MANOVA

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MANOVA
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MANOVA

• 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).
• It is strongly related to discriminant function
  analysis
• MANOVA DA are strongly related but
  conceptually distinct ? Math is virtually
  identical -- but direction of prediction or
  understanding is switched

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Categorical IV, Continuous DV
DV
Single DV
Multiple DVs
Hotellings T2
Single IV 2 groups
t test
IV
Single IV gt 2 groups
One-way ANOVA
MANOVA
Factorial ANOVA
Multiple IVs
MANOVA
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MANOVA vs. Discriminant Analysis

• MANOVA
• Similar to ANOVA but deals with multiple
  dependent variables at the same time
• Can deal with multiple factors (e.g, A, B, AXB
  design)
• Hypothesis testing procedure
• Discriminant Analysis
• Uses multiple variables to identify group
  membership in categories
• Used for single categorical grouping variable
• Identifies dimensionality among groups

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Assumptions of MANOVA

• DVs are multivariate normal
• Robust against modest violation
• Lack of normality reflected in failing Boxs M
  test
• Population covariance matrices equal
  (homogeneity)? Boxs M test
• Robust to modest violation if groups are of equal
  size
• Linear relationships
• Multicollinearity between DVs should not be too
  high,
• Observations independent (no correlated error)
• Sensitive to outliers

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Violation of Assumptions

• If you violate assumptions of homogeneity of
  covariance matrices you can
• Discard outliers
• Discard groups
• Combine groups
• Drop a DV or combination of DVs
• Transform DVs

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Why use MANOVA?

• Multiple DVs -- how to analyze this?
• Problems with multiple ANOVAs
• Inflated Type I error rate (e.g., 5 DVs, a .05,
  Type I error rate 23)
• Doesnt take into account intercorrelation among
  DVs
• MANOVA is a simultaneous test of an ANOVA with
  multiple DVs
• Reduces Type I error rates
• Takes into account intercorrelation among DVs
  (optimal linear combinations of DVs)
• Nonsignificant results for many DVs may become
  significant when combined
• Multivariate DVs may be conceptually meaningful

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ANOVA Review

• ANOVA
• Ho m1 m2 ... mk
• Tested by SSbtwn / SSwithin
• SStotal SSbtwn SSwithin

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MANOVA
Tested by MAX B / Werror T B W
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MANOVA

• Creates linear combinations of DVs that optimally
  discriminate among groups
• Goal maximizes discrimination among groups
• Each linear combination is orthogonal
• Number of linear combinations extracted for each
  hypothesis test is equal to df for hypothesis or
  number of DVs, whichever is smaller (different
  numbers of linear combinations for different
  hypothesis tests)

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Overall MV Significance Tests

• Each MV test provides an approximate F test for a
  particular effect on all of the DVs taken
  together Tests made of different combinations of
  matrices, but often yield same result
• Wilks Lambda (L)
• Depends on multiplication of lis (differences
  across various dimensions)
• Pillais Trace (V)
• Depends on summation of lis (differences across
  various dimensions)
• Most robust against violations of MV normality
  and homogeneity of covariance matrices
• More robust when sample size low or unequal cell
  sizes appear

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Overall MV Significance Tests

• Hotelling-Lawley Trace (T)
• Depends on summation of l (differences across
  various dimensions)
• Roys Greatest Root (q)
• Only focuses on first discriminant function
  (largest l)
• Works best when theres only one underlying
  component or factor
• When these conditions are met, the most powerful
  statistic

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MANOVA Interpretation

• An overall MV significant effect suggests that
  the groups are significantly different on one or
  more linear combinations of the DVs
• Follow-up Tests
• Univariate ANOVAs performed only if MANOVA
  significant (protected univariate F test)
• Ignores intercorrelations
• Completely partialled F tests (residuals of the
  DVs)
• Bonferroni adjusted univariate ANOVAs performed
  to test specific Hs regardless of whether overall
  MANOVA significant

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MANOVA Interpretation

• Can use discriminant weights to interpret
• Like b weights in regression
• Susceptible to same problems as b weights
  (intercorrelation, cross-validation)
• Can use discriminant loadings to interpret
  results
• AKA structure coefficients or canonical variate
  correlations
• Reporting MANOVA
• Describe MV test statistic used
• Approximate F test and df
• Effect size

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MANOVA

• Assume you have high performing employees that
  exhibit different trends of performance
  (improving, maintaining, declining) that are due
  to different causes (ability, effort, ease of
  job)
• You have four DVs
• Pay (change in pay)
• Promotion (likelihood to promote)
• Expect (expected future performance)
• Affect (your feelings toward the employee)
• Design is 3 (trend) by 3 (cause) ANOVA with four
  DVs

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MANOVA
CAUSE
TREND
Four DVs (1) Pay (change in pay) (2) Promotion
(likelihood to promote) (3) Expect (expected
future performance) (4) Affect (your feelings
toward the employee)
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MANOVA SPSS Commands
MANOVA pay promote expect affect BY inform(1 3)
trend(1 3) /DISCRIM RAW STAN ESTIM CORR
ROTATE(VARIMAX) ALPHA(1) /PRINT SIGNIF(MULT
UNIV EIGN DIMENR) SIGNIF(EFSIZE)
HOMOGENEITY(BARTLETT COCHRAN BOXM) /NOPRINT
PARAM(ESTIM) /POWER T(.05) F(.05) /OMEANS
TABLES( inform trend ) /PMEANS TABLES( inform
trend ) /METHODUNIQUE /ERROR
WITHINRESIDUAL /DESIGN
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Results - Trend
EFFECT .. TREND Multivariate Tests of
Significance (S 2, M 1/2, N 372 1/2) Test
Name Value Appr. F Hyp. DF Err DF
Sig. of F Pillais .067 6.52
8.00 1496.00 .000 Hotellings
.071 6.60 8.00 1492.00
.000 Wilks .933 6.56
8.00 1494.00 .000 Roys
.055 Multivariate Effect Size and Observed Power
at .0500 Level TEST NAME Effect Size
Noncent. Power Pillais .034
52.175 1.00 Hotellings
.034 52.765 1.00
Wilks .034 52.470
1.00
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Results - Trend
Univariate F-tests with (2,750) D. F. Variable
Hyp. SS Err SS Hyp. MS Err MS F
Sig. of F PAY 24.34 28316.6
12.17 37.76 .322 .725 PROMOTE
40.94 12750.9 20.47 17.00 1.204
.301 EXPECT 270.76 9526.0 135.38
12.70 10.659 .000 AFFECT 79.17
12532.4 39.59 16.71 2.369
.094 EFFECT .. TREND (Cont.) Univariate F-tests
with (2,750) D. F. (Cont.) Variable ETA
Square Noncent. Power PAY
.00086 .64475 .10554
PROMOTE .00320 2.40782
.26224 EXPECT .02764 21.31717
.98943 AFFECT .00628
4.73821 .47909
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Results - Trend
VARIMAX rotated correlations between canonical
and DVs Can. Var.
DEP. VAR. 1 2 PAY
.048 -.139 PROMOTE .318
.157 EXPECT .824 .076 AFFECT
.140 .587
These are the loadings. What are variate 1 and
variate 2 comprised of?
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Results
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DF2
0,.20
X Maintaining
X Declining
0,0
DF1
-.20,0
.20,0
X Improving
DF1 Expect,Promote DF2 Affect ( Pay
disappears due to its intercorrelations w/ other
DVs)
0,-.20
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Interaction Results

• A n a l y s i s o f V a r i a n c
  e -- design 1
• EFFECT .. CAUSE BY TREND
• Multivariate Tests of Significance (S 4, M
  -1/2, N 372 1/2)
• Test Name Value Appr. F Hyp. DF Error DF
  Sig. of F
• Pillais .58748 32.28 16.00
  3000.00 .000
• Hotellings 1.06873 49.80 16.00
  2982.00 .000
• Wilks .46039 41.24 16.00
  2282.76 .000
• Roys .49049
• Multivariate Effect Size and Observed Power at
  .0500 Level
• TEST NAME Effect Size Noncent. Power
• Pillais .147 516.462
  1.00
• Hotellings .211 796.738
  1.00
• Wilks .176 488.515
  1.00

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Interaction Results

• EFFECT .. INFORM BY TREND (Cont.)
• Univariate F-tests with (4,750) D. F.
• Variable Hyp. SS Err SS Hyp. MS Err MS
  F Sig. of F
• PAY 15569.8 28316.6 3892.4
  37.76 103.10 .000
• PROMOT 8819.4 12750.9 2204.9 17.00
  129.69 .000
• EXPECT 3400.2 9526.0 850.0 12.70
  66.93 .000
• AFFECT 9053.6 12532.4 2263.4 16.71
  135.45 .000
• EFFECT .. INFORM BY TREND (Cont.)
• Univariate F-tests with (4,750) D. F. (Cont.)
• Variable ETA Square Noncent. Power
• PAY .355 412.39
  1.00000
• PROMOTE .409 518.75
  1.00000
• EXPECT .263 267.70
  1.00000
• AFFECT .419 541.82
  1.00000

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Interaction Results
VARIMAX rotated correlations between canonical
and DEPENDENT variables Can. Var.
DEP. VAR. 1 2 3
4 PAY .168 .263
.890 .331 PROMOTE .957
.184 .146 .171 EXPECT
.207 .912 .250 .250 AFFECT
.213 .280 .357 .865
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Results
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Results
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MANOVA Problems

• No guarantee that the linear combinations of DVs
  will make sense
• Rotation of discriminant function can help
• Significance tests on each of DVs can yield
  conflicting results when compared to the overall
  MV significance test
• Capitalization on chance
• Cross-validation is crucial
• Intercorrelation creates problems with
  discriminant weights and their interpretation
• Washing out effect (including many nonsig DVs
  with only a few signif DVs)
• Low power
• Power generally declines as number of DVs
  increases

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Example
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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Example Contd
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
SYNTAX manova stand_v, stand_q, comp_v, comp_q by
group(1,4) /print cellinfo(all) parameters
signif(singledf) homogeneity error /power
exact /design .
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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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Multivariate Analyses Variables Treated as Linear
Combinations that Maximize Group Separation
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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.
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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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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.
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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.
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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.
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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.
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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
Weights
Loadings
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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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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 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.
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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.
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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.
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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
66
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.
67
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
68
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.
69
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
70
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.
71
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