More sophisticated ANOVA applications

1. More sophisticated ANOVA applications Repeated measures and factorial PSY295-001 SP2003

2. Major Topics • What are repeated-measures? • An example • Assumptions • Advantages and disadvantages • Review questions

3. Effects of Counseling For Post-Traumatic Stress Disorder • Foa, et al. (1991) • Provided supportive counseling (and other therapies) to victims of rape • Do number of symptoms change with time? • Point out lack of control group • Not a test of effectiveness of supportive counseling • Foa actually had controls. Cont.

4. Effect of Counseling--cont. • 9 subjects measured before therapy, after therapy, and 3 months later • We are ignoring Foa’s other treatment conditions.

5. Therapy for PTSD • Dependent variable = number of reported symptoms. • Question--Do number of symptoms decrease over therapy and remain low? • Data on next slide

6. The Data

7. Plot of the Data

8. Preliminary Observations • Notice that subjects differ from each other. • Between-subjects variability • Notice that means decrease over time • Faster at first, and then slower • Within-subjects variability

9. Partitioning Variability Total Variability Between-subj. variability Within-subj. variability Time Error This partitioning is reflected in the summary table.

10. Summary Table

11. Interpretation • Note parallel with diagram • Note subject differences not in error term • Note MSerror is denominator for F on Time • Note SStime measures what we are interested in studying

12. Assumptions • Correlations between trials are all equal • Actually more than necessary, but close • Matrix shown below Cont.

13. Assumptions--cont. • Previous matrix might look like we violated assumptions • Only 9 subjects • Minor violations are not too serious. • Greenhouse and Geisser (1959) correction • Adjusts degrees of freedom

14. Multiple Comparisons • With few means: • t test with Bonferroni corrections • Limit to important comparisons • With more means: • Require specialized techniques • Trend analysis

15. Advantages of Repeated-Measures Designs • Eliminate subject differences from error term • Greater power • Fewer subjects needed • Often only way to address the problem • This example illustrates that case.

16. Disadvantages • Carry-over effects • Counter-balancing • May tip off subjects

17. Major Points • What is a factorial design? • An example • Main effects • Interactions • Simple effects Cont.

18. Major Points-cont. • Unequal sample sizes • Magnitude of effect • Review questions

19. What is a Factorial • At least two independent variables • All combinations of each variable • R X C factorial • Cells

20. Video Violence • Bushman study • Two independent variables • Two kinds of videos • Male and female subjects • See following diagram

21. 2 X 2 Factorial

22. Bushman’s Study-cont. • Dependent variable = number of aggessive associates • 50 observations in each cell • We will work with means and st. dev., instead of raw data. • This illustrates important concepts.

23. The Data (cell means and standard deviations)

24. Plotting Results

25. Effects to be estimated • Differences due to videos • Violent appear greater than nonviolent • Differences due to gender • Males appear higher than females • Interaction of video and gender • What is an interaction? • Does violence affect males and females equally? Cont.

26. Estimated Effects--cont. • Error • average within-cell variance • Sum of squares and mean squares • Extension of the same concepts in the one-way

27. Summary Table

28. Conclusions • Main effects • Significant difference due to video • More aggressive associates following violent video • Significant difference due to gender • Males have more aggressive associates than females. Cont.

29. Conclusions--cont. • Interaction • No interaction between video and gender • Difference between violent and nonviolent video is the same for males (1.5) as it is for females (1.4) • We could see this in the graph of the data.

30. Elaborate on Interactions • Diagrammed on next slide as line graph • Note parallelism of lines • Means video differences did not depend on gender • A significant interaction would have nonparallel lines • Ordinal and disordinal interactions

31. Line Graph of Interaction

32. Simple Effects • Effect of one independent variable at one level of the other. • e.g. Difference between males and females for only violent video • Difference between males and females for only nonviolent video

33. Unequal Sample Sizes • A serious problem for hand calculations • Can be computed easily using computer software • Can make the interpretation difficult • Depends, in part, on why the data are missing.

34. Analysis of Variance for AGGASSOC Source DF SS MS F P GENDER 1 66.1 66.1 4.49 0.035 VIDEO 1 105.1 105.1 7.14 0.008 Interaction 1 0.1 0.1 0.01 0.927 Error 196 2885.6 14.7 Total 199 3057.0 Minitab Example Cont.

35. Minitab--cont. Individual 95% CI GENDER Mean --------+---------+---------+---------+--- 1 6.95 (----------*----------) 2 5.80 (----------*----------) --------+---------+---------+---------+--- 5.60 6.30 7.00 7.70 Individual 95% CI VIDEO Mean ---------+---------+---------+---------+-- 1 7.10 (---------*--------) 2 5.65 (---------*--------) ---------+---------+---------+---------+-- 5.60 6.40 7.20 8.00