2. Xuhua Xia Slide 1 MANOVA All statistical methods we have learned so far have only one continuous DV and one or more IVs which may be continuous or categorical There are cases with multiple DVs. All statistical methods that deal with multiple DVs or multiple variables without DV/IV specification are grouped into multivariate statistics Multivariate analysis of variance (MANOVA) as an example Why MANOVA instead of multiple ANOVA for each DV? –Experimentwise error rate –See what univariate analysis cannot see The power of MANOVA test generally decreases with the number of variables that do not differ among groups. So be cautious in including variables in MANOVA (or any other multivariate statistical methods). There are two ways in science to increase the statistical power. Everyone knows the first, i.e., to increase sample size, but many ignore the second, i.e., to formulate specific and explicit hypotheses.

3. Slide 2 GenderHeightWeight Male6970 Male6368 Male7173 Female7279 Female8284 Female7879 Male6874 Male7274 Male7073 Female6465 Female6968 Female7071 Male7580 Male6366 Male5659 Female7474 Female7676 Female6061 Male7885 Male7176 Male7783 Female7275 Female6865 Male6868 Male7278 Gender Difference:

4. Xuhua Xia Slide 3 Advantage of MANOVA md<-read.table("manovaex1.txt",header=T) attach(md) fitANOVA1<-aov(Height~Gender) summary(fitANOVA1) fitANOVA2<-aov(Weight~Gender) summary(fitANOVA2) anova(fitANOVA2) plot(Height[Gender=="Female"],Weight[Gender=="Female"],pch=16,co l="red",xlim=c(min(Height),max(Height)),ylim=c(min(Weight),max(W eight))) points(Height[Gender=="Male"],Weight[Gender=="Male"],pch=16) Y <- cbind(Height, Weight) fit <- manova(Y ~ Gender) summary(fit, test="Pillai")

5. Xuhua Xia Slide 4 Fisher Iris Data Collected by Dr. Edgar Anderson, published in Fisher (1936) Sepal length and width, petal length and width (in cm) of fifty plants for each of three types of iris –Iris setosa, diploid with 38 chromosomes –Iris versicolor, hexaploid (108 chromosomes) –Iris virginica, tetroploid Fisher, R.A. (1936). "The Use of Multiple Measurements in Taxonomic Problems". Annals of Eugenics 7: 179–188. Saved in "MANOVAex2.txt"

6. Xuhua Xia Slide 5 Advantage of MANOVA md<-read.table("MANOVAex2.txt",header=T) attach(md) fitANOVA1<-aov(SepalLen~Species) summary(fitANOVA1) fitANOVA2<-aov(SepalWid~Species) summary(fitANOVA2) fitANOVA3<-aov(PetalLen~Species) summary(fitANOVA3) fitANOVA4<-aov(PetalWid~Species) summary(fitANOVA4) Y <- cbind(SepalLen, SepalWid, PetalLen, PetalWid) fit <- manova(Y ~ Species) summary(fit, test="Pillai")

7. Xuhua Xia Slide 6 Interpretation of MANOVA If the multivariate test is –not significant, report no group differences among the mean vectors –significant, perform univariate ANOVA and relevant contrasts –Correlation among variables that may lead to significant MANOVA test but no significant ANOVA test. Contrasts –Prior (planned): Certain theory predicts which treatments should be different –Post hoc (unplanned): Not sure which treatments should be different

8. Xuhua Xia Slide 7 MANOVA Assumptions Independence assumption: All observations are independent (residuals are uncorrelated) Multivariate normality Sphericity assumption in repeated measures Homoscedasticity (equal variance and covariance) assumption: Each sample (group) has the same covariance matrix (compound symmetry) Linearity assumption: Relationship among variables are linear.