1. Multivariate vs Univariate ANOVA: Assumptions

2. Outline of Today’s Discussion 1.Within Subject ANOVAs in SPSS 2.Within Subject ANOVAs: Sphericity Post Hoc Tests

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. Spehericity – In Detail Sphericity Assumption - The difference scores across all possible conditions have comparable variances. The variance of A scores minus B scores, is equal to the variance of A scores minus C scores, which is equal to the variance of B scores minus C scores, etc.. A univariate ANAOVA has one DV, measured many times, so we should assume sphericity on that ONE DV. This is because a given participant’s score s/b correlated across repeated measures of a given DV. (Some people are trans-situationally lazy, or motivated, or whatever.) By contrast, a multivariate test has multiple DVs that could be independent of each other. So there’s no reason to ASSUME that the difference scores across the (multivariate) DVs would have similar variances. The independent DVs could be measured on scales that differ from each other by orders of magnitude.

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. Multivariate Output The Repeated Measures In SPSS Multivariate Box For a repeated measures ANOVA, you can skip the sphericity assumption, and instead go to the multi-variate tests box.

20. Multivariate Output The Repeated Measures In SPSS Multivariate Box Multivariate tests do NOT assume sphericity. This is because the DVs are presumably different from each other (not necessarily correlated).

21. Multivariate Output The Repeated Measures In SPSS Multivariate Box Use The Wilk’s Lambda information to report the proportion of variance NOT explained!

22. Multivariate Output The Repeated Measures In SPSS Multivariate Box Let’s compare the Multivariate DFs to the DFs in a standard repeated measures ANOVA.

23. Multivariate Output The Repeated Measures In SPSS Test of Within-Subject Effects DFs in a standard repeated measures ANOVA.

24. Multivariate Output The Repeated Measures In SPSS Test of Within-Subject Effects As DFs increase, so does the sensitivity of the test. So, a (univariate) repeated measures ANOVA may be either More or less sensitive than the multivariate approach.

25. Multivariate Output The Repeated Measures In SPSS Test of Within-Subject Effects If the univariate ANOVA has a large sphericity violation, the univariate DFs can be sharply reduced, rendering the multivariate approach comparatively more sensitive.

26. 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. :( ……

27. 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?

28. 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.

29. 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?

30. 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?

31. 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?

32. 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?

33. 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?