeine a stimulating substance when you drink

stimulating substance when you drink coffee the caffeine enters the body

coffee the caffeine enters the body affects the central nervous system and

affects the central nervous system and leads to increased alerted drinking

leads to increased alerted drinking coffee is the cause of the feeling of

coffee is the cause of the feeling of alerted that comes afterwards without

alerted that comes afterwards without drinking coffee the effect I.E the

drinking coffee the effect I.E the feeling of alerted would not occur but

feeling of alerted would not occur but causality is not always so easy to

causality is not always so easy to determine clear requirements must be met

determine clear requirements must be met in order to speak of a causal

in order to speak of a causal relationship but more about that later

relationship but more about that later so what is the difference between

so what is the difference between correlation and causality a correlation

correlation and causality a correlation tells us that there is a relationship

tells us that there is a relationship between two variables

between two variables example there is a positive correlation

example there is a positive correlation between ice cream sales and the number

between ice cream sales and the number of sunburns however an existing

of sunburns however an existing correlation cannot tell us which

correlation cannot tell us which variable influences which or whether a

variable influences which or whether a third variable is responsible for the

third variable is responsible for the correlation in our example both

correlation in our example both variables are influenced by a common

variables are influenced by a common cause namely sunny weather on sunny days

cause namely sunny weather on sunny days people buy more ice cream and and spend

people buy more ice cream and and spend more time Outdoors this can lead to an

more time Outdoors this can lead to an increased risk of sunburns causality

increased risk of sunburns causality means that there is a clear cause effect

means that there is a clear cause effect relationship between two variables

relationship between two variables causality exists when you can say with

causality exists when you can say with certainity which variable influences

certainity which variable influences which however a common mistake in the

which however a common mistake in the interpretation of Statistics is that a

interpretation of Statistics is that a correlation is immediately assumed to be

correlation is immediately assumed to be a causal relationship here is an example

a causal relationship here is an example example the American statistician Daryl

example the American statistician Daryl Huff found a negative correlation

Huff found a negative correlation between the number of headli and the

between the number of headli and the body temperature of the inhabitants of

body temperature of the inhabitants of an island a negative correlation means

an island a negative correlation means that people with many head lies

that people with many head lies generally have a lower body temperature

generally have a lower body temperature and people with few head lies generally

and people with few head lies generally have a higher body temperature the

have a higher body temperature the Islanders concluded that head lies were

Islanders concluded that head lies were good for health because they reduced

good for health because they reduced fever so their assumption was that

fever so their assumption was that headlights have an effect on the

headlights have an effect on the temperature of the body in reality the

temperature of the body in reality the correct conclusion is the other way

correct conclusion is the other way around in an experiment it was possible

around in an experiment it was possible to prove that high fever drives away the

to prove that high fever drives away the lies so the high body temperature is the

lies so the high body temperature is the cause not the effect what are the

cause not the effect what are the conditions for talking about causality

conditions for talking about causality there are two conditions for causality

there are two conditions for causality number one there is a significant

number one there is a significant correlation between the variables this

correlation between the variables this is easy to check we simply check whether

is easy to check we simply check whether the correlation coefficient is

the correlation coefficient is significantly different from zero number

significantly different from zero number two the second condition can be met in

two the second condition can be met in three ways first chronological sequence

three ways first chronological sequence there is a chronological sequence and

there is a chronological sequence and the results of one variable occurred

the results of one variable occurred before the results of the other variable

before the results of the other variable second experiment a controlled

second experiment a controlled experiment was conducted in which the

experiment was conducted in which the two variables can be specifically

two variables can be specifically influenced and number three Theory there

influenced and number three Theory there is a well-founded and plausible theory

is a well-founded and plausible theory in which direction the causal

in which direction the causal relationship goes if there is only a

relationship goes if there is only a significant correlation but none of the

significant correlation but none of the other three conditions are met we can

other three conditions are met we can only speak of correlation never of

only speak of correlation never of causality after examining how

causality after examining how correlation analysis helps us to

correlation analysis helps us to determine the extent to which variable

determine the extent to which variable are related we now move on into the

are related we now move on into the field of

field of regression first we'll start with an

regression first we'll start with an overview of regression analysis where we

overview of regression analysis where we will break down the fundamentals and

will break down the fundamentals and explore Its Real World application next

explore Its Real World application next we'll dive into simple linear regression

we'll dive into simple linear regression where you will learn how to model the

where you will learn how to model the relationship between two variables then

relationship between two variables then we will move on to the multiple linear

we will move on to the multiple linear regression where we extend the model to

regression where we extend the model to include multiple predictors making our

include multiple predictors making our predictions more powerful and finally

predictions more powerful and finally we'll cover logistic regression which is

we'll cover logistic regression which is essential when working with categorical

essential when working with categorical variables like predicting whether

variables like predicting whether something will happen or not so let's

something will happen or not so let's get started with the first question what

get started with the first question what is a regression analysis A regression

is a regression analysis A regression analysis is a method for modeling

analysis is a method for modeling relationships between variables it makes

relationships between variables it makes it possible to infer or predict a

it possible to infer or predict a variable based on one or more other

variable based on one or more other variables let's say you want to find out

variables let's say you want to find out what influences a person's salary for

what influences a person's salary for example you could take the highest level

example you could take the highest level of Education the weekly working hours

of Education the weekly working hours and the age of a person you could now

and the age of a person you could now investigate whether these three

investigate whether these three variables have an influence on the

variables have an influence on the salary of a person if they do you can

salary of a person if they do you can predict a person's salary by taking the

predict a person's salary by taking the highest level of Education weekly

highest level of Education weekly working hours and a person's age the

working hours and a person's age the variable we want to inere or predict is

variable we want to inere or predict is called the dependent variable the

called the dependent variable the variables used for prediction are called

variables used for prediction are called independent variables depending on your

independent variables depending on your field independent variables may also be

field independent variables may also be called predictor variables or input

called predictor variables or input variables while the dependent variable

variables while the dependent variable might be referred to as the response

might be referred to as the response output or Target variable okay but when

output or Target variable okay but when do we use a regression analysis

do we use a regression analysis regression analysis can be used to

regression analysis can be used to achieve two goals you can measure the

achieve two goals you can measure the influence of one variable or several

influence of one variable or several variables on another variable or you can

variables on another variable or you can predict a variable based on other

predict a variable based on other variables let's go through some examples

variables let's go through some examples let's start by measuring the influence

let's start by measuring the influence of one or more variables on another in

of one or more variables on another in the context of your research you may be

the context of your research you may be interested in understanding the factors

interested in understanding the factors that influence children 's ability to

that influence children 's ability to concentrate specifically you aim to

concentrate specifically you aim to determine whether certain parameters

determine whether certain parameters have a positive or negative impact on

have a positive or negative impact on their concentration but in this case

their concentration but in this case you're not interested in predicting

you're not interested in predicting children's ability to concentrate or you

children's ability to concentrate or you could investigate whether the

could investigate whether the educational level of the parents and the

educational level of the parents and the place of residence have an influence on

place of residence have an influence on the future educational level of children

the future educational level of children this area is therefore very research

this area is therefore very research based and has many applications in

based and has many applications in Social and economic Sciences the second

Social and economic Sciences the second area using regression for predictions is

area using regression for predictions is more application oriented to get the

more application oriented to get the most out of Hospital occupancy you might

most out of Hospital occupancy you might be interested in how long a patient will

be interested in how long a patient will stay in the hospital so based on the

stay in the hospital so based on the characteristics of the prospective

characteristics of the prospective patient such as age reason for stay and

patient such as age reason for stay and pre-existing conditions you want to know

pre-existing conditions you want to know how long that person is like likely to

how long that person is like likely to stay in the hospital based on this

stay in the hospital based on this prediction bath planning can then be

prediction bath planning can then be optimized or as an operator of an online

optimized or as an operator of an online store you are very interested in which

store you are very interested in which product a person is most likely to buy

product a person is most likely to buy you want to suggest this product to the

you want to suggest this product to the visitor in order to increase the sales

visitor in order to increase the sales of the online store second point is

of the online store second point is highly application oriented focusing on

highly application oriented focusing on making predictions to enhance efficiency

making predictions to enhance efficiency okay great now there are different types

okay great now there are different types of regression analysis there is the

of regression analysis there is the simple linear multiple linear and

simple linear multiple linear and logistic regression in simple linear

logistic regression in simple linear regression we use just one independent

regression we use just one independent variable to predict the dependent

variable to predict the dependent variable for example if we want to

variable for example if we want to predict a person's salary we use only

predict a person's salary we use only one variable either if a person has

one variable either if a person has studied or not the weekly working hours

studied or not the weekly working hours or the age of a person multiple linear

or the age of a person multiple linear regret on the other hand uses several

regret on the other hand uses several independent variables to predict or

independent variables to predict or inere the dependent variable I.E the

inere the dependent variable I.E the highest level of edication the number of

highest level of edication the number of hours worked per week and the age of the

hours worked per week and the age of the person therefore the difference between

person therefore the difference between a simple and the multiple regression is

a simple and the multiple regression is that in one case only one independent

that in one case only one independent variable is used and in the other case

variable is used and in the other case several both have in common that the

several both have in common that the dependent variable is matric matric

dependent variable is matric matric variable are for example the salary of a

variable are for example the salary of a person the body size or the electricity

person the body size or the electricity consumption in contrast logistic

consumption in contrast logistic regression is used when you have a

regression is used when you have a categorical dependent variable

categorical dependent variable categorical variables are for example if

categorical variables are for example if a person is at risk of burnout or not if

a person is at risk of burnout or not if a person is diseased or not or type of

a person is diseased or not or type of animal however the most common form of

animal however the most common form of logistic regression is the so-called

logistic regression is the so-called binary logistic regression

binary logistic regression in this case the outcome variable is

in this case the outcome variable is binary meaning it has two possible

binary meaning it has two possible values like yes or no success of failure

values like yes or no success of failure or diseased and not diseased therefore

or diseased and not diseased therefore in linear regression the dependent

in linear regression the dependent variable is a metric variable while in

variable is a metric variable while in logistic regression it is a categorical

logistic regression it is a categorical variable also known as nominal variable

variable also known as nominal variable but what about the independent variables

but what about the independent variables in all cases the level of measure M of

in all cases the level of measure M of the independent variables can be nominal

the independent variables can be nominal ordinal or metric okay actually in

ordinal or metric okay actually in regression you can only use categorical

regression you can only use categorical variables with two categories or levels

variables with two categories or levels such as gender with male and female in

such as gender with male and female in this case we can code one category with

this case we can code one category with zero and the other with one however if a

zero and the other with one however if a variable has more than two categories

variable has more than two categories like vehicle type there's an easy

like vehicle type there's an easy solution we create dummy variables don't

solution we create dummy variables don't worry we'll explain more about dummy

worry we'll explain more about dummy variables later in this playlist okay a

variables later in this playlist okay a quick recap there is the simple linear

quick recap there is the simple linear regression a question could be does the

regression a question could be does the weekly working time have an impact on

weekly working time have an impact on the hourly wage of employees here we

the hourly wage of employees here we only have one independent

only have one independent variable there is the multiple linear

variable there is the multiple linear regression do the weekly working hours

regression do the weekly working hours and the age of employees have an

and the age of employees have an influence on the hourly wage here we

influence on the hourly wage here we have at least two independent variables

have at least two independent variables in this case weekly working hours and

in this case weekly working hours and age and the last case logistic

age and the last case logistic regression du the weekly working hours

regression du the weekly working hours and the age of employees have an

and the age of employees have an influence on the probability that they

influence on the probability that they are at risk of burnout where burnout at

are at risk of burnout where burnout at risk has the categories yes or no now

risk has the categories yes or no now that we've covered the basics of

that we've covered the basics of regression analysis and its applications

regression analysis and its applications let's take a closer look at the simple

let's take a closer look at the simple linear regression this technique helps

linear regression this technique helps us to understand the relationship

us to understand the relationship between two variables allowing us to

between two variables allowing us to predict one based on the other in the

predict one based on the other in the next section we'll break down how it

next section we'll break down how it works and go through a real example and

works and go through a real example and show you how to interpret the results

show you how to interpret the results let's get started what is a simple

let's get started what is a simple linear regression simple linear

linear regression simple linear regression is a method to understand the

regression is a method to understand the relationship between two variables you

relationship between two variables you can infer or predict a variable based on

can infer or predict a variable based on another variable for example You can

another variable for example You can predict the annual salary of a person

predict the annual salary of a person based on the years of work experience so

based on the years of work experience so a simple linear regression can help us

a simple linear regression can help us understand how salary changes with an

understand how salary changes with an increase in years of experience the

increase in years of experience the variable we want to inere or predict is

variable we want to inere or predict is called the dependent variable the

called the dependent variable the variable we use for prediction is called

variable we use for prediction is called independent variable let's look at an

independent variable let's look at an example imagine we want to predict house

example imagine we want to predict house prices the dependent variable the one we

prices the dependent variable the one we want to predict is of course the price

want to predict is of course the price of the house the independent variable

of the house the independent variable the one we used to make the prediction

the one we used to make the prediction could be the size of the house in square

could be the size of the house in square feet of course you can also use more

feet of course you can also use more than one independent variable for

than one independent variable for example the construction year or

example the construction year or features like whether the house has a

features like whether the house has a swimming pool the number of bathrooms

swimming pool the number of bathrooms and the house size but in this case it

and the house size but in this case it would be a multiple linear regression

would be a multiple linear regression because you have more than one

because you have more than one independent variable more on that in my

independent variable more on that in my video about multiple linear regression

video about multiple linear regression okay but how do we calculate a simple

okay but how do we calculate a simple linear regression first of all we need

linear regression first of all we need data so we collect information from 10

data so we collect information from 10 houses including their size in square

houses including their size in square feet and the price they were sold for

feet and the price they were sold for now we can use use this data to

now we can use use this data to calculate our regression model here Y is

calculate our regression model here Y is the dependent variable house price and X

the dependent variable house price and X is the independent variable house size

is the independent variable house size and we want to use our data to determine

and we want to use our data to determine the coefficient b and a but how do we do

the coefficient b and a but how do we do that let's visualize our data using a

that let's visualize our data using a scatter plot on the xais we plot our

scatter plot on the xais we plot our independent variable the house size and

independent variable the house size and on the Y AIS we plot the dependent VAR

on the Y AIS we plot the dependent VAR variable the house price each point is

variable the house price each point is therefore one house with the respective

therefore one house with the respective house size and the house price okay now

house size and the house price okay now we want to summarize all this data using

we want to summarize all this data using a simple linear regression to do this we

a simple linear regression to do this we draw a straight line through the points

draw a straight line through the points on the scatter plot but the line we draw

on the scatter plot but the line we draw isn't just any random line it's the line

isn't just any random line it's the line that tries to minimize the error or the

that tries to minimize the error or the distance between the actual data points

distance between the actual data points and the line line itself if we add up

and the line line itself if we add up the length of all the red lines we get

the length of all the red lines we get the total error our goal is to find the

the total error our goal is to find the regression line that minimizes this

regression line that minimizes this error but how do we actually calculate

error but how do we actually calculate this line This is where the equation of

this line This is where the equation of the linear regression comes into play in

the linear regression comes into play in the equation B is the slope of the line

the equation B is the slope of the line the slope shows how much the house price

the slope shows how much the house price changes if the house size increases by

changes if the house size increases by one square squ food a is the Y intercept

one square squ food a is the Y intercept telling us where the line crosses the Y

telling us where the line crosses the Y AIS so if we have a house with a size of

AIS so if we have a house with a size of zero the model will predict a house

zero the model will predict a house price of a of course predicting the

price of a of course predicting the price of a house with zero size doesn't

price of a house with zero size doesn't make sense however every model is a

make sense however every model is a simplification of the real world and in

simplification of the real world and in the case of simple linear regression our

the case of simple linear regression our model is defined by a regression line

model is defined by a regression line with a specific slope B B and an

with a specific slope B B and an intercept a let's look at this example

intercept a let's look at this example in that case our intercept is 100 so we

in that case our intercept is 100 so we enter 100 for a but how do we read the

enter 100 for a but how do we read the slope for this we take a one unit step

slope for this we take a one unit step in the independent variable for example

in the independent variable for example if we move from 1 to two then we observe

if we move from 1 to two then we observe how much the dependent variable changes

how much the dependent variable changes with this one unit increase in this case

with this one unit increase in this case if the independent variable increases by

if the independent variable increases by one unit the dependent variable

one unit the dependent variable increases by 50 units so our B is

increases by 50 units so our B is 50 okay but how do we calculate b and a

50 okay but how do we calculate b and a there are two ways to do this we can

there are two ways to do this we can calculate them by hand or using

calculate them by hand or using statistical software like data tab let's

statistical software like data tab let's look at this example how can we

look at this example how can we calculate b and a by hand to calculate

calculate b and a by hand to calculate the slope B we use this formula R is the

the slope B we use this formula R is the correlation coefficient between X and Y

correlation coefficient between X and Y so in our case the correlation between

so in our case the correlation between house size and house price we get a

house size and house price we get a correlation coefficient of

correlation coefficient of 0.92 s y is the standard deviation of

0.92 s y is the standard deviation of the dependent variable house price and

the dependent variable house price and SX is the standard deviation of the

SX is the standard deviation of the independent variable so house size so in

independent variable so house size so in this case our B is 108

this case our B is 108 35 here's a quick side note for anyone

35 here's a quick side note for anyone who wants to follow along and

who wants to follow along and recalculate the example there are two

recalculate the example there are two slightly different formulas for standard

slightly different formulas for standard deviation one divides by n and the other

deviation one divides by n and the other by n minus one without diving into the

by n minus one without diving into the details now almost all statistical

details now almost all statistical software uses the formula with n minus1

software uses the formula with n minus1 to calculate the standard deviation

to calculate the standard deviation however for calculating the regression

however for calculating the regression coefficient B we use the formula that

coefficient B we use the formula that divides by n if you like a more detailed

divides by n if you like a more detailed explanation of the standard deviation

explanation of the standard deviation feel free to check out my video on that

feel free to check out my video on that topic all right once we've calculated B

topic all right once we've calculated B we can find The Intercept a using this

we can find The Intercept a using this formula here Y Bar represents the mean

formula here Y Bar represents the mean of the house prices B is the slope which

of the house prices B is the slope which is calculated and xar is the mean of the

is calculated and xar is the mean of the house sizes substituting these values

house sizes substituting these values the inter ccept a comes out to be

6,919 44 So based on this data we have now

44 So based on this data we have now calculated the coefficient b and a if we

calculated the coefficient b and a if we insert the numbers for b and a we get

insert the numbers for b and a we get this equation if we enter Z for X the

this equation if we enter Z for X the house size we get

house size we get 6,919 44 which is our intercept if we

6,919 44 which is our intercept if we increase the house size by One S square

increase the house size by One S square foot each time we get a house price that

foot each time we get a house price that is

$18.35 higher each time okay before we start with the last important topic the

start with the last important topic the assumptions for a simple linear

assumptions for a simple linear regression let's check the results with

regression let's check the results with data tab if you like you can load this

data tab if you like you can load this sample data set the link is in the video

sample data set the link is in the video description we want to calculate a

description we want to calculate a regression so we click on regression

regression so we click on regression here we can now select our dependent

here we can now select our dependent variable

variable house price and the independent variable

house price and the independent variable house size now let's look at the results

house size now let's look at the results we will focus on this table which shows

we will focus on this table which shows the key information we need if you're

the key information we need if you're curious about the other tables you can

curious about the other tables you can click on AI interpretation for the

click on AI interpretation for the corresponding table or check out our

corresponding table or check out our video on multiple linear regression

video on multiple linear regression where we explain these details in depth

where we explain these details in depth okay in this table we can see the

okay in this table we can see the calculated regression coefficient

calculated regression coefficient for the constant we called it intercept

for the constant we called it intercept and house size the values match exactly

and house size the values match exactly with the ones we calculated by hand the

with the ones we calculated by hand the intercept and the slope in the results

intercept and the slope in the results table for a linear regression you'll

table for a linear regression you'll also see the P value what does the P

also see the P value what does the P value tell us the P value helps

value tell us the P value helps determine whether the relationship

determine whether the relationship between the independent variable and a

between the independent variable and a dependent variable is statistically

dependent variable is statistically significant to to test whether the

significant to to test whether the relationship we observe is Meaningful or

relationship we observe is Meaningful or just due to random Chens we start by

just due to random Chens we start by stating the null hypothesis there is no

stating the null hypothesis there is no relationship between the independent

relationship between the independent variable and the dependent variable if

variable and the dependent variable if the P value is small typically smaller

the P value is small typically smaller than

than 0.05 we reject the null hypothesis this

0.05 we reject the null hypothesis this is suggesting a significant relationship

is suggesting a significant relationship between the variables if the P value is

between the variables if the P value is large typically great later than

large typically great later than 0.05 we fail to reject the null

0.05 we fail to reject the null hypothesis indicating The observed data

hypothesis indicating The observed data may have occurred by chance with no

may have occurred by chance with no strong evidence for a relationship so in

strong evidence for a relationship so in our case the P value is highly

our case the P value is highly significant indicating strong evidence

significant indicating strong evidence of a relationship between house price

of a relationship between house price and size all right in this example it's

and size all right in this example it's pretty obvious a bigger house size

pretty obvious a bigger house size typically costs more however there are

typically costs more however there are cases where the relationship isn't that

cases where the relationship isn't that clear and what about the assumptions

clear and what about the assumptions here are the key assumptions number one

here are the key assumptions number one linear

linear relationship in linear regression a

relationship in linear regression a straight line is drawn through the data

straight line is drawn through the data this straight line should represent All

this straight line should represent All Points as good as possible if the

Points as good as possible if the relation is nonlinear the straight line

relation is nonlinear the straight line cannot fulfill this requirement number

cannot fulfill this requirement number two independence of Errors the error so

two independence of Errors the error so the differences between actual and

the differences between actual and predicted values should be independent

predicted values should be independent of each other this means that the error

of each other this means that the error of one point doesn't affect another

of one point doesn't affect another number three

number three homoscedasticity or equal variance of

homoscedasticity or equal variance of Errors if we plot the errors on the Y

Errors if we plot the errors on the Y AIS and the dependent variable on the

AIS and the dependent variable on the xaxis their spread should be roughly the

xaxis their spread should be roughly the same across all values of X in other

same across all values of X in other words the variance of the error should

words the variance of the error should remain constant in this case the

remain constant in this case the assumption is fulfilled but what about

assumption is fulfilled but what about that case here we observe unequal

that case here we observe unequal variant at low values of X the errors

variant at low values of X the errors are small while at high values the

are small while at high values the variance of the errors becomes much

variance of the errors becomes much larger number four normally distributed

larger number four normally distributed errors the arrows should be normally

errors the arrows should be normally distributed the normality of the arrow

distributed the normality of the arrow can be tested both analytically and

can be tested both analytically and graphically however be cautious with

graphically however be cautious with analytical tests for small samples they

analytical tests for small samples they often indicate normality and with large

often indicate normality and with large samples they quickly become significant

samples they quickly become significant because of these limitations graphical

because of these limitations graphical methods such as the QQ plot are more

methods such as the QQ plot are more commonly used today if you use data tab

commonly used today if you use data tab you just need to click here to check the

you just need to click here to check the assumptions so let's just go through how

assumptions so let's just go through how the assumptions are checked in practice

the assumptions are checked in practice to check for a linear relationship you

to check for a linear relationship you can use a scatter plot plot the

can use a scatter plot plot the independent variable against the

independent variable against the dependent variable if the points form a

dependent variable if the points form a clear straight line pattern a linear

clear straight line pattern a linear relationship exists if not the

relationship exists if not the relationship is likely nonlinear in our

relationship is likely nonlinear in our case we observe a clear linear

case we observe a clear linear relationship to test if the errors are

relationship to test if the errors are normally distributed you can use a QQ

normally distributed you can use a QQ plot or one of the several analytical

plot or one of the several analytical tests with a QQ plot the residuals

tests with a QQ plot the residuals should fall roughly along a straight

should fall roughly along a straight line indicating normality if you use an

line indicating normality if you use an analytical test check whether the

analytical test check whether the calculated P value is greater than

calculated P value is greater than 0.05 if it is not there is evidence that

0.05 if it is not there is evidence that the data are not normally distributed

the data are not normally distributed the choice of test often depends on your

the choice of test often depends on your field of research however as mentioned

field of research however as mentioned the QQ plot is increasingly preferred as

the QQ plot is increasingly preferred as a visual and intuitive way to assess

a visual and intuitive way to assess normality independence of erors can be

normality independence of erors can be tested using the Durban Watson test

tested using the Durban Watson test which checks for autocorrelation in the

which checks for autocorrelation in the residuals if the calculated P value is

residuals if the calculated P value is greater than

greater than 0.05 it indicates that there is no

0.05 it indicates that there is no significant autocorrelation in the

significant autocorrelation in the residuals and the independence

residuals and the independence assumption is likely satisfied

assumption is likely satisfied homoscedasticity can be checked using a

homoscedasticity can be checked using a residual plot where the predicted values

residual plot where the predicted values are plotted on the xais and the

are plotted on the xais and the residuals or arrow on the Y AIS the

residuals or arrow on the Y AIS the residuals should show a consistent

residuals should show a consistent spread across the plot a funnel shape

spread across the plot a funnel shape indicates hat Tois causticity meaning

indicates hat Tois causticity meaning the variance is not constant in our case

the variance is not constant in our case the plot looks acceptable though not

the plot looks acceptable though not perfect if these assumptions are

perfect if these assumptions are violated the regression results might

violated the regression results might not be reliable or meaningful and the

not be reliable or meaningful and the predictions could be inaccurate so

predictions could be inaccurate so always check these assumptions before

always check these assumptions before drawing conclusions from a regression

drawing conclusions from a regression model so far we've seen how simple

model so far we've seen how simple linear regression helps us model the

linear regression helps us model the relationship between two variables one

relationship between two variables one dependent and one independent but what

dependent and one independent but what if we have more than one factor

if we have more than one factor influencing our outcome that's where

influencing our outcome that's where multiple linear regression comes in in

multiple linear regression comes in in the next section we'll extend our

the next section we'll extend our regression model to including multiple

regression model to including multiple predictors making our predictions more

predictors making our predictions more accurate and realistic let's dive in

accurate and realistic let's dive in what is a multiple linear regression a

what is a multiple linear regression a multiple linear regression is a method

multiple linear regression is a method for modeling relationships between

for modeling relationships between variables it makes it possible to inere

variables it makes it possible to inere or predict a variable based on other

or predict a variable based on other variables an example let's say you want

variables an example let's say you want to find out what influence uences a

to find out what influence uences a person's salary you take the highest

person's salary you take the highest level of Education the weekly working

level of Education the weekly working hours and the age of a person you now

hours and the age of a person you now investigate whether these three

investigate whether these three variables have an influence on the

variables have an influence on the salary of a person if they do you can

salary of a person if they do you can predict a person's salary by taking the

predict a person's salary by taking the highest educational level the weekly

highest educational level the weekly working hours and a person's age the

working hours and a person's age the variables we use for prediction are

variables we use for prediction are called independent variables the Vari we

called independent variables the Vari we want to inere or predict is called the

want to inere or predict is called the dependent variable but what is the

dependent variable but what is the difference between a simple linear and

difference between a simple linear and the multiple linear regression as we

the multiple linear regression as we know from the previous video in simple

know from the previous video in simple linear regression we use just one

linear regression we use just one independent variable to predict the

independent variable to predict the dependent variable for example if we

dependent variable for example if we want to predict a person's salary we use

want to predict a person's salary we use either if a person has studied or not

either if a person has studied or not the weekly working hours or the age of a

the weekly working hours or the age of a person multiple linear regression on the

person multiple linear regression on the other hand uses several independent

other hand uses several independent variables to predict or inere the

variables to predict or inere the dependent variable therefore the

dependent variable therefore the difference between a simple and a

difference between a simple and a multiple linear regression is that in

multiple linear regression is that in one case only one independent variable

one case only one independent variable is used and in the other case several

is used and in the other case several both have in common that the dependent

both have in common that the dependent variable is matric matric variables are

variable is matric matric variables are for example the salary of a person the

for example the salary of a person the body size or the electricity consumption

body size or the electricity consumption so unlike simple linear regression

so unlike simple linear regression multiple linear regression can include

multiple linear regression can include two or more independent variables but

two or more independent variables but what impact does that have on the

what impact does that have on the regression equation in the case of

regression equation in the case of linear regression this was our equation

linear regression this was our equation we had one dependent variable Y and one

we had one dependent variable Y and one independent variable X now in multiple

independent variable X now in multiple linear regression we have more than one

linear regression we have more than one dependent variable but don't worry the

dependent variable but don't worry the coefficients b and a are interpreted

coefficients b and a are interpreted similarly to those in a simple linear

similarly to those in a simple linear regression if all independent variables

regression if all independent variables are zero the value a is obtained so we

are zero the value a is obtained so we get a value of a for the dependent

get a value of a for the dependent variable y furthermore if an independent

variable y furthermore if an independent variable increases by one unit the

variable increases by one unit the associated coefficient B indicates the

associated coefficient B indicates the corresponding change in the dependent

corresponding change in the dependent variable

variable okay let's make one small adjustment

okay let's make one small adjustment going forward instead of Y we'll use y

going forward instead of Y we'll use y head but why in the previous video we

head but why in the previous video we learned that regression aims to model

learned that regression aims to model the dependent variable as accurately as

the dependent variable as accurately as possible however when working with real

possible however when working with real world data there's always some error in

world data there's always some error in other words the True Values often differ

other words the True Values often differ from the predictions now y hat

from the predictions now y hat represents the predicted value values

represents the predicted value values from the regression model While y

from the regression model While y denotes The observed actual values great

denotes The observed actual values great you're gradually becoming an expert now

you're gradually becoming an expert now there are four topics to cover the

there are four topics to cover the assumptions of regression how to

assumptions of regression how to calculate a regression with data tab how

calculate a regression with data tab how to interpret the results and finally how

to interpret the results and finally how to handle categorical variables in

to handle categorical variables in regression by creating dami variables

regression by creating dami variables let's start with the first topic so what

let's start with the first topic so what are the assumptions the first four

are the assumptions the first four assumptions of multiple linear

assumptions of multiple linear regression are similar to those of

regression are similar to those of simple linear regression but there is an

simple linear regression but there is an additional fifth assumption let's

additional fifth assumption let's briefly recap the first four assumptions

briefly recap the first four assumptions and then we'll go into more details on

and then we'll go into more details on the fifth assumption let's start with

the fifth assumption let's start with the first one linear relationship in the

the first one linear relationship in the case of the simple linear regression we

case of the simple linear regression we were able to test this assumption easily

were able to test this assumption easily a straight line is drawn through the

a straight line is drawn through the data this straight line should represent

data this straight line should represent All Points as good as possible if the

All Points as good as possible if the relationship is nonlinear the straight

relationship is nonlinear the straight line cannot fulfill this requirement in

line cannot fulfill this requirement in simple linear regression we have one

simple linear regression we have one independent variable and one dependent

independent variable and one dependent variable making it straightforward to

variable making it straightforward to plot the data points and the regression

plot the data points and the regression line in contrast multiple linear

line in contrast multiple linear regression involves multiple independent

regression involves multiple independent variables which complicate Ates the

variables which complicate Ates the visualization however you can still plot

visualization however you can still plot each independent variable separately

each independent variable separately against the dependent variable to gain

against the dependent variable to gain an initial sense of whether a linear

an initial sense of whether a linear relationship might exist number two

relationship might exist number two independence of Errors the errors so the

independence of Errors the errors so the differences between actual and predicted

differences between actual and predicted values should be independent of each

values should be independent of each other this means that the arrow of one

other this means that the arrow of one point doesn't affect another one we can

point doesn't affect another one we can test this with the Durban Watson test

test this with the Durban Watson test number three

number three homoscedasticity or equal variance of

homoscedasticity or equal variance of Errors if we plot the errors on the Y

Errors if we plot the errors on the Y AIS and the predicted values from the

AIS and the predicted values from the regression model on the xaxis their

regression model on the xaxis their spread should be roughly the same across

spread should be roughly the same across all values of X in other words the

all values of X in other words the variance of the error should remain

variance of the error should remain constant in this this case the

constant in this this case the assumption is fulfilled but what about

assumption is fulfilled but what about that case here we observe unequal

that case here we observe unequal variance at low values of X the errors

variance at low values of X the errors are small while at high values the

are small while at high values the variance of the errors becomes much

variance of the errors becomes much larger number four normally distributed

larger number four normally distributed errors the arrrow should be normally

errors the arrrow should be normally distributed we can test this with a QQ

distributed we can test this with a QQ plot or with analytical tests if you

plot or with analytical tests if you like you can check out my video test for

like you can check out my video test for normal distribution for a deeper dive

normal distribution for a deeper dive and what about the fifth assumption no

and what about the fifth assumption no multicolinearity first of all what is

multicolinearity first of all what is multicolinearity in regression

multicolinearity in regression multicolinearity means that two or more

multicolinearity means that two or more independent variables are highly

independent variables are highly correlated with each other as a result

correlated with each other as a result the effect of individual variables

the effect of individual variables cannot be clearly separated why is that

cannot be clearly separated why is that a problem let's look at the regression

a problem let's look at the regression equation again we have here the

equation again we have here the dependent variable and there the

dependent variable and there the independent variables with the

independent variables with the respective coefficients for example if

respective coefficients for example if there is a high correlation between X1

there is a high correlation between X1 and X2 or if these two variables are

and X2 or if these two variables are almost equal then it is difficult to

almost equal then it is difficult to determine B1 and B2 if both are

determine B1 and B2 if both are completely equal the regression model

completely equal the regression model cannot determine how large B1 and how

cannot determine how large B1 and how large B2 should be this means that one

large B2 should be this means that one independent variable can be predicted

independent variable can be predicted from the others with a high degree of

from the others with a high degree of accuracy an example imagine you're

accuracy an example imagine you're trying to predict the price of a house

trying to predict the price of a house to do this you use the size of the house

to do this you use the size of the house the number of rooms and some other

the number of rooms and some other variables usually the size of the house

variables usually the size of the house is related to the number of rooms large

is related to the number of rooms large houses tend to have more rooms so these

houses tend to have more rooms so these two variables are correlated if we now

two variables are correlated if we now include both in our regression model the

include both in our regression model the model will struggle to decide how much

model will struggle to decide how much of the price is influenced by size and

of the price is influenced by size and how much is influenced by number of

how much is influenced by number of rooms this is because they overlap in

rooms this is because they overlap in the information they provide and this is

the information they provide and this is multicolinearity in this case it becomes

multicolinearity in this case it becomes impossible to reliably determine the

impossible to reliably determine the regression coefficients if you just want

regression coefficients if you just want to use the regression model for a

to use the regression model for a prediction the presence of

prediction the presence of multicolinearity is less critical in

multicolinearity is less critical in this context the focus is on how

this context the focus is on how accurate the prediction is rather than

accurate the prediction is rather than on understanding the influence of the

on understanding the influence of the individual

individual variables however if the regression

variables however if the regression model is used to assess the influence of

model is used to assess the influence of independent variables on the dependent

independent variables on the dependent variable there should be no

variable there should be no multicolinearity

multicolinearity okay but how do we detect m

okay but how do we detect m multicolinearity if we look at the

multicolinearity if we look at the regression equation again we have the

regression equation again we have the variable X1 X2 and up to variable XK we

variable X1 X2 and up to variable XK we now want to determine whether X1 is

now want to determine whether X1 is nearly identical to any other variable

nearly identical to any other variable or a combination of the other variables

or a combination of the other variables for this we simply set up a new

for this we simply set up a new regression model in this regression

regression model in this regression model we take X1 as the new dependent

model we take X1 as the new dependent variable and keep the others as

variable and keep the others as independent variables if we can predict

independent variables if we can predict X1 accurately using the other

X1 accurately using the other independent variables X1 becomes

independent variables X1 becomes unnecessary its information is already

unnecessary its information is already captured by the other variables we can

captured by the other variables we can now do this for all other variables so

now do this for all other variables so we estimate X1 by using the other

we estimate X1 by using the other independent variables we estimate X2 by

independent variables we estimate X2 by using the other variables and we

using the other variables and we estimate XK by using the other

estimate XK by using the other independent variables okay but what is a

independent variables okay but what is a method to detect multical linearity for

method to detect multical linearity for all K regression models we calculate R

all K regression models we calculate R sared which is the so-called coefficient

sared which is the so-called coefficient of determination what is the coefficient

of determination what is the coefficient of determination R sared if X1 is the

of determination R sared if X1 is the dependent variable and the other

dependent variable and the other independent variables are used as

independent variables are used as predictors R squar explains how well the

predictors R squar explains how well the independent variables explain the

independent variables explain the variability of the dependent variable

variability of the dependent variable therefore a high r squared in this

therefore a high r squared in this context suggests that X1 is highly

context suggests that X1 is highly correlated with the other independent

correlated with the other independent variables this is a sign of

variables this is a sign of multicolinearity

multicolinearity using R squar we can calculate the

using R squar we can calculate the tolerance and the variance inflation

tolerance and the variance inflation Factor short

Factor short VI basically the vi is one divided by

VI basically the vi is one divided by the tolerance if the tolerance is less

the tolerance if the tolerance is less than

than 0.1 it indicates potential

0.1 it indicates potential multicolinearity and caution is required

multicolinearity and caution is required on the other hand a vif value greater

on the other hand a vif value greater than 10 is a warning sign of

than 10 is a warning sign of multicolinearity

multicolinearity requiring further investigation

requiring further investigation typically statistical programs like data

typically statistical programs like data tab provide the tolerance and variance

tab provide the tolerance and variance inflation Factor via F values for each

inflation Factor via F values for each independent variable okay but how to

independent variable okay but how to address multicolinearity there are two

address multicolinearity there are two common ways to address multicolinearity

common ways to address multicolinearity number one remove one of the correlated

number one remove one of the correlated variables so choose the variable that is

variables so choose the variable that is less significant and remove it or number

less significant and remove it or number two combine variables create a single

two combine variables create a single variable by combining the correlated

variable by combining the correlated variables e taking an average if you're

variables e taking an average if you're using data Tab and calculate the

using data Tab and calculate the regression you just need to click on

regression you just need to click on test

test assumptions here you can see the table

assumptions here you can see the table with the tolerance and the

with the tolerance and the vif all right let's work through an

vif all right let's work through an example on how to calculate a multip

example on how to calculate a multip linear regression and then look at how

linear regression and then look at how to interpret the results our goal is to

to interpret the results our goal is to analyze the influence of age weight and

analyze the influence of age weight and cholesterol level on blood pressure so

cholesterol level on blood pressure so blood pressure is our dependent variable

blood pressure is our dependent variable while age weight and cholesterol level

while age weight and cholesterol level are our independent variables to

are our independent variables to calculate the regression we just go to

calculate the regression we just go to data.net and copy our data into the

data.net and copy our data into the table if you like you can load the

table if you like you can load the sample data set using the link in the

sample data set using the link in the video description we want to calculate a

video description we want to calculate a regression so we click on regression now

regression so we click on regression now we simply click on BL pressure under

we simply click on BL pressure under dependent variable and age and weight

dependent variable and age and weight and cholesterol level under independent

and cholesterol level under independent variable afterwards we automatically get

variable afterwards we automatically get the results of the regression we will

the results of the regression we will now discuss how to interpret the

now discuss how to interpret the individual tables if you need an

individual tables if you need an interpretation of your individual data

interpretation of your individual data you can just click on AI interpretation

you can just click on AI interpretation at each table and you will get a

at each table and you will get a detailed explanation of your results and

detailed explanation of your results and if you want to test assumptions just

if you want to test assumptions just click here but back to the results let's

click here but back to the results let's start with the most important table the

start with the most important table the table with the regression coefficients

table with the regression coefficients and then take a closer look at the model

and then take a closer look at the model summary table we will focus on these

summary table we will focus on these three columns here we can see the three

three columns here we can see the three independent variables age weight and

independent variables age weight and cholesterol the first row represents the

cholesterol the first row represents the constant so in our regression equation

constant so in our regression equation let's replace X1 X2 and X3 with their

let's replace X1 X2 and X3 with their corresponding names so we want to

corresponding names so we want to predict blood pressure based on a

predict blood pressure based on a person's age weight and cholesterol

person's age weight and cholesterol levels okay in the First Column we see

levels okay in the First Column we see the unstandardized regression

the unstandardized regression coefficients these are our coefficients

coefficients these are our coefficients from the regression equation now we can

from the regression equation now we can calculate the blood pressure for a given

calculate the blood pressure for a given person let's say a person is 55 years

person let's say a person is 55 years old has a weight of 95 kg and a

old has a weight of 95 kg and a cholesterol level of

cholesterol level of 180 then our model would predict a blood

180 then our model would predict a blood pressure of 91 so for example if we look

pressure of 91 so for example if we look at the variable H

at the variable H 0.26 means that for each additional year

0.26 means that for each additional year of age the blood pressure increases by

of age the blood pressure increases by 0.26 units assuming other variables

0.26 units assuming other variables remain constant and what about the

remain constant and what about the standardized coefficients the the

standardized coefficients the the standardized coefficient tells us the

standardized coefficient tells us the relative importance of each independent

relative importance of each independent variable after standardizing the

variable after standardizing the variables to the same scale why is this

variables to the same scale why is this useful our model includes variables

useful our model includes variables measured in different units such as age

measured in different units such as age in years weight in kilograms comparing

in years weight in kilograms comparing their unstandardized coefficients can be

their unstandardized coefficients can be misleading because these coefficients

misleading because these coefficients are influenced by the units of

are influenced by the units of measurement for in instance if weight is

measurement for in instance if weight is measured in tons the coefficient would

measured in tons the coefficient would be larger if we measured in grams it

be larger if we measured in grams it would be smaller Additionally the values

would be smaller Additionally the values for age and years are generally smaller

for age and years are generally smaller than the values for cholesterol so you

than the values for cholesterol so you cannot directly compare their

cannot directly compare their unstandardized coefficients with each

unstandardized coefficients with each other in contrast their standardized

other in contrast their standardized coefficient remains consistent

coefficient remains consistent regardless of the units this allows for

regardless of the units this allows for direct comparison of the relative

direct comparison of the relative effects of different variables for

effects of different variables for example we can see that cholesterol

example we can see that cholesterol level has the largest standardized

level has the largest standardized coefficient indicating that it has the

coefficient indicating that it has the strongest influence on blood pressure in

strongest influence on blood pressure in our video about simple linear regression

our video about simple linear regression we explained the P value in detail the

we explained the P value in detail the interpretation is similar in multiple

interpretation is similar in multiple linear regression to summarize the P

linear regression to summarize the P value shows whether the corresponding

value shows whether the corresponding coefficient is significantly different

coefficient is significantly different from zero in other words it tells us if

from zero in other words it tells us if a variable has a real influence or if

a variable has a real influence or if the result could just be due to chance

the result could just be due to chance if the P value is smaller than

if the P value is smaller than 0.05 it means the difference is

0.05 it means the difference is significant in our case all P values are

significant in our case all P values are smaller than

smaller than 0.05 so all variables have a significant

0.05 so all variables have a significant influence perfect let's move on to the

influence perfect let's move on to the next table the model summary table first

next table the model summary table first we get the multiple correlation

we get the multiple correlation coefficient r R measures the correlation

coefficient r R measures the correlation between the dependent variable and the

between the dependent variable and the combination of the independent variables

combination of the independent variables what does that mean here we have the

what does that mean here we have the equation for linear regression once the

equation for linear regression once the coefficients are determined we can sum

coefficients are determined we can sum all up and calculate the predicted

all up and calculate the predicted values y heads of the dependent variable

values y heads of the dependent variable so if we use our example data we have

so if we use our example data we have the real blood pressure data and we can

the real blood pressure data and we can predict the blood pressure data with the

predict the blood pressure data with the regression model the multiple

regression model the multiple correlation coefficient R is now the

correlation coefficient R is now the correlation between the predicted values

correlation between the predicted values y hat and the actual values Y in other

y hat and the actual values Y in other words the multiple correlation

words the multiple correlation coefficient R indicates the strength of

coefficient R indicates the strength of the correlation between the actual

the correlation between the actual dependent variable and its estimated

dependent variable and its estimated values therefore the greater the

values therefore the greater the correlation the better the regression

correlation the better the regression model in our case an R value of

model in our case an R value of 0.27 indicates a strong positive

0.27 indicates a strong positive relationship okay and what about r

relationship okay and what about r squared r squared is called the

squared r squared is called the coefficient of determination r squared

coefficient of determination r squared indicates the proportion of variance in

indicates the proportion of variance in the dependent variable that is explained

the dependent variable that is explained by the independent variables the greater

by the independent variables the greater the explained variance the better the

the explained variance the better the model's performance for example an r

model's performance for example an r squared value of one would mean that the

squared value of one would mean that the entire variation in blood pressure can

entire variation in blood pressure can be perfectly explained by the variables

be perfectly explained by the variables age weight and cholesterol level however

age weight and cholesterol level however in reality this is really the case an r

in reality this is really the case an r s of

s of 0.52 means that 52% of the variation in

0.52 means that 52% of the variation in blood pressure is explained by the model

blood pressure is explained by the model what is the adjusted r squared the

what is the adjusted r squared the adjusted R squar Accounts for the number

adjusted R squar Accounts for the number of independent variables in the model

of independent variables in the model this provides a more accurate measure of

this provides a more accurate measure of explanatory power when a model includes

explanatory power when a model includes many independent variables the regular R

many independent variables the regular R squar can overestimate how well the

squar can overestimate how well the model explains the data in such cases it

model explains the data in such cases it is recommended to consider the adjusted

is recommended to consider the adjusted R squar to avoid

R squar to avoid overestimation okay and what about the

overestimation okay and what about the standard error of the estimate the

standard error of the estimate the standard error of the estimate measur me

standard error of the estimate measur me the average distance between the

the average distance between the observed data points and the regression

observed data points and the regression line a standard aror of the estimate of

line a standard aror of the estimate of 6.6 indicates that on average the

6.6 indicates that on average the model's predictions deviate from the

model's predictions deviate from the actual values by 6.6 units so if we

actual values by 6.6 units so if we predict a person's blood pressure using

predict a person's blood pressure using their age weight and cholesterol level

their age weight and cholesterol level our prediction will on average deviate

our prediction will on average deviate by 6.6 units from from the person's

by 6.6 units from from the person's actual plot pressure okay if you want an

actual plot pressure okay if you want an interpretation of the other tables

interpretation of the other tables simply click on AI

interpretation earlier in this video I mentioned that independent variables in

mentioned that independent variables in regression analysis can be nominal okay

regression analysis can be nominal okay but what are nominal variables nominal

but what are nominal variables nominal variables are variables with different

variables are variables with different categories like gender with male and

categories like gender with male and female or vehicle type but how do we use

female or vehicle type but how do we use nominal variables in a regression model

nominal variables in a regression model as independent variables let's keep

as independent variables let's keep things simple let's start with variables

things simple let's start with variables with two categories imagine we have the

with two categories imagine we have the variable gender with the categories male

variable gender with the categories male and female now we can code female as

and female now we can code female as zero and male as one the category coded

zero and male as one the category coded with zero is our so-called reference

with zero is our so-called reference category all right let's take a look at

category all right let's take a look at the regression equation suppose the

the regression equation suppose the variable X1 represents gender then B1 is

variable X1 represents gender then B1 is the regression coefficient for gender

the regression coefficient for gender but how do we interpret B1 we said zero

but how do we interpret B1 we said zero is female and one is male so let's just

is female and one is male so let's just insert this for X1 for a female

insert this for X1 for a female individual we have Z multiplied by B1

individual we have Z multiplied by B1 and for a male individual we have one

and for a male individual we have one multiplied by B1 accordingly B1

multiplied by B1 accordingly B1 represents the difference between males

represents the difference between males and females now that we've discussed how

and females now that we've discussed how to handle variables with two values

to handle variables with two values let's explore how to approach variables

let's explore how to approach variables with more than two values let's say we

with more than two values let's say we want to predict the fuel consumption of

want to predict the fuel consumption of a car based on its horsepower and

a car based on its horsepower and vehicle type to keep it simple let's say

vehicle type to keep it simple let's say there are only three vehicle types cedan

there are only three vehicle types cedan sports car and family van thus we have a

sports car and family van thus we have a variable vehicle type with more than two

variable vehicle type with more than two categories however as we know in

categories however as we know in regression we can only include variables

regression we can only include variables with two categories so what's the

with two categories so what's the solution this is where dumy variables

solution this is where dumy variables come into play dumy variables are

come into play dumy variables are artificial variables that make it

artificial variables that make it possible to handle variables with more

possible to handle variables with more than two categories for the variable

than two categories for the variable vehicle type we create a total of three

vehicle type we create a total of three dummy variables is sedan is sports car

dummy variables is sedan is sports car and is family van each of these dummy

and is family van each of these dummy variables has only two possible values

variables has only two possible values zero or one a value of one indicates the

zero or one a value of one indicates the presence of the specific category while

presence of the specific category while a value of zero indicates its absence

a value of zero indicates its absence instead of having one variable with

instead of having one variable with three categories we now have three

three categories we now have three variables with two categories each these

variables with two categories each these newly created dummy variables can be

newly created dummy variables can be included in the regression model okay

included in the regression model okay but what does this mean for our data

but what does this mean for our data preparation initially we have one column

preparation initially we have one column labeled vehicle type where the

labeled vehicle type where the individual vehicle types from our sample

individual vehicle types from our sample are listed the first entry is see then

are listed the first entry is see then the second is also San the third is a

the second is also San the third is a sports car and so on from this column we

sports car and so on from this column we create three new variables for the first

create three new variables for the first vehicle which is the sedan we assign one

vehicle which is the sedan we assign one under the sedan and zero under the

under the sedan and zero under the others as it's neither a sports car nor

others as it's neither a sports car nor a family van similarly the second

a family van similarly the second vehicle is also a seeden the Third

vehicle is also a seeden the Third vehicle however is a sports car so you

vehicle however is a sports car so you assign a one under sports car and zero

assign a one under sports car and zero under the others by doing this we've

under the others by doing this we've successfully created our dummy variables

successfully created our dummy variables one important thing to note the number

one important thing to note the number of dummy variables you create will

of dummy variables you create will always be the number of categories minus

always be the number of categories minus one so in our case we have three

one so in our case we have three categories so we basically only need two

categories so we basically only need two dummy variables why is that the case if

dummy variables why is that the case if we know a vehicle is a sedan we

we know a vehicle is a sedan we automatically know it is neither a

automatically know it is neither a sports car nor a family van similarily

sports car nor a family van similarily if we know it's a sports car we can

if we know it's a sports car we can inere that it's not a seeden or a family

inere that it's not a seeden or a family van finally if it's neither a Sean nor a

van finally if it's neither a Sean nor a sports car we know it must be a family

sports car we know it must be a family van this means we can express the same

van this means we can express the same information with just two variables

information with just two variables instead of three including all three

instead of three including all three variables would make the regression

variables would make the regression model

model overdetermined okay but don't worry if

overdetermined okay but don't worry if you're using data tab it will

you're using data tab it will automatically create the dummy variables

automatically create the dummy variables for you for example if we select fuel

for you for example if we select fuel consumption as the dependent variable

consumption as the dependent variable and horsepower and vehicle type as the

and horsepower and vehicle type as the independent variables we can then see

independent variables we can then see the three categories here and and can

the three categories here and and can select which one to use as the reference

select which one to use as the reference category for example if we choose sedan

category for example if we choose sedan as the reference dummy variables will be

as the reference dummy variables will be created for sports car and family van

created for sports car and family van when we examine the results we will see

when we examine the results we will see the two variables vehicle type sports

the two variables vehicle type sports car and vehicle type family van along

car and vehicle type family van along with

with horsepower so far we focused on linear

horsepower so far we focused on linear regression where our goal is to predict

regression where our goal is to predict a continuous variable like sales or

a continuous variable like sales or house prices but what if we need to

house prices but what if we need to predict categories instead like whether

predict categories instead like whether a costomer will buy a product or not or

a costomer will buy a product or not or whether an email is a spam that's where

whether an email is a spam that's where logistic regression comes into play so

logistic regression comes into play so in the next section we'll explore how

in the next section we'll explore how logistic regression helps us model

logistic regression helps us model binary outcomes and make probability

binary outcomes and make probability based predictions let's jump in what is

based predictions let's jump in what is a logistic Reg regression how is it

a logistic Reg regression how is it calculated and most importantly how are

calculated and most importantly how are the results interpreted let's start with

the results interpreted let's start with the first question what is a regression

the first question what is a regression in a regression analysis you want to

in a regression analysis you want to infer or predict an outcome variable

infer or predict an outcome variable based on one or more other variables

based on one or more other variables okay so what about logistic regression a

okay so what about logistic regression a binary logistic regression is now a type

binary logistic regression is now a type of regression analysis used when the

of regression analysis used when the outcome variable is binary meaning it

outcome variable is binary meaning it has two possible values like yes or no

has two possible values like yes or no success or failure let's look at an

success or failure let's look at an example let's say we are researchers and

example let's say we are researchers and we want to know whether a particular

we want to know whether a particular medication and a person's age have an

medication and a person's age have an influence on whether a person gets a

influence on whether a person gets a certain disease or not so the outcome

certain disease or not so the outcome we're interested in is whether the

we're interested in is whether the patients developed the disease or did

patients developed the disease or did not develop it and our independent

not develop it and our independent variables are medication and age of a

variables are medication and age of a person now with the help of a logistic

person now with the help of a logistic regression we want to inere or predict

regression we want to inere or predict the outcome variable based on the

the outcome variable based on the independent variables okay but what is

independent variables okay but what is the difference between a linear and a

the difference between a linear and a logistic regression in a linear

logistic regression in a linear regression the dependent variable is a

regression the dependent variable is a metric variable it salary or electricity

metric variable it salary or electricity consumption in a logistic regression the

consumption in a logistic regression the dependent variable is a binary variable

dependent variable is a binary variable so with the help of logistic regression

so with the help of logistic regression we can determine what has an influence

we can determine what has an influence on whether a certain disease is present

on whether a certain disease is present or not for example we could study the

or not for example we could study the influence of age gender and smoking

influence of age gender and smoking status on that particular disease in

status on that particular disease in this case one stands for diseased and

this case one stands for diseased and zero for not diseased we now want to

zero for not diseased we now want to estimate the probability that a person

estimate the probability that a person is diseased so our data set might look

is diseased so our data set might look like this here we have the independent

like this here we have the independent variables and there the dependent

variables and there the dependent variable with zero and one we could now

variable with zero and one we could now investigate what influence the

investigate what influence the independent variables have on the

independent variables have on the disease if there is an influence we can

disease if there is an influence we can predict How likely someone is to have a

predict How likely someone is to have a disease okay but why do we need logistic

disease okay but why do we need logistic regression in this case why can't we

regression in this case why can't we just use linear regression a quick recap

just use linear regression a quick recap in linear regression this is our

in linear regression this is our regression equation we have the

regression equation we have the dependent variable the independent

dependent variable the independent variables and the regression

variables and the regression coefficients however our dependent

coefficients however our dependent variable is now binary taking on the

variable is now binary taking on the value of either zero or one regardless

value of either zero or one regardless of the values of the independent

of the values of the independent variables the outcome will always be

variables the outcome will always be zero or one a linear regression would

zero or one a linear regression would now simply put a straight line through

now simply put a straight line through the points we can now see that in the

the points we can now see that in the case of linear regression values between

case of linear regression values between minus and plus infinity can occur

minus and plus infinity can occur however the goal of logistic regression

however the goal of logistic regression is to estimate the probability of

is to estimate the probability of occurrence the value range for the

occurrence the value range for the prediction should therefore be between

prediction should therefore be between zero and one so we need a function that

zero and one so we need a function that only takes values between zero and one

only takes values between zero and one and that is exactly what the logistic

and that is exactly what the logistic function does no matter where we are on

function does no matter where we are on the X AIS between minus and plus

the X AIS between minus and plus infinity only values between 0er and one

infinity only values between 0er and one result and that is exactly what we want

result and that is exactly what we want the equation for the logistic function

the equation for the logistic function looks like this logistic regression now

looks like this logistic regression now uses the logistic function for that the

uses the logistic function for that the equation of the linear regression is now

equation of the linear regression is now simply used this gives us that equation

simply used this gives us that equation this equation gives us the probability

this equation gives us the probability of the dependent variable equal one

of the dependent variable equal one given specific values of the independent

given specific values of the independent variables hm what does this look like

variables hm what does this look like for our example now in our example the

for our example now in our example the probability of having a certain disease

probability of having a certain disease is a function of age gender and smoking

is a function of age gender and smoking status next we need to determine the

status next we need to determine the coefficients that help our model best

coefficients that help our model best fit the given data this is done using

fit the given data this is done using the maximum likelihood method for this

the maximum likelihood method for this there are numerical methods that can

there are numerical methods that can solve the problem effectively a

solve the problem effectively a statistics program such as data tab

statistics program such as data tab therefore calculates the coefficients

therefore calculates the coefficients all right let's work through this

all right let's work through this example on how to calculate a logistic

example on how to calculate a logistic regression and then look at how to

regression and then look at how to interpret the results to calculate the

interpret the results to calculate the regression we just go to data.net and

regression we just go to data.net and copy our data into this table if you

copy our data into this table if you like you can load the sample data using

like you can load the sample data using the link in the video description we

the link in the video description we want to calculate a logistic regression

want to calculate a logistic regression so we just click on regression we choose

so we just click on regression we choose disease as the dependent variable and

disease as the dependent variable and age gender and smoking status as the

age gender and smoking status as the independent variables datab now

independent variables datab now calculates a logistic regression for us

calculates a logistic regression for us depending on how our dependent variable

depending on how our dependent variable is scaled data will calculate either a

is scaled data will calculate either a logistic or a linear regression under

logistic or a linear regression under the tab

the tab regression since we have two categorical

regression since we have two categorical variables we can set the reference

variables we can set the reference category we will just use female and

category we will just use female and nonsmoker as reference now we can choose

nonsmoker as reference now we can choose for which category we want to build the

for which category we want to build the regression model so we can decide if we

regression model so we can decide if we want to predict if a person is diseased

want to predict if a person is diseased or not diseased instead of diseased and

or not diseased instead of diseased and not deceased we could of course also

not deceased we could of course also have one and zero okay he before we go

have one and zero okay he before we go into detail about the different results

into detail about the different results a little tip if you don't know how to

a little tip if you don't know how to interpret the results you can also just

interpret the results you can also just click on summary inverts a logistic

click on summary inverts a logistic regression analysis was performed to

regression analysis was performed to examine the influence of age gender fale

examine the influence of age gender fale and smoking status smoker on the

and smoking status smoker on the variable disease to predict the value

variable disease to predict the value diseased logistic regression analysis

diseased logistic regression analysis showed that the model as a whole was

showed that the model as a whole was significant ific and then comes the

significant ific and then comes the interpretation of the different

interpretation of the different independent variables further you can

independent variables further you can click on AI interpretation at the

click on AI interpretation at the different tables we will now carefully

different tables we will now carefully go through each table step by step to

go through each table step by step to ensure everything is clear to you let's

ensure everything is clear to you let's begin at the top first we get the result

begin at the top first we get the result table here we can see that a total of 36

table here we can see that a total of 36 people were examined with the help of

people were examined with the help of the regression model of these 36 person

the regression model of these 36 person 26 could be correctly assigned that is

26 could be correctly assigned that is 72.22% next is the classification table

72.22% next is the classification table this table shows how often the

this table shows how often the categories not deceased and deceased

categories not deceased and deceased were observed and how frequently they

were observed and how frequently they were predicted in total not deceased was

were predicted in total not deceased was observed 16 times among these 16

observed 16 times among these 16 individuals the regression model

individuals the regression model correctly classified 11 as not deceased

correctly classified 11 as not deceased while misclassifying five as deceased of

while misclassifying five as deceased of the 20 deceased individuals the

the 20 deceased individuals the regression model misclassified five as

regression model misclassified five as not deceased and correctly classified 15

not deceased and correctly classified 15 as deceased but how do we determine

as deceased but how do we determine whether a person is classified as

whether a person is classified as diseased or not as mentioned earlier

diseased or not as mentioned earlier logistic regression provides the

logistic regression provides the probability of a person being diseased

probability of a person being diseased so we obtain values ranging from 0 to

so we obtain values ranging from 0 to 100% now we simply set a threshold of

100% now we simply set a threshold of 50% if a value exceeds 50% the person is

50% if a value exceeds 50% the person is classified as diseased otherwise they

classified as diseased otherwise they are classified as not deceased of course

are classified as not deceased of course you can choose a threshold other than

you can choose a threshold other than 50% to learn more about this check out

50% to learn more about this check out our video on the r curve so let's have a

our video on the r curve so let's have a look at the next table the kai Square

look at the next table the kai Square test evaluates whether the model as a

test evaluates whether the model as a whole is statistically significant for

whole is statistically significant for this two models are compared in one

this two models are compared in one model all independent variables are used

model all independent variables are used and in the other model the independent

and in the other model the independent variables are not used now we can

variables are not used now we can compare how good the prediction is when

compare how good the prediction is when the independent variables are used and

the independent variables are used and how good it is when the independent

how good it is when the independent variables are not used the kai Square

variables are not used the kai Square test now tells us if there is a

test now tells us if there is a significant difference between these two

significant difference between these two results the null hypothesis is that both

results the null hypothesis is that both models are the same if the P value is

models are the same if the P value is less than

less than 0.05 this null hypothesis is rejected in

0.05 this null hypothesis is rejected in our example the P value is less than

our example the P value is less than 0.05 and we assume that there is a

0.05 and we assume that there is a significant difference between the

significant difference between the models thus the model as a whole is

models thus the model as a whole is significant next comes the model model

significant next comes the model model summary in this table we can see on the

summary in this table we can see on the one hand the minus 2 log likelihood

one hand the minus 2 log likelihood value and on the other hand we are given

value and on the other hand we are given different coefficients of the

different coefficients of the determination r squar r squ is used to

determination r squar r squ is used to find out how well the regression model

find out how well the regression model explains the dependent variable in a

explains the dependent variable in a linear regression the r squ indicates

linear regression the r squ indicates the proportion of the variance that can

the proportion of the variance that can be explained by the independent

be explained by the independent variables the more variance can be

variables the more variance can be explained the better the regression

explained the better the regression model in a logistic regression however

model in a logistic regression however its interpretation differ and multiple

its interpretation differ and multiple methods exist to calculate R squar

methods exist to calculate R squar unfortunately there's no consensus yet

unfortunately there's no consensus yet on which method is considered the best

on which method is considered the best data daab gives you the r squared

data daab gives you the r squared according to Cox and Snell according to

according to Cox and Snell according to Nagle ker and according to MC feton and

Nagle ker and according to MC feton and now comes the most important table the

now comes the most important table the table with the model coefficients the

table with the model coefficients the most important parameters are the

most important parameters are the coefficient B the P value and the Ys

coefficient B the P value and the Ys ratio we'll now discuss all three

ratio we'll now discuss all three columns in the First Column we can read

columns in the First Column we can read the calculated coefficients from our

the calculated coefficients from our model we can insert these into the

model we can insert these into the regression equation so we get the

regression equation so we get the coefficients for age gender smoker and

coefficients for age gender smoker and the constant for example for a person

the constant for example for a person which is 55 years old is male and is

which is 55 years old is male and is nonsmoker we get a probability of

nonsmoker we get a probability of 36% thus it is 36% likely that a

36% thus it is 36% likely that a 55-year-old male nonsmoker is diseased

55-year-old male nonsmoker is diseased in reality that would certainly be many

in reality that would certainly be many other and different independent

other and different independent variables okay but what about the P

variables okay but what about the P value so the P value shows whether the

value so the P value shows whether the corresponding coefficient is

corresponding coefficient is significantly different from zero in

significantly different from zero in other words it tells us if a variable

other words it tells us if a variable has a real influence or if the result

has a real influence or if the result could just be due to chance if the P

could just be due to chance if the P value is smaller than

value is smaller than 0.05 it means the difference is

0.05 it means the difference is significant in our case all P values are

significant in our case all P values are greater than

greater than 0.05 indicating that none of the

0.05 indicating that none of the variables have a significant influence

variables have a significant influence and finally the odds ratio but what are

and finally the odds ratio but what are odds and and what is the odds ratio

odds and and what is the odds ratio let's start with the odds let's say we

let's start with the odds let's say we have two possible outcomes of something

have two possible outcomes of something success and failure for example if a

success and failure for example if a therapy is successful or not let's say

therapy is successful or not let's say that the probability that the therapy is

that the probability that the therapy is successful is 0.7 so

successful is 0.7 so 70% and thus the probability of failure

70% and thus the probability of failure is 1us 0.7 so

is 1us 0.7 so 0.3 okay but what about the odds odds

0.3 okay but what about the odds odds are defined as the ratio of the

are defined as the ratio of the probability of success and the

probability of success and the probability of failure or in other words

probability of failure or in other words odds represent the ratio of the

odds represent the ratio of the probability of an event happening to the

probability of an event happening to the probability of it not happening if we

probability of it not happening if we look at our example the odds are

look at our example the odds are 0.7 divided by

0.7 divided by 0.3 which equals

0.3 which equals 2.33 this means the event success

2.33 this means the event success is

is 2.33 times more likely to happen than

2.33 times more likely to happen than not so odds give us a measure of the

not so odds give us a measure of the likelihood of an event happening versus

likelihood of an event happening versus it not happening in this case we've

it not happening in this case we've calculated the odds of success of course

calculated the odds of success of course we can also calculate the odds of

we can also calculate the odds of failure all right now that we understand

failure all right now that we understand odds let's talk about odds ratios so

odds let's talk about odds ratios so what are odds ratios let's look at the

what are odds ratios let's look at the example from the beginning we're

example from the beginning we're studying a new medication to reduce the

studying a new medication to reduce the risk of a certain disease so we have a

risk of a certain disease so we have a group a patients with medication and the

group a patients with medication and the group b patients without medication

group b patients without medication let's say in group a we calculated a

let's say in group a we calculated a probability of 60% or 0.6 of getting

probability of 60% or 0.6 of getting diseased so the odds of getting diseased

diseased so the odds of getting diseased is

is 0.6 ided 0.4 which is 1.5 five again

0.6 ided 0.4 which is 1.5 five again odds just represent the ratio of the

odds just represent the ratio of the probability of an event happening to the

probability of an event happening to the probability of it not happening in our

probability of it not happening in our case in group a the likelihood of being

case in group a the likelihood of being diseased is 1.5 time higher than the

diseased is 1.5 time higher than the likelihood of not being diseased let's

likelihood of not being diseased let's say in group b where the patients didn't

say in group b where the patients didn't get the medication the probability of

get the medication the probability of getting deceased is 80% or

getting deceased is 80% or 0.8 so the odds in group b of getting

0.8 so the odds in group b of getting diseased are 0.8 divided by 0.2 so four

diseased are 0.8 divided by 0.2 so four therefore in group b the likelihood of

therefore in group b the likelihood of being diseased is four times higher than

being diseased is four times higher than the likelihood of not being diseased

the likelihood of not being diseased what about the odds ratio with the odds

what about the odds ratio with the odds ratio we can now compare the two groups

ratio we can now compare the two groups to do this we can compare the odds of

to do this we can compare the odds of getting the disease in group a relative

getting the disease in group a relative to the odds of getting the disease in

to the odds of getting the disease in group b so the odds ratio is simply

group b so the odds ratio is simply calculated by dividing the odds in group

calculated by dividing the odds in group a by the odds in group b this results in

a by the odds in group b this results in an odds ratio of

an odds ratio of 0.38 the odds ratio of 0.38 means that

0.38 the odds ratio of 0.38 means that the odds of being deceased in group a

the odds of being deceased in group a are

are 0.38 times the odds of being deceased in

0.38 times the odds of being deceased in group b of course we can also switch the

group b of course we can also switch the order then the odds ratio would be the

order then the odds ratio would be the odds in Group B divided by the odds in

odds in Group B divided by the odds in group a in this case the odds ratio of

group a in this case the odds ratio of approximately 2.67 means that the odds

approximately 2.67 means that the odds of being deceased in group b are 2.67

of being deceased in group b are 2.67 times higher than the odds of being

times higher than the odds of being deceased in group a so an odds ratio is

deceased in group a so an odds ratio is simply a comparison of the odds of an

simply a comparison of the odds of an event occurring in two different groups

event occurring in two different groups the odds ratio indicates how much more

the odds ratio indicates how much more likely the event is to occur in one

likely the event is to occur in one group group compared to the other group

group group compared to the other group if the odds ratio is greater than one

if the odds ratio is greater than one the event is more likely to occur in the

the event is more likely to occur in the first group if it is less than one the

first group if it is less than one the event is less likely in the first group

event is less likely in the first group okay now let's put it all together and

okay now let's put it all together and look at how to interpret the odds ratio

look at how to interpret the odds ratio in logistic regression let's get started

in logistic regression let's get started first of all to calculate a logistic

first of all to calculate a logistic regression we need data let's say we

regression we need data let's say we have data from 50 patients our our

have data from 50 patients our our outcome variable is disease which is

outcome variable is disease which is coded as zero for not diseased and one

coded as zero for not diseased and one for diseased and we have two independent

for diseased and we have two independent variables medication and age now we can

variables medication and age now we can use this data to calculate a logistic

use this data to calculate a logistic regression you can find a link to the

regression you can find a link to the data set in the video description in the

data set in the video description in the First Column we can see the coefficients

First Column we can see the coefficients that Define our model these coefficients

that Define our model these coefficients can be entered into the logistic

can be entered into the logistic regression formula here we can see the

regression formula here we can see the coefficients from the table the constant

coefficients from the table the constant the coefficients for medication and for

the coefficients for medication and for age now we just need to enter a value

age now we just need to enter a value for medication such as one indicating

for medication such as one indicating the patient received medication and a

the patient received medication and a value for age for example 50 then we can

value for age for example 50 then we can calculate the probability in this case

calculate the probability in this case the probability of being diseased is

the probability of being diseased is 0.55 or

0.55 or 55% so for a patient who took the

55% so for a patient who took the medication and is 50 years old the

medication and is 50 years old the probability of being deceased is

probability of being deceased is 55% of course we can simply use data to

55% of course we can simply use data to calculate this probability to do this

calculate this probability to do this just enter one here and 50 there we will

just enter one here and 50 there we will then also get a probability of

then also get a probability of 0.55 and data dep further gives us the

0.55 and data dep further gives us the odds as we know the odds are calculated

odds as we know the odds are calculated by the probability that a certain event

by the probability that a certain event will happen divided by the probability

will happen divided by the probability ility that the event will not happen we

ility that the event will not happen we therefore get

therefore get 0.55 / 1 -

0.55 / 1 - 0.55 which equals

0.55 which equals 1.22 okay but we are not interested in

1.22 okay but we are not interested in the odds alone we're interested in the

the odds alone we're interested in the odds ratio again the odds ratio is

odds ratio again the odds ratio is simply a comparison of the odds of an

simply a comparison of the odds of an event occurring in two different groups

event occurring in two different groups the two groups could be persons who took

the two groups could be persons who took the medication and persons who did not

the medication and persons who did not take the medication therefore if we're

take the medication therefore if we're going back to data tab we just need to

going back to data tab we just need to compare the odds of a person who took

compare the odds of a person who took the

the medication with the odds of a person who

medication with the odds of a person who did not take the medication so to get

did not take the medication so to get the odds ratio we just need to divide

the odds ratio we just need to divide the odds of a person who took the

the odds of a person who took the medication by the odds of a person who

medication by the odds of a person who did not take the medication this results

did not take the medication this results in an odds ratio of

in an odds ratio of 0.64 and surprise the calculated value

0.64 and surprise the calculated value matches the odds ratio listed for the

matches the odds ratio listed for the variable medication the odds ratio of

variable medication the odds ratio of 0.64 for medication indicates that for

0.64 for medication indicates that for individuals who took the medication the

individuals who took the medication the odds of the outcome diseased are

odds of the outcome diseased are 0.64 times the odds of those who did not

0.64 times the odds of those who did not take the medication all right with

take the medication all right with medication we have two groups to compare

medication we have two groups to compare but what about a continuous variable

but what about a continuous variable like AG in this case we simply look at

like AG in this case we simply look at what happens when we increase age by one

what happens when we increase age by one unit for example we might compare the

unit for example we might compare the odds of the outcome for someone aged 50

odds of the outcome for someone aged 50 versus someone aged 51 this allows us to

versus someone aged 51 this allows us to calculate the odds ratio by comparing

calculate the odds ratio by comparing the two odds in this case we get an odds

the two odds in this case we get an odds ratio of

ratio of 1.04 so for each one year increase in

1.04 so for each one year increase in age the odds of the outcome deceased

age the odds of the outcome deceased increase by a factor of

increase by a factor of 1.04 but there's one thing I haven't

1.04 but there's one thing I haven't told you yet the odds ratio can actually

told you yet the odds ratio can actually be calculated simply by exponentiating

be calculated simply by exponentiating each coefficient so e to the power of

each coefficient so e to the power of 0.45 is

0.45 is 0.64 which is the odds ratio of

0.64 which is the odds ratio of medication and e to the power of

medication and e to the power of 0.04 is 1.04

0.04 is 1.04 which is the odds ratio for H to sum it

which is the odds ratio for H to sum it up odds are simply the ratio of the

up odds are simply the ratio of the probability of an event happening to the

probability of an event happening to the probability of it not happening the odds

probability of it not happening the odds ratio is now the ratio of the odds that

ratio is now the ratio of the odds that an event occurs in two different groups

an event occurs in two different groups we just talked about regression where we

we just talked about regression where we predict values based on known

predict values based on known relationships let's now shift our Focus

relationships let's now shift our Focus to discovering hidden patterns in data

to discovering hidden patterns in data with no apparent relationship this

with no apparent relationship this brings us to class analysis specifically

brings us to class analysis specifically the K means clustering technique K means

the K means clustering technique K means clustering is a powerful method used to

clustering is a powerful method used to identify hidden groups or clusters

identify hidden groups or clusters within our data let's explore how that

within our data let's explore how that works and how it can enhance our

works and how it can enhance our understanding of complex data sets in

understanding of complex data sets in this video I will explain to you

this video I will explain to you everything you need to know about clust

everything you need to know about clust analysis I will start with the question

analysis I will start with the question what is the K mean claster and then I

what is the K mean claster and then I will show you how it can easily

will show you how it can easily calculated online with data Tab and now

calculated online with data Tab and now let's start with the question what is

let's start with the question what is the C's clust analysis the K means class

the C's clust analysis the K means class analysis is one of the simplest and most

analysis is one of the simplest and most common methods for class analysis by

common methods for class analysis by using the K means method you can cluster

using the K means method you can cluster your data by a given number of clusters

your data by a given number of clusters so you already need to Define beforehand

so you already need to Define beforehand the number of clusters for example you

the number of clusters for example you have a data set and you want to Cluster

have a data set and you want to Cluster your cases into three clusters this can

your cases into three clusters this can be done with the C's clust analysis for

be done with the C's clust analysis for example you could have a data set with

example you could have a data set with 15 European countries and you want to

15 European countries and you want to Cluster them into three country groups

Cluster them into three country groups so now the question is how does the

so now the question is how does the kin's clust analysis work there are five

kin's clust analysis work there are five simple steps required let's start with

simple steps required let's start with the first step first you have to define

the first step first you have to define the number of clusters to find the

the number of clusters to find the groups or clusters the number of

groups or clusters the number of clusters is the K in K means in our case

clusters is the K in K means in our case we simply select three classs so in this

we simply select three classs so in this example K was selected equal to three

example K was selected equal to three the second step now is to set the

the second step now is to set the cluster centers random each of the

cluster centers random each of the centes now represent

centes now represent one cluster let's come to step three now

one cluster let's come to step three now we have selected the number of clusters

we have selected the number of clusters and we set the claster centers randomly

and we set the claster centers randomly now we assign each element to one

now we assign each element to one claster so for example we assign each

claster so for example we assign each country to one cluster let's start with

country to one cluster let's start with one element and now the distance from

one element and now the distance from the first element to each of the claster

the first element to each of the claster centroids is calculated so for example

centroids is calculated so for example we calc calate the distance from this

we calc calate the distance from this element to each cluster centroid

element to each cluster centroid afterwards each element is assigned to

afterwards each element is assigned to the claster to which it has the smallest

the claster to which it has the smallest distance in our example the distance

distance in our example the distance between this element and the claster

between this element and the claster centroid is the smallest so we will

centroid is the smallest so we will assign this element to the yellow

assign this element to the yellow cluster now this step is repeated for

cluster now this step is repeated for all further elements so at the end we

all further elements so at the end we have one yellow cluster one red cluster

have one yellow cluster one red cluster and one green cluster and then all

and one green cluster and then all points are initially assigned to a

points are initially assigned to a cluster so let's summarize it again we

cluster so let's summarize it again we first have defined the number of

first have defined the number of clusters we have then assigned the

clusters we have then assigned the cluster centroid randomly and we have

cluster centroid randomly and we have assigned each element in step four we

assigned each element in step four we now calculate the center of each cluster

now calculate the center of each cluster so for the green elements for the yellow

so for the green elements for the yellow elements and for the red elements the

elements and for the red elements the center of each clust is calculated these

center of each clust is calculated these centers are the new claster centroids

centers are the new claster centroids now this means that we simply shift the

now this means that we simply shift the centroids into the cluster Center so the

centroids into the cluster Center so the cluster centroids are moved to the

cluster centroids are moved to the cluster centers now in Step number five

cluster centers now in Step number five we assign the elements to the new

we assign the elements to the new clusters since the centroids now can be

clusters since the centroids now can be located at a different element each

located at a different element each element is assigned to the claster that

element is assigned to the claster that is closest to it

is closest to it now we have finished all steps and from

now we have finished all steps and from now on step four and five are repeated

now on step four and five are repeated until the claster solution does not

until the claster solution does not change anymore then the classing

change anymore then the classing procedure is over one disadvantage of

procedure is over one disadvantage of the K means method is that the final

the K means method is that the final results depends very much on which

results depends very much on which initial clusters we used to take this

initial clusters we used to take this into account the whole procedure is

into account the whole procedure is carried out several times and different

carried out several times and different randomly chosen starting points are used

randomly chosen starting points are used for each of the

for each of the calculations each time we use different

calculations each time we use different starting points it could be that the

starting points it could be that the outcome is different so we do the whole

outcome is different so we do the whole class analysis several times in order to

class analysis several times in order to get the best possible result if you use

get the best possible result if you use data to calculate the class analysis the

data to calculate the class analysis the analysis is for example done 10 times

analysis is for example done 10 times with 10 different randomly chosen sty

with 10 different randomly chosen sty starting points and then at the end the

starting points and then at the end the best cluster solution is chosen so the

best cluster solution is chosen so the next question is what is the optimal

next question is what is the optimal claster number with each new cluster the

claster number with each new cluster the sum distance in the Clusters gets

sum distance in the Clusters gets smaller and smaller so if we have a look

smaller and smaller so if we have a look at this picture where we have two classs

at this picture where we have two classs and that picture where we have three

and that picture where we have three classers for sure these three clusters

classers for sure these three clusters fit the data better than these two

fit the data better than these two clusters the distance between the

clusters the distance between the elements and the Clusters is in this

elements and the Clusters is in this case higher than in that case so the

case higher than in that case so the question now is how many clusters should

question now is how many clusters should be used in order to answer this question

be used in order to answer this question we use the elbow method with each

we use the elbow method with each additional cluster the summed distance

additional cluster the summed distance between the elements and the claster

between the elements and the claster center becomes smaller and smaller

center becomes smaller and smaller however there is a claster number from

however there is a claster number from which each additional cluster reduces

which each additional cluster reduces the sum distance only slightly this

the sum distance only slightly this point is used as the number of clusters

point is used as the number of clusters so if we have a look at this plot we can

so if we have a look at this plot we can see that there is a big gap between

see that there is a big gap between cluster number one and two and there's

cluster number one and two and there's also a big gap between the number of

also a big gap between the number of clusters two and three but there's only

clusters two and three but there's only a small gap between number of clusters

a small gap between number of clusters three and four so in this case we select

three and four so in this case we select a number of clusters of three clusters

a number of clusters of three clusters now I'd like to show you how you can

now I'd like to show you how you can easily calculate a k means class

easily calculate a k means class analysis online with data tab to do this

analysis online with data tab to do this please visit data

please visit data t.net and click on the statistics

t.net and click on the statistics calculator in order to calculate a class

calculator in order to calculate a class analysis you simply choose the tab

analysis you simply choose the tab claster if you want to use your own data

claster if you want to use your own data you can clear the table and then copy

you can clear the table and then copy your own data into this table I will

your own data into this table I will simply use the example data now so I

simply use the example data now so I want to calculate a k means clust

want to calculate a k means clust analysis let's say we want to clust a

analysis let's say we want to clust a group of people by salary and age first

group of people by salary and age first we can Define the number of clusters and

we can Define the number of clusters and we enter the number of clusters data

we enter the number of clusters data deab will now calculate everything for

deab will now calculate everything for you and you get your results right away

you and you get your results right away so here you can see the three clusters

so here you can see the three clusters with the centroids and there you can see

with the centroids and there you can see the elbow method in this case the

the elbow method in this case the results indicate that we should use a

results indicate that we should use a solution with two two clusters we have

solution with two two clusters we have selected three clusters before so we can

selected three clusters before so we can change it in order to get the best

change it in order to get the best suitable number of clusters but let's go

suitable number of clusters but let's go through the results now step by step

through the results now step by step here we can see how many elements are

here we can see how many elements are assigned to the different clusters and

assigned to the different clusters and here we get the plot with the different

here we get the plot with the different clusters here we can see one cluster we

clusters here we can see one cluster we see one cluster there and we see another

see one cluster there and we see another cluster here further we get a table

cluster here further we get a table where each element is allocated to a

where each element is allocated to a cluster if we now choose two as the

cluster if we now choose two as the number of clusters we get new results

number of clusters we get new results and we can see them here in this plot we

and we can see them here in this plot we have one claster here and one claster

have one claster here and one claster there more over we can see that the two

there more over we can see that the two clusters fit the data quite well there's

clusters fit the data quite well there's one more important concept we need to

one more important concept we need to discuss confidence intervals in the next

discuss confidence intervals in the next video we'll break down what confidence

video we'll break down what confidence intervals are why they matter in

intervals are why they matter in statistical analysis and how to

statistical analysis and how to interpret them properly let's dive in in

interpret them properly let's dive in in this video we'll uncover the true

this video we'll uncover the true definition of a confidence interval we

definition of a confidence interval we will clear up some common misconceptions

will clear up some common misconceptions and explain the difference between the

and explain the difference between the incorrect and the correct interpretation

incorrect and the correct interpretation so let's get started first of all why do

so let's get started first of all why do we need confidence intervals in

we need confidence intervals in statistics par parameters of the

statistics par parameters of the population are often estimated based on

population are often estimated based on a sample therefore on the one hand you

a sample therefore on the one hand you have the population but since in most

have the population but since in most cases you cannot serve the entire

cases you cannot serve the entire population you draw a sample now we want

population you draw a sample now we want to use this sample to estimate a

to use this sample to estimate a parameter of the population parameters

parameter of the population parameters that can be estimated are for example

that can be estimated are for example the mean or the variance let's look at

the mean or the variance let's look at an example you want to know the height

an example you want to know the height of all profession basketball players in

of all profession basketball players in the US in order to figure this out you

the US in order to figure this out you draw a sample the mean of the sample is

draw a sample the mean of the sample is most likely different from the mean of

most likely different from the mean of the population let's assume that we draw

the population let's assume that we draw not just one but several samples which

not just one but several samples which of course you don't actually do in

of course you don't actually do in practice each sample is likely to show a

practice each sample is likely to show a different mean so in the first sample we

different mean so in the first sample we have one mean in the second sample we

have one mean in the second sample we most likely have another mean and again

most likely have another mean and again in another sample we have another mean

in another sample we have another mean of course it is also possible that

of course it is also possible that purely by chance two or more samples

purely by chance two or more samples have means that are exactly the same but

have means that are exactly the same but this is very unlikely now it would be

this is very unlikely now it would be extremely valuable to have a range that

extremely valuable to have a range that we expect to capture the true parameter

we expect to capture the true parameter with a certain level of confidence and

with a certain level of confidence and this is precisely where the

this is precisely where the misconception about confidence intervals

misconception about confidence intervals comes in in fact published Studies have

comes in in fact published Studies have shown that scientists frequently

shown that scientists frequently misinterpret conf confidence intervals

misinterpret conf confidence intervals let's dive in and break down exactly

let's dive in and break down exactly what a confidence interval means and

what a confidence interval means and just as importantly what it does not

just as importantly what it does not mean there are two common ways to

mean there are two common ways to explain the confidence interval on the

explain the confidence interval on the one hand there is a simpler explanation

one hand there is a simpler explanation of the confidence interval but it's not

of the confidence interval but it's not correct when feuded from a frequenti

correct when feuded from a frequenti statistics perspective on the other hand

statistics perspective on the other hand there's a slightly more complex

there's a slightly more complex explanation that is actually true to

explanation that is actually true to make the difference clear we'll start

make the difference clear we'll start with the simple but wrong interpretation

with the simple but wrong interpretation then explain why it falls short and

then explain why it falls short and finally lead us to a clearer

finally lead us to a clearer understanding of the correct

understanding of the correct interpretation to keep things simple

interpretation to keep things simple let's focus on the 95% confidence

let's focus on the 95% confidence interval but the same goes for the

interval but the same goes for the others of

others of course so let's address the simple but

course so let's address the simple but incorrect interpretation this

incorrect interpretation this interpretation goes like this there is a

interpretation goes like this there is a 95% chance that the true parameter lies

95% chance that the true parameter lies within a calculated confidence interval

within a calculated confidence interval so what does this actually mean imagine

so what does this actually mean imagine we have a population with a true mean

we have a population with a true mean value this true mean value is the one we

value this true mean value is the one we want to estimate although we don't know

want to estimate although we don't know this true mean we can make an educated

this true mean we can make an educated guess by taking a sample from the

guess by taking a sample from the population from this sample we calculate

population from this sample we calculate both the sample mean and the 95%

both the sample mean and the 95% confidence interval the simplified

confidence interval the simplified interpretation is to say the confidence

interpretation is to say the confidence interval provides a range within which

interval provides a range within which the true mean lies with a certain

the true mean lies with a certain probability or in case of the 95%

probability or in case of the 95% confidence interval we would say there

confidence interval we would say there is a 95% chance that the True Value

is a 95% chance that the True Value Falls within this interval however this

Falls within this interval however this interpretation isn't accurate but why in

interpretation isn't accurate but why in frequent statistics the true parameter

frequent statistics the true parameter in our case the true mean is treated as

in our case the true mean is treated as a fixed but unknown quantity so the true

a fixed but unknown quantity so the true parameter does not move around it is

parameter does not move around it is fixed if we now draw a sample and

fixed if we now draw a sample and calculate the confidence interval the

calculate the confidence interval the True Value either lies inside the

True Value either lies inside the interval or it doesn't in this case the

interval or it doesn't in this case the confidence interval contains the True

confidence interval contains the True Value therefore there's no probability

Value therefore there's no probability associated with the parameter being

associated with the parameter being within this specific interval but why

within this specific interval but why because probabilities in frequentist

because probabilities in frequentist terms only apply to events that are

terms only apply to events that are subject to variability and again the

subject to variability and again the true parameter is fixed and cannot

true parameter is fixed and cannot change the only thing that varies is the

change the only thing that varies is the sample data we collect every time we

sample data we collect every time we draw a new sample we have new data and

draw a new sample we have new data and consequently a new mean and confidence

consequently a new mean and confidence interval so for example in this sample

interval so for example in this sample the True Value Falls within a confidence

the True Value Falls within a confidence interval if we take a second sample

interval if we take a second sample maybe the confidence interval will not

maybe the confidence interval will not include the True Value therefore there's

include the True Value therefore there's no probability associated with the

no probability associated with the parameter being within this specific

parameter being within this specific interval but why the reason is that

interval but why the reason is that probabilities in frequencies terms only

probabilities in frequencies terms only apply to events that are subject to

apply to events that are subject to variability and again the true parameter

variability and again the true parameter is fixed and cannot change therefore you

is fixed and cannot change therefore you cannot assign a probability to the true

cannot assign a probability to the true parameter being in a given interval

parameter being in a given interval parameter is either inside the interval

parameter is either inside the interval or it's not the only thing that varies

or it's not the only thing that varies is the sample data we collect every time

is the sample data we collect every time we draw a new sample we have new data

we draw a new sample we have new data and consequently a new mean and

and consequently a new mean and confidence interval so for example in

confidence interval so for example in all these samples the True Value Falls

all these samples the True Value Falls within the confidence interval while in

within the confidence interval while in those samples it doesn't in summary you

those samples it doesn't in summary you can say that there is a 95% chance that

can say that there is a 95% chance that this interval contains the true

this interval contains the true parameter because once the interval is

parameter because once the interval is calculated it either contains the

calculated it either contains the parameter or it doesn't and there is no

parameter or it doesn't and there is no probability left to assign in the

probability left to assign in the frequen sense but what is the correct

frequen sense but what is the correct interpretation let's say we took a lot

interpretation let's say we took a lot of random samples and we calculated the

of random samples and we calculated the mean value and the confidence interval

mean value and the confidence interval of each sample the confidence interval

of each sample the confidence interval can now be be interpreted in the

can now be be interpreted in the following way if we were to take an

following way if we were to take an extremely large number of random samples

extremely large number of random samples and construct a confidence interval for

and construct a confidence interval for each sample 95% of those intervals would

each sample 95% of those intervals would contain the True Value while 5% would

contain the True Value while 5% would not in other words if we were to take

not in other words if we were to take 100 random samples we would expect that

100 random samples we would expect that on average 95 of the confidence

on average 95 of the confidence intervals would contain the True Value

intervals would contain the True Value while five would not you can also see it

while five would not you can also see it the other way around the confidence

the other way around the confidence interval can be defined in terms of

interval can be defined in terms of probability with respect to a single

probability with respect to a single theoretical sample that has yet to be

theoretical sample that has yet to be realized therefore if you haven't drawn

realized therefore if you haven't drawn the sample yet you can be 95% sure that

the sample yet you can be 95% sure that the interval from the next sample you

the interval from the next sample you draw will contain the True Value but if

draw will contain the True Value but if you have taken the sample the true value

you have taken the sample the true value is either in the interval or not

is either in the interval or not therefore cont confidence is about the

therefore cont confidence is about the method not the specific interval the 95%

method not the specific interval the 95% confidence refers to the long run

confidence refers to the long run reliability of the method you use to

reliability of the method you use to construct the interval it means that if

construct the interval it means that if you use this method repeatedly on

you use this method repeatedly on different samples you expect to capture

different samples you expect to capture the true parameter

the true parameter 95% of the time but once you've applied

95% of the time but once you've applied it and obtained a specific interval you

it and obtained a specific interval you then cannot make a probability statement

then cannot make a probability statement about whether this interval contains the

about whether this interval contains the fixed true parameter or not a side note

fixed true parameter or not a side note in statistics there are two distinct

in statistics there are two distinct approaches or Frameworks the frequentist

approaches or Frameworks the frequentist and the basan the confidence interval is

and the basan the confidence interval is a method used in the frequentist

a method used in the frequentist approach in AAS approach we would treat

approach in AAS approach we would treat the parameter as a random variable with

the parameter as a random variable with its own probability distribution

its own probability distribution reflecting our uncertainity about it in

reflecting our uncertainity about it in that frame work it would make sense to

that frame work it would make sense to say that given our data there is a

say that given our data there is a certain probability that the parameter

certain probability that the parameter will fall within a certain range but

will fall within a certain range but compared to the frequentist

compared to the frequentist interpretation this is a fundamentally

interpretation this is a fundamentally different way of thinking in the basan

different way of thinking in the basan approach there is a concept known as the

approach there is a concept known as the credible interval which serves as the

credible interval which serves as the counterpart to the confidence interval

counterpart to the confidence interval in frequented statistics but

in frequented statistics but unfortunately there's also critics of

unfortunately there's also critics of the basan way in short patient

the basan way in short patient statistics require the use of a

statistics require the use of a so-called prior distribution the main

so-called prior distribution the main criticism is that credible intervals may

criticism is that credible intervals may not be entirely objective as they are

not be entirely objective as they are influenced by the choice of the prior

influenced by the choice of the prior distribution this makes the results

distribution this makes the results potentially sensitive to subjective

potentially sensitive to subjective inputs however this same feature can

inputs however this same feature can also be seen as a strength as it allows

also be seen as a strength as it allows for incorporating prior knowledge into

for incorporating prior knowledge into the analysis in a principled way okay

the analysis in a principled way okay but now to the easiest part how is the

but now to the easiest part how is the confidence interval for the mean

confidence interval for the mean calculated if you data are normally

calculated if you data are normally distributed the confidence interval for

distributed the confidence interval for the mean can be calculated with this

the mean can be calculated with this formula the confidence interval CI is

formula the confidence interval CI is xar plus or minus Z * s / by the root of

xar plus or minus Z * s / by the root of n here xar is the mean set is the set

n here xar is the mean set is the set value for the respective confidence

value for the respective confidence level n is the sample size and S is the

level n is the sample size and S is the standard deviation plus minus results

standard deviation plus minus results from the fact that we have once the

from the fact that we have once the upper limit with plus and once the lower

upper limit with plus and once the lower limit with minus where do we obtain the

limit with minus where do we obtain the set value the set value for a given

set value the set value for a given confidence interval can be found in a

confidence interval can be found in a standard normal distribution table which

standard normal distribution table which lists set values corresponding to the

lists set values corresponding to the different confidence levels for example

different confidence levels for example at a 95% confidence level the set value

at a 95% confidence level the set value is

is 1.96 using this the confidence interval

1.96 using this the confidence interval can be expressed as the sample mean plus

can be expressed as the sample mean plus -

- 1.96 times the standard deviation

1.96 times the standard deviation divided by the square root of the sample

divided by the square root of the sample size the confidence interval can of

size the confidence interval can of course be calculated for many

course be calculated for many statistical parameters not only for the

statistical parameters not only for the mean value thanks for watching and I

mean value thanks for watching and I hope you enjoyed the video bye

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YouTube Transcript:Statistics - A Full Lecture to learn Data Science (2025 Version)

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Statistics - A Full Lecture to learn Data Science (2025 Version)