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.

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 SSamong groups / SSwithin groups

• CV1 = .53 Phyattr - .05 Happy +.005 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 =

groupswithin

groupsamong

SS

SS

_

_ 1.7672 for root 1.1677 for root 2

Canonical Correlations2

• .639 for the first canonical variate• .144 for the second

total

groupsamong

SS

SS _

Testing Sig. of Can. Corrs

• H0: 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

H0: 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-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 arenot significantly different.

t Grouping Mean N PA

A 8.2821 39 Beautiful

B 3.4737 38 Average

B      

B 2.8919 37 Unattr

Happiness

Means with the same letter arenot significantly different.

 Grouping Mean N PA

A 5.7895 38 Average

A      

A 5.3590 39 Beautiful

B 4.0000 37 Unattr

Independence

Means with the same letter arenot significantly different.

 Grouping Mean N PA

A 6.9474 38 Average

A      

A 6.4103 39 Beautiful

B 5.0000 37 Unattr

Sophistication

Means with the same letter arenot significantly different.

Grouping Mean N pa

A 6.0769 39 Beautiful

B 4.7368 38 Average

B      

B 4.3243 37 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=1.63582926*z_phyattr - 0.1512594*z_happy +.0122376*z_indepen+.0965477*z_sophist;

• CV2=-0.05808645*z_phyattr+0.70694469*z_happy +0.71902789*z_indepen-0.33710555*z_sophist;

ANOVA on CV1

• SSAmong/SSWithin = eigenvalue for root 1

• SSAmong/SSTotal = first canonical correlation2

Source DF Sum of Squares

Mean Square F Value Pr > F

Model 2 196.1586479 98.0793240 98.08 <.0001

Error 111 110.9999952 1.0000000    

Corrected Total 113 307.1586432      

LSD on CV1

Means with the same letter arenot significantly different.

Grouping Mean N PA

A 1.8145 39 Beautiful

B -0.8303 38 Average

B      

B -1.0599 37 Unattr

ANOVA on CV2

Source DF Sum of Squares

Mean Square F Value Pr > F

Model 2 18.6108887 9.3054443 9.31 0.0002

Error 111 111.0000540 1.0000005    

Corrected Total

113 129.6109426      

High CV2 = Happy and Independent

LSD on CV2

Means with the same letter arenot significantly different.

Grouping Mean N PA

A 0.5110 38 Average

B -0.0398 39 Beautiful

B      

B -0.4829 37 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.