Multivariate Statistics

1. Multivariate Statistics Discriminant Function Analysis MANOVA

2. Discriminant Function Analysis • You wish to predict group membership from a set of two or more continuous variables. • Example: The IRS wants to classify tax returns as OK or fraudulent. • The have data on many predictor variables from audits conducted in past years.

3. The Discriminant Function • Is a weighted linear combination of the predictors • The weights are selected so that the two groups differ as much as possible on the discriminant function.

4. Eigenvalue and Canonical r • Compute a discriminant score, Di, for each case. • Use ANOVA to compare the groups on Di • SSBetween Groups/ SSWithin Groups=eigenvalue

5. Classification • The analysis includes a classification function. • This allows one to predict group membership for any case on which you have data on the predictors. • Those who are predicted to have submitted fraudulent returns are audited.

6. Two or More Discriminant Functions • You may be able to obtain more than one discriminant function. • The maximum number you can obtain is the smaller of • The number of predictor variables • One less than the number of groups • Each function is orthogonal to the others. • The first will have the greatest eigenvalue, the second the next greatest, etc.

7. Labeling Discriminant Functions • You may wish to name these things you have created or discovered. • As when naming factors from a factor analysis, look at the loadings (correlations between Di and the predictor variables) • Look at the standardized discriminant function coefficients (weights).

8. Predicting Jurors’ Verdict Selections • Poulson, Braithwaite, Brondino, and Wuensch (1997). • Subjects watch a simulated trial. • Defendant accused of murder. • There is no doubt that he did the crime. • He is pleading insanity. • What verdict does the juror recommend?

9. The Verdict Choices • Guilty • GBMI (Guilty But Mentally Ill) • NGRI (Not Guilty By Reason of Insanity) • The jurors are not allowed to know the consequences of these different verdicts.

10. Eight Predictor Variables • Attitude about crime control • Attitude about the insanity defense • Attitude about the death penalty • Attitude about the prosecuting attorneys • Attitude about the defense attorneys • Assessment of the expert testimony • Assessment of mental status of defendant. • Can the defendant be rehabilitated?

11. Multicollinearity • This is a problem that arises when one predictor can be nearly perfectly predicted by a weighted combination of the others. • It creates problems with the analysis. • One solution is to drop one or more of the predictors. • If two predictors are so highly correlated, what is to be lost by dropping one of them?

12. But I Do Not Want To Drop Any • The lead researcher did not want to drop any of the predictors. • He considered them all theoretically important. • So we did a little magic to evade the multicollinearity problem.

13. Principal Components • We used principal components analysis to repackage the variance in the predictors into eight orthogonal components. • We used those components as predictors in a discriminant function analysis. • And then transformed the results back into the metric of the original predictor variables.

14. The First Discriminant Function • Separated those selecting NGRI from those selecting Guilty. • Those selecting NGRI: • Believed the defendant mentally ill • Believed the defense expert testimony more than the prosecution expert testimony • Were receptive to the insanity defense • Opposed the death penalty • Thought the defendant could be rehabilitated • Favored lenient treatment over strict crime control.

15. The Second Discriminant Function • Separated those selecting GBMI from those selecting NGRI or Guilty. • Those selecting GBMI: • Distrust attorneys (especially prosecution) • Think rehabilitation likely • Oppose lenient treatment • Are not receptive to the insanity defense • Do not oppose the death penalty.

16. MANOVA • This is just a DFA in reverse. • You predict a set of continuous variables from one or more grouping variables. • Often used in an attempt to control familywise error when there are multiple outcome variables. • This approach is questionable, but popular.

17. MANOVA First, ANOVA Second • 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.

18. 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

19. 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

20. 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

21. Results, Sentencing • Female jurors gave longer sentences, but only with American defendants • Attractiveness lowered the sentence for American burglars • But increased the sentence for American swindlers • Female jurors gave shorter sentences to female defendants

22. Results, Ratings • The following were rated more favorably • Physically attractive defendants • American defendants • Swindlers

23. Canonical Variates • For each effect (actually each treatment df) there is a different set of weights applied to the outcome variables. • The weights are those that make the effect as large as possible. • The resulting linear combination is called a canonical variate. • Again, one canonical variate per treatment degree of freedom.

24. Labeling the Canonical Variates • Look at the loadings • Look at the standardized weights (standardized discriminant function coefficients)

25. Sexual Harassment Trial: Manipulation Check • Moore, Wuensch, Hedges, and Castellow (1994) • Physical attractiveness (PA) of defendant, manipulated. • Social desirability (SD) of defendant, manipulated. • Sex/gender of mock juror. • Ratings of the litigants on 19 attributes. • Experiment 2: manipulated PA and SD of plaintiff.

26. Experiment 1: Ratings of Defendant • Social Desirabililty and Physical Attractiveness manipulations significant. • CVSocial Desirability loaded most heavily on sociability, intelligence, warmth, sensitivity, and kindness. • CVPhysical Attractiveness loaded well on only the physical attractiveness ratings.

27. Experiment 2: Ratings of Plaintiff • Social Desirabililty and Physical Attractiveness manipulations significant. • CVSocial Desirability loaded most heavily on intelligence, poise, sensitivity, kindness, genuineness, warmth, and sociability. • CVPhysical Attractiveness loaded well on only the physical attractiveness ratings.