MANOVA

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MANOVA
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Multivariate (multiple) analysis of variance
(MANOVA) represents a blend of univariate
analysis of variance principles and canonical
correlation analysis. It is understood best
against the backdrop of basic univariate analysis
of variance (ANOVA), especially its matrix form
and relationships to linear combinations. The
ways in which ANOVA represents the construction
of different linear combinations provides a very
flexible approach to data analysis that
translates well to MANOVA.
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• The basic ANOVA model can be captured in this
  basic matrix formulation in which
• X represents a Groups x Variables matrix of means
• L represents an L x Groups matrix of contrast
  codes that represent group comparisons of
  interest
• M represents a Variables x M matrix of contrast
  codes that represent comparisons among variables
  of interest
• K represents an L x M matrix of null hypothesis
  values, usually 0

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• This formulation works for the raw data matrix as
  well. In that case
• X represents a Cases x Variables matrix of scores
• L represents an L x Cases matrix of contrast
  codes that represent group comparisons of
  interest, but taking sample size into account.
• M represents a Variables x M matrix of contrast
  codes that represent comparisons among variables
  of interest
• K represents an L x M matrix of null hypothesis
  values, usually 0

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An L matrix
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Variance of the mean
Variance of scores
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A K matrix
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Sample 1
Sample 2
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An L matrix
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Another L matrix
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An M matrix
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Pre-multiplication by a diagonal matrix
multiplies the rows by the corresponding diagonal
element.
Post-multiplication by a diagonal matrix
multiplies the columns by the corresponding
diagonal element.
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LXM
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Transformation of the original variance-covariance
matrix to get the variance of the new linear
combination
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LXM
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The transformation of the dependent variables is
the same the group comparison has changed.
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A new M matrix
The variance of the difference score
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Sample 1
Sample 2
Sample 3
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Another L matrix
Another M matrix
What are these transformations testing?
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• The basic ANOVA model can be captured in this
  basic matrix formulation in which
• X represents a Groups x Variables matrix of means
• L represents an L x Groups matrix of contrast
  codes that represent group comparisons of
  interest
• M represents a Variables x M matrix of contrast
  codes that represent comparisons among variables
  of interest
• K represents an L x M matrix of null hypothesis
  values, usually 0

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SPSS can be used off the shelf to produce
default comparisons or the L Matrix, M Matrix,
and K Matrix can be specified to control
precisely the kinds of comparisons made. Command
for the GLM procedure to produce the default
ANOVA
GLM T1 T2 T3 BY GROUP /PRINTDESCRIPTIVES,TEST(LMA
TRIX,MMATRIX) HOMOGENEITY ETASQ OPOWER parameter
rsscp.
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In repeated measures designs and in multivariate
tests, the variance-covariance matrices are
assumed to be equal. Ok here, but the sample size
is small.
Homogeneous?
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This assumption is required for repeated measures
tests with more than one degree of freedom.
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Individual tests for equality of variances, an
assumption for univariate tests.
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Univariate tests of significance. These are
omnibus tests for any group differences.
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These are the default single degree of freedom
comparisons that SPSS provides. They test Groups
1 and 2 against Group 3. The intercept is the
mean for Group 3.
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The default L matrix
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The default M matrixjust an identity matrix, so
no transformation of the dependent variables.
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This command uses the default L matrix but now
specifies the M matrix
GLM T1 T2 T3 BY GROUP /MMATRIX T1 1 T2 1 T3 1 T1
1 T2 0 T3 -1 T1 1 T2 -2 T3 1 /PRINTDESCRIPTIVES,
TEST(LMATRIX,MMATRIX) HOMOGENEITY ETASQ OPOWER
parameter rsscp.
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The specified M matrix. What is being tested here?
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Tests of the intercept are tests of the grand
mean for each contrast, here defined by the
specified M matrix.
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These F tests correspond to tests that the grand
mean for each transformed variable (from the M
Matrix) is equal to zero. The first transformed
variable (the sum) is not an interesting
comparison. The scale does not include zero, so
this mean must be greater than zero. The other
two transformed variables represent differences
and so the mean compared to zero is sensible and
informative.
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This command uses the full power of matrix
specification to design particular group
comparisons and particular linear combinations of
the original variables
GLM T1 T2 T3 BY GROUP /LMATRIX GROUP 1 -1
0 /LMATRIX GROUP 1 1 -2 /MMATRIX T1 1 T2 1 T3
1 T1 1 T2 0 T3 -1 T1 1 T2 -2 T3
1 /PRINTDESCRIPTIVES,TEST(LMATRIX,MMATRIX)
HOMOGENEITY ETASQ OPOWER parameter rsscp.
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Custom Hypothesis Test 1
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The specified M matrix
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These tests combine the specified L matrix and
the specified M matrix to provide a set of very
particular comparisons.
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Custom Hypothesis Tests 2
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MANOVA approaches the analysis of multiple
dependent variables in a different way. It
represents the application of canonical
correlation analysis procedures to find the
optimal M matrix for separating means defined by
the L matrix.
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• The basic ANOVA model can be captured in this
  basic matrix formulation in which
• X represents a Groups x Variables matrix of means
• L is an L x Groups matrix of contrast codes that
  represent group comparisons of interest
• M is a Variables x M matrix of contrast codes
  that represent comparisons among variables of
  interest
• K represents an L x M matrix of null hypothesis
  values, usually 0

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One hundred students, preparing to take the
Graduate Record Exam, were randomly assigned to
one of four training conditions Group 1 No
special training Group 2 Standard book and
paper training Group 3 Computer-based
training Group 4 Standard and computer-based
training
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At the end of the study, all students complete a
paper-and-pencil version of the Verbal and
Quantitative scales of the GRE. All students also
completed computer-administered parallel forms of
the paper-and-pencil versions. The order of
administration of the four outcome measures was
counterbalanced.
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Univariate Analyses Each Variable Examined
Separately
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manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /print cellinfo(all) parameters
signif(singledf) homogeneity error /power
exact /design .
In these analyses, no structure is imposed on the
Group variable. By default, SPSS will use
deviation contrasts (effects coding) to partition
the degrees of freedom into single df effects.
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Cell Means and Standard Deviations Variable ..
STAND_V Standard Measure of Verbal
Ability FACTOR CODE
Mean Std. Dev. N GROUP
No Train 47.855 10.588
25 GROUP Standard
61.863 12.841 25 GROUP
Computer 24.169 11.089
25 GROUP Both
92.450 5.766 25 For entire sample
56.584 26.860
100 - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - Variable ..
STAND_Q Standard Measure of Quantitative
Ability FACTOR CODE
Mean Std. Dev. N GROUP
No Train 47.517 9.985
25 GROUP Standard
71.831 10.873 25 GROUP
Computer 32.781 9.353
25 GROUP Both
81.931 8.764 25 For entire sample
58.515 21.764
100 - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - Variable .. COMP_V
Computer Measure of Verbal Ability
FACTOR CODE Mean
Std. Dev. N GROUP No Train
45.720 10.843 25
GROUP Standard 48.774
10.277 25 GROUP Computer
53.363 10.302 25 GROUP
Both 82.434
8.784 25 For entire sample
57.573 17.723 100 Variable
.. COMP_Q Computer Measure of
Quantitative Ability FACTOR CODE
Mean Std. Dev. N
GROUP No Train 46.284
10.699 25 GROUP Standard
49.652 10.972 25 GROUP
Computer 60.613
9.005 25 GROUP Both
91.507 6.262 25 For
entire sample 62.014
20.182 100
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Univariate Homogeneity of Variance Tests
Variable .. STAND_V Standard Measure
of Verbal Ability Cochrans C(24,4)
.38062, P .099 (approx.)
Bartlett-Box F(3,16589)
4.71516, P .003 Variable .. STAND_Q
Standard Measure of Quantitative Ability
Cochrans C(24,4)
.30932, P .677 (approx.) Bartlett-Box
F(3,16589) .40322, P
.751 Variable .. COMP_V Computer
Measure of Verbal Ability Cochrans
C(24,4) .28928, P
1.000 (approx.) Bartlett-Box F(3,16589)
.37396, P .772 Variable ..
COMP_Q Computer Measure of
Quantitative Ability Cochrans C(24,4)
.33895, P .333
(approx.) Bartlett-Box F(3,16589)
2.74884, P .041
One assumption underlying ANOVA is homogeneity of
variance. Cochrans test is preferred over
Bartletts test. No real problem here.
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WITHIN CELLS Correlations with Std. Devs. on
Diagonal STAND_V STAND_Q
COMP_V COMP_Q STAND_V 10.407
STAND_Q .814 9.775 COMP_V
.598 .710 10.081 COMP_Q
.573 .659 .828 9.423
The multiple outcomes are highly related,
especially the different abilities measured by
the same method.
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EFFECT .. GROUP (Cont.) Univariate F-tests with
(3,96) D. F. Variable Hypoth. SS Error SS
Hypoth. MS Error MS F Sig. of F
STAND_V 61029.1914 10396.7654 20343.0638
108.29964 187.84055 .000 STAND_Q
37721.0301 9173.31497 12573.6767 95.55536
131.58525 .000 COMP_V 21342.3217
9755.21167 7114.10723 101.61679 70.00917
.000 COMP_Q 31801.2382 8523.41291 10600.4127
88.78555 119.39344 .000
These omnibus F tests indicate that there are
significant group differences for each of the
dependent measures. They do not indicate where
those differences exist, but there is little
doubt that difference do exist.
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EFFECT .. 1ST Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 24858.3523
10396.7654 24858.3523 108.29964 229.53310
.000 STAND_Q 14804.6423 9173.31497 14804.6423
95.55536 154.93261 .000 COMP_V
16848.6252 9755.21167 16848.6252 101.61679
165.80553 .000 COMP_Q 25563.6948
8523.41291 25563.6948 88.78555 287.92630
.000
By default, SPSS uses effects coding for the
Groups variable, which when unique sums of
squares are tested, is a test of each group
against the grand mean (except for the last
group). The first parameter is thus a test of
Group 1 against the grand mean of all groups, for
each outcome variable.
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EFFECT .. 2ND Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 1145.38103
10396.7654 1145.38103 108.29964 10.57604
.002 STAND_Q 841.89842 9173.31497 841.89842
95.55536 8.81058 .004 COMP_V
3902.96775 9755.21167 3902.96775 101.61679
38.40869 .000 COMP_Q 6172.09410
8523.41291 6172.09410 88.78555 69.51688
.000
The second parameter is a test of Group 2 against
the grand mean.
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EFFECT .. 3RD Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 35025.4580
10396.7654 35025.4580 108.29964 323.41251
.000 STAND_Q 22074.4893 9173.31497 22074.4893
95.55536 231.01256 .000 COMP_V
590.72871 9755.21167 590.72871 101.61679
5.81330 .018 COMP_Q 65.44923
8523.41291 65.44923 88.78555 .73716
.393
The third parameter is a test of Group 3 against
the grand mean. This parameter exhausts the 3
degrees of freedom for the Group effect.
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manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /contrast(group)special(1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1
1) /print cellinfo(means) parameters
signif(singledf) /power exact /design .
The grand mean across groups
In this analysis, the implicit 2 x 2 structure in
the between-subjects part of the design is
explicitly coded. This allows separating the 3
degrees of freedom associated with the Group
effect into sensible single degree of freedom
tests.
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EFFECT .. GROUP (Cont.) Univariate F-tests with
(3,96) D. F. Variable Hypoth. SS Error SS
Hypoth. MS Error MS F Sig. of F
STAND_V 61029.1914 10396.7654 20343.0638
108.29964 187.84055 .000 STAND_Q
37721.0301 9173.31497 12573.6767 95.55536
131.58525 .000 COMP_V 21342.3217
9755.21167 7114.10723 101.61679 70.00917
.000 COMP_Q 31801.2382 8523.41291 10600.4127
88.78555 119.39344 .000
The omnibus tests will not change. The total
variability accounted for by Groups is the same
no matter how we decide to partition it into
separate effects.
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EFFECT .. 1ST Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 297.60172
10396.7654 297.60172 108.29964 2.74795
.101 STAND_Q 134.29369 9173.31497 134.29369
95.55536 1.40540 .239 COMP_V
10661.8848 9755.21167 10661.8848 101.61679
104.92247 .000 COMP_Q 19728.7607
8523.41291 19728.7607 88.78555 222.20688
.000
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EFFECT .. 2ND Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 42321.2505
10396.7654 42321.2505 108.29964 390.77924
.000 STAND_Q 33731.7308 9173.31497 33731.7308
95.55536 353.00719 .000 COMP_V
6449.97000 9755.21167 6449.97000 101.61679
63.47347 .000 COMP_Q 7336.70340
8523.41291 7336.70340 88.78555 82.63398
.000
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EFFECT .. 3RD Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 18410.3391
10396.7654 18410.3391 108.29964 169.99446
.000 STAND_Q 3855.00556 9173.31497 3855.00556
95.55536 40.34316 .000 COMP_V
4230.46688 9755.21167 4230.46688 101.61679
41.63157 .000 COMP_Q 4735.77408
8523.41291 4735.77408 88.78555 53.33947
.000
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manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /contrast(group)special(1 1 1 1
-3 1 1 1
0 2 -1
-1 0 0
1 -1 ) /print cellinfo(means) parameters
signif(singledf) /power exact /design .
Any set of contrasts could be tested. This set
compares the control group to all other
conditions, the standard training condition to
the computer training conditions, and the
combined training condition to the computer only
training condition.
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EFFECT .. GROUP (Cont.) Univariate F-tests with
(3,96) D. F. Variable Hypoth. SS Error SS
Hypoth. MS Error MS F Sig. of F
STAND_V 61029.1914 10396.7654 20343.0638
108.29964 187.84055 .000 STAND_Q
37721.0301 9173.31497 12573.6767 95.55536
131.58525 .000 COMP_V 21342.3217
9755.21167 7114.10723 101.61679 70.00917
.000 COMP_Q 31801.2382 8523.41291 10600.4127
88.78555 119.39344 .000
As before, the particular method of partitioning
does not affect the overall analysis.
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EFFECT .. 1ST Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 2539.71566
10396.7654 2539.71566 108.29964 23.45082
.000 STAND_Q 4032.21838 9173.31497 4032.21838
95.55536 42.19772 .000 COMP_V
4682.80969 9755.21167 4682.80969 101.61679
46.08303 .000 COMP_Q 8247.41778
8523.41291 8247.41778 88.78555 92.89144
.000
Any training is better than no training, for each
of the outcome variables.
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EFFECT .. 2ND Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 210.44792
10396.7654 210.44792 108.29964 1.94320
.167 STAND_Q 3492.11022 9173.31497 3492.11022
95.55536 36.54541 .000 COMP_V
6095.65647 9755.21167 6095.65647 101.61679
59.98671 .000 COMP_Q 11623.0953
8523.41291 11623.0953 88.78555 130.91201
.000
Any computer training is better than no computer
training for computer administered measures.
Standard training is better for standard format
math test.
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EFFECT .. 3RD Parameter of GROUP (Cont.)
Univariate F-tests with (1,96) D. F. Variable
Hypoth. SS Error SS Hypoth. MS Error MS
F Sig. of F STAND_V 58279.0278
10396.7654 58279.0278 108.29964 538.12763
.000 STAND_Q 30196.7015 9173.31497 30196.7015
95.55536 316.01262 .000 COMP_V
10563.8555 9755.21167 10563.8555 101.61679
103.95778 .000 COMP_Q 11930.7251
8523.41291 11930.7251 88.78555 134.37688
.000
Combined training is better than computer
training only, for all measures.
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Univariate Analyses Variables Treated as a
Repeated Measure
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manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /wsfactorsoutcomes(4) /print
cellinfo(means) parameters signif(singledf)
homogeneity error(cor) transform /power
exact /design .
In this analysis, no structure is imposed on the
groups or on the repeated measures. By default,
SPSS will used effects coding for the groups and
polynomial contrasts for the repeated measures.
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Orthonormalized Transformation Matrix
(Transposed) T1 T2
T3 T4 STAND_V .500
-.671 .500 -.224 STAND_Q
.500 -.224 -.500 .671 COMP_V
.500 .224 -.500 -.671
COMP_Q .500 .671 .500
.224
The transformation imposed by SPSS by default
will create the sum of the outcome measures (for
the between-subjects part of the design) and the
linear, quadratic, and cubic effects for the
within-subjects part of the design. The latter do
not represent particularly interesting ways to
partition the within-subjects variance for this
problem, but might be important in a longitudinal
design.
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Tests of Between-Subjects Effects. Tests of
Significance for T1 using UNIQUE sums of squares
Source of Variation SS DF MS
F Sig of F WITHIN CELLS
29196.56 96 304.13 GROUP
120145.66 3 40048.55 131.68 .000
1ST Parameter 80948.33 1 80948.33
266.16 .000 2ND Parameter
5319.11 1 5319.11 17.49 .000
3RD Parameter 33878.22 1 33878.22
111.39 .000
The between-subjects analysis indicates that
average performance across the four measures is
different among the four groups. Each of the
first three groups is also significantly
different from the grand mean.
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Tests involving 'OUTCOMES' Within-Subject
Effect. Mauchly sphericity test, W
.58337 Chi-square approx. 51.04857
with 5 D. F. Significance
.000 Greenhouse-Geisser Epsilon .71922
Huynh-Feldt Epsilon .75931
Lower-bound Epsilon
.33333 AVERAGED Tests of Significance that
follow multivariate tests are equivalent
to univariate or split-plot or mixed-model
approach to repeated measures. Epsilons may be
used to adjust d.f. for the AVERAGED results.
When the sphericity assumption is violated,
multiple degree of freedom tests in the
within-subject design can be biased. One
correction is to adjust the degrees of freedom
using the epsilon values.
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Tests involving 'OUTCOMES' Within-Subject
Effect. AVERAGED Tests of Significance for
MEAS.1 using UNIQUE sums of squares Source of
Variation SS DF MS F
Sig of F WITHIN CELLS 8652.14
288 30.04 OUTCOMES 1675.96
3 558.65 18.60 .000 GROUP BY
OUTCOMES 31748.12 9 3527.57
117.42 .000 1ST Parameter 1126.98
3 375.66 12.50 .000 2ND
Parameter 6743.24 3 2247.75
74.82 .000 3RD Parameter 23877.91
3 7959.30 264.94 .000
The overall Outcomes test indicates that there
are significant differences among the outcome
measures when averaged across the groups. The
Group x Outcome interactions indicate that the
pattern of these differences changes across
groups. The parameter tests further indicate that
the outcome patterns are different when the first
three groups are each compared to the grand mean.
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Estimates for T2 --- Individual univariate
.9500 confidence intervals --- two-tailed
observed power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
3.43171375 .72394 4.74032 .00001
1.99470 4.86873 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
-4.8873338 1.25390 -3.89769 .00018
-7.37631 -2.39835 3 -16.778771
1.25390 -13.38122 .00000 -19.26775
-14.28979 4 25.6179424 1.25390
20.43054 .00000 23.12896 28.10692
The tests for T2 are tests for the Linear effect.
The parameter effects for Groups are the
defaultseach group but the last compared to the
grand mean.
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Overall linear effectacross groups
Estimates for T2 --- Individual univariate
.9500 confidence intervals --- two-tailed
observed power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
3.43171375 .72394 4.74032 .00001
1.99470 4.86873 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
-4.8873338 1.25390 -3.89769 .00018
-7.37631 -2.39835 3 -16.778771
1.25390 -13.38122 .00000 -19.26775
-14.28979 4 25.6179424 1.25390
20.43054 .00000 23.12896 28.10692
Group comparisons for linear effect.
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Estimates for T3 --- Individual univariate .9500
confidence intervals --- two-tailed observed
power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
1.25497490 .43736 2.86946 .00506
.38683 2.12312 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
-.80367829 .75752 -1.06093 .29138
-2.30735 .69999 3 -5.8005573
.75752 -7.65728 .00000 -7.30423
-4.29689 4 -1.9365362 .75752
-2.55641 .01214 -3.44021 -.43287
The tests for T3 test for the quadratic effect.
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Estimates for T4 --- Individual univariate .9500
confidence intervals --- two-tailed observed
power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
1.84606749 .43115 4.28170 .00004
.99024 2.70190 GROUP BY OUTCOMES Parameter
Coeff. Std. Err. t-Value Sig. t Lower
-95 CL- Upper 2 -.99248228
.74678 -1.32902 .18699 -2.47483
.48986 3 10.8905171 .74678
14.58332 .00000 9.40817 12.37286
4 -7.5038674 .74678 -10.04831 .00000
-8.98621 -6.02152
The tests for T4 examine the cubic effect.
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manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /wsfactorsoutcomes(4) /contrast(outcom
es)special(1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1
1) /renameaverage format content
interact /contrast(group)special(1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1
1) /print cellinfo(means) parameters
signif(singledf) homogeneity error(cor)
transform /power exact /design .
M
L
This design partitions the data into the implicit
2 x 2 x 2 x 2 factorial design.
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Orthonormalized Transformation Matrix
(Transposed) AVERAGE FORMAT
CONTENT INTERACT STAND_V .500
.500 .500 .500 STAND_Q .500
.500 -.500 -.500 COMP_V
.500 -.500 .500 -.500 COMP_Q
.500 -.500 -.500 .500
SPSS automatically normalizes the transformation
matrix (M) for the dependent variables.
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Tests of Between-Subjects Effects. Tests of
Significance for AVERAGE using UNIQUE sums of
squares Source of Variation SS DF
MS F Sig of F WITHIN CELLS
29196.56 96 304.13 GROUP
120145.66 3 40048.55 131.68
.000 1ST Parameter 15547.36 1
15547.36 51.12 .000 2ND Parameter
77103.28 1 77103.28 253.52
.000 3RD Parameter 27495.01 1
27495.01 90.41 .000
The overall between-subjects test for Group will
duplicate the previous analysis, but the
parameter tests now reflect the specified
contraststhe main effect of computer training,
the main effect of standard training, and the
interaction.
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Tests involving 'OUTCOMES' Within-Subject
Effect. AVERAGED Tests of Significance for
MEAS.1 using UNIQUE sums of squares Source of
Variation SS DF MS F
Sig of F WITHIN CELLS 8652.14
288 30.04 OUTCOMES 1675.96
3 558.65 18.60 .000 GROUP BY
OUTCOMES 31748.12 9 3527.57
117.42 .000 1ST Parameter 15275.18
3 5091.73 169.49 .000 2ND
Parameter 12736.37 3 4245.46
141.32 .000 3RD Parameter 3736.57
3 1245.52 41.46 .000
The overall test for Outcomes does not change,
but the parameter tests now reflect how the
outcome patterns change as a function of the 2 x
2 between-subjects design.
109
Estimates for FORMAT --- Individual univariate
.9500 confidence intervals --- two-tailed
observed power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
-2.2438316 .73827 -3.03929 .00306
-3.70930 -.77837 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
47.6105778 2.95310 16.12224 .00000
41.74872 53.47243 3 -44.683406
2.95310 -15.13102 .00000 -50.54526
-38.82155 4 12.7828792 2.95310
4.32863 .00004 6.92102 18.64474
The within-subjects design was partitioned into a
2 x 2 design as well. These are tests for the
Format variable.
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Estimates for CONTENT --- Individual univariate
.9500 confidence intervals --- two-tailed
observed power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
-3.1858820 .40612 -7.84464 .00000
-3.99203 -2.37974 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
1.67261026 1.62449 1.02962 .30577
-1.55197 4.89719 3 -3.3432877
1.62449 -2.05806 .04229 -6.56787
-.11870 4 13.9642185 1.62449
8.59607 .00000 10.73963 17.18880
The within-subjects design was partitioned into a
2 x 2 design as well. These are tests for the
Content variable.
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Estimates for INTERACT --- Individual univariate
.9500 confidence intervals --- two-tailed
observed power taken at .0500 level OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 1
1.25497490 .43736 2.86946 .00506
.38683 2.12312 GROUP BY OUTCOMES
Parameter Coeff. Std. Err. t-Value
Sig. t Lower -95 CL- Upper 2
-13.208471 1.74942 -7.55019 .00000
-16.68105 -9.73589 3 -5.4804290
1.74942 -3.13271 .00230 -8.95301
-2.00785 4 15.4741870 1.74942
8.84531 .00000 12.00161 18.94676
The within-subjects design was partitioned into a
2 x 2 design as well. These are tests for the
interaction of Content and Format.
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115
Multivariate Analyses Variables Treated as Linear
Combinations that Maximize Group Separation
116
Multivariate analysis of variance can be thought
of as addressing the question of whether any
linear combination among dependent variables can
produce a significant separation of groups. In
this sense it is similar to canonical correlation
analysis in that the linear combination of
variables that produces the biggest difference
between groups is formed, and if possible,
subsequent linear combinations are formed that
are independent of the first and that also
produce the largest group separation possible.
117
The significance of these linear combinations can
be gauged in several ways. Four common tests of
significance represent generalizations of the
univariate approach to significance testing. In
the univariate model, an F test gauges the amount
of between-groups variability to within-groups
variability.
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119
In the multivariate model, the single sums of
squares are replaced by matrices of sums of
squares and cross-products (H and E). Different
combinations of these matrices define the four
most common tests of significance Roys test is a
function of the largest eigenvalue of HE-1 Lawley
and Hotellings Trace (T) tr(HE-1) Pillais
test is a function of (V) tr(H(HE)-1) Wilks
likelihood test is based on L E/HE The
latter three usually agree because they use much
the same information.
120
The tests of significance can also be defined in
terms of the eigenvalues (and canonical
correlations) that corresponds to the linear
combinations that are formed. li the ith
eigenvalue of HE-1 li ri2/(1-ri2) Roys test
is a l1/(1l1) Lawley and Hotellings Trace (T)
S li Pillais test is V S li/(1li) Wilks L
P li/(1li)
121
manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /print cellinfo(means) parameters
signif(singledf multiv dimenr eigen univ hypoth)
homogeneity error(cor sscp) transform /discrim
stan corr alpha(1) /power exact /design .
One multivariate approach to these data attempts
to find the linear combinations of the four
outcome variables that best separate the groups,
with no structure imposed on the groups. This
would be the most exploratory version.
122
Pooled within-cells Variance-Covariance matrix
STAND_V STAND_Q COMP_V
COMP_Q STAND_V 108.300 STAND_Q
82.840 95.555 COMP_V 62.760
69.970 101.617 COMP_Q 56.160
60.735 78.667 88.786
Multivariate test for Homogeneity of Dispersion
matrices Boxs M
100.94212 F WITH (30,25338) DF
3.10957, P .000 (Approx.) Chi-Square with 30
DF 93.40651, P .000 (Approx.)
This is an assumption underlying MANOVA.
123
WITHIN CELLS Sum-of-Squares and Cross-Products
STAND_V STAND_Q COMP_V
COMP_Q STAND_V 10396.765 STAND_Q
7952.665 9173.315 COMP_V 6025.008
6717.116 9755.212 COMP_Q 5391.382
5830.579 7552.022 8523.413 - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - - A n a l y s i s o f V
a r i a n c e -- design 1 EFFECT
.. GROUP Adjusted Hypothesis Sum-of-Squares and
Cross-Products STAND_V STAND_Q
COMP_V COMP_Q STAND_V 61029.191
STAND_Q 46007.755 37721.030 COMP_V
27128.328 17591.995 21342.322 COMP_Q
29381.455 18376.530 25858.361 31801.238
E
H
These are the two major matrices that are used in
the calculation of the significance tests.
124
EFFECT .. GROUP Multivariate Tests of
Significance (S 3, M 0, N 45 1/2) Test
Name Value Approx. F Hypoth. DF Error
DF Sig. of F Pillais 2.06382
52.35748 12.00 285.00 .000
Hotellings 12.07879 92.26856 12.00
275.00 .000 Wilks .01408
82.31218 12.00 246.35 .000 Roys
.86325
As in canonical correlation analysis, this
overall test simply indicates whether there are
any linear combinations of the outcome variables
that can discriminate the groups significantly.
It does not indicate how many linear combinations
there are. The rationale for using this omnibus
test as a Type I error protection approach is
that included among the possible linear
combinations are those in which each outcome
variable is the only variable receiving a weight.
125
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 6.313 52.263
52.263 .929 2 5.199
43.042 95.305 .916 3
.567 4.695 100.000 .602 - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - Dimension Reduction Analysis
Roots Wilks L. F Hypoth. DF
Error DF Sig. of F 1 TO 3 .01408
82.31218 12.00 246.35 .000 2 TO 3
.10294 66.32659 6.00 188.00
.000 3 TO 3 .63811 26.93858
2.00 95.00 .000
With four groups and four variables there are
three possible linear combinations that could be
made (limited by the degrees of freedom for
groups). All three are providing significant and
independent separation of the groups.
126
EFFECT .. GROUP (Cont.) Standardized
discriminant function coefficients
Function No. Variable 1 2
3 STAND_V .804 .713
1.328 STAND_Q .589 -1.219
-1.429 COMP_V -.477 .121
.598 COMP_Q -.188 1.070
-.852 A n a l y s i s o f V a r
i a n c e -- design 1 EFFECT ..
GROUP (Cont.) Correlations between DEPENDENT and
canonical variables Canonical
Variable Variable 1 2
3 STAND_V .891 .405
.035 STAND_Q .782 .153
-.484 COMP_V .267 .568
-.327 COMP_Q .266 .775 -.538
The canonical variates and loadings are used in
the same way here as they were in canonical
correlation analysis. What are these linear
combinations?
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131
EFFECT .. 1ST Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.78886 86.86668 4.00 93.00
.000 Hotellings 3.73620 86.86668
4.00 93.00 .000 Wilks
.21114 86.86668 4.00 93.00
.000 Roys .78886 Note.. F
statistics are exact.
The default group parameters are effects codes,
indicating the extent to which groups are
different from the grand mean. This more refined
test indicates whether any linear combinations of
the outcome variables can discriminate the first
group from the grand mean.
132
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 3.736 100.000
100.000 .888
Because this is inherently the comparison of two
groups, there is only one way the
discrimination can be made.
133
EFFECT .. 1ST Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V -.701 STAND_Q .338
COMP_V .298 COMP_Q -.964
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 1ST
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V -.800 STAND_Q -.657
COMP_V -.680 COMP_Q -.896
Just a single linear combination can be formed to
make the discrimination.
134
EFFECT .. 2ND Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.74710 68.68199 4.00 93.00
.000 Hotellings 2.95406 68.68199
4.00 93.00 .000 Wilks
.25290 68.68199 4.00 93.00
.000 Roys .74710 Note.. F
statistics are exact.
A similar test can be made for discriminating the
second group from the grand mean.
135
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 2.954 100.000
100.000 .864
Here too a single linear combination is possible.
136
EFFECT .. 2ND Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V -.894 STAND_Q 1.696
COMP_V -.399 COMP_Q -.771
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 2ND
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V -.193 STAND_Q .176
COMP_V -.368 COMP_Q -.495
137
EFFECT .. 3RD Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.84347 125.28328 4.00 93.00
.000 Hotellings 5.38853 125.28328
4.00 93.00 .000 Wilks
.15653 125.28328 4.00 93.00
.000 Roys .84347 Note.. F
statistics are exact.
The last group parameter is a test of the third
group against the grand mean. Significant
discrimination is possible here too.
138
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 5.389 100.000
100.000 .918
A single linear combination is possible.
139
EFFECT .. 3RD Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V .810 STAND_Q .632
COMP_V -.411 COMP_Q -.502
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 3RD
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V .791 STAND_Q .668
COMP_V .106 COMP_Q .038
140
manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /contrast(group)special(1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1
1) /print cellinfo(means) parameters
signif(singledf multiv dimenr eigen univ hypoth)
homogeneity error(cor sscp) transform /discrim
stan corr alpha(1) /power exact /design .
A potentially more revealing analysis would
specify the 2 x 2 structure for the Groups
variable. Then the linear combinations that are
sought would be directed toward making those
specified distinctions.
141
EFFECT .. GROUP Multivariate Tests of
Significance (S 3, M 0, N 45 1/2) Test
Name Value Approx. F Hypoth. DF Error
DF Sig. of F Pillais 2.06382
52.35748 12.00 285.00 .000
Hotellings 12.07879 92.26856 12.00
275.00 .000 Wilks .01408
82.31218 12.00 246.35 .000 Roys
.86325
As with the univariate analyses, the omnibus test
for the multivariate analysis does not change. It
simply gauges if any discrimination is possible.
142
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 6.313 52.263
52.263 .929 2 5.199
43.042 95.305 .916 3
.567 4.695 100.000 .602 - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - Dimension Reduction Analysis
Roots Wilks L. F Hypoth. DF
Error DF Sig. of F 1 TO 3 .01408
82.31218 12.00 246.35 .000 2 TO 3
.10294 66.32659 6.00 188.00
.000 3 TO 3 .63811 26.93858
2.00 95.00 .000
These are the same as well. They are the number
of possible linear combinations that could be
extracted.
143
EFFECT .. GROUP (Cont.) Standardized
discriminant function coefficients
Function No. Variable 1 2
3 STAND_V .804 .713
1.328 STAND_Q .589 -1.219
-1.429 COMP_V -.477 .121
.598 COMP_Q -.188 1.070
-.852 A n a l y s i s o f V a r
i a n c e -- design 1 EFFECT ..
GROUP (Cont.) Correlations between DEPENDENT and
canonical variables Canonical
Variable Variable 1 2
3 STAND_V .891 .405
.035 STAND_Q .782 .153
-.484 COMP_V .267 .568
-.327 COMP_Q .266 .775
-.538
144
EFFECT .. 1ST Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.82398 108.83866 4.00 93.00
.000 Hotellings 4.68123 108.83866
4.00 93.00 .000 Wilks
.17602 108.83866 4.00 93.00
.000 Roys .82398 Note.. F
statistics are exact.
Now the first parameter reflects the structure
imposed on the Groups variable. This tests
whether it is possible to form a linear
combination of the outcome variables that
separates the average of the computer-trained
groups from the average of the groups that did
not receive any computer training.
145
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 4.681 100.000
100.000 .908
EFFECT .. 1ST Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V -.231 STAND_Q 1.155
COMP_V -.196 COMP_Q -1.170
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 1ST
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V -.078 STAND_Q .056
COMP_V -.483 COMP_Q -.703
146
EFFECT .. 2ND Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.83243 115.49713 4.00 93.00
.000 Hotellings 4.96762 115.49713
4.00 93.00 .000 Wilks
.16757 115.49713 4.00 93.00
.000 Roys .83243 Note.. F
statistics are exact.
This tests whether it is possible to form a
linear combination that separates those who
received standard training from those who did not
receive standard training.
147
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 4.968 100.000
100.000 .912
EFFECT .. 2ND Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V .635 STAND_Q .707
COMP_V -.556 COMP_Q .047
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 2ND
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V .905 STAND_Q .860
COMP_V .365 COMP_Q .416
148
EFFECT .. 3RD Parameter of GROUP Multivariate
Tests of Significance (S 1, M 1 , N 45
1/2) Test Name Value Exact F Hypoth.
DF Error DF Sig. of F Pillais
.70845 56.49617 4.00 93.00
.000 Hotellings 2.42994 56.49617
4.00 93.00 .000 Wilks
.29155 56.49617 4.00 93.00
.000 Roys .70845 Note.. F
statistics are exact.
The remaining parameter is the interaction. It
can be thought of as test of the No Training and
Complete Training groups compared to the groups
that received just one kind of training.
149
Eigenvalues and Canonical Correlations Root No.
Eigenvalue Pct. Cum. Pct. Canon
Cor. 1 2.430 100.000
100.000 .842
EFFECT .. 3RD Parameter of GROUP (Cont.)
Standardized discriminant function coefficients
Function No. Variable 1
STAND_V -1.501 STAND_Q .983
COMP_V -.005 COMP_Q -.263
A n a l y s i s o f V a r i a n c e
-- design 1 EFFECT .. 3RD
Parameter of GROUP (Cont.) Correlations between
DEPENDENT and canonical variables
Canonical Variable Variable 1
STAND_V -.854 STAND_Q -.416
COMP_V -.422 COMP_Q -.478
150
Full understanding of the matrix form for the
ANOVA model allows complete control over the
hypotheses tested. This includes linear
combinations of groups (L), and, linear
combinations of variables (M).