What is a one-way repeated measures ANOVA?

Repeated Measures (ANOVA)

Conceptual Explanation

How did you get here?

How did you get here?So, you have decided to use a Repeated Measures ANOVA.

How did you get here?So, you have decided to use a Repeated Measures ANOVA.Let’s consider the decisions you made to get here.

First of all, you must have noticed the problem to be solved deals with generalizing from a smaller sample to a larger population.



Sample of 30


Sample of 30

Generalizes to


Large Population of 30,000

Sample of 30

Generalizes to


Therefore, you would determine that the problem deals with inferential not descriptive statistics.

Large Population of 30,000

Sample of 30

Generalizes to



Double check your problem to see if that is the case


Inferential Descriptive


You would have also noticed that the problem dealt with questions of difference not Relationships, Independence nor Goodness of Fit. Inferential Descriptive

You would have also noticed that the problem dealt with questions of difference not Relationships, Independence nor Goodness of Fit.



Difference




Difference Relationship




DifferenceDifference Relationship




Difference Goodness of FitDifference Relationship

After checking the data, you noticed that the data was ratio/interval rather than extreme ordinal (1st, 2nd, 3rd place) or nominal (male, female)



Difference Goodness of FitDifference Relationship




Difference Goodness of Fit

Ratio/Interval






OrdinalRatio/Interval






NominalOrdinalRatio/Interval


The distribution was more or less normal rather than skewed or kurtotic.








Skewed







Skewed Kurtotic







Skewed Kurtotic Normal



Only one Dependent Variable (DV) rather than two or more exist.


DV #1

Chemistry Test Scores


DV #1 DV #2


Class Attendance


DV #1 DV #2 DV #3


Class Attendance

Homework Completed









Descriptive



1 DV


Inferential





Difference Relationship Difference Goodness of Fit

Ratio/Interval Ordinal Nominal


1 DV 2+ DV


Only one Independent Variable (DV) rather than two or more exist.


IV #1

Use of Innovative eBook


IV #1 IV #2


Doing Homework to Classical Music


IV #1 IV #2 IV #3


Doing Homework to Classical Music Gender


IV #1 IV #2 IV #3


Doing Homework to Classical Music Gender


Only one Independent Variable (DV) rather than two or more exist. Descriptive



1 DV 2+ DV

Inferential



Only one Independent Variable (DV) rather than two or more exist. Inferential Descriptive



1 DV 2+ DV

1 IV

Inferential





Nominal


1 DV 2+ DV

1 IV 2+ IV

Inferential


Difference Relationship Difference




1 DV 2+ DV

1 IV 2+ IV


Inferential


Difference Relationship Difference

There are three levels of the Independent Variable (IV) rather than just two levels. Note – even though repeated measures ANOVA can analyze just two levels, this is generally analyzed using a paired sample t-test.

There are three levels of the Independent Variable (DV) rather than just two levels. Note – even though repeated measures ANOVA can analyze just two levels, this is generally analyzed using a paired sample t-test.

Level 1

Before using the innovative ebook


Level 1 Level 2


Using the innovative ebook

for 2 months


Level 1 Level 2 Level 3



for 2 months


for 4 months

Descriptive



1 DV 2+ DVs

2+ IVs

Inferential



2 levels 3+ levels

1 IV

Difference

The samples are repeated rather than independent. Notice that the same class (Chem 100 section 003) is repeatedly tested.

The samples are repeated rather than independent. Notice that the same class (Chem 100 section 003) is repeatedly tested.

Chem 100 Section 003

January


March


May

Before using the innovative

ebook


for 2 months


for 4 months

Descriptive



1 DV 2+ DVs

2+ IVs

Inferential



2 levels 3+ levels

1 IV

Difference

RepeatedIndependent

If this was the appropriate path for your problem then you have correctly selected Repeated-measures ANOVA to solve the problem you have been presented.

Repeated Measures ANOVA –

Repeated Measures ANOVA –Another use of analysis of variance is to test whether a single group of people change over time.

Repeated Measures ANOVA –Another use of analysis of variance is to test whether a single group of people change over time.

In this case, the distributions that are compared to each other are not from different groups


versus

Group 1 Group 2


versus

Group 1 Group 2


But from different times.

versus

Group 1 Group 2


But from different times.

versus

Group 1 Group 2

Group 1 Group 1: Two Months Later

versus

For example, an instructor might administer the same test three times throughout the semester to ascertain whether students are improving in their skills.


January FebruaryApril

Exam 1Exam 2

Exam 3


The overall F-ratio will reveal whether there are differences somewhere among three time periods.


Exam 1Exam 2

Exam 3




Exam 1Exam 2

Exam 3




Exam 1Exam 2

Exam 3

Average Score

Average Score

Average Score




Exam 1Exam 2

Exam 3

Average Score

Average Score

Average Score




Exam 1Exam 2

Exam 3

Average Score

Average Score

Average Score

There is a difference but

we don’t know where

Post hoc tests will reveal exactly where the differences occurred.



Exam 1Exam 2

Exam 3

Average Score 35

Average Score 38

Average Score 40



Exam 1Exam 2

Exam 3

Average Score 35

Average Score 38

Average Score 40

There is a statistically significant

difference only between Exam 1

and Exam 3

In contrast, with the One-way analysis of Variance (ANOVA) we were attempting to determine if there was a statistical difference between 2 or more (generally 3 or more) groups.

In contrast, with the One-way analysis of Variance (ANOVA) we were attempting to determine if there was a statistical difference between 2 or more (generally 3 or more) groups.In our One-way ANOVA example in another presentation we attempted to determine if there was any statistically significant difference in the amount of Pizza Slices consumed by three different player types (football, basketball, and soccer).

The data would be set up thus:

The data would be set up thus:Football Players

Pizza Slices

Consumed

Basketball Players

Pizza Slices Consumed

Soccer Players


Ben 5 Cam 6 Dan 5

Bob 7 Colby 4 Denzel 8

Bud 8 Conner 8 Dilbert 8

Bubba 9 Custer 4 Don 1

Burt 10 Cyan 2 Dylan 2


Notice how the individuals in these groups are different (hence different names)

Football Players

Pizza Slices

Consumed

Basketball Players


Soccer Players


Ben 5 Cam 6 Dan 5






Notice how the individuals in these groups are different (hence different names)

Football Players

Pizza Slices

Consumed

Basketball Players


Soccer Players


Ben 5 Cam 6 Dan 5






Notice how the individuals in these groups are different (hence different names)A Repeated Measures ANOVA is different than a One-Way ANOVA in one simply way: Only one group of person or observations is being measured, but they are measured more than one time.

Football Players

Pizza Slices

Consumed

Basketball Players


Soccer Players


Ben 5 Ben 6 Ben 5

Bob 7 Bob 4 Bob 8

Bud 8 Bud 8 Bud 8

Bubba 9 Bubba 4 Bubba 1

Burt 10 Burt 2 Burt 2


Notice how the individuals in these groups are different (hence different names)A Repeated Measures ANOVA is different than a One-Way ANOVA in one simply way: Only one group of persons or observations is being measured, but they are measured more than one time.

Football Players

Pizza Slices

Consumed

Basketball Players


Soccer Players


Ben 5 Ben 6 Ben 5

Bob 7 Bob 4 Bob 8

Bud 8 Bud 8 Bud 8



Notice the different times football player pizza consumption is being measured.

Football Players

Pizza Slices

Consumed



Ben 5 Ben 6 Ben 5

Bob 7 Bob 4 Bob 8

Bud 8 Bud 8 Bud 8



Notice the different times football player pizza consumption is being measured.

Football Players

Pizza Slices

ConsumedBefore the

Season


During the Season


After the Season

Ben 5 Ben 6 Ben 5

Bob 7 Bob 4 Bob 8

Bud 8 Bud 8 Bud 8



Since only one group is being measured 3 times, each time is dependent on the previous time. By dependent we mean there is a relationship.


Pizza Slices ConsumedFootball Players Before the

SeasonDuring the

SeasonAfter the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


The relationship between the scores is that we are comparing the same person across multiple observations.


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

So, Ben’s before-season and during-season and after-season scores have one important thing in common:

So, Ben’s before-season and during-season and after-season scores have one important thing in common:


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

So, Ben’s before-season and during-season and after-season scores have one important thing in common: THESE SCORES ALL BELONG TO BEN.


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

So, Ben’s before-season and during-season and after-season scores have one important thing in common: THESE SCORES ALL BELONG TO BEN.

They are subject to all the factors that are special to Ben when consuming pizza, including how much he likes or dislikes, the toppings that are available, the eating atmosphere, etc.


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

What we want to find out is – how much the BEFORE, DURING, and AFTER season pizza consuming sessions differ.

What we want to find out is – how much the BEFORE, DURING, and AFTER season pizza consuming sessions differ.But we have to find a way to eliminate the variability that is caused by individual differences that linger across all three eating sessions. Once again we are not interested in the things that make Ben, Ben while eating pizza (like he’s a picky eater). We are interested in the effect of where we are in the season (BEFORE, DURING, and AFTER on Pizza consumption.)



That way we can focus just on the differences that are related to WHEN the pizza eating occurred.

That way we can focus just on the differences that are related to WHEN the pizza eating occurred. After running a repeated-measures ANOVA, this is the output that we will get:

That way we can focus just on the differences that are related to WHEN the pizza eating occurred. After running a repeated-measures ANOVA, this is the output that we will get:

Tests of Within-Subjects Effects

Measure: Pizza slices

Source

Type III Sum of

Squares dfMean

Square F Sig.

Between Subjects 21.333 4

Between Groups 19.733 2 9.867 9.548 .008

Error 8.267 8 1.033

Total 49.333 14

This output will help us determine if we reject the null hypothesis:

This output will help us determine if we reject the null hypothesis:There is no significant difference in the amount of pizza consumed by football players before,

during, and/or after the season.


during, and/or after the season.Or accept the alternative hypothesis:


during, and/or after the season.Or accept the alternative hypothesis:There is a significant difference in the amount of

pizza consumed by football players before, during, and/or after the season.

To do so, let’s focus on the value .008

To do so, let’s focus on the value .008Tests of Within-Subjects Effects

Measure: Pizza slices consumed

SourceType III Sum of Squares df

Mean Square F Sig.

Between Subjects 21.333 4Between Groups 19.733 2 9.867 9.548 .008

Error 8.267 8 1.033

Total 49.333 14

To do so, let’s focus on the value .008Tests of Within-Subjects Effects


Source

Type III Sum of

Squares dfMean

Square F Sig.Between Subjects 21.333 4Between Groups 19.733 2 9.867 9.548 .008Error 8.267 8 1.033Total 49.333 14


This means that if we were to reject the null hypothesis, the probability that we would be wrong is 8 times out of 1000. As you remember, if that were to happen, it would be called a Type 1 error.



Source

Type III Sum of

Squares dfMean



This means that if we were to reject the null hypothesis, the probability that we would be wrong is 8 times out of 1000. As you remember, if that were to happen, it would be called a Type 1 error.



Source

Type III Sum of

Squares dfMean


But it is so unlikely, that we would be willing to take that risk and hence reject the null hypothesis.

But it is so unlikely, that we would be willing to take that risk and hence we reject the null hypothesis.

There IS NO statistically significant difference between the number of slices of pizza consumed

by football players before, during, or after the football season.

But it is so unlikely, that we would be willing to take that risk and hence we reject the null hypothesis.

There IS NO statistically significant difference between the number of slices of pizza consumed

by football players before, during, or after the football season. REJE

CT

And accept the alternative hypothesis:


There IS A statistically significant difference between the number of slices of pizza consumed

by football players before, during, or after the football season.


There IS A statistically significant difference between the number of slices of pizza consumed

by football players before, during, or after the football season. ACCEPT

Now we do not know which of the three are significantly different from one another or if all three are different. We just know that a difference exists.



SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6



SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


Later, we can run what is called a “Post-hoc” test to determine where the difference lies.


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

From this point on – we will delve into the actual calculations and formulas that produce a Repeated-measures ANOVA. If such detail is of interest or a necessity to know, please continue.

How was a significance value of .008 calculated?

How was a significance value of .008 calculated?Let’s begin with the calculation of the various sources of Sums of Squares

How was a significance value of .008 calculated?Let’s begin with the calculation of the various sources of Sums of Squares



Source

Type III Sum of

Squares dfMean


We do this so that we can explain what is causing the scores to vary or deviate.

We do this so that we can explain what is causing the scores to vary or deviate.• Is it error?

We do this so that we can explain what is causing the scores to vary or deviate.• Is it error?• Is it differences between times (before,

during, and after)?

We do this so that we can explain what is causing the scores to vary or deviate.• Is it error?• Is it differences between times (before,

during, and after)?Remember, the full name for sum of squares is the sum of squared deviations about the mean. This will help us determine the amount of variation from each of the possible sources.

Let’s begin by calculating the total sums of squares.


𝑆𝑆𝑡𝑜𝑡𝑎𝑙=Σ(𝑋 𝑖𝑗− �́� )2





This means one pizza eating observation for person “I” (e.g., Ben) on

time “j” (e.g., before)

For example:

For example: Pizza Slices Consumed

Football Players Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example: Pizza Slices Consumed

Football Players Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

OR


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

OR


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

OR


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

OR


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

OR


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

For example:

ETC


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6



This means the average of all of the

observations



observationsThis means one pizza eating observation for

person “I” (e.g., Ben) on time “j” (e.g., before)



observations


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

This means one pizza eating observation for




observations


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Average of All Observations




This means sum or add

everything up



everything up

This means the average of

all of the observations

�́�𝑿



everything up

This means the average of

all of the observations



Let’s calculate total sums of squares with this data set:

Let’s calculate total sums of squares with this data set:


SeasonDuring the


Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

To do so we will rearrange the data like so:

To do so we will rearrange the data like so:Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt


BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season

Ben BeforeBob BeforeBud Before

Bubba BeforeBurt BeforeBen DuringBob DuringBud During

Bubba DuringBurt DuringBen AfterBob AfterBud After

Bubba AfterBurt After


BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players

Season Slices of Pizza

Ben Before 5Bob Before 7Bud Before 8

Bubba Before 9Burt Before 10Ben During 4Bob During 5Bud During 7

Bubba During 8Burt During 7Ben After 4Bob After 5Bud After 6

Bubba After 4Burt After 6

To do so we will rearrange the data like so:We will

subtract each of these values from

the grand mean, square the

result and sum them all up.

Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players










Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players











Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players







Each observation




result and sum them all up.Here is how we

compute the Grand Mean =

Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players











Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players











Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Pizza Slices ConsumedFootball Players

Before the Season

During the Season

After the Season

Ben 5 4 4Bob 7 5 5Bud 8 7 6

Bubba 9 8 4Burt 10 7 6






Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players







Before the Season

During the Season

After the Season

Ben 5 4 4Bob 7 5 5Bud 8 7 6


Average of All Observations =

6.3






Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Football Players


Ben Before 5 -Bob Before 7 -Bud Before 8 -

Bubba Before 9 -Burt Before 10 -Ben During 4 -Bob During 5 -Bud During 7 -

Bubba During 8 -Burt During 7 -Ben After 4 -Bob After 5 -Bud After 6 -

Bubba After 4 -Burt After 6 -





Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Football Players








To do so we will rearrange the data like so:We

will subtract each of these values from the

grand mean, square the result and sum

them all up.

Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Football Players






Football Players


Grand Mean

Ben Before 5 - 6.3Bob Before 7 - 6.3Bud Before 8 - 6.3

Bubba Before 9 - 6.3Burt Before 10 - 6.3Ben During 4 - 6.3Bob During 5 - 6.3Bud During 7 - 6.3

Bubba During 8 - 6.3Burt During 7 - 6.3Ben After 4 - 6.3Bob After 5 - 6.3Bud After 6 - 6.3

Bubba After 4 - 6.3Burt After 6 - 6.3


We will subtract each of these values

from the grand mean,

square the result and sum them all up.

Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Football Players






Football Players


Grand Mean





Football Players


Grand Mean

Ben Before 5 - 6.3 =Bob Before 7 - 6.3 =Bud Before 8 - 6.3 =

Bubba Before 9 - 6.3 =Burt Before 10 - 6.3 =Ben During 4 - 6.3 =Bob During 5 - 6.3 =Bud During 7 - 6.3 =

Bubba During 8 - 6.3 =Burt During 7 - 6.3 =Ben After 4 - 6.3 =Bob After 5 - 6.3 =Bud After 6 - 6.3 =

Bubba After 4 - 6.3 =Burt After 6 - 6.3 =


We will subtract each of these values

from the grand mean,

square the result and sum them all up.

Football Players

BenBobBud

BubbaBurtBenBobBud

BubbaBurtBenBobBud

BubbaBurt

Football Players

Season





Football Players






Football Players






Football Players


Grand Mean





Football Players


Grand Mean

Ben Before 5 - 6.3 =Bob Before 7 - 6.3 =Bud Before 8 - 6.3 =

Bubba Before 9 - 6.3 =Burt Before 10 - 6.3 =Ben During 4 - 6.3 =Bob During 5 - 6.3 =Bud During 7 - 6.3 =

Bubba During 8 - 6.3 =Burt During 7 - 6.3 =Ben After 4 - 6.3 =Bob After 5 - 6.3 =Bud After 6 - 6.3 =

Bubba After 4 - 6.3 =Burt After 6 - 6.3 =

Football Players


Grand Mean

Deviation

Ben Before 5 - 6.3 = -1.3Bob Before 7 - 6.3 = 0.7Bud Before 8 - 6.3 = 1.7

Bubba Before 9 - 6.3 = 2.7Burt Before 10 - 6.3 = 3.7Ben During 4 - 6.3 = -2.3Bob During 5 - 6.3 = -1.3Bud During 7 - 6.3 = 0.7

Bubba During 8 - 6.3 = 1.7Burt During 7 - 6.3 = 0.7Ben After 4 - 6.3 = -2.3Bob After 5 - 6.3 = -1.3Bud After 6 - 6.3 = -0.3

Bubba After 4 - 6.3 = -2.3Burt After 6 - 6.3 = -0.3


We will subtract each of these values from the grand mean, square the result and sum them all up.

Football Players


Grand Mean

Deviation Squared

Ben Before 5 - 6.3 = -1.3 1.8Bob Before 7 - 6.3 = 0.7 0.4Bud Before 8 - 6.3 = 1.7 2.8

Bubba Before 9 - 6.3 = 2.7 7.1Burt Before 10 - 6.3 = 3.7 13.4Ben During 4 - 6.3 = -2.3 5.4Bob During 5 - 6.3 = -1.3 1.8Bud During 7 - 6.3 = 0.7 0.4

Bubba During 8 - 6.3 = 1.7 2.8Burt During 7 - 6.3 = 0.7 0.4Ben After 4 - 6.3 = -2.3 5.4Bob After 5 - 6.3 = -1.3 1.8Bud After 6 - 6.3 = -0.3 0.1

Bubba After 4 - 6.3 = -2.3 5.4Burt After 6 - 6.3 = -0.3 0.1

Football Players


Grand Mean

Deviation Squared






We will subtract each of these values from the grand mean, square the result and sum them all up.

Football Players


Grand Mean

Deviation Squared





= 49.3

Football Players


Grand Mean

Deviation Squared






Then –

Football Players


Grand Mean

Deviation Squared





= 49.3


Then – we place the total sums of squares result in the ANOVA table.

Football Players


Grand Mean

Deviation Squared





Football Players


Grand Mean

Deviation Squared





= 49.3

Football Players


Grand Mean

Deviation Squared







Football Players


Grand Mean

Deviation Squared





= 49.3


Football Players


Grand Mean

Deviation Squared





= 49.3


Football Players


Grand Mean

Deviation Squared





= 49.3




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

We have now calculated the total sums of squares. This is a good starting point. Because now we want to know of that total sums of squares how many sums of squares are generated from the following sources:

We have now calculated the total sums of squares. This is a good starting point. Because now we want to know of that total sums of squares how many sums of squares are generated from the following sources:• Between subjects (this is the variance we

want to eliminate)


want to eliminate)• Between Groups (this would be between

BEFORE, DURING, AFTER)


want to eliminate)• Between Groups (this would be between

BEFORE, DURING, AFTER)• Error (the variance that we cannot explain

with our design)

With these sums of squares we will be able to compute our F ratio value and then statistical significance.





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


Let’s calculate the sums of squares between subjects.




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

Remember if we were just computing a one way ANOVA the table would go from this:





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


To this:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


To this:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14




Mean Square F Sig.

Between Groups 19.733 2 9.867 2.669 .078

Error 29.600 8 3.700

Total 49.333 14


To this:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14




Mean Square F Sig.

Between Groups 19.733 2 9.867 2.669 .078

Error 29.600 8 3.700

Total 49.333 14

All of that variability goes into the error or within groups sums of squares (29.600) which makes the F statistic smaller (from 9.548 to 2.669), the significance value no longer significant (.008 to .078).

All of that variability goes into the error or within groups sums of squares (29.600) which makes the F statistic smaller (from 9.548 to 2.669), the significance value no longer significant (.008 to .078).But the difference in within groups variability is not a function of error, it is a function of Ben, Bob, Bud, Bubba, and Burt’s being different in terms of the amount of slices they eat regardless of when they eat!

All of that variability goes into the error or within groups sums of squares (29.600) which makes the F statistic smaller (from 9.548 to 2.669), the significance value no longer significant (.008 to .078).But the difference in within groups variability is not a function of error, it is a function of Ben, Bob, Bud, Bubba, and Burt’s being different in terms of the amount of slices they eat regardless of when they eat!

Pizza Slices Consumed Football Players

Before the Season

During the Season

After the Season

Average

Ben 5 4 4 4.3Bob 7 5 5 5.7Bud 8 7 6 7.0

Bubba 9 8 4 7.0Burt 10 7 6 7.7

Here is a data set where there are not between group differences, but there is a lot of difference based on when the group eats their pizza:



Before the Season

During the Season

After the Season

Average

Ben 1 5 9 5.0Bob 2 5 8 5.0Bud 3 5 7 5.0

Bubba 1 5 9 5.0Burt 2 5 8 5.0


There is no variability between subjects (they are all 5.0).


Before the Season

During the Season

After the Season

Average

Ben 1 5 9 5.0Bob 2 5 8 5.0Bud 3 5 7 5.0

Bubba 1 5 9 5.0Burt 2 5 8 5.0

Look at the variability between groups:

Look at the variability between groups: Pizza Slices Consumed

Football Players

Before the Season

During the Season

After the Season

Average

Ben 1 5 9 5.0Bob 2 5 8 5.0Bud 3 5 7 5.0

Bubba 1 5 9 5.0Burt 2 5 8 5.0

1.8 5.0 8.2

Look at the variability between groups:

They are very different from one another.


Before the Season

During the Season

After the Season

Average

Ben 1 5 9 5.0Bob 2 5 8 5.0Bud 3 5 7 5.0

Bubba 1 5 9 5.0Burt 2 5 8 5.0

1.8 5.0 8.2

Here is what the ANOVA table would look like:

Here is what the ANOVA table would look like:Tests of Within-Subjects Effects



Mean Square F Sig.


Error 5.600 8 0.700

Total 49.333 14


Notice how there are no sum of squares values for the between subjects source of variability!




Mean Square F Sig.


Error 5.600 8 0.700

Total 49.333 14


Notice how there are no sum of squares values for the between subjects source of variability!But there is a lot of sum of squares values for the between groups.




Mean Square F Sig.


Error 5.600 8 0.700

Total 49.333 14


Notice how there are no sum of squares values for the between subjects source of variability!But there is a lot of sum of squares values for the between groups.




Mean Square F Sig.


Error 5.600 8 0.700

Total 49.333 14




Mean Square F Sig.


Error 5.600 8 0.700

Total 49.333 14

What would the data set look like if there was very little between groups (by season) variability and a great deal of between subjects variability:

What would the data set look like if there was very little between groups (by season) variability and a great deal of between subjects variability:Here it is:

What would the data set look like if there was very little between groups (by season) variability and a great deal of between subjects variability:Here it is:


Football Players

Before the Season

During the Season

After the Season

Average

Ben 3 3 3 3.0

Bob 5 5 5 5.0

Bud 7 7 7 7.0

Bubba 8 8 8 8.0

Burt 12 12 13 12.3

Between Subjects

In this case the between subjects (Ben, Bob, Bud . . .), are very different.

In this case the between subjects (Ben, Bob, Bud . . .), are very different.When you see between SUBJECTS averages that far away, you know that the sums of squares for between groups will be very large.

In this case the between subjects (Ben, Bob, Bud . . .), are very different.When you see between SUBJECTS averages that far away, you know that the sums of squares for between groups will be very large.




Mean Square F Sig.


Error 0.533 8 0.067

Total 148.933 14

Notice, in contrast, as we compute the between group (seasons) average how close they are.



Before the Season

During the Season

After the Season

Average

Ben 3 3 3 3.0Bob 5 5 5 5.0Bud 7 7 7 7.0

Bubba 8 8 8 8.0Burt 12 12 13 12.3

7.0 7.0 7.2



Before the Season

During the Season

After the Season

Average

Ben 3 3 3 3.0Bob 5 5 5 5.0Bud 7 7 7 7.0

Bubba 8 8 8 8.0Burt 12 12 13 12.3

7.0 7.0 7.2

Between Groups



Before the Season

During the Season

After the Season

Average

Ben 3 3 3 3.0Bob 5 5 5 5.0Bud 7 7 7 7.0

Bubba 8 8 8 8.0Burt 12 12 13 12.3

7.0 7.0 7.2

Between Groups

When you see between group averages this close you know that the sums of squares for between groups will be very small.





Mean Square F Sig.


Error 0.533 8 0.067

Total 148.933 14


Now that we have conceptually considered the sources of variability as described by the sum of squares, let’s begin calculating between subjects, between groups, and the error sources.




Mean Square F Sig.


Error 0.533 8 0.067

Total 148.933 14

We will begin with calculating Between Subjects sum of squares.

We will begin with calculating Between Subjects sum of squares.To do so, let’s return to our original data set:



Before the Season

During the Season

After the Season

Ben 5 4 4Bob 7 5 5Bud 8 7 6



Here is the formula for calculating SS between subjects.


Before the Season

During the Season

After the Season

Ben 5 4 4Bob 7 5 5Bud 8 7 6



Here is the formula for calculating SS between subjects.


Before the Season

During the Season

After the Season

Ben 5 4 4Bob 7 5 5Bud 8 7 6


𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠=𝑘∗ Σ(𝑋𝑏𝑠− �́� )2

𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠=𝑘∗ Σ(𝑿𝒃𝒔− �́� )2

𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠=𝑘∗ Σ(𝑿𝒃𝒔− �́� )2 Pizza Slices Consumed

Football Players

Before the Season

During the Season

After the Season

Average

Ben 5 4 4 4.3

Bob 7 5 5 5.7

Bud 8 7 6 7.0

Bubba 9 8 4 7.0

Burt 10 7 6 7.7


Football Players

Before the Season

During the Season

After the Season

Average

Ben 5 4 4 4.3

Bob 7 5 5 5.7

Bud 8 7 6 7.0

Bubba 9 8 4 7.0

Burt 10 7 6 7.7

𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠=𝑘∗ Σ(𝑿𝒃𝒔− �́� )2This means the average of between

subjects


Football Players

Before the Season

During the Season

After the Season

Average minus

Ben 5 4 4 4.3 -

Bob 7 5 5 5.7 -

Bud 8 7 6 7.0 -

Bubba 9 8 4 7.0 -

Burt 10 7 6 7.7 -

𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠=𝑘∗ Σ(𝑿𝒃𝒔− �́� )2

This means the average of all of the observations

Here is how we calculate the grand mean again:

Here is how we calculate the grand mean again: Pizza Slices Consumed

Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Average of All Observations =

6.3

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.


Football Players

Before the Season

During the Season

After the Season

Average minus Grand Mean

Ben 5 4 4 4.3 - 6.3

Bob 7 5 5 5.7 - 6.3

Bud 8 7 6 7.0 - 6.3

Bubba 9 8 4 7.0 - 6.3

Burt 10 7 6 7.7 - 6.3

This means the average of all of the observations

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean.

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean.


Football Players

Before the Season

During the Season

After the Season


Deviation

Ben 5 4 4 4.3 - 6.3 -2.0

Bob 7 5 5 5.7 - 6.3 -0.6

Bud 8 7 6 7.0 - 6.3 0.7

Bubba 9 8 4 7.0 - 6.3 0.7

Burt 10 7 6 7.7 - 6.3 1.4

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean. Now we will square the deviations.

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean. Now we will square the deviations.

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean. Now we will square the deviations


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean. Now we will square the deviations.Then we sum all of these squared deviations.




Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Sum up

Here is how we calculate the grand mean again:Now we subtract each subject or person average from the Grand Mean.This gives us the person’s average score deviation from the total or grand mean. Now we will square the deviations.Then we sum all of these squared deviations.Finally, we multiply the sum all of these squared deviations by the number of groups:


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups

Sum of Squares Between Subjects 21.3


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups


Number of conditions


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups



Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups



Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups


1 2 3


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups


1 2 3



Source Type III Sum of Squares df

Mean Square F Sig.

Between Subjects 21.333 4 Between Groups 19.733 2 9.867 9.548 .008

Error 8.267 8 1.033

Total 49.333 14


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups


1 2 3


Football Players

Before the

Season

During the Season

After the Season


Deviation Squared

Ben 5 4 4 4.3 - 6.3 -2.0 3.9

Bob 7 5 5 5.7 - 6.3 -0.6 0.4

Bud 8 7 6 7.0 - 6.3 0.7 0.5

Bubba 9 8 4 7.0 - 6.3 0.7 0.5

Burt 10 7 6 7.7 - 6.3 1.4 1.9

7.1

Times 3 groups


1 2 3




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

Now it is time to compute the between groups (seasons) sum of squares.

Now it is time to compute the between groups’ (seasons) sum of squares.

Here is the equation we will use to compute it:

Now it is time to compute the between groups’ (seasons) sum of squares.

Here is the equation we will use to compute it:

Let’s break this down with our data set:




Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

We begin by computing the mean of each condition (k)


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

Then subtract each condition mean from the grand mean.


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -

Grand Mean

6.3 6.3 6.3



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -

Grand Mean

6.3 6.3 6.3

equals

Deviation 1.5 -0.1 -1.3


Square the deviation.


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -

Grand Mean

6.3 6.3 6.3

equals

Deviation 1.5 -0.1 -1.3

Squared Deviation

2.2 0.0 1.8

Sum the Squared Deviations:



Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -

Grand Mean

6.3 6.3 6.3

equals

Deviation 1.5 -0.1 -1.3

Squared Deviation

2.2 0.0 1.8

Sum


Bubba 9 8 4

Burt 10 7 6

Condition Mean

7.8 6.2 5.0

minus - - -

Grand Mean

6.3 6.3 6.3

equals

Deviation 1.5 -0.1 -1.3

Squared Deviation

2.2 0.0 1.8

Sum


3.95

Sum of Squared Deviations

Multiply by the number of observations per condition (number of pizza eating slices across before, during, and after).


3.95



3.95



3.95


5

Number of observations


3.95


5



3.95


5


19.7Weighted Sum of

Squared Deviations

Let’s return to the ANOVA table and put the weighted sum of squared deviations.



Measure: Pizza_slices


Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

3.95


5


19.7Weighted Sum of

Squared Deviations





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

3.95


5


19.7Weighted Sum of

Squared Deviations

So far we have calculated Total Sum of Squares along with Sum of Squares for Between Subjects, and Between Groups.

So far we have calculated Total Sum of Squares along with Sum of Squares along with Sum of Squares for Between Subjects, Between Groups.




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

Now we will calculate the sum of squares associated with Error.

Now we will calculate the sum of squares associated with Error.


Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

To do this we simply add the between subjects and between groups sums of squares.



Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

21.333

Between Subjects Sum of Squares

19.733

Between Groups Sum of Squares

41.600

Between Subjects & Groups Sum of

Squares Combined

Then we subtract the Between Subjects & Group Sum of Squares Combined (41.600) from the Total Sum of Squares (49.333)


49.333

Total Sum of Squares

41.600 Between Subjects &

Groups Sum of Squares Combined

8.267

Sum of Squares Attributed to Error

or Unexplained


49.333

Total Sum of Squares

41.600 Between Subjects &

Groups Sum of Squares Combined

8.267

Sum of Squares Attributed to Error

or Unexplained


Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

Now we have all of the information necessary to determine if there is a statistically significant difference between pizza slices consumed by football players between three different eating occasions (before, during or after the season).

Now we have all of the information necessary to determine if there is a statistically significant difference between pizza slices consumed by football players between three different eating occasions (before, during or after the season).


Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

To calculate the significance level

To calculate the significance levelSource

Type III Sum of Squares df

Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14

We must calculate the F ratio

We must calculate the F ratioSource


Mean Square F Sig.

Between Subjects 21.333 4Between Groups 19.733 2 9.867 9.548 .008Error 8.267 8 1.033

Total 49.333 14

Which is calculated by dividing the Between Groups Mean Square value (9.867) by the Error Mean Square value (1.033).

Which is calculated by dividing the Between Groups Mean Square value (9.867) by the Error Mean Square value (1.033).


Mean Square F Sig.


Total 49.333 14

=

Which is calculated by dividing the sum of squares between groups by its degrees of freedom, as shown below:



Mean Square F Sig.

Between Subjects 21.333 4 Between Groups 19.733 2 9.867 9.548 .008Error 8.267 8 1.033

Total 49.333 14

=


And


Mean Square F Sig.


Total 49.333 14

=


And


Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14 =

=


And

Now we need to figure out how we calculate degrees of freedom for each source of sums of squares.


Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14 =

=

Let’s begin with determining the degrees of freedom Between Subjects.




Mean Square F Sig.


Total 49.333 14


We take the number of subjects which, in this case, is 5 – 1 = 4


Mean Square F Sig.


Total 49.333 14




Mean Square F Sig.


Total 49.333 14




Mean Square F Sig.


Total 49.333 14


Football Players

Before the Season

During the Season

After the Season

Average

Ben 3 3 3 3.0

Bob 5 5 5 5.0

Bud 7 7 7 7.0

Bubba 8 8 8 8.0

Burt 12 12 13 12.3

Between Subjects

1

2

3

4

5

Now – onto Between Groups Degrees of Freedom (df)



Mean Square F Sig.


Total 49.333 14


We take the number of groups which in this case is 3 – 1 = 2


Mean Square F Sig.


Total 49.333 14




Mean Square F Sig.


Total 49.333 14




Mean Square F Sig.


Total 49.333 14


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

1 2 3




Mean Square F Sig.


Total 49.333 14


Football Players

Before the Season

During the Season

After the Season

Ben 5 4 4

Bob 7 5 5

Bud 8 7 6

Bubba 9 8 4

Burt 10 7 6

1 2 3

The error degrees of freedom are calculated by multiplying the between subjects by the between groups degrees of freedom.


4

Between Subjects Degrees of Freedom

2

Between Groups Degrees of Freedom

8Error Degrees of

Freedom


4


2


8Error Degrees of

Freedom


Mean Square F Sig.


Total 49.333 14


4


2


8Error Degrees of

Freedom


Mean Square F Sig.


Total 49.333 14

The degrees of freedom for total sum of squares is calculated by adding all of the degrees of freedom from the other three sources.


4 2 8 14


4 2 8 14


Mean Square F Sig.


Total 49.333 14


4 2 8 14


Mean Square F Sig.


Total 49.333 14

We will compute the Mean Square values for just the Between Groups and Error. We are not interested in what is happening with Between Subjects. We calculated the Between Subjects sum of squares only take out any differences that are a function of differences that would exist regardless of what group we were looking at.

Once again, if we had not pulled out Between Subjects sums of squares, then the Between Subjects would be absorbed in the error value:





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047Within Groups 29.600 8 1.033

Total 49.333 14





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


Mean Square F Sig.


Total 49.333 14





Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14


Mean Square F Sig.


Total 49.333 14

Within Groups is another way of

saying Error

And that would have created a larger error mean square value:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14

Which in turn would have created a smaller F value:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14

=

=

Which in turn would have created a larger significance value:




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14




Mean Square F Sig.


Error 8.267 8 1.033

Total 49.333 14



Mean Square F Sig.

Between Groups 19.733 2 9.867 4.000 .047

Error 29.600 12 2.467

Total 49.333 14

=

=

With a larger significance value it makes it less likely to reject the null hypothesis.

With a larger significance value it makes it less likely to reject the null hypothesis.It is for that reason that we calculate the Between Subjects sums of squares and pull it out of the error sums of squares to get an uncontaminated error value…



Mean Square F Sig.


Total 49.333 14



Mean Square F Sig.


Total 49.333 14


And a more accurate F value…


Mean Square F Sig.


Total 49.333 14


And a more accurate F value…


Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14

…as well as a more accurate Significance value…

…as well as a more accurate Significance value…Source


Mean Square F Sig.


Total 49.333 14


Therefore, we will only focus on mean square values for Between Groups and Error:


Mean Square F Sig.


Total 49.333 14


Therefore, we will only focus on mean square values for Between Groups and Error:


Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14

As previously demonstrated, let’s continue with our calculations by dividing the Between Groups mean square value (9.867) by the Error mean square value (1.033).



Mean Square F Sig.


Total 49.333 14


Which gives us an F value of 9.548


Mean Square F Sig.


Total 49.333 14

=

Because we are using statistical software we will also get a significance value of .008. This means that is we were to theoretically run this experiment 1000 times we would be wrong to reject the null hypothesis 8 times this incurring a type 1 error.

Because we are using statistical software we will also get a significance value of .008. This means that is we were to theoretically run this experiment 1000 times we would be wrong to reject the null hypothesis 8 times this incurring a type 1 error.If we are willing to live with those odds of failure (8 out of 1000) then we would reject the null hypothesis.

If we had set our alpha cut off at .05 that would mean we would be willing to take the risk of being wrong 50 out of 1000 or 5 out of 100 times.

If we had set our alpha cut off at .05 that would mean we would be willing to take the risk of being wrong 50 out of 1000 or 5 out of 100 times.If we do not get a significance value (.008) then we could go to the F table to determine if our F value of 9.548 exceeds the F critical value in the F table.

This F critical value is located using the degrees of freedom for error (8) and the degrees of freedom for between groups (2).



Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14

Error df




Mean Square F Sig.


Total 49.333 14


Mean Square F Sig.


Total 49.333 14

BG df

Now let’s put them together:

Now let’s put them together:Source


Mean Square F Sig.


Total 49.333 14



Mean Square F Sig.


Total 49.333 14

BG df

Error df



Mean Square F Sig.


Total 49.333 14

BG df

Error df



Mean Square F Sig.


Total 49.333 14

BG df

Error df


Mean Square F Sig.


Total 49.333 14



Since 9.548 exceeds 4.46 at the .05 alpha level, we will reject the null hypothesis.


Mean Square F Sig.


Total 49.333 14


Since 9.548 exceeds 4.46 at the .05 alpha level, we will reject the null hypothesis.


Mean Square F Sig.


Total 49.333 14


Since 9.548 exceeds 4.46 at the .05 alpha level, we will reject the null hypothesis.Once again, we only show you the table as another way to determine if you have statistical significance.


Mean Square F Sig.


Total 49.333 14


Since 9.548 exceeds 4.46 at the .05 alpha level, we will reject the null hypothesis.Once again, we only show you the table as another way to determine if you have statistical significance.That’s it. You have now seen the inner workings of Repeated Measures ANOVA.


Mean Square F Sig.


Total 49.333 14

Education

What is a one-way repeated measures ANOVA?