Transcript MANOVA - Texas A&M University

LECTURE 17
MANOVA
Known population covariance. For a two group experimental design with several outcomes in
which the variances and covariances are known and assumed to be identical for both groups, yy,
the null hypothesis is
H0: E - C = 0
with alternative hypothesis
H1: E - C  0 .
The test statistic
_ _
_ _
X2 = [ nE nC / ( nE + nC)] ( yE. – yC.)-1yy ( yE. – yC.)
is chi-square distributed with df = p.
Unknown population covariance. For most experimental and observational
studies the population covariance matrix is not known. This matrix, labeled W
for “within” is simply the covariance matrix “unstandardized” by the sample
size
SSCPyy = Wyy = (n1 –1) Syy1 + (n2 - 1) Syy2 .
An assumption of the model is that the population covariance matrices for all
groups are homogeneous. The following statistic is computed for groups with
sample size nE and nC :
_
_
_
_
T2 = [ nE nC (nE + nC -2) / (nE + nC ) ] ( yE – yC)W-1yy ( yE. – yC) ,
Hotelling’s T2 , distributed as an F-statistic with df = p, n1 + n2 – p -1 :
F = [ (n-p)/p(n-1] T2 .
Wilk’s lambda:
 = W / B + W
The matrix W is the pooled group SSCP matrices as above. This plays the same role as
within-group sum of squares does in ANOVA. The matrix B + W is the equivalent of a
total sum of squares. It is more easily represented as the covariance matrix of all data
ignoring group membership.
The between-groups sum of squares and cross products, is not usually computed directly
but as the difference between T and W. Thus, for a single outcome Wilk’s lambda can
be shown to be
W=1 = SSe / SST = SSe / [SStreat + SSe ]
= 1 – R2
Other Measures
Pillai-Bartlett trace, V
Multiple discriminant analysis (MDA) is the part of MANOVA where
canonical roots are calculated. Each significant root is a dimension on which
the vector of group means is differentiated. The Pillai-Bartlett trace is the sum
of explained variances on the discriminant variates, which are the variables
which are computed based on the canonical coefficients for a given root. Olson
(1976) found V to be the most robust of the four tests and is sometimes preferred for this reason.
Roy's greatest characteristic root (GCR
is similar to the Pillai-Bartlett trace but is based only on the first (and hence
most important) root.Specifically, let lambda be the largest eigenvalue, then
GCR = lambda/(1 + lambda). Note that Roy's largest root is sometimes also
equated with the largest eigenvalue, as in SPSS's GLM procedure (however,
SPSS reports GCR for MANOVA). GCR is less robust than the other tests in
the face of violations of the assumption of multivariate normality.
Confidence
interval for
outcome y2
centered at 0
_ _
d2 = yE2 – yC2
t-test for y2
is not
significant
1 -  confidence
ellipsoid
_ _
d1 = yE1 – yC1
0,0
t-test for y1
is not
significant
Confidence
interval for
outcome y1
centered at 0
Fig. 15.2: Spatial representation of two group MANOVA difference vector and confidence
ellipsoid for two independent outcomes
Dependent variable contrasts
Just as one can specify contrasts concerning levels of independent
variables, it is possible to specify contrasts among multivariate outcomes.
In many research situations this is both feasible and of interest, because the
outcomes form a theoretical hierarchy or are related to each other in
theoretically interesting ways. In MANOVA a matrix of the contrasts is
developed in which the rows are the contrast coefficients and columns
represent the outcomes.
In a study of gender differences in five outcomes, suppose that the outcomes are separable into two
theoretical orientations, internal and external. Along with a global MANOVA analysis of whether
males differ from females on all five outcomes, we might be interested in whether they differ between
the internal and external orientation, and within each orientation. They M matrix might have five
nonorthogonal contrasts for those questions:
OUTCOME:
M=
Y1
Y2
Y3
Y4
Y5
3
3
-2
-2
-2
-1
1
0
0
0
0
0
1
-1
0
0
0
1
0
-1
0
0
0
1
-1
Clearly, the set of difference that the last three contrasts represent are an arbitrary set for pairwise
difference. Instead, two orthogonal contrasts might be used if there were theoretical reasons for
particular comparisons. In SAS PROC GLM, the MANOVA command, the matrix is specified as
shown except for placing commas after each contrast set and a semi-colon at the end.
Confidence
interval for
outcome y2
centered at 0
_ _
d2 = yE2 – yC2
1 -  confidence
ellipsoid
t-test for y2
is not
significant
_ _
d1 = yE1 – yC1
0,0 0,0
t-test for y1
is significant
Confidence
interval for
outcome y1
centered at 0
Fig. 15.3: Spatial representation of two group MANOVA difference vector and confidence
ellipsoid for two correlated outcomes
C1
y1
11
1
11
12
1
C2
21
12
y2
13
2
32
31
13
C3
y3
3
Fig. 15.4: SEM representation of MANOVA with 4 groups and 3 dependent variables
C1
1
y1
11*
11*
12
y2
C2
12*
1
32
21*
31*
13*
C3
2
2
y3
13
3
--- fixed value
From 1st model
Fig. 15.4: Canonical SEM representation of MANOVA with 4 groups and 3 dependent
variables
A
y1
11
1
11
12
1
B
21
12
y2
32
31
13
AB
13
2
y3
Fig. 15.6: MANOVA 2 x 2 factorial design with three outcomes
3
Assumptions- Homoscedasticity
• Box's M: Box's M tests MANOVA's assumption of
homoscedasticity using the F distribution. If p(M)<.05,
then the covariances are significantly different. Thus we
want M not to be significant, rejecting the null hypothesis
that the covariances are not homogeneous. That is, the
probability value of this F should be greater than .05 to
demonstrate that the assumption of homoscedasticity is
upheld. Note, however, that Box's M is extremely sensitive
to violations of the assumption of normality, making the
Box's M test less useful than might otherwise appear. For
this reason, some researchers test at the p=.001 level,
especially when sample sizes are unequal.
Assumptions-Normality
Multivariate normal distribution. For
purposes of significance testing, variables
follow multivariate normal distributions. In
practice, it is common to assume
multivariate normality if each variable
considered separately follows a normal
distribution. MANOVA is robust in the face
of most violations of this assumption if
sample size is not small (ex., <20).
DISCRIMINANT ANALYSIS
The inverse of the MANOVA
problem
Discriminant functions. The problem of discriminant analysis is to produce a score D that is a linear
combination of the predictors
D = a1y1 + a2y2 + …apyp
that maximally separates the groups. In effect, this is an ANOVA problem with D as the dependent
variable, and the criterion used to select the regression weights a 1 …ap is
C = [ SSgroups / SStotal ] for all possible D.
y2
Group 1 mean
*
D = a1y1 + a2y2
*
*
Group 2 mean
y1
Group 3 mean
Fig. 15.8: First discriminant function in predictor space
y2
Group 1 mean
*
D = a1y1 + a2y2
*
*
Group 2 mean
y1
Group 3 mean
Fig. 15.8: Second discriminant function in predictor space
Bartlett’s V statistic:
V = - [ N –1 – (I + p)/2 ] ln  , a chi-square statistic with (I-1)p degrees of freedom,
and
V1 = - [ N –1 – (I + p)/2 ] ln (1 + 1 ).
If V is not significant it is assumed that V1 is not significant. If V is significant, then Vr is tested:
Vr = V – V1 , a chi-square statistic with (I-2)(p-1) degrees of freedom.
If Vr is not significant it is concluded that only the first function is signficant. If Vr is signficant, the
second function statistic V2 is computed and subtracted from Vr and the remainder is tested, an iteration of
the first procedure. This continues for all functions
D2
Mean of Group 1 on D2
Mean of Group 2 on D2
Group 2
* Group 1
*
D1
Mean of Group 2 on D2
Mean of Group 1 on D1
Mean of Group 3 on D1
Group 3
*
Mean of Group 3 on D2
Fig. 15.10: Centroids for 3 groups in two discriminant function space
Canonical Discriminant Function 2
-6.0
-4.0
-2.0
.0
2.0
4.0
6.0
+---------+---------+---------+---------+---------+---------+
6.0 +
32
+
I
32
I
I
32
I
I
32
I
I
32
I
I
32
I
4.0 +
+
+
32 +
+
+
+
I
32
I
I
32
I
I
32
I
I
32
I
I
32
I
2.0 +
+
+
32 +
+
+
+
I
32
I
I
32
I
I
3112
I
I
31 12
I
*31
I
122*
I
31 *14*22
+ Canonical Discriminant Function 1
31
14 44222
I
31
14
44422
I
31
14
4422
I
31
14
44222
I
31
14
44422
I
+
31
14+
+
44222 +
+
31
14
44422
I
31
14
4422
I
31
14
44222
I
31
14
44422
I
31
14
44222I
+
31 +
14+
+
+
444+
31
14
I
31
14
I
31
14
I
31
14
I
31
14
I
31
14
+
+---------+---------+---------+---------+---------+---------+
-6.0
-4.0
-2.0
.0
2.0
4.0
6.0
Canonical Discriminant Function 1
_
.0 +
I
I
I
I
I
-2.0 +
I
I
I
I
I
-4.0 +
I
I
I
I
I
-6.0 +
+
+
Symbols used in territorial map
------
Symbol Group Label
----- -------------------1
2
3
4
*
2
3
4
5
Indicates a group centroid
Fig. 15.11: Territorial map for four groups in two-discriminant function space
e1
y1
a1 (r1 = w1/211a1)
Sex
D
e2
y2
a2 (r2 = w1/222a2)
Note: Structure
coefficients in
parenthesis
Fig. 15.13: Discriminant function path diagram showing regression and structure coefficients
y2
1 12 111 1
1 2 222 22 2 2
2
1 111 1 1
2 1 22212222222 2 1 2
1
111 111 111 2 2 111 2221 2 21
1 12 111 11 2 2 121 2222222 22 2
1 111 1 2 1111 2212112222 2222 2 2
1 111 112111211 1222122 2 2 2222
11 111111111 22 11 22222222 2 22
y1
1 1 11122 111 2 1222 2222222
2
1 111211 1 11112212111222 22 2 2 2 2 1222
1 11 11211 121111212212222 2 2 2
1
11
111121 2 212222 2 222
11 111 111 111111 21111 122222 22 2
1 1 111 11 1 1111 21122 2 22
2
2111 111 112111 1 11121222 12222 2 2
1 1 111 1 2211 11 2111 222 2 2
y2
1 11 1 22222222222 22 11111
11 1 11 2 2222 22122 11211 11
211 1 21 22 22122 12211111 1 2
1 1 12 2222 22 1221211 1 1 1
1 1 211 1 1 222 2 2 12211 1 1 1
1 11121 12122 2 2 221 1 2 1 111
1 1 211 22122 11 1112111 111
1
1211 222 12 11 1 1211 1
1 1 1 2 11222 111211 1 1 1
1 1 1 11111 1 1211221211111
1 1 111 11111 11 1112
11 111 1 11 1
1 11111 11211
1 1 1 1111111 1111
Fig. 15.14: Scatterplots for two predictors of two groups with approximate curves
separating the groups for classification
DISCRIMINANT ANALYSIS AS A STRUCTURAL
EQUATION DIAGRAM
1
Y1
e1
1
3 groups
Y2
e2
Y3
e3
2
2