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MANOVA

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

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

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

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

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

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

ANOVA Review

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

MANOVA

Tested by MAX B / Werror T B W

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)

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

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

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

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

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

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)

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

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

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

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?

Results

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

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

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

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

Results

Results

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

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

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 .

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.

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.

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.

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.

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.

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.

Multivariate Analyses Variables Treated as Linear

Combinations that Maximize Group Separation

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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

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.

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.

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.

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

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.

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

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.

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

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.

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