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 =
SSamong _ groups
SSwithin _ groups
1.7672 for root 1
.1677 for root 2
Canonical Correlations2
SSamong _ groups
SStotal
• .639 for the first canonical variate
• .144 for the second
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 are
not significantly different.
t Grouping
Mean
N
PA
A
8.2821
39
Beautiful
B
3.4737
38
Average
2.8919
37
Unattr
B
B
Happiness
Means with the same letter are
not significantly different.
Grouping
Mean
N
PA
A
5.7895
38
Average
A
5.3590
39
Beautiful
B
4.0000
37
Unattr
A
Independence
Means with the same letter are
not significantly different.
Grouping
Mean
N
PA
A
6.9474
38
Average
A
6.4103
39
Beautiful
B
5.0000
37
Unattr
A
Sophistication
Means with the same letter are
not significantly different.
Grouping
Mean
N
pa
A
6.0769
39
Beautiful
B
4.7368
38
Average
4.3243
37
Unattr
B
B
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
Model
DF
Sum of Mean Square
Squares
2 196.1586479
Error
111
110.9999952
Corrected Total
113 307.1586432
98.0793240
1.0000000
F Value
Pr > F
98.08
<.0001
LSD on CV1
Means with the same letter are
not significantly different.
Grouping
Mean
N
PA
A
1.8145
39
Beautiful
B
-0.8303
38
Average
-1.0599
37
Unattr
B
B
ANOVA on CV2
Source
Model
DF
Sum of Mean Square
Squares
2
18.6108887
9.3054443
Error
111
111.0000540
1.0000005
Corrected
Total
113 129.6109426
F Value
Pr > F
9.31
0.0002
High CV2 = Happy and Independent
LSD on CV2
Means with the same letter are
not significantly different.
Grouping
Mean
N
PA
A
0.5110
38
Average
B
-0.0398
39
Beautiful
-0.4829
37
Unattr
B
B
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.