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.

Slide 2

Gender Height WeightMale 69 70Male 63 68Male 71 73Female 72 79Female 82 84Female 78 79Male 68 74Male 72 74Male 70 73Female 64 65Female 69 68Female 70 71Male 75 80Male 63 66Male 56 59Female 74 74Female 76 76Female 60 61Male 78 85Male 71 76Male 77 83Female 72 75Female 68 65Male 68 68Male 72 7855 60 65 70 75 80

60

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Height[Gender == "Female"]

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Gender Difference:

Xuhua Xia Slide 3

Advantage of MANOVAmd<-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,col="red",xlim=c(min(Height),max(Height)),ylim=c(min(Weight),max(Weight)))points(Height[Gender=="Male"],Weight[Gender=="Male"],pch=16)Y <- cbind(Height, Weight)fit <- manova(Y ~ Gender)summary(fit, test="Pillai")

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"

Xuhua Xia Slide 5

Advantage of MANOVAmd<-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")

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

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.