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 Single DV Multiple DVs Single IV 2 groups Hotelling’s T2 t test IV Single IV > 2 groups One-way ANOVA MANOVA Factorial ANOVA Multiple IVs MANOVA

MANOVA vs. Discriminant Analysis 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 Box’s M test Population covariance matrices equal (homogeneity) Box’s 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%) Doesn’t 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) Pillai’s 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) Roy’s Greatest Root (q) Only focuses on first discriminant function (largest l) Works best when there’s 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 Ability Effort Ease of Job Improving Maintaining Declining CAUSE   Ability Effort Ease of Job Improving Maintaining Declining 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 /DESIGN 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 ) /METHOD=UNIQUE /ERROR WITHIN+RESIDUAL /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 Meaning: If there is improvement, maintenance or decline in their performance influences the expectations for future performance.

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 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 Cont’d… 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.

Standard Quantitative Computer Verbal Computer Quantitative Standard Verbal Standard Quantitative Computer Verbal Computer Quantitative No Training Standard Training Computer Training Standard and Computer Training

Each Variable Examined Separately 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 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 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 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

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. Cochran’s test is preferred over Bartlett’s 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.

The second parameter is a test of Group 2 against the grand mean. 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.

This is an assumption underlying MANOVA. 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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?

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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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.

Here too a single linear combination is possible. 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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.

A single linear combination is possible. 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable 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 * * * * * * Correlations between DEPENDENT and canonical variables Canonical Variable STAND_V -.854 STAND_Q -.416 COMP_V -.422 COMP_Q -.478