1. Within Subject ANOVAs: Assumptions & Post Hoc Tests

2. Outline of Today’s Discussion 1.Within Subject ANOVAs in SPSS 2.Within Subject ANOVAs: Assumptions & Post Hoc Tests 3.In Class Exercise: Applying our knowledge to 200-level Research Courses

3. The Research Cycle Real World Research Representation Research Results Research Conclusions Abstraction Data Analysis MethodologyGeneralization ***

4. Part 1 Within Subject ANOVAs in SPSS

5. 1.Fun Fact: It can be shown that there is a formal mathematical relationship between ANOVA and linear correlations! 2.Any ANOVA is considered a special case of a “linear model”, to mathematicians. (We won’t bother with the details here.) 3.Here are the SPSS steps for the within-subjects ANOVA: Analyze  General Linear Model  Repeated Measures

6. Within Subject ANOVAs in SPSS 1.You will then be prompted by a box… “Repeated Measures Define Factor(s)” 2.For each variable in your ANOVA, you will be prompted for a Factor Name (of your choosing), and the number of levels. 3.You can click ADD after each variable is entered…then click DEFINE….

7. Within Subject ANOVAs in SPSS 1.Finally, you should slide the variables in the left box over to the “Within-Subjects Variables” box on the right. 2.Note: SPSS does NOT conduct Post Hoc tests on Within Subjects variables. (Say it with me)

8. Part 2 Within Subject ANOVAs: Assumptions & Post Hocs

9. Between-Subjects ANOVA  Equal Variance Assumption The “Sig.” value here is > 0.05, so we retain the equal variance assumption. (The ANOVA is a fair test of this data set.)

10. Assumptions & Post Hocs The repeated measures ANOVA is based on the “Sphericity Assumption” (say it with me)

11. Assumptions & Post Hocs Sphericity Assumption - The correlations among scores in the various conditions are equal (or close enough!). Correlation between A & B, is equal to the correlation between A & C, which is equal to the correlation between B & C, etc.. The sphericity assumption is a bit more complicated than that, but that will do!

12. Assumptions & Post Hocs Great News! SPSS automatically conducts a test (Mauchly’s Test of Sphericity) to indicate whether the sphericity assumption should be retained or rejected. Remember: SPSS did the same for us in the between-subjects case with Levene’s statistic.

13. Assumptions & Post Hocs Within-Subjects ANOVA Because this “Sig.” value is < 0.05, we “reject something”! …namely, the sphericity assumption.

14. Assumptions & Post Hocs Within-Subjects ANOVA If this “Sig.” value had been >0.05, we could use the F-Value listed in the row labeled “Sphericity Assumed”….

15. Assumptions & Post Hocs Within-Subjects ANOVA If we retain the sphericity assumption, use the df an F values in the top row(s).

16. Assumptions & Post Hocs Within-Subjects ANOVA If we reject the sphericity assumption, use the “Greenhouse-Geisser” row(s)…

17. Assumptions & Post Hocs Within-Subjects ANOVA When sphericity is not assumed, the degrees of freedom are adjusted according to these epsilon values (coefficients).

18. Assumptions & Post Hocs Within-Subjects ANOVA Could someone walk us through the relationship between the DF & epsilon values here?

19. Assumptions & Post Hocs Review Question: What were the two reasons for using post hoc tests? Unfortunately, SPSS does not perform post hoc tests for the within-subjects ANOVAs. :( ……

20. Assumptions & Post Hocs To isolate which means differ from which in a within-subjects ANOVA, we can use “lots of little” repeated measures t-tests. Of course, this raises the problem of cumulative type 1 error. What was cumulative type 1 error, again?

21. Assumptions & Post Hocs The Bonferroni post hoc adjustment controls cumulative type 1 error among the repeated measures t-tests by multiplying each observed alpha level (“sig” value) by the number of t-tests we’ve run. Example: If we run 2 t-tests (post hoc), we would multiply each observed alpha level (“sig” value) by 2, and compare it to 0.05 (as always). Now, the new Bonferroni-adjusted ‘sig’ value for a particular t-test in SPSS would have to be lower than 0.05 for us to claim statistical significance.

22. Assumptions & Post Hocs Let’s get some practice with this idea. Let’s say we ran 5 t-tests (post hoc). If a particular t-test had a “sig” value of 0.015, would we retain or reject?

23. Assumptions & Post Hocs Let’s get some practice with this idea. Let’s say we ran 4 t-tests (post hoc). If a particular t-test had a “sig” value of 0.015, would we retain or reject?

24. Assumptions & Post Hocs Let’s get some practice with this idea. Let’s say we ran 3 t-tests (post hoc). If a particular t-test had a “sig” value of 0.015, would we retain or reject?

25. Assumptions & Post Hocs Let’s get some practice with this idea. Let’s say we ran 2 t-tests (post hoc). If a particular t-test had a “sig” value of 0.04, would we retain or reject?

26. Assumptions & Post Hocs Let’s get some practice with this idea. Let’s say we ran 2 t-tests (post hoc). If a particular t-test had a “sig” value of 0.015, would we retain or reject?

27. Part 3 In Class Exercise: Applying Our Methods To 200-Level Research Courses