Presentation on theme: "Multivariate Statistics"— Presentation transcript:
1 Multivariate Statistics Discriminant Function AnalysisMANOVA
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 predictorsThe 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 DiSSBetween 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 ofThe number of predictor variablesOne less than the number of groupsEach 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 controlAttitude about the insanity defenseAttitude about the death penaltyAttitude about the prosecuting attorneysAttitude about the defense attorneysAssessment of the expert testimonyAssessment of mental status of defendant.Can the defendant be rehabilitated?
11 MulticollinearityThis 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 ComponentsWe 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 illBelieved the defense expert testimony more than the prosecution expert testimonyWere receptive to the insanity defenseOpposed the death penaltyThought the defendant could be rehabilitatedFavored 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 likelyOppose lenient treatmentAre not receptive to the insanity defenseDo 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 TaiwanGrouping variablesDefendant physically attractive or notSex of defendantType of crime: Swindle or burglaryDefendant American or ChineseSex of juror
19 Dependent Variables One set of two variables Length of recommended sentenceRated seriousness of the crimeA second set of 12 variables, ratings of the defendant on attributes such asPhysical attractivenessIntelligenceSociability
20 Type I Boogeyman If we did a five-way ANOVA on one DV We would do 27 F testsAnd that is just for the omnibus analysisIf we do that for each of the 14 DVsThat is 378 F testsAnd the Boogeyman is licking his chops
21 Results, SentencingFemale jurors gave longer sentences, but only with American defendantsAttractiveness lowered the sentence for American burglarsBut increased the sentence for American swindlersFemale jurors gave shorter sentences to female defendants
22 Results, Ratings The following were rated more favorably Physically attractive defendantsAmerican defendantsSwindlers
23 Canonical VariatesFor 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 loadingsLook 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.