factorial anova

Factorial Analysis of Variance

How did we get here?

First we begin with a research question

Who eats more slices of pizza in one sitting: football, basketball or soccer players. What effect does age (comparing adults to teenagers) have on the results?

First we begin with a research question


Inferential DescriptiveIs this an / question?


Inferential DescriptiveIs this an / question?

It’s inferential because the question does not

specify a specific group of football, basketball, or

soccer players.


Inferential Descriptive

Is this a question?DifferenceRelationship Goodness of Fit

Independence



Is this a DifferenceRelationship Goodness of Fit

Independence question?



Is this a DifferenceRelationship Goodness of Fit

Independence question?

It’s difference in terms of amount of pizza

consumed across player types and age.



Is this data set

DifferenceRelationship Goodness of Fit

Independence

?Normal Skewed or Kurtotic



Is the data set


Independence

?

Normal Skewed or Kurtotic

Ratio/IntervalOrdinalNominal



Are there


Independence

?


1 Dependent Variable 2 or more Dependent Variables




Are there


Independence

?




Amount of Pizza Slices Consumed in one sitting



Are there


Independence




1 Independent Variable 2 or more Independent Variables ?



Are there


Independence




1 Independent Variable 2 or more Independent Variables ?

Player Type and Age



Are there


Independence




1 Independent Variable 2 or more Independent Variables

?2 levels 3 or more levels



Are there


Independence





?3 or more levels2 levels



Are there


Independence





?3 or more levels

3 levels for player type: football, basketball, soccer2 levels for Age: teenager / adult

2 levels




Independence





2 levels 3 or more levels

Are the samples ?Independent Repeated




Independence






Are the samples ?Independent Repeated

The same individuals are not being measured repeatedly and therefore are independent




Independence






Independent Repeated

Will covariates no covariates be analyzed?




Independence







Will covariates no covariates be analyzed?

For example, we will not be analyzing the difference between athletes after eliminating the influence of age

(that would have made age a covariate)




Independence







covariates no covariates

The appropriate analytical method based our answers to these questions is . . .




Independence







covariates no covariates

The appropriate analytical method based our answers to these questions is . . .

Factorial ANOVA

Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels

One dependent variable

Dependent Variable: Amount of pizza eaten



One independent variable






Independent Variable: Athletes




Categorized into several levels









Level 1:Football Player








Level 2: Basketball Player









Level 3: Soccer Player

We can consider the effect of multiple independent variables on a single dependent variable.


For example:


For example:

First Independent Variable: Athletes





For example:





Second Independent Variable: Age


For example:





Second Independent Variable: Age

Level 1:Adults

Level 2: Teenagers


For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers).


For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers). Now, rather than comparing only 3 groups, we will be comparing 6 groups (3 levels of athlete x 2 levels of age groups).


For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers). Now, rather than comparing only 3 groups, we will be comparing 6 groups (3 levels of athlete x 2 levels of age groups).

Let’s see what this data set might look like.

First we list our three levels of athletes

First we list our three levels of athletes

Athletes

Football Player 1

Football Player 2

Football Player 3

Football Player 4

Football Player 5

Football Player 6

Basketball Player 1

Basketball Player 2

Basketball Player 3

Basketball Player 4

Basketball Player 5

Basketball Player 6

Soccer Player 1

Soccer Player 2

Soccer Player 3

Soccer Player 4

Soccer Player 5

Soccer Player 6

Then our two age groups

Athletes

Football Player 1

Football Player 2

Football Player 3

Football Player 4

Football Player 5

Football Player 6

Basketball Player 1

Basketball Player 2

Basketball Player 3

Basketball Player 4

Basketball Player 5

Basketball Player 6

Soccer Player 1

Soccer Player 2

Soccer Player 3

Soccer Player 4

Soccer Player 5

Soccer Player 6

Then our two age groups

Athletes Adults Teenagers

Football Player 1

Football Player 2

Football Player 3

Football Player 4

Football Player 5

Football Player 6

Basketball Player 1

Basketball Player 2

Basketball Player 3

Basketball Player 4

Basketball Player 5

Basketball Player 6

Soccer Player 1

Soccer Player 2

Soccer Player 3

Soccer Player 4

Soccer Player 5

Soccer Player 6

Now we add our dependent variable - pizza consumed


Football Player 1

Football Player 2

Football Player 3

Football Player 4

Football Player 5

Football Player 6

Basketball Player 1

Basketball Player 2

Basketball Player 3

Basketball Player 4

Basketball Player 5

Basketball Player 6

Soccer Player 1

Soccer Player 2

Soccer Player 3

Soccer Player 4

Soccer Player 5

Soccer Player 6

Now we add our dependent variable - pizza consumed


Football Player 1 9

Football Player 2 10





Basketball Player 1 1






Soccer Player 1 1

Soccer Player 2 2

Soccer Player 3 3

Soccer Player 4 2

Soccer Player 5 3

Soccer Player 6 5

The procedure by which we analyze the sums of squares among the 6 groups based on 2 independent variables (Age Group and Athlete Category) is called Factorial ANOVA.

The procedure by which we analyze the sums of squares among the 6 groups based on 2 independent variables (Age Group and Athlete Category) is called Factorial ANOVA.

sums of squares between groups

sums of squares within groups

degrees of freedom

means square

F ratio & F critical

hypothesis testing

one-way ANOVA

factorialANOVA

Factorial ANOVA partitions the total sums of squares into the unexplained variable and the variance explained by the main effects of each of the independent variables and the interaction of the independent variables.

Factorial ANOVA partitions the total sums of squares into the unexplained variable and the variance explained by the main effects of each of the independent variables and the interaction of the independent variables.

Main Effect Interaction Effect Error

Explained Variance Type of Athlete

Age group

Type of Athlete by

Age Group

Unexplained Variance Within Groups

Continuing our example:


• The type of athlete may have an effect on the number of slices of pizza eaten.



• But also the age group might as well have an effect on the number of slices eaten.




• And the interaction of type of athlete and age group may have an effect on slices eaten as well




• And the interaction of type of athlete and age group may have an effect on slices eaten as well

In other words, some age groups within different athlete categories may consume different amounts of pizza. For example, maybe football and basketball adults eat much more than football and basketball teenagers, while adult soccer players eat much less than teenage soccer players.

In that case, the soccer players did not follow the trend of the football and basketball players. This would be considered an interaction effect between age group and type of athlete.


Of course, there are 6 (3 x 2) possible combinations of age groups and types of athletes any one of which may not follow the direct main effect trend of age group or type of athlete.


Of course, there are 6 (3 x 2) possible combinations of age groups and types of athletes any one of which may not follow the direct main effect trend of age group or type of athlete.

• Adult Football Player

• Teenage Football Player

• Adult Basketball Player

• Teenage Basketball Player

• Adult Soccer Player

• Teenage Soccer Player

You could also order them this way:


The order doesn’t really matter.







When subgroups respond differently under different conditions, we say that an interaction has occurred.


Adult Football Players eat 19 slices on average Teenage Football Players

eat 12 slices on average


Adult Football Players eat 19 slices on average

Adult Basketball Players eat 14 slices on average

Teenage Football Players eat 12 slices on average

Teenage Basketball Players eat 10 slices on average


Do you see the trend here?







• Football players consume more pizza slices in one sitting than do basketball players








• And adults consume more pizza slices than do teenagers









Now let’s add the soccer players









Now let’s add the soccer players





Adult Soccer Players eat 6 slices on average

Teenage Soccer Players eat 8 slices on average

Because the soccer players do not follow the trend of the other two groups, this is called an interaction effect between type of athlete and age group.

So in the case below there would be no interaction effect because all of the trends are the same:






Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat

6 slices on average


• As you get older you eat more slices of pizza

• If you play football you eat more than basketball and soccer players

• etc.






6 slices on average

But in our first case there is an interaction effect because one of the subgroups is not following the trend:


• Soccer players do not follow the trend of the older you are the more pizza you eat.







A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three.


Here they are:


Here they are:

• Main Effect for Age Group: There is no significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting.


Here they are:


• Main Effect for Type of Athlete: There is no significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.


Here they are:


• Main Effect for Type of Athlete: There is no significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.

• Interaction Effect Between Age Group and Type of Athlete: There is no significant interaction between the amount of pizza eaten by football, basketball and soccer players in one sitting.

Let’s begin with the main effect for Age Group


Adultseat 13 slices on average Teenagers



So adults eat 3 slices on average more than teenagers. Is this a statistically significant difference? That’s what we will find out using sums of squares logic.

Adultseat 13 slices on average Teenagers


Now let’s look at main effect for Type of Athlete


Football Playerseat 15.5 slices on average

Basketball Playerseat 10 slices on average

Soccer Playerseat 7slices on average


So Football Players eat on average 5.5 slices more than Basketball Players; Basketball Players eat 3 more slices on average than Soccer Players; and Football Players eat 8.5 slices on average more than Soccer Players.





So Football Players eat on average 5.5 slices more than Basketball Players; Basketball Players eat 3 more slices on average than Soccer Players; and Football Players eat 8.5 slices on average more than Soccer Players. Is this a statistically significant difference? That’s what we will find out using sums of squares logic.




Finally let’s consider the interaction effect


As noted in this example earlier, it appears that there will be an interaction effect between Age Group and Types of Athletes.







So how do we test these possibilities statistically?


Factorial ANOVA will produce an F-ratio for each main effect and for each interaction.



• Main effect: Age Group



• Main effect: Age Group – F ratio.




• Main effect: Type of Athlete




• Main effect: Type of Athlete – F ratio.





• Interaction effect: Age Group by Type of Athlete





• Interaction effect: Age Group by Type of Athlete – F ratio





• Interaction effect: Age Group by Type of Athlete – F ratio

Each of these F ratios will be compared with their individual F-critical values on the F distribution table to determine if the null hypothesis will be retained or rejected.

Always interpret the F-ratio for the interactions effect first, before considering the F-ratio for the main effects.


If the F-ratio for the interaction is significant, the results for the main effects may be moot.







If the interaction is significant, it is extremely helpful to plot the interaction to determine where the effect is occurring.

If the interaction is significant, it is extremely helpful to plot the interaction to determine where the effect is occurring.

Notice how you can tell visually that soccer players are not following the age trend as is the case with football and basketball

players.

This looks a lot like our earlier image:

There are many possible combinations of effects that can render a significant F-ratio for the interaction. In our example, one of the 6 groups might respond very differently than the others …

There are many possible combinations of effects that can render a significant F-ratio for the interaction. In our example, one of the 6 groups might respond very differently than the others … or 2, or 3, or … it can be very complex.

If the interaction is significant, it is the primary focus of interpretation.


However, sometimes the main effects may be significant and meaningful; even the presence of the significant interaction. The plot will help you decide if it is meaningful.


However, sometimes the main effects may be significant and meaningful; even the presence of the significant interaction. The plot will help you decide if it is meaningful.

For example, if all players increase in pizza consumption as they age but some increase much faster in than others, both the interaction and the main effect for age may be important.

If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward,

If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward, as would be the case in this example:

If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward, as would be the case in this example:






6 slices on average

You will now see how to calculate a Factorial ANOVA by hand. Normally you will use a statistical software package to do this calculation. That being said, it is important to see what is going on behind the scenes.

Here is the data set we will be working with:

Here is the data set we will be working with:

Age Group Slices of Pizza Eaten Type of Player

Adult 17 Football Player



Adult 13 Basketball Player



Adult 2 Soccer Player



Teenage 11 Football Player



Teenage 8 Basketball Player



Teenage 7 Soccer Player



First we will compute the between group sums of squares for Age Group




















Then we will compute the between group sums of squares for Type of Player




















And then the sums of squares for the interaction effect




















Then, we’ll round it off with the total sums of squares.


Once we have all of the sums of squares we can produce an ANOVA table …



Dependent Variable: Pizza_Slices

Source Type III Sum of Squares df Mean Square F Sig.

Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00

Age_Group * Type of Player 73.444 2 36.722 10.84 0.00

Error 40.667 12 3.389

Total 386.278 17

Tests of Between-Subjects Effects



… that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We begin with calculating Age Group Sums of Squares



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We begin with calculating Age Group Sums of Squares

Here’s how we do it:



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We organize the data set with Age Groups in the headers,

We organize the data set with Age Groups in the headers,

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

We organize the data set with Age Groups in the headers, then calculate the mean for each age group

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00

Then calculate the grand mean (which is the average of all of the data)

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00

grand mean


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00

grand mean 11.39


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00

grand mean 11.39 11.39

We subtract the grand mean from each age group mean to get the deviation score

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00



Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

Then we square the deviations

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev.


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

Then multiply each squared deviation by the number of persons (9). This is called weighting the squared deviations. The more person, the heavier the weighting, or larger the weighted squared deviation values.

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

wt. sq. dev.


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

wt. sq. dev. 17.36


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

wt. sq. dev. 17.36 17.36

Finally, sum up the weighted squared deviations to get the sums of squares for age group.

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

wt. sq. dev. 17.36 17.36

Finally, sum up the weighted squared deviations to get the sums of squares for age group.

Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

mean 12.78 10.00


dev.score 1.39 - 1.39

sq.dev. 1.93 1.93

wt. sq. dev. 17.36 17.36 34.722

Note – this is the value from the ANOVA Table shown previously:

Note – this is the value from the ANOVA Table shown previously:



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Next we calculate the Type of Player Sums of Squares

Next we calculate the Type of Player Sums of Squares



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We reorder the data so that we can calculate sums of squares for Type of Player

We reorder the data so that we can calculate sums of squares for Type of Player

Football Basketball Soccer

17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

Calculate the mean for each Type of Player


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

Calculate the mean for each Type of Player


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

Calculate the grand mean (average of all of the scores)


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

Calculate the grand mean (average of all of the scores)


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

Calculate the deviation between each group mean and the grand mean(subtract grand mean from each mean).


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

Calculate the deviation between each group mean and the grand mean(subtract grand mean from each mean).


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

Square the deviations


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72



17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

sq.dev. 16.9 0.4 22.3

Weight the squared deviations


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

sq.dev. 16.9 0.4 22.3

Weight the squared deviations


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

sq.dev. 16.9 0.4 22.3

wt. sq. dev. 101.4 2.2 133.8

Sum the weighted squared deviations


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

sq.dev. 16.9 0.4 22.3

wt. sq. dev. 101.4 2.2 133.8

Sum the weighted squared deviations


17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

mean 15.50 12.00 6.67

grand mean 11.4 11.4 11.4

dev.score 4.11 0.61 - 4.72

sq.dev. 16.9 0.4 22.3

wt. sq. dev. 101.4 2.2 133.8 237.444

Here is the ANOVA table again:

Here is the ANOVA table again:



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Here is how we reorder the data to calculate the within groups sums of squares

Here is how we reorder the data to calculate the within groups sums of squares

Type of Player Age Group Slices of Pizza Eaten

Football Player Adult 17Football Player Adult 19Football Player Adult 21Football Player Teenage 11Football Player Teenage 12Football Player Teenage 13Basketball Player Adult 13Basketball Player Adult 14Basketball Player Adult 15Basketball Player Teenage 8Basketball Player Teenage 10Basketball Player Teenage 12Soccer Player Adult 2Soccer Player Adult 6Soccer Player Adult 8Soccer Player Teenage 7Soccer Player Teenage 8Soccer Player Teenage 9

Calculate the mean for each subgroup

Type of Player Age Group Slices of Pizza Eaten

Football Player Adult 17Football Player Adult 19Football Player Adult 21Football Player Teenage 11Football Player Teenage 12Football Player Teenage 13Basketball Player Adult 13Basketball Player Adult 14Basketball Player Adult 15Basketball Player Teenage 8Basketball Player Teenage 10Basketball Player Teenage 12Soccer Player Adult 2Soccer Player Adult 6Soccer Player Adult 8Soccer Player Teenage 7Soccer Player Teenage 8Soccer Player Teenage 9

Calculate the mean for each subgroup

Type of Player Age Group Slices of Pizza Eaten Group Average

Football Player Adult 17 19



Football Player Teenage 11 12



Basketball Player Adult 13 14



Basketball Player Teenage 8 10



Soccer Player Adult 2 5



Soccer Player Teenage 7 8



Calculate the deviations by subtracting the group average from each athlete’s pizza eaten:

Type of Player Age Group Slices of Pizza Eaten Group Average



















Calculate the deviations by subtracting the group average from each athlete’s pizza eaten:

Type of Player Age Group Slices of Pizza Eaten Group Average Deviations

Football Player Adult 17 19 - 2.0

Football Player Adult 19 19 0

Football Player Adult 21 19 2.0

Football Player Teenage 11 12 - 1.0

Football Player Teenage 12 12 0

Football Player Teenage 13 12 1.0

Basketball Player Adult 13 14 - 1.0

Basketball Player Adult 14 14 0

Basketball Player Adult 15 14 1.0

Basketball Player Teenage 8 10 - 2.0

Basketball Player Teenage 10 10 0

Basketball Player Teenage 12 10 2.0

Soccer Player Adult 2 5 - 3.3

Soccer Player Adult 6 5 0.7


Soccer Player Teenage 7 8 - 1.0

Soccer Player Teenage 8 8 0

Soccer Player Teenage 9 8 1.0


Type of Player Age Group Slices of Pizza Eaten Group Average Deviations

Football Player Adult 17 19 - 2.0

Football Player Adult 19 19 0

Football Player Adult 21 19 2.0

Football Player Teenage 11 12 - 1.0

Football Player Teenage 12 12 0

Football Player Teenage 13 12 1.0

Basketball Player Adult 13 14 - 1.0

Basketball Player Adult 14 14 0

Basketball Player Adult 15 14 1.0

Basketball Player Teenage 8 10 - 2.0

Basketball Player Teenage 10 10 0

Basketball Player Teenage 12 10 2.0

Soccer Player Adult 2 5 - 3.3



Soccer Player Teenage 7 8 - 1.0

Soccer Player Teenage 8 8 0

Soccer Player Teenage 9 8 1.0


Type of Player Age Group Slices of Pizza Eaten Group Average Deviations Squared

Football Player Adult 17 19 - 2.0 4.0

Football Player Adult 19 19 0 0

Football Player Adult 21 19 2.0 4.0

Football Player Teenage 11 12 - 1.0 1.0

Football Player Teenage 12 12 0 0

Football Player Teenage 13 12 1.0 1.0

Basketball Player Adult 13 14 - 1.0 1.0

Basketball Player Adult 14 14 0 0

Basketball Player Adult 15 14 1.0 1.0

Basketball Player Teenage 8 10 - 2.0 4.0

Basketball Player Teenage 10 10 0 0

Basketball Player Teenage 12 10 2.0 4.0

Soccer Player Adult 2 5 - 3.3 11.1

Soccer Player Adult 6 5 0.7 0.4


Soccer Player Teenage 7 8 - 1.0 1.0

Soccer Player Teenage 8 8 0 0

Soccer Player Teenage 9 8 1.0 1.0

Sum the squared deviations








































sum of squares





















40.7sum of squares




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Here is a simple way we go about calculating sums of squares for the interaction between type of athlete and age group

Here is a simple way we go about calculating sums of squares for the interaction between type of athlete and age group



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We simply sum up the total sums of squares and then subtract it from the other sums of squares



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Total Age Type of Player Error Age * Player

– – – =




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – – – =




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – – =




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – 237.444 – =




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – 237.444 – 40.667 =




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – 237.444 – 40.667 = 73.444




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – 237.444 – 40.667 = 73.444

Interaction Effect

So here is how we calculate sums of squares:

We line up our data in one column:

Slices of Pizza Eaten

17

19

21

13

14

15

2

6

8

11

12

13

8

10

12

7

8

9

Then we compute the grand mean (which the average of all of the scores) and subtract the grand mean from each of the scores.Slices of Pizza Eaten

17

19

21

13

14

15

2

6

8

11

12

13

8

10

12

7

8

9

Then we compute the grand mean (which the average of all of the scores) and subtract the grand mean from each of the scores.Slices of Pizza Eaten Grand Mean

17 – 11.4

19 – 11.4

21 – 11.4

13 – 11.4

14 – 11.4

15 – 11.4

2 – 11.4

6 – 11.4

8 – 11.4

11 – 11.4

12 – 11.4

13 – 11.4

8 – 11.4

10 – 11.4

12 – 11.4

7 – 11.4

8 – 11.4

9 – 11.4

This gives us the deviation scores between each score and the grand mean

Slices of Pizza Eaten Grand Mean

17 – 11.4

19 – 11.4

21 – 11.4

13 – 11.4

14 – 11.4

15 – 11.4

2 – 11.4

6 – 11.4

8 – 11.4

11 – 11.4

12 – 11.4

13 – 11.4

8 – 11.4

10 – 11.4

12 – 11.4

7 – 11.4

8 – 11.4

9 – 11.4

This gives us the deviation scores between each score and the grand mean

Slices of Pizza Eaten Grand Mean Deviations

17 – 11.4 = 5.6

19 – 11.4 = 7.6

21 – 11.4 = 9.6

13 – 11.4 = 1.6

14 – 11.4 = 2.6

15 – 11.4 = 3.6

2 – 11.4 = - 9.4

6 – 11.4 = - 5.4

8 – 11.4 = - 3.4

11 – 11.4 = - 0.4

12 – 11.4 = 0.6

13 – 11.4 = 1.6

8 – 11.4 = - 3.4

10 – 11.4 = - 1.4

12 – 11.4 = 0.6

7 – 11.4 = - 4.4

8 – 11.4 = - 3.4

9 – 11.4 = - 2.4

Then square the deviations

Slices of Pizza Eaten Grand Mean Deviations

17 – 11.4 = 5.6

19 – 11.4 = 7.6

21 – 11.4 = 9.6

13 – 11.4 = 1.6

14 – 11.4 = 2.6

15 – 11.4 = 3.6

2 – 11.4 = - 9.4

6 – 11.4 = - 5.4

8 – 11.4 = - 3.4

11 – 11.4 = - 0.4

12 – 11.4 = 0.6

13 – 11.4 = 1.6

8 – 11.4 = - 3.4

10 – 11.4 = - 1.4

12 – 11.4 = 0.6

7 – 11.4 = - 4.4

8 – 11.4 = - 3.4

9 – 11.4 = - 2.4

Then square the deviations

Slices of Pizza Eaten Grand Mean Deviations Squared

17 – 11.4 = 5.6 2 = 31.5

19 – 11.4 = 7.6 2 = 57.9

21 – 11.4 = 9.6 2 = 92.4

13 – 11.4 = 1.6 2 = 2.6

14 – 11.4 = 2.6 2 = 6.8

15 – 11.4 = 3.6 2 = 13.0

2 – 11.4 = - 9.4 2 = 88.2

6 – 11.4 = - 5.4 2 = 29.0

8 – 11.4 = - 3.4 2 = 11.5

11 – 11.4 = - 0.4 2 = 0.2

12 – 11.4 = 0.6 2 = 0.4

13 – 11.4 = 1.6 2 = 2.6

8 – 11.4 = - 3.4 2 = 11.5

10 – 11.4 = - 1.4 2 = 1.9

12 – 11.4 = 0.6 2 = 0.4

7 – 11.4 = - 4.4 2 = 19.3

8 – 11.4 = - 3.4 2 = 11.5

9 – 11.4 = - 2.4 2 = 5.7

And sum the deviations


17 – 11.4 = 5.6 2 = 31.5

19 – 11.4 = 7.6 2 = 57.9

21 – 11.4 = 9.6 2 = 92.4

13 – 11.4 = 1.6 2 = 2.6

14 – 11.4 = 2.6 2 = 6.8

15 – 11.4 = 3.6 2 = 13.0

2 – 11.4 = - 9.4 2 = 88.2

6 – 11.4 = - 5.4 2 = 29.0

8 – 11.4 = - 3.4 2 = 11.5

11 – 11.4 = - 0.4 2 = 0.2

12 – 11.4 = 0.6 2 = 0.4

13 – 11.4 = 1.6 2 = 2.6

8 – 11.4 = - 3.4 2 = 11.5

10 – 11.4 = - 1.4 2 = 1.9

12 – 11.4 = 0.6 2 = 0.4

7 – 11.4 = - 4.4 2 = 19.3

8 – 11.4 = - 3.4 2 = 11.5

9 – 11.4 = - 2.4 2 = 5.7



17 – 11.4 = 5.6 2 = 31.5

19 – 11.4 = 7.6 2 = 57.9

21 – 11.4 = 9.6 2 = 92.4

13 – 11.4 = 1.6 2 = 2.6

14 – 11.4 = 2.6 2 = 6.8

15 – 11.4 = 3.6 2 = 13.0

2 – 11.4 = - 9.4 2 = 88.2

6 – 11.4 = - 5.4 2 = 29.0

8 – 11.4 = - 3.4 2 = 11.5

11 – 11.4 = - 0.4 2 = 0.2

12 – 11.4 = 0.6 2 = 0.4

13 – 11.4 = 1.6 2 = 2.6

8 – 11.4 = - 3.4 2 = 11.5

10 – 11.4 = - 1.4 2 = 1.9

12 – 11.4 = 0.6 2 = 0.4

7 – 11.4 = - 4.4 2 = 19.3

8 – 11.4 = - 3.4 2 = 11.5

9 – 11.4 = - 2.4 2 = 5.7

total sums of squares



17 – 11.4 = 5.6 2 = 31.5

19 – 11.4 = 7.6 2 = 57.9

21 – 11.4 = 9.6 2 = 92.4

13 – 11.4 = 1.6 2 = 2.6

14 – 11.4 = 2.6 2 = 6.8

15 – 11.4 = 3.6 2 = 13.0

2 – 11.4 = - 9.4 2 = 88.2

6 – 11.4 = - 5.4 2 = 29.0

8 – 11.4 = - 3.4 2 = 11.5

11 – 11.4 = - 0.4 2 = 0.2

12 – 11.4 = 0.6 2 = 0.4

13 – 11.4 = 1.6 2 = 2.6

8 – 11.4 = - 3.4 2 = 11.5

10 – 11.4 = - 1.4 2 = 1.9

12 – 11.4 = 0.6 2 = 0.4

7 – 11.4 = - 4.4 2 = 19.3

8 – 11.4 = - 3.4 2 = 11.5

9 – 11.4 = - 2.4 2 = 5.7

386.278total sums of squares

And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player.

And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



386.278 – 34.722 – 237.444 – 40.667 = 73.444

We then determine the degrees of freedom for each source of variance:



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Why do we need to determine the degrees of freedom?

Why do we need to determine the degrees of freedom? Because this will make it possible to test our three null hypotheses:


• Main effect for Age Group: There is NO significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting.



• Main effect for Type of Player: There is NO significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.



• Main effect for Type of Player: There is NO significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.

• Interaction effect between Age Group and Type of Athlete: There is NO significant interaction between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.

By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical.




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



If the F ratio is greater than the F critical, we would reject the null hypothesis and determine that the result is statistically significant.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17



If the F ratio is greater than the F critical, we would reject the null hypothesis and determine that the result is statistically significant. If the F ratio is smallerthan the F critical then we would fail to reject the null hypothesis.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Most statistical packages report statistical significance.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Most statistical packages report statistical significance.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


This means that if we took 1000 samples we

would be wrong 1 time. We just don’t know if

this is that time.

Most statistical packages report statistical significance. But it is important to know where this value came from.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


This means that if we took 1000 samples we

would be wrong 1 time. We just don’t know if

this is that time.

So let’s calculate the number of degrees of freedom beginning with Age_Group.

So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one.

So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there?


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9


Adults Teens

17 11

19 12

21 13

13 8

14 10

15 12

2 7

6 8

8 9

2 – 1 = 1 degree of freedom for age




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Now we determine the degrees of freedom for Type of Player.

Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?



17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9



17 13 2

19 14 6

21 15 8

11 8 7

12 10 8

13 12 9

3 – 1 = 2 degrees of freedom for type of player




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


To determine the degrees of freedom for the interaction effect between age and type of player you multiply the degrees of freedom for age by the degrees of freedom for type of player.


1 * 2 = 2 degrees of freedom for interaction effect




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We now determine the degrees of freedom for error.


Here we take the number of subjects (18) and subtract that number by the number of subgroups (6):









18 – 6 = 12 degrees of freedom for error





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


To determine the total degrees of freedom we simply add up all of the other degrees of freedom

To determine the total degrees of freedom we simply add up all of the other degrees of freedom



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We now calculate the mean square.

We now calculate the mean square. The reason this value is called mean square because it represents the average squared deviation of scores from the mean.

We now calculate the mean square. The reason this value is called mean square because it represents the average squared deviation of scores from the mean. You will notice that this is actually the definition for variance.

So the mean square is a variance.

So the mean square is a variance.• The mean square for Age_Group is the variance between the two ages

(adult and teenager) and the grand mean. (This is explained variance or variance explained by whether you are an adult or a teenager)



• The mean square for Type of Player is the variance between the three types of player (football, basketball, and soccer) and the grand mean. (This is explained variance or variance explained by whether you are a football, basketball, or soccer player)




• The mean square for the interaction effect represents the variance between each subgroup and the grand mean. (This is explained variance or variance explained by the interaction between Age and Type of Player effects)




• The mean square for the interaction effect represents the variance between each subgroup and the grand mean. (This is explained variance or variance explained by the interaction between Age and Type of Player effects)

• The mean square for the error or within groups scores represents the variance between each individual and the grand mean. (This is unexplained variance or variance that is not explained by what group subjects are in or how they interact)

The mean square is calculated by dividing the sums of squares by the degrees of freedom.

The mean square is calculated by dividing the sums of squares by the degrees of freedom.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


We are now ready to calculate the F ratio. It is called the F ratio because it is a ratio between variance that is explained (e.g., by age, type of player or the interaction between the two) and the error variance (or variance that is not explained).




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


Another name for variance




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


First we will calculate the F ratio for Age_Group by dividing mean square for Age_Group (34.722) by the mean square for error (3.389)




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


10.25


And we get an F ratio of 10.25 for Age_Group



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


10.25

The significance value of 0.01 means that if we were to take 100 samples with the same Factorial Design and analyze the results we would be wrong to reject the null hypothesis 1 time.

The significance value of 0.01 means that if we were to take 100 samples with the same Factorial Design and analyze the results we would be wrong to reject the null hypothesis 1 time. Because we are probably comfortable with those odds, we will reject the null hypothesis that age group has no effect on pizza consumption.

Next, we will calculate the F ratio for type of player by dividing mean square for type of player (118.722) by the mean square for error (3.389).




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


35.03


And we get an F ratio of 35.03 for type of player.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


35.03

The significance value of 0.00 (which, let’s say, is 0.002) means that if we were to take 1000 samples with the same factorial design and analyze the results we would be wrong to reject the null hypothesis 2 times.

The significance value of 0.00 (which, let’s say, is 0.002) means that if we were to take 1000 samples with the same factorial design and analyze the results we would be wrong to reject the null hypothesis 2 times. Because we are probably comfortable with those odds, we will reject the null hypothesis that type of player has no effect on pizza consumption.

Finally, we will calculate the F ratio for the interactioneffect of age group and type of player by dividing mean square for Age_Group * Type of Player (36.722) by the mean square for error (3.389)




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17





Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


10.84


And we get an F ratio of 10.84 for Age_Group * Type of Player



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


10.84

The significance value of 0.00 (which, let’s say, is .003) means that if we were to take 1000 samples with the same factorial design and analyze the results we would be wrong to reject the null hypothesis 3 times.

The significance value of 0.00 (which, let’s say, is .003) means that if we were to take 1000 samples with the same factorial design and analyze the results we would be wrong to reject the null hypothesis 3 times. Because we are probably comfortable with those odds, we will reject the null hypothesis that Age_Group * Type of Player has no interaction effect on pizza consumption.

The significance value of 0.00 (which, let’s say, is .003) means that if we were to take 1000 samples with the same factorial design and analyze the results we would be wrong to reject the null hypothesis 3 times. Because we are probably comfortable with those odds, we will reject the null hypothesis that Age_Group * Type of Player has no interaction effect on pizza consumption.

Once again, this means that one of the subgroups is not acting like one or more other subgroups.

means







In summary:

In summary:

As you can see, it took a lot of work to get the sums of squares values.

In summary:




Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


In summary:


But once we have the sums of squares values and the degrees of freedom we use simple division to calculate the mean square.



Age_Group 34.722 1 34.722 10.25 0.01

Type of Player 237.444 2 118.722 35.03 0.00


Error 40.667 12 3.389

Total 386.278 17


End of Presentation

factorial anova

Education

slices of pizza

soccer players

specific group of football

research questionwho

player types