Advanced Insights into Multivariate Analysis of Variance (MANOVA)

Multivariate Analysis of Variance (MANOVA) Prof. Andy Field

Aims When and why do we Use MANOVA? Issues for MANOVA analysis Theory behind MANOVA: SSCP matrices Discriminant variates MANOVA test statistics Slide 2

When and Why To test for differences between groups When we have several outcome variables (DVs) Better than multiple ANOVA Controls familywise error rate (Type I error) Takes account of relationships between DVs Slide 3

Issues in MANOVA Selecting dependent variables Test statistics (choice of four) Power Power depends upon correlations between dependent variables Conflicting evidence about test power Slide 4

Theory: An Example Efficacy of psychotherapy on OCD Three groups: cognitive behaviour therapy (CBT) behaviour therapy (BT) no treatment (NT) Two outcome variables (DVs): obsession-related actions obsession-related thoughts Slide 5

Theory of ANOVA (Revision) SST Total Variance in the Data SSM SSR Improvement Due to the Model Error in Model F SSM/SSR Slide 6

Total Cross-Products Slide 7

Model Cross-Products Slide 8

Residual Cross-Products Slide 9

Sums of Squares Slide 10

MANOVA Matrices: Total SSCP (T) Slide 11

MANOVA Matrices: Residual SSCP (E) Slide 12

MANOVA Matrices: Model SSCP (M) Slide 13

An Equivalent to F SSCP SSCP F M R H E SSCP SSCP = = -1 HE M R Slide 14

The Result 0.0202 0.0021 0.0021 0.0084 = 1 E 0.2273 0.1930 0.0852 0.1794 = 1 HE Slide 15

Discriminant Variates = + V Actions 1 b b Thoughts 1 2 0.603 0.335 0.425 0.339 = = eigenvector eigenvector 1 2 = V . 0 = 603 Actions . 0 + 335 Thoughts 1 V . 0 425 Actions . 0 339 Thoughts 2 Slide 16

Eigenvalues (i) 0.335 0.000 0.000 0.073 HE = 1 variates Slide 17

PillaiBartlett Trace s = i = V i 1 + i 1 Slide 18

Hotellings Trace s = i = T i 1 Slide 19

Wilkss Lambda s = i 1 = + 1 i 1 Slide 20

Roys Largest Root = largest root Slide 21

Which Test Statistic? Issues to consider Power Robustness Equality of sample sizes Pillai s trace Robust when sample sizes are equal Slide 22

Exploring Data

Setting contrasts For this example it makes sense to compare each of the treatment groups to the no- treatment control group. The no-treatment control group was coded as the last category, so we could set this contrast by executing: contrasts(ocdData$Group)<-contr.treatment(3, base = 3) Slide 24

Setting contrasts Alternatively, we could set the contrasts by executing: CBT_vs_NT<-c(1, 0, 0) BT_vs_NT <-c(0, 1, 0) contrasts(ocdData$Group)<-cbind(CBT_vs_NT, BT_vs_NT)

The MANOVA model In the case of MANOVA there are several outcomes so the model becomes outcomes ~ predictor(s) . To put multiple outcomes into the model, we have to bind the variables together into a single entity using the cbind() function: outcome<-cbind(ocdData$Actions, ocdData$Thoughts) We use this new object as the outcome in our model, and specify any predictors as we have in previous chapters: ocdModel<-manova(outcome ~ Group, data = ocdData) Slide 26

Obtaining the Output To see the output of the model we use the summary command; by default, R produces Pillai s trace (which is a sensible choice), but we can see the other test statistics by including the test = option. For example, to see all four test statistics we would need to execute: summary(ocdModel, intercept = TRUE) summary(ocdModel, intercept = TRUE, test = "Wilks") summary(ocdModel, intercept = TRUE, test = "Hotelling") summary(ocdModel, intercept = TRUE, test = "Roy")

Follow-Up Analysis: Univariate Test Statistics To follow up the analysis with univariate analyses of the individual outcome measures, execute: summary.aov(ocdModel)

Contrasts The contrasts are not part of the MANOVA model and so to generate the output for them you have to create separate linear models for each outcome measure. actionModel<-lm(Actions ~ Group, data = ocdData) thoughtsModel<-lm(Thoughts ~ Group, data = ocdData) summary.lm(actionModel) summary.lm(thoughtsModel)

Contrasts

Robust MANOVA Factor A: Treatment Group Group Actions Thoughts 1 CBT 5 14 2 CBT 5 11 3 CBT 4 16 4 CBT 4 13 5 CBT 5 12 6 CBT 3 14 7 CBT 7 12 8 CBT 6 15 9 CBT 6 16 10 CBT 4 11 11 BT 4 14 12 BT 4 15 13 BT 1 13 14 BT 1 14 15 BT 4 15 16 BT 6 19 17 BT 5 13 18 BT 5 18 19 BT 2 14 20 BT 5 17 21 NT 4 13 22 NT 5 15 23 NT 5 14 24 NT 4 14 25 NT 6 13 26 NT 4 20 27 NT 7 13 28 NT 4 16 29 NT 6 14 30 NT 5 18 CBT BT NT Actions 5 5 4 4 5 3 7 6 6 4 Thoughts 14 11 16 13 12 14 12 15 16 11 Actions 4 4 1 1 4 6 5 5 2 5 Thoughts 14 15 13 14 15 19 13 18 14 17 Actions 4 5 5 4 6 4 7 4 6 5 Thoughts 13 15 14 14 13 20 13 16 14 18 Original Data Restructured Data

Robust MANOVA ocdData$row<-rep(1:10, 3) ocdMelt<-melt(ocdData, id = c("Group", "row"), measured = c("Actions", "Thoughts")) names(ocdMelt)<-c("Group", "row", "Outcome_Measure", "Frequency") ocdRobust<-cast(ocdMelt, row ~ Group + Outcome_Measure, value = "Frequency") ocdRobust$row<-NULL ocdRobust

Robust MANOVA mulrank(3, 2, ocdRobust) cmanova(3, 2, ocdRobust)

Following MANOVA with Discriminant Function Analysis Discriminant analysis is quite straightforward in R: you use the lda() function from the MASS package. The basic format of this function is: newModel<-lda(Group ~ Predictor(s), data = dataFrame, prior = prior probabilities, na.action = "na.omit") For the current data, we could, therefore, execute: ocdDFA<-lda(Group ~ Actions + Thoughts, data = ocdData) ocdDFA Slide 36

Discriminant Analysis Output

Discriminant Analysis Output

Discriminant Function Analysis BT 2 BT CBT BT CBT 1 CBT CBT NT CBT BT NT BT BT BT LD2 CBT NT 0 NT CBT CBT NT NT NT NT CBT -1 BT CBT BT NT NT -2 BT -2 -1 0 LD1 1 2 BT BT 2 2 BT BT CBT CBT BT BT CBT CBT 1 1 CBT CBT CBT NT CBT NT CBT CBT BT NT BT NT BT BT BT BT BT BT LD2 LD2 CBT NT CBT NT 0 0 NT NT CBT CBT CBT NT CBT NT NT NT NT NT NT NT CBT CBT -1 -1 BT BT CBT CBT BT NT BT NT NT NT -2 -2 BT BT -2 -1 0 LD1 1 2 -2 -1 0 LD1 1 2