1. Mixed ANOVA Models combining between and within

2. Mixed ANOVA models We have examined One-way and Factorial designs that use: We have examined One-way and Factorial designs that use: – a single between-subjects IV – multiple between-subjects IVs – a single within-subjects IV – multiple within-subjects IVs

3. Mixed ANOVA models Mixed ANOVA models Mixed ANOVA models – contain at least one between- subjects IV and one within-subjects IV – two-way, three-way, or higher order factorial designs can be created using any combination of between and within subjects IVs

4. Two-Way Mixed ANOVA One between-subjects IV One between-subjects IV One within-subjects IV One within-subjects IV Commonly used design Commonly used design Very useful for addressing frequently occurring research questions Very useful for addressing frequently occurring research questions Often called split-plot design from origins in agricultural applications Often called split-plot design from origins in agricultural applications

5. Split-plot designs Two crops are compared Two crops are compared Each crop is exposed to three fertilizer conditions Each crop is exposed to three fertilizer conditions The combined effect of crop and fertilizer is examined The combined effect of crop and fertilizer is examined

6. Two-Way Mixed ANOVA The within-subjects IV can take all three forms: The within-subjects IV can take all three forms: – the same subjects are measured on 3 or more occasions – the same subjects are exposed to 3 or more treatments – the same subjects provide three or more ratings that are measured on the same scale

7. Two-Way Mixed ANOVA The between-subjects IV can be: The between-subjects IV can be: – randomly assigned - treatment vs. control – attribute variable - gender, grade, age group, etc. The most common use involves: The most common use involves: – between-subjects IV – treatment or control condition – within-subjects IV - growth over time

8. Examples Treatment and control groups are are assessed on pre, mid, and post treatment occasions. Treatment and control groups are are assessed on pre, mid, and post treatment occasions. Males and females are given three different types of medication. Males and females are given three different types of medication. Tenured and non-tenured teachers rate three different aspects of school climate. Tenured and non-tenured teachers rate three different aspects of school climate.

9. Examples Children are randomly assigned to get the treatment (Head Start) or not (At home & daycare), AND are assessed on pre, mid, and post treatment occasions. Children are randomly assigned to get the treatment (Head Start) or not (At home & daycare), AND are assessed on pre, mid, and post treatment occasions.

10. Examples Males and females rate the same three reasons for teaching in a private religious school. Males and females rate the same three reasons for teaching in a private religious school.

11. Two-Way Mixed ANOVA Both the between-subjects IV and the within-subjects IV can have any number of levels (2+). Both the between-subjects IV and the within-subjects IV can have any number of levels (2+). Three research questions Three research questions Three sets of null and alternative hypotheses Three sets of null and alternative hypotheses Two main effects, one interaction Two main effects, one interaction

12. Two-Way Mixed ANOVA The question and hypotheses for the between-subjects IV will follow the same patterns we have used before. The question and hypotheses for the between-subjects IV will follow the same patterns we have used before. The question and hypotheses for the within-subjects IV will also follow the same patterns we have used before. The question and hypotheses for the within-subjects IV will also follow the same patterns we have used before.

13. Two-Way Mixed ANOVA Interpret the interaction effect first. Interpret the interaction effect first. Follow the same interpretation strategies we have used for other types of factorial designs. Follow the same interpretation strategies we have used for other types of factorial designs. Graphing is particularly helpful. Graphing is particularly helpful.

14. Profile Analysis Approach Uses Multivariate Approach Uses Multivariate Approach No sphericity assumption No sphericity assumption Homogeneity of Variance - Covariance Homogeneity of Variance - Covariance Main Effect for Group Main Effect for Group – Height Main Effect for Time Main Effect for Time – Slope Group X Time Interaction Group X Time Interaction – Parallelism

15. Examples The Mixed ANOVA approach is the best way to analyze the data we have been working with all semester. The Mixed ANOVA approach is the best way to analyze the data we have been working with all semester.

16. Steps for Interpretation Follow the same interpretation guidelines as other Factorial designs Follow the same interpretation guidelines as other Factorial designs Use the Tukey Spreadsheet on the web Use the Tukey Spreadsheet on the web Calculate the appropriate effects sizes that “tell the story” Calculate the appropriate effects sizes that “tell the story”

17. Steps for Interpretation Step 1 – Interpret the interaction term Step 1 – Interpret the interaction term Step 2 – Interpret the main effects Step 2 – Interpret the main effects Step 3 – Graph the data “both ways”, meaning exchange the row and column variables to determine which picture is most useful Step 3 – Graph the data “both ways”, meaning exchange the row and column variables to determine which picture is most useful

18. Steps for Interpretation Typically it is most helpful to illustrate “change over time”, or whatever the within-subjects variable is, on the X axis Typically it is most helpful to illustrate “change over time”, or whatever the within-subjects variable is, on the X axis Typically it is most helpful to put the group variable, or whatever the between-subjects term is, as the separate lines variable. Typically it is most helpful to put the group variable, or whatever the between-subjects term is, as the separate lines variable.

19. Time on X, Groups as Lines

20. Steps for Interpretation Step 4 – If the interaction term is statistically significant, qualify the interpretation of the main effects. Step 4 – If the interaction term is statistically significant, qualify the interpretation of the main effects. Step 5 – If there is a statistically significant main effect with only two levels, no more analyses are needed for that effect. Simply examine the two marginal means (row or column totals). Step 5 – If there is a statistically significant main effect with only two levels, no more analyses are needed for that effect. Simply examine the two marginal means (row or column totals).

21. Steps for Interpretation Step 6 – If there is a main effect with more than two levels, perform post hoc comparisons among the marginal means (row or column totals). Step 6 – If there is a main effect with more than two levels, perform post hoc comparisons among the marginal means (row or column totals). This may require running additional analyses as SPSS only gives you Post Hoc comparisons for Between-Subjects terms. This may require running additional analyses as SPSS only gives you Post Hoc comparisons for Between-Subjects terms.

22. Steps for Interpretation Step 7 – Next, turn to the interaction effects. There is not one rule that fits all situations. The exact comparisons needed to make interpretations will vary from analysis to analysis. Step 7 – Next, turn to the interaction effects. There is not one rule that fits all situations. The exact comparisons needed to make interpretations will vary from analysis to analysis. Look for the portion of your graphs where the lines are non-parallel. Look for the portion of your graphs where the lines are non-parallel. Next use the Tukey spreadsheet. Next use the Tukey spreadsheet.

23. Steps for Interpretation Step 8 – Consider Simple Effects first. This means look at the pattern of differences with rows or columns in your design first. If they are different, then you have your answer about where the interaction is coming from. Step 8 – Consider Simple Effects first. This means look at the pattern of differences with rows or columns in your design first. If they are different, then you have your answer about where the interaction is coming from. If this does not completely explain the interaction, then consider looking at cell mean comparisons across rows and columns. If this does not completely explain the interaction, then consider looking at cell mean comparisons across rows and columns.

24. Steps for Interpretation Step 9 – Effect size calculations. Again, there is no one rule that will fit every situation. Your job is to illustrate the findings from your study with the effect sizes that fit the pattern in the results. Step 9 – Effect size calculations. Again, there is no one rule that will fit every situation. Your job is to illustrate the findings from your study with the effect sizes that fit the pattern in the results. Within-subjects terms = Dependent Case Within-subjects terms = Dependent Case Between-subjects terms = Independent Case Between-subjects terms = Independent Case

25. Steps for Interpretation Center, p<.005 Time, p<.001 Interaction, p<.001 Now what?