Split Plot ANOVA

• Another application of ANOVA is mixed design or “split plot” ANOVA.

• Another application of ANOVA is mixed design or “split plot” ANOVA.

• Split plot ANOVA is a special instance of factorial ANOVA. Recall that in factorial ANOVA, 2 or more independent variables are tested for possible interaction effects on a single dependent variable.

• For example, if we compare the number of pizza slices consumed in one sitting between football, basketball, and soccer players we would run a One-Way ANOVA.

• For example, if we compare the number of pizza slices consumed in one sitting between football, basketball, and soccer players we would run a One-Way ANOVA.

• Here is how we would input the data:

• For example, if we compare the number of pizza slices consumed in one sitting between football, basketball, and soccer players we would run a One-Way ANOVA.

• Here is how we would input the data:Player Type Pizza Slices Consumed1 = Football Player 81 = Football Player 91 = Football Player 11

1 = Football Player 122 = Basketball Player 42 = Basketball Player 52 = Basketball Player 72 = Basketball Player 83 = Soccer Player 13 = Soccer Player 23 = Soccer Player 43 = Soccer Player 5

• For example, if we compare the number of pizza slices consumed in one sitting between football, basketball, and soccer players we would run a One-Way ANOVA.

• Here is how we would input the data:Player Type Pizza Slices Consumed1 = Football Player 81 = Football Player 91 = Football Player 11

1 = Football Player 122 = Basketball Player 42 = Basketball Player 52 = Basketball Player 72 = Basketball Player 83 = Soccer Player 13 = Soccer Player 23 = Soccer Player 43 = Soccer Player 5

• A Factorial ANOVA tests at least two independent variables or main effects (1. Player Type / 2. Team Type) along with the interaction between them (Player Type & Team Type).

• A Factorial ANOVA tests at least two independent variables or main effects (1. Player Type / 2. Team Type) along with the interaction between them (Player Type & Team Type).

• Here is how we would input the data for a simple factorial ANOVA:

• A Factorial ANOVA tests at least two independent variables or main effects (1. Player Type / 2. Team Type) along with the interaction between them (Player Type & Team Type).

• Here is how we would input the data for a simple factorial ANOVA:

Player Type Team Type Pizza Slices Consumed1 = Football Player 1 = Junior Varsity 81 = Football Player 1 = Junior Varsity 91 = Football Player 2 = Varsity 11

1 = Football Player 2 = Varsity 122 = Basketball Player 1 = Junior Varsity 42 = Basketball Player 1 = Junior Varsity 52 = Basketball Player 2 = Varsity 72 = Basketball Player 2 = Varsity 83 = Soccer Player 1 = Junior Varsity 13 = Soccer Player 1 = Junior Varsity 23 = Soccer Player 2 = Varsity 43 = Soccer Player 2 = Varsity 5

• A Factorial ANOVA tests at least two independent variables or main effects (1. Player Type / 2. Team Type) along with the interaction between them (Player Type & Team Type).

• Here is how we would input the data for a simple factorial ANOVA:

Player Type Team Type Pizza Slices Consumed1 = Football Player 1 = Junior Varsity 81 = Football Player 1 = Junior Varsity 91 = Football Player 2 = Varsity 11

1 = Football Player 2 = Varsity 122 = Basketball Player 1 = Junior Varsity 42 = Basketball Player 1 = Junior Varsity 52 = Basketball Player 2 = Varsity 72 = Basketball Player 2 = Varsity 83 = Soccer Player 1 = Junior Varsity 13 = Soccer Player 1 = Junior Varsity 23 = Soccer Player 2 = Varsity 43 = Soccer Player 2 = Varsity 5

• A Factorial ANOVA tests at least two independent variables or main effects (1. Player Type / 2. Team Type) along with the interaction between them (Player Type & Team Type).

• Here is how we would input the data for a simple factorial ANOVA:

Player Type Team Type Pizza Slices Consumed1 = Football Player 1 = Junior Varsity 81 = Football Player 1 = Junior Varsity 91 = Football Player 2 = Varsity 11

1 = Football Player 2 = Varsity 122 = Basketball Player 1 = Junior Varsity 42 = Basketball Player 1 = Junior Varsity 52 = Basketball Player 2 = Varsity 72 = Basketball Player 2 = Varsity 83 = Soccer Player 1 = Junior Varsity 13 = Soccer Player 1 = Junior Varsity 23 = Soccer Player 2 = Varsity 43 = Soccer Player 2 = Varsity 5

Independent Samples

Another set of Independent Samples

• Split plot ANOVA tests for interactions in the same way. However, in split plot ANOVA one of the independent variables is a fixed factor such as group membership (e.g., player type) and the other independent variable is a repeated measures variable (e.g., before and after the season).

• Split plot ANOVA tests for interactions in the same way. However, in split plot ANOVA one of the independent variables is a fixed factor such as group membership (e.g., player type) and the other independent variable is a repeated measures variable (e.g., before and after the season).

Player Type Before or After the Season Pizza Slices Consumed 1 = Football Player 1 = Before 81 = Football Player 1 = Before 91 = Football Player 2 = After 11

1 = Football Player 2 = After 122 = Basketball Player 1 = Before 42 = Basketball Player 1 = Before 52 = Basketball Player 2 = After 72 = Basketball Player 2 = After 83 = Soccer Player 1 = Before 23 = Soccer Player 1 = Before 33 = Soccer Player 2 = After 43 = Soccer Player 2 = After 5

• Split plot ANOVA tests for interactions in the same way. However, in split plot ANOVA one of the independent variables is a fixed factor such as group membership (e.g., player type) and the other independent variable is a repeated measures variable (e.g., before and after the season).

Player Type Before or After the Season Pizza Slices Consumed 1 = Football Player 1 = Before 81 = Football Player 1 = Before 91 = Football Player 2 = After 11 1 = Football Player 2 = After 122 = Basketball Player 1 = Before 42 = Basketball Player 1 = Before 52 = Basketball Player 2 = After 72 = Basketball Player 2 = After 83 = Soccer Player 1 = Before 23 = Soccer Player 1 = Before 33 = Soccer Player 2 = After 43 = Soccer Player 2 = After 5

• Split plot ANOVA tests for interactions in the same way. However, in split plot ANOVA one of the independent variables is a fixed factor such as group membership (e.g., player type) and the other independent variable is a repeated measures variable (e.g., before and after the season).

Player Type Before or After the Season Pizza Slices Consumed 1 = Football Player 1 = Before 81 = Football Player 1 = Before 91 = Football Player 2 = After 11 1 = Football Player 2 = After 122 = Basketball Player 1 = Before 42 = Basketball Player 1 = Before 52 = Basketball Player 2 = After 72 = Basketball Player 2 = After 83 = Soccer Player 1 = Before 23 = Soccer Player 1 = Before 33 = Soccer Player 2 = After 43 = Soccer Player 2 = After 5

• Split plot ANOVA tests for interactions in the same way. However, in split plot ANOVA one of the independent variables is a fixed factor such as group membership (e.g., player type) and the other independent variable is a repeated measures variable (e.g., before and after the season).

Player Type Before or After the Season Pizza Slices Consumed 1 = Football Player 1 = Before 81 = Football Player 1 = Before 91 = Football Player 2 = After 11 1 = Football Player 2 = After 122 = Basketball Player 1 = Before 42 = Basketball Player 1 = Before 52 = Basketball Player 2 = After 72 = Basketball Player 2 = After 83 = Soccer Player 1 = Before 23 = Soccer Player 1 = Before 33 = Soccer Player 2 = After 43 = Soccer Player 2 = After 5

Independent samples

Repeated samples

• Split-plot ANOVA very effectively tests whether groups change differently over time.

• Split-plot ANOVA very effectively tests whether groups change differently over time.

Pizza Slices Before the Season

After the Season

121110987654321

• Split-plot ANOVA very effectively tests whether groups change differently over time.

Pizza Slices Before the Season

After the Season

121110987654321

• Split-plot ANOVA very effectively tests whether groups change differently over time.

Pizza Slices Before the Season

After the Season

121110987654321

Football Players

Basketball Players

Soccer Players

2.5 average slices

4.5 average slices

4.5 average slices

6.5 average slices

8.5 average slices

10.5 average slices

• Split-plot ANOVA very effectively tests whether groups change differently over time.

• For example, a treatment group may change more rapidly (or in a different direction) from pre-test to post-test than a non-treatment control group

• Think of the example of a class that receives innovative instruction (treatment group) and a class that does not (non-treatment control group). The pre-test scores and post-test scores are seen below:

• Think of the example of a class that receives innovative instruction (treatment group) and a class that does not (non-treatment control group). The pre-test scores and post-test scores are seen below:

Treatment – Non Treatment Pre-test scores Post-test scores 1 = Treatment Group 5 121 = Treatment Group 6 13

1 = Treatment Group 5 14

1 = Treatment Group 6 12

1 = Treatment Group 4 142 = Nontreatment Control Group 6 82 = Nontreatment Control Group 5 7

2 = Nontreatment Control Group 4 8

2 = Nontreatment Control Group 5 7

2 = Nontreatment Control Group 6 7

• Think of the example of a class that receives innovative instruction (treatment group) and a class that does not (non-treatment control group). The pre-test scores and post-test scores are seen below:

Treatment – Non Treatment Pre-test scores Post-test scores 1 = Treatment Group 5 121 = Treatment Group 6 13

1 = Treatment Group 5 14

1 = Treatment Group 6 12

1 = Treatment Group 4 142 = Nontreatment Control Group 6 82 = Nontreatment Control Group 5 7

2 = Nontreatment Control Group 4 8

2 = Nontreatment Control Group 5 7

2 = Nontreatment Control Group 6 7

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• In our previous example the main effect for groups would be the average scores between the treatment and the non-treatment control group:

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• In our previous example the main effect for groups would be the average scores between the treatment and the non-treatment control group:– Average scores for the treatment group – 9.1– Average scores for the non-treatment group – 6.3

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• In our previous example the main effect for groups would be the average scores between the treatment and the non-treatment control group:– Average scores for the treatment group – 9.1– Average scores for the non-treatment group – 6.3

• This difference is impressive and tells the story that the treatment scored higher on average than the non-treatment group.

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• The second main effect is between pre and post-tests.

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• The second main effect is between pre and post-tests. – Average pre-test score – 5.2– Average post-test score – 10.2

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• The second main effect is between pre and post-tests. – Average pre-test score – 5.2– Average post-test score – 10.2

• This difference is also impressive.

• In a split-plot ANOVA there will be a main effect for groups, a main effect for time, and an interaction between group and time.

• The second main effect is between pre and post-tests. – Average pre-test score – 5.2– Average post-test score – 10.2

• This difference is also impressive. • But what we don’t know is how different their

growth trajectory is across time.

• The interaction term will reveal whether there is differential change over time according to group membership. If it is significant, then plotting the interaction will reveal the nature of the differential change.

• The interaction term will reveal whether there is differential change over time according to group membership. If it is significant, then plotting the interaction will reveal the nature of the differential change.

• Here is a graph that shows the interaction effect or compares the growth or decay trajectory over time:

Scores Pre-test Post-test

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Scores Pre-test Post-test

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Treatment Group

Non-Treatment Control Group

5.2 points

7.4 points

5.2 points

13.0 points

Scores Pre-test Post-test

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13.0 points

• In this case the interaction effect is very impressive.

Scores Pre-test Post-test

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Treatment Group

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5.2 points

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5.2 points

13.0 points

• In this case the interaction effect is very impressive.

⁻ Pre-post differential for treatment group (5.2 – 13 = 7.8 absolute value)

Scores Pre-test Post-test

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Treatment Group

Non-Treatment Control Group

5.2 points

7.4 points

5.2 points

13.0 points

• In this case the interaction effect is very impressive.

⁻ Pre-post differential for treatment group (5.2 – 13 = 7.8 absolute value)

⁻ Pre-post differential for non-treatment control group (5.2 – 7.4 = 2.2 absolute value)

• Now we see that the growth differential between the two groups is vastly different. The non-treatment control group increased by only 2.2 points between pre and post-tests. The treatment group increased by 7.8 points. This adds a more informative piece to the puzzle we are trying to put together.

• Now we see that the growth differential between the two groups is vastly different. The non-treatment control group increased by only 2.2 points between pre and post-tests. The treatment group increased by 7.8 points. This adds a more informative piece to the puzzle we are trying to put together.

• If the interaction term is not significant, then an interpretation of the main effects may be informative.

Scores Pre-test Post-test

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Scores Pre-test Post-test

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Treatment Group

Non-Treatment Control Group

5.2 points

7.4 points

10.8 points

13.0 points

• For example, in the table the groups have the same measured difference at the beginning as they do at the end, but their growth is identical.

Scores Pre-test Post-test

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Treatment Group

Non-Treatment Control Group

5.2 points

7.4 points

10.8 points

13.0 points

• For example, in the table the groups have the same measured difference at the beginning as they do at the end, but their growth is identical.

• In this case there is no interaction effect because their growth rates are similar.

Scores Pre-test Post-test

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Treatment Group

Non-Treatment Control Group

5.2 points

7.4 points

10.8 points

13.0 points

• For example, in the table the groups have the same measured difference at the beginning as they do at the end, but their growth is identical.

• In this case there is no interaction effect because their growth rates are similar.

• Therefore, the main effect (pre-post difference) is the only difference we are interested in.