MANOVA One-Way

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.

Michelle Plaster’s Thesis Male participants were shown a picture of one of three young women. Pilot work had indicated that the one woman was beautiful another of average physical attractiveness, and the third unattractive.

Manipulation Check Participants rated the woman they saw on each of twelve attributes. Here we shall use only four of those physical attractiveness (PHYATTR), happiness (HAPPY), INDEPENdence, and SOPHISTication.

SAS Code Proc ANOVA; class PA; model phyattr happy indepen sophist = pa / Nouni; MANOVA h = PA / Canonical; –PA is the physical attractiveness manipulation –Nouni suppresses univariate ANOVA output –Canonical produces statistics related to the canonical variates

Canonical Variates AKA, roots, discriminant functions How many will there be? The lesser of –The number of Dependent Variables –The number of Groups minus 1. Here we shall have 3-1 = 2 roots.

The First Canonical Variate The weighted linear combination of the Ys that maximizes SS among groups / SS within groups CV1 =.53 Phyattr -.05 Happy Indep +.04 Sophis High CV1 = High Physical Attractiveness Standardized weights = 1.64, -.15,.01,.10 Loadings =.99,.08,.08,.24

The Second Canonical Variate Orthogonal to the first. CV2 = -.02 Phyattr +.28 Happy +.28 Indepen -.14 Sophis High CV2 = Happy and Independent Standardized weights = -.06,.71,.72, -.34 Loadings =.08,.72,.81,.12

Eigenvalues If we were to compute, for each case, canonical variate scores And then do ANOVAs comparing the groups on the canonical variates Each root would have an eigenvalue = for root for root 2

Canonical Correlations for the first canonical variate.144 for the second

Testing Sig. of Can. Corrs H 0 : The population canonical correlation for this root and all subsequent roots are zero. The test statistics are likelihood ratios transformed to F values. Roots 1 & 2 simultaneously tested: p <.0001 Root 2: p =.0007

H 0 : PA Manipulation Has No Effect Wilks' Lambda (  ) = error/(error + treatment) The smaller , the greater the evidence against the null.  is identical to the likelihood ratio for testing the first and all subsequent roots.  =.309, p <.0001

Other Test Statistics Pillai’s Trace – more robust than the other test statistics Hotelling-Lawley Trace – the sum of the eigenvalues Roy’s Maximum Root – tests only the first root. For our data, p <.0001 with each of the four tests.

Strength of Effect Estimate See multivariate omega-squaredmultivariate omega-squared For Plaster’s data,  2 =.684 and the adjusted  2 =.665.

Univariate ANOVAs & LSD For the physical attractiveness ratings Means with the same letter are not significantly different. t GroupingMeanNPA A Beautiful B Average B B Unattr

Happiness Means with the same letter are not significantly different. GroupingMeanNPA A Average A A Beautiful B Unattr

Independence Means with the same letter are not significantly different. GroupingMeanNPA A Average A A Beautiful B Unattr

Sophistication Means with the same letter are not significantly different. GroupingMeanNpa A Beautiful B Average B B Unattr

ANOVAs on Canonical Variates Following the significant MANOVA with univariate ANOVAs is common practice But it ignores the canonical variates. I shall show you how to do ANOVAs on the canonical variates. I have never seen anybody else do this, but I think it makes good sense.

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.

Create the Canonical Variate Scores I standardized each of the dependent variables. Then I used the standardized weights to compute the standardized canonical variate scores. CV1= *z_phyattr *z_happy *z_indepen *z_sophist; CV2= *z_phyattr *z_happy *z_indepen *z_sophist;

ANOVA on CV1 SS Among /SS Within = eigenvalue for root 1 SS Among /SS Total = first canonical correlation 2 SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total

LSD on CV1 Means with the same letter are not significantly different. GroupingMeanNPA A Beautiful B Average B B Unattr

ANOVA on CV2 SourceDF Sum of Squares Mean SquareF ValuePr > F Model Error Corrected Total High CV2 = Happy and Independent

LSD on CV2 Means with the same letter are not significantly different. GroupingMeanNPA A Average B Beautiful B B Unattr

MANOVA and DFA I also conducted a DFA with these data. If you look at the DFA output, you will see that the eigenvalues, canonical correlations, loadings, and canonical coefficients are identical to those obtained with the MANOVA

SPSS Please see my handout for instructions on how to use SPSS to do MANOVA.