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Multivariate Statistics An Introduction & Multidimensional Contingency Tables What Are Multivariate Stats? • Univariate = one variable (mean) • Bivariate = two variables (Pearson r) • Multivariate = three or more variables simultaneously analyzed One-Way ANOVA • Could consider bivariate – one grouping variable, one continuous variable. • Could consider multivariate – predict Y from the set of k-1 dichotomous dummy variables coding the grouping variable. Factorial ANOVA • I consider it multivariate – one continuous variable and two or more grouping variables. • Some call it univariate, as in “univariate ANOVA.” Here the focus is on how many comparison variables there are (only one Y). • If there were more than one Y, they would call it MANOVA and consider it multivariate. Independent and Dependent Variables • Data analyzed with multivariate techniques are most often nonexperimental. • You know how I feel about using the terms “independent variable” and “dependent variable” in that case. • But others use these terms more loosely. • Independent = grouping, prior, known, thought to be the cause. • Dependent = continuous, later, predicted, thought to be the effect. Descriptive vs. Inferential • Like univariate and bivariate stats, multivariate stats can be used descriptively. • In this case, there are no assumptions. • If you use 2, t, or F, then there are assumptions. Rank Data/Scale of Measurement • Only God knows if your data are interval rather than merely ordinal, and she is not saying. • Ordinal data may be normally distributed. • Interval data may not be normally distributed. • Ranks are not normally distributed, but may be close enough to normal. Why Use Multivariate Stats? • • • • To impress your friends. To obfuscate. Because SPSS makes it so easy to do. To statistically hold constant the effects of confounding variables in nonexperimental research. Why NOT use Multivariate Stats? • You may be able adequately to address your research question with more simple analysis. • One may be able to get pretty much any damn results she wishes, so why bother? • Do you really understand what is going on out there in hyperspace? I am already confused enough in three dimensional space. Multidimensional Contingency Table Analysis • Chapter 17 in Howell. • Have three or more dimensions in the contingency table. All variables are categorical. • Moore, Wuensch, Hedges, & Castellow (1994) • Simulated civil case, sexual harassment. • Female plaintiff, male defendant. The Design • Physical attractiveness (PA) of defendant, manipulated. • Social desirability (SD) of defendant, manipulated. • Sex/gender of mock juror. • Verdict recommended by juror (dependent). • Experiment 2: manipulated PA and SD of litigant. Logit Analysis • This is a special case. • One variable is identified as dependent. • We are interested only in effects that involve the dependent variable. Earlier Research • Physically attractive litigants are better treated by the jurors. No Social Desirability manipulation. • But jurors rated the physically attractive litigants as more socially desirable (intelligent, sincere, and so on). • Which is directly affecting the verdict, PA or inferred SD ? More Earlier Research • Follow-up to that just described. • Manipulated only the SD of the litigants. • Socially desirable litigants were treated better by the jurors. • But the jurors rated the (never seen) socially desirable litigants as more physically attractive. • Still do not know if it is PA or SD that directly affects the verdict. Experiment 1(manipulate characteristics of defendant) • Guilty verdicts were more likely when – Juror was female – Defendant was socially undesirable • Gender x PA Interaction: Female jurors: – Judged the physically attractive defendants more harshly – Maybe they thought the defendants used their PA to take advantage of the plaintiff. • No significant effect of PA among male jurors. Experiment 1(manipulate characteristics of plaintiff) • Judgments in favor of plaintiff more frequent when she was socially desirable. • No other effects were significant. • Strength of effect estimates in both experiments showed effect of SD much greater than effect of PA. Conclusions • When jurors have no relevant info on SD, they infer that the beautiful are good, and that affects their verdicts. • When jurors do have relevant info on SD, the PA of the litigants is of little importance.