1. MANOVA Factorial

2. The Beautiful Criminal Wuensch, Chia, Castellow, Chuang, & Cheng (1993) Data collected in Taiwan Grouping variables –Defendant physically attractive or not –Sex of defendant –Type of crime: Swindle or burglary –Defendant American or Chinese –Sex of juror

3. Dependent Variables One set of two variables –Length of recommended sentence –Rated seriousness of the crime A second set of 12 variables, ratings of the defendant on attributes such as –Physical attractiveness –Intelligence –Sociability

4. Type I Boogeyman If we did a five-way ANOVA on one DV –We would do 27 F tests –And that is just for the omnibus analysis If we do that for each of the 14 DVs –That is 378 F tests –And the Boogeyman is licking his chops

6. Common Tactic Suppose you have an A x B factorial design. You have five dependent variables. You worry that the Type I boogeyman will get you if you just do five A x B ANOVAs. You do an A x B factorial MANOVA first. For any effect that is significant (A, B, A x B) in MANOVA, you do five ANOVAs, but only interpret effects that were significant in the MANOVA.

7. Plaster’s Thesis A similar but less complex design. The female defendant was attractive, average, or unattractive. The crime was a burglary or a swindle. All mock jurors were male. Dependent variables were –Years sentenced –Rated seriousness of the crime.

8. The SAS Code See it here.See it here In the data step I created canonical variable scores using the standardized coefficients from the MANOVA. Then I used these as dependent variables in univariate ANOVAs.

9. Main Effect of PA Manipulation For first root, r can =.275 For second root, r can =.099 First/Second root  =.915, p =.049 Second root  =.990, p =.30 Roy’s greatest root =.082, F(2, 108) = 4.43, p =.014.F(2, 108) = 4.43, p =.014 The groups differ significantly on the first canonical variate for the main effect of PA.

10. First Root of Main Effect of PA Betas: 1.20 Years -.69 Seriousness Loadings:.80 Years, -.08 Seriousness Notice that seriousness is a suppressor CV1 is increased by greater Years Sentenced and decreased by greater Perceived Seriousness of the Crime. High CV1 = juror gives a long sentence despite thinking the crime not serious.

11. Main Effect of Type of Crime r can =.275  =.995, p =.76 The groups do not differ significantly on the weighted linear combination of sentence and severity.

12. Crime x Attractiveness For first root, r can =.218 For second root, r can =.015 First/Second root  =.952, p =.26 Second root  =.9998, p =.87 Roy’s greatest root =.0502, F(2, 108) = 2.71), p =.071.F(2, 108) = 2.71), p =.071 For pedagogical purposes, I shall relax alpha to.10.

13. First Root of PA x Crime Betas: 1.20 Years -.56 Seriousness Loadings:.88 Years, -.05 Seriousness Very similar to the first root for the main effect of crime. High CV1 = juror gives a long sentence despite thinking the crime not serious.

14. ANOVA on CV_PA1 SourceDFType III SS Mean Square F ValuePr > F PA28.859183304.429591654.430.0142 Crime10.51473376 0.510.4746 PA*Crime25.308185802.654092902.650.0750 The PA groups differ significantly on the weighted linear combination of sentence and seriousness. Roy’s root had F = 4.43 and p =.0142.

15. LSD on CV_PA1 Means with the same letter are not significantly different. GroupingMeanNPA A0.382437Unattractive B-0.157338Average B B-0.209539Attractive Remember that high scores on the canonical variate indicate the juror gave a long sentence despite not thinking the crime very serious.

16. ANOVA on CV_INT1 SourceDFType III SS Mean Square F ValuePr > F PA28.710064764.355032384.360.0152 Crime10.46250976 0.460.4979 PA*Crime25.411253892.705626942.710.0714 The Interaction is significant, with relaxed alpha, for the weighted combination of sentence and seriousness that makes as large as possible the interaction. Roy’s root had F = 2.71 and p =.0714.

18. Simple Main Effects, CV_INT1 PA*Crime Effect Sliced by PA for CV_INT1 PADF Sum of Squares Mean Square F ValuePr > F Attractive10.075443 0.080.7841 Average10.690913 0.690.4077 Unattractive15.052687 5.050.0266 The “long sentence despite not high seriousness” mean was significantly higher for swindlers than for burglars, but only when the defendant was unattractive.

19. ANCOV Seriousness of the crime is the covariate. Holding constant the seriousness of the crime, does physical attractiveness affect sentencing? SourceDFType III SSMean SquareF ValuePr > F Serious1379.4891539 41.14<.0001 PA280.417011940.20850594.360.0151 Crime14.4277539 0.480.4899 PA*Crime249.297642524.64882132.670.0737

20. PAYears LSMEANLSMEAN Number Attractive4.080331551 Average4.207686112 Unattractive5.939306953 Least Squares Means for effect PA Pr > |t| for H0: LSMean(i)=LSMean(j) Dependent Variable: Years i/j123 1 0.85540.0092 20.8554 0.0154 30.00920.0154 Adjusted for perceived seriousness of the crime, years sentenced is significantly higher for the unattractive defendant than for the average and attractive defendants.

21. What Follow-ups are Appropriate? My tactic is to do follow-up analyses on the canonical variates that were found to be significant in the MANOVA. The usual tactic is to do univariate ANOVA on the original dependent variables for any effect that was significant in the MANOVA. Let us see what happens if we use that usual tactic.

22. PA Manipulation There was a significant MANOVA main effect of the physical attractiveness manipulation on the first canonical variate combining sentence and seriousness. I’ll conduct two ANOVAs, looking at main effect of the PA manipulation –One on the sentence variable –Another on the seriousness variable

23. Years Sentenced SourceDFType III SS Mean Square F ValuePr > F PA276.98001738.4900083.040.0518 Crime12.49705132.49705120.200.6577 PA*Crime252.76379226.3818962.090.1293 Falls short of significance.

24. Seriousness of the Crime SourceDFType III SS Mean Square F ValuePr > F PA25.62951262.81475630.560.5721 Crime10.3929989 0.080.7800 PA*Crime20.20842160.10421080.020.9794 PA has a significant effect in the MANOVA, but no significant univariate ANOVA effects. How annoying is that?

25. Alternative Strategy First, conduct a PCA reducing the number of observed variables to a smaller number of principal components. Then conduct ANOVA on each of those principal components. With the Plaster data, there is only one component with eigenvalue > 1, and ANOVA on it produces no significant results.