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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
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?
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?
Inferential DescriptiveIs this an / 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?
Inferential DescriptiveIs this an / 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?
Inferential DescriptiveIs this an / question?
It’s inferential because the question does not
specify a specific group of football, basketball, or
soccer players.
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?
Inferential Descriptive
Is this a question?DifferenceRelationship Goodness of Fit
Independence
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?
Inferential Descriptive
Is this a DifferenceRelationship Goodness of Fit
Independence 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?
Inferential Descriptive
Is this a DifferenceRelationship Goodness of Fit
Independence question?
It’s difference in terms of amount of pizza
consumed across player types and age.
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?
Inferential Descriptive
Is this data set
DifferenceRelationship Goodness of Fit
Independence
?Normal Skewed or Kurtotic
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?
Inferential Descriptive
Is this data set
DifferenceRelationship Goodness of Fit
Independence
?Normal Skewed or Kurtotic
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?
Inferential Descriptive
Is the data set
DifferenceRelationship Goodness of Fit
Independence
?
Normal Skewed or Kurtotic
Ratio/IntervalOrdinalNominal
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?
Inferential Descriptive
Is the data set
DifferenceRelationship Goodness of Fit
Independence
?
Normal Skewed or Kurtotic
Ratio/IntervalOrdinalNominal
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
?
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
?
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
Amount of Pizza Slices Consumed in one sitting
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables ?
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables ?
Player Type and Age
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
?2 levels 3 or more levels
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
?3 or more levels2 levels
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?
Inferential Descriptive
Are there
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
?3 or more levels
3 levels for player type: football, basketball, soccer2 levels for Age: teenager / adult
2 levels
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Are the samples ?Independent Repeated
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Are the samples ?Independent Repeated
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Are the samples ?Independent Repeated
The same individuals are not being measured repeatedly and therefore are independent
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Independent Repeated
Will covariates no covariates be analyzed?
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Independent Repeated
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)
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Independent Repeated
covariates no covariates
The appropriate analytical method based our answers to these questions is . . .
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?
Inferential Descriptive
DifferenceRelationship Goodness of Fit
Independence
Normal Skewed or Kurtotic
1 Dependent Variable 2 or more Dependent Variables
Ratio/IntervalOrdinalNominal
1 Independent Variable 2 or more Independent Variables
2 levels 3 or more levels
Independent Repeated
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
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Dependent Variable: Amount of pizza eaten
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Dependent Variable: Amount of pizza eaten
Independent Variable: Athletes
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Categorized into several levels
Dependent Variable: Amount of pizza eaten
Independent Variable: Athletes
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Categorized into several levels
Dependent Variable: Amount of pizza eaten
Independent Variable: Athletes
Level 1:Football Player
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Categorized into several levels
Dependent Variable: Amount of pizza eaten
Independent Variable: Athletes
Level 1:Football Player
Level 2: Basketball Player
Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
One dependent variable
One independent variable
Categorized into several levels
Dependent Variable: Amount of pizza eaten
Independent Variable: Athletes
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.
We can consider the effect of multiple independent variables on a single dependent variable.
For example:
We can consider the effect of multiple independent variables on a single dependent variable.
For example:
First Independent Variable: Athletes
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:
First Independent Variable: Athletes
Level 1:Football Player
Level 2: Basketball Player
Level 3: Soccer Player
Second Independent Variable: Age
We can consider the effect of multiple independent variables on a single dependent variable.
For example:
First Independent Variable: Athletes
Level 1:Football Player
Level 2: Basketball Player
Level 3: Soccer Player
Second Independent Variable: Age
Level 1:Adults
Level 2: Teenagers
We can consider the effect of multiple independent variables on a single dependent variable.
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).
We can consider the effect of multiple independent variables on a single dependent variable.
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).
We can consider the effect of multiple independent variables on a single dependent variable.
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
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
Athletes Adults Teenagers
Football Player 1 9
Football Player 2 10
Football Player 3 12
Football Player 4 12
Football Player 5 15
Football Player 6 17
Basketball Player 1 1
Basketball Player 2 5
Basketball Player 3 9
Basketball Player 4 3
Basketball Player 5 6
Basketball Player 6 8
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:
Continuing our example:
• The type of athlete may have an effect on the number of slices of pizza eaten.
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.
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
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
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.
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.
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.
• 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:
You could also order them this way:
• 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.
• Adult Football Player
• Teenage Football Player
• Adult Basketball Player
• Teenage Basketball Player
• Adult Soccer Player
• Teenage Soccer Player
When subgroups respond differently under different conditions, we say that an interaction has occurred.
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
Do you see the trend here?
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
Do you see the trend here?
• Football players consume more pizza slices in one sitting than do basketball players
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
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
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
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
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
When subgroups respond differently under different conditions, we say that an interaction has occurred.
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
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
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:
So in the case below there would be no interaction effect because all of the trends are the same:
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
Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat
6 slices on average
So in the case below there would be no interaction effect because all of the trends are the same:
• As you get older you eat more slices of pizza
• If you play football you eat more than basketball and soccer players
• etc.
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
Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat
6 slices on average
But in our first case there is an interaction effect because one of the subgroups is not following the trend:
But in our first case there is an interaction effect because one of the subgroups is not following the trend:
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 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.
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three.
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:
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:
• Main Effect for Age Group: There is no significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting.
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:
• 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 Athlete: There is no significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.
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:
• 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 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
Let’s begin with the main effect for Age Group
Adultseat 13 slices on average Teenagers
eat 11 slices on average
Let’s begin with the main effect for Age Group
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
eat 11 slices on average
Now let’s look at main effect for Type of Athlete
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
Now let’s look at main effect for Type of Athlete
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.
Football Playerseat 15.5 slices on average
Basketball Playerseat 10 slices on average
Soccer Playerseat 7slices on average
Now let’s look at main effect for Type of Athlete
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.
Football Playerseat 15.5 slices on average
Basketball Playerseat 10 slices on average
Soccer Playerseat 7slices on average
Finally let’s consider the interaction effect
Finally let’s consider the interaction effect
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
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.
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
So how do we test these possibilities statistically?
So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main effect and for each interaction.
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
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 – F ratio.
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 – F ratio.
• Main effect: Type of Athlete
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 – F ratio.
• Main effect: Type of Athlete – F ratio.
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 – F ratio.
• Main effect: Type of Athlete – F ratio.
• Interaction effect: Age Group by Type of Athlete
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 – F ratio.
• Main effect: Type of Athlete – F ratio.
• Interaction effect: Age Group by Type of Athlete – F ratio
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 – F ratio.
• Main effect: 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.
Always interpret the F-ratio for the interactions effect first, before considering the F-ratio for the main effects.
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
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.
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
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.
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:
This looks a lot like our earlier image:
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
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.
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.
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.
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:
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
Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat
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 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
First we will compute the between group sums of squares for Age Group
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
First we will compute the between group sums of squares for Age Group
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
Then we will compute the between group sums of squares for Type of Player
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
Then we will compute the between group sums of squares for Type of Player
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
And then the sums of squares for the interaction effect
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
And then the sums of squares for the interaction effect
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
Then, we’ll round it off with the total sums of squares.
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 …
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
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
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 …
… that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
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
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 …
… that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
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
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 …
… that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
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
We begin with calculating Age Group Sums of Squares
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
We begin with calculating Age Group Sums of Squares
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
We begin with calculating Age Group Sums of Squares
Here’s how we do it:
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
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
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
mean
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
mean 12.78
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
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
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
grand mean
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
grand mean 11.39
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
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
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
grand mean 11.39 11.39
dev.score
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
grand mean 11.39 11.39
dev.score 1.39
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev.
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
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev.
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
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
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
grand mean 11.39 11.39
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:
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
Next we calculate the Type of Player Sums of Squares
Next we calculate the Type of Player Sums of Squares
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
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
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
Football Basketball Soccer
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)
Football Basketball Soccer
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)
Football Basketball Soccer
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).
Football Basketball Soccer
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).
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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
Football Basketball Soccer
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:
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
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 Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 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
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 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
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 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
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 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 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 Adult 8 5 2.7
Soccer Player Teenage 7 8 - 1.0
Soccer Player Teenage 8 8 0
Soccer Player Teenage 9 8 1.0
Square the deviations
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 Adult 8 5 2.7
Soccer Player Teenage 7 8 - 1.0
Soccer Player Teenage 8 8 0
Soccer Player Teenage 9 8 1.0
Square the deviations
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 Adult 8 5 2.7 7.1
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
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 Adult 8 5 2.7 7.1
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
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 Adult 8 5 2.7 7.1
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 of squares
Sum the squared deviations
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 Adult 8 5 2.7 7.1
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
40.7sum of squares
Sum the squared deviations
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
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
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
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
– – – =
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – – – =
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – – =
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – =
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 =
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
We simply sum up the total sums of squares and then subtract it from the other sums of squares
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
Total Age Type of Player Error Age * Player
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
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
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
total sums of squares
And sum 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
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.
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player.
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player.
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
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
We then determine the degrees of freedom for each source of variance:
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
We then determine the degrees of freedom for each source of variance:
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
We then determine the degrees of freedom for each source of variance:
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
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:
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.
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.
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.
• 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.
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.
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
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.
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
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.
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
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.
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
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.
If the F ratio is greater than the F critical, we would reject the null hypothesis and determine that the result is statistically significant.
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
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.
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.
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
Most statistical packages report statistical significance.
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
Most statistical packages report statistical significance.
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
Most statistical packages report statistical significance.
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
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.
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
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?
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
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
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
2 – 1 = 1 degree of freedom for age
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?
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
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?
Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?
Football Basketball Soccer
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
Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?
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
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.
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
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.
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
We now determine the degrees of freedom for error.
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):
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):
• Adult Football Player
• Teenage Football Player
• Adult Basketball Player
• Teenage Basketball Player
• Adult Soccer Player
• Teenage Soccer Player
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):
• Adult Football Player
• Teenage Football Player
• Adult Basketball Player
• Teenage Basketball Player
• Adult Soccer Player
• Teenage Soccer Player
18 – 6 = 12 degrees of freedom for error
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):
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
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
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
To determine the total degrees of freedom we simply add up all of the other degrees of freedom
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
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)
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)
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)
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 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.
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
The mean square is calculated by dividing the sums of squares by the degrees of freedom.
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
The mean square is calculated by dividing the sums of squares by the degrees of freedom.
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
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).
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).
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
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).
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
Another name for variance
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).
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
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).
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
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)
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)
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
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)
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
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)
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
10.25
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)
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
10.25
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)
And we get an F ratio of 10.25 for Age_Group
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
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).
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).
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
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).
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
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).
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
35.03
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).
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
35.03
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).
And we get an F ratio of 35.03 for type of player.
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
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)
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)
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
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)
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
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)
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
10.84
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)
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
10.84
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)
And we get an F ratio of 10.84 for Age_Group * Type of Player
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
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
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
means
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
Adult Soccer Players eat 6 slices on average
Teenage Soccer Players eat 8 slices on average
In summary:
In summary:
As you can see, it took a lot of work to get the sums of squares values.
In summary:
As you can see, it took a lot of work to get the sums of squares values.
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
In summary:
As you can see, it took a lot of work to get the sums of squares values.
But once we have the sums of squares values and the degrees of freedom we use simple division to calculate the mean square.
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
In summary:
As you can see, it took a lot of work to get the sums of squares values.
But once we have the sums of squares values and the degrees of freedom we use simple division to calculate the mean square.
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
End of Presentation