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Chapter 8: MANOVA

Chapter 8: MANOVA. Hotelling T 2. Q = (X-m)'S -1 (X-m )        T 2 = n Q = n (X-m)' S -1 (X-m ) . One DV and two DVs. Maximizing the difference between two groups.

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Chapter 8: MANOVA

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  1. Chapter 8: MANOVA

  2. Hotelling T2 • Q = (X-m)'S-1(X-m )  •       T2 = nQ = n (X-m)'S-1(X-m ) 

  3. One DV and two DVs

  4. Maximizing the difference between two groups formation of linear combinations of y1 and y2 in order to (a) correct for redundancy and (b) maximally discriminate amongst the groups. The discriminant weights of this composite (d1 and d2) are mathematically derived such that if we were to calculate a composite score for each participant, then the weights yield composite scores that will give the maximum F Ratio value. This minimizes within-group variance and maximizes between-group variance. The new composite dimension can be plotted against y1 and y2 to examine the univariate frequency distributions of the groups were overlap is minimized

  5. Multivariate Analysis of Variance (MANOVA) In MANOVA, there are in general g groups of observations, of sizes of n1, n2,…, ng , and p variables X1, X2,…, Xp  that describe N observations.  It is useful to express the variables as deviations, x’s, from the grand mean or centroid (over all groups).  The vector of observations of the p variables, for the ith observation in the kth group is Xkj and these values can be decomposed into two components: where (mk-m) is the deviation between the centroid of the kth group and the grand centroid, and (xki-mk)  is the deviation between the ith observation in the kth group and the centroid for that group.  The first term here could be thought of as analogous to the systematic component of some data, while the second term can be though of as the irregular or unpredictable component.

  6. Calculation As in univariate analysis of variance, the total sum of squares of the dependent variables (the x’s) can be decomposed into two parts: Each of the individual terms is a matrix, e.g. T is the “total” sums-of-squares matrix,:

  7. Sum of Squares is the “among-groups” sum-of-squares matrix, and is the “within-groups” sum-of squares matrix, and so T=A+W is a statistic that can be used to test the null hypothesis that the individual group centroids (the mk’s ) are all equal is Wilk’s Lambda,

  8. Wilk’s Lambda Λ=|W|/|T| Where |W| and |T| are the determinants of the within-group and total sums-of-squares matrices, respectively.  As the within-groups sums-of-squares gets smaller relative to the total sums-of-squares, the value of Λ decreases, which in practice also signals a decrease in the P-value of Λ. In other words, as Λ decreases we should be more inclined to reject the null hypothesis that the individual group centroids (the mk’s ) are all equal.

  9. Wilk’s Lambda can be converted into an F statistic using Rao’s approximation: As Λ gets smaller, F gets larger.

  10. SAS One Way MANOVA • Example: A researcher randomly assigns 33 subjects to one of three groups. The first group receives technical dietary information interactively from an on-line website. Group 2 receives the same information in from a nurse practitioner, while group 3 receives the information from a video tape made by the same nurse practitioner. The researcher looks at three different ratings of the presentation, difficulty, useful and importance, to determine if there is a difference in the modes of presentation. In particular, the researcher is interested in whether the interactive website is superior because that is the most cost-effective way of delivering the information: 3 DV, one IV

  11. SAS Program: get basic information about DV proc means n mean std min max data="c:\data\manova"; var useful difficulty importance; run; The MEANS Procedure Variable N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ USEFUL 33 16.3303030 3.2924615 11.8999996 24.2999992 DIFFICULTY 33 5.7151515 2.0175978 2.4000001 10.2500000 IMPORTANCE 33 6.4757576 3.9851309 0.2000000 18.7999992 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

  12. Information about I. V proc freq data="c:\data\manova"; table group; run; The FREQ Procedure Cumulative Cumulative GROUP Frequency Percent Frequency Percent 1 11 33.33 11 33.33 2 11 33.33 22 66.67 3 11 33.33 33 100.00

  13. Information of DV by IV proc means n mean std min max data="c:\data\manova"; class group; var useful difficulty importance; run; The MEANS Procedure N GROUP Obs Variable N Mean Std Dev Minimum Maximum 1 11 USEFUL 11 18.1181817 3.9037974 13.0000000 24.2999992 DIFFICULTY 11 6.1909091 1.8997129 3.7500000 10.2500000 IMPORTANCE 11 8.6818181 4.8630890 3.3000000 18.7999992 2 11 USEFUL 11 15.5272729 2.0756162 12.8000002 19.7000008 DIFFICULTY 11 5.5818183 2.4342631 2.4000001 9.8500004 IMPORTANCE 11 5.1090909 2.5311873 0.2000000 8.5000000 3 11 USEFUL 11 15.3454545 3.1382682 11.8999996 19.7999992 DIFFICULTY 11 5.3727273 1.7590287 2.6500001 8.7500000 IMPORTANCE 11 5.6363637 3.5469065 0.7000000 10.3000002

  14. Dependency of DV proc corr data="c:\data\manova"; var useful difficulty importance; run; The CORR Procedure 3 Variables: USEFUL DIFFICULTY IMPORTANCE Pearson Correlation Coefficients, N = 33 Prob > |r| under H0: Rho=0 USEFUL DIFFICULTY IMPORTANCE USEFUL 1.00000 0.09783 -0.34112 0.5881 0.0520 DIFFICULTY 0.09783 1.00000 0.19782 0.5881 0.2698 IMPORTANCE -0.34112 0.19782 1.00000 0.0520 0.2698

  15. SAS One-way Manova We will use the SAS proc glm procedure to do the one-way manova. proc glm data="c:\data\manova"; class group; model useful difficulty importance = group / ss3; contrast '1 vs 2&3' group 2 -1 -1; contrast '2 vs 3' group 0 1 -1; manova h=_all_; manova h=group m=(1 0 1); run; quit; Type I SS (sum of squares) measures incremental sums of squares for the model as each variable is added. Type II SS is the sum of squares for a balanced test of each effect, adjusted for every other effect. Type III (orthogonal) :reduction in error SS due to adding the term after all other terms have been added to the model Type IV (balanced) :variation explained by balanced comparison of averages of cell means .

  16. SS I-IV Source Type I SS Type II SS Type III or IV SS A SS(A|u) SS(A|u,B) SS(A|B,AB) B SS(B|u,A) SS(B|u,A) SS(B|A,AB) A*B SS(A*B|u,A,B) SS(A*B|u,A,B) SS(AB|A,B)

  17. Results of Manova Although this is a multivariate analysis, we will begin with the output for the separate univariate anovas to get a feel for what is happening with the data The GLM Procedure Class Level Information Class Levels Values GROUP 3 1 2 3 Number of Observations Read 33 Number of Observations Used 33

  18. For USEFUL Dependent Variable: USEFUL Sum of Source DF Squares Mean Square F Value Pr > F Model 2 52.9242378 26.4621189 2.70 0.0835 Error 30 293.9654425 9.7988481 Corrected Total 32 346.8896803 R-Square Coeff Var Root MSE USEFUL Mean 0.152568 19.16873 3.130311 16.33030 Source DF Type III SS Mean Square F Value Pr > F GROUP 2 52.92423783 26.46211891 2.70 0.0835 Contrast DF Contrast SS Mean Square F Value Pr > F 1 vs 2&3 1 52.74241913 52.74241913 5.38 0.0273 2 vs 3 1 0.18181870 0.18181870 0.02 0.8926

  19. For DIFFICULTY Dependent Variable: DIFFICULTY Sum of Source DF Squares Mean Square F Value Pr > F Model 2 3.9751512 1.9875756 0.47 0.6282 Error 30 126.2872767 4.2095759 Corrected Total 32 130.2624279 R-Square Coeff Var Root MSE DIFFICULTY Mean 0.030516 35.89975 2.051725 5.715152 Source DF Type III SS Mean Square F Value Pr > F GROUP 2 3.97515121 1.98757560 0.47 0.6282 Contrast DF Contrast SS Mean Square F Value Pr > F 1 vs 2&3 1 3.73469643 3.73469643 0.89 0.3538 2 vs 3 1 0.24045478 0.24045478 0.06 0.8127

  20. For IMPORTANCE Dependent Variable: IMPORTANCE Sum of Source DF Squares Mean Square F Value Pr > F Model 2 81.8296936 40.9148468 2.88 0.0718 Error 30 426.3708962 14.2123632 Corrected Total 32 508.2005898 R-Square Coeff Var Root MSE IMPORTANCE Mean 0.161018 58.21603 3.769929 6.475758 Source DF Type III SS Mean Square F Value Pr > F GROUP 2 81.82969356 40.91484678 2.88 0.0718 Contrast DF Contrast SS Mean Square F Value Pr > F 1 vs 2&3 1 80.30060224 80.30060224 5.65 0.0240 2 vs 3 1 1.52909132 1.52909132 0.11 0.7452 none of the three anovas were statistically significant at the alpha = .05 level, in particular, the anova for difficulty was less than 1.

  21. Overall the manova itself Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for GROUP E = Error SSCP Matrix Characteristic Characteristic Vector V'EV=1 Root Percent USEFUL DIFFICULTY IMPORTANCE 0.89198790 99.42 0.06410227 -0.00186162 0.05375069 0.00524207 0.58 0.01442655 0.06888878 -0.02620577 0.00000000 0.00 -0.03149580 0.05943387 0.01270798 MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall GROUP Effect H = Type III SSCP Matrix for GROUP E = Error SSCP Matrix S=2 M=0 N=13 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.52578838 3.54 6 56 0.0049 Pillai's Trace 0.47667013 3.02 6 58 0.0122 Hotelling-Lawley Trace 0.89722998 4.12 6 35.61 0.0031 Roy's Greatest Root 0.89198790 8.62 3 29 0.0003

  22. Using Contrast The overall multivariate test is significant, we will follow up with several post-hoc tests beginning with multivariate test of group 1 versus the average of group 2 and 3. /* contrast '1 vs 2&3' group 2 -1 -1; manova h-_all_; */ MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall 1 vs 2&3 Effect H = Contrast SSCP Matrix for 1 vs 2&3 E = Error SSCP Matrix S=1 M=0.5 N=13 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.52899035 8.31 3 28 0.0004 Pillai's Trace 0.47100965 8.31 3 28 0.0004 Hotelling-Lawley Trace 0.89039367 8.31 3 28 0.0004 Roy's Greatest Root 0.89039367 8.31 3 28 0.0004 Here is the multivariate test of group 2 versus group 3.

  23. Multivariate test of group 2 versus group 3. /* contrast '2 vs 3' group 0 1 -1; manova h-_all_; */ MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall 2 vs 3 Effect H = Contrast SSCP Matrix for 2 vs 3 E = Error SSCP Matrix S=1 M=0.5 N=13 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.99321011 0.06 3 28 0.9785 Pillai's Trace 0.00678989 0.06 3 28 0.9785 Hotelling-Lawley Trace 0.00683631 0.06 3 28 0.9785 Roy's Greatest Root 0.00683631 0.06 3 28 0.9785

  24. Combination of Useful & Importance From the univariate tests above that difficulty by itself was clearly not significant. This next test does the multivariate test using the combination of useful and importance. /* manova h=group m=(1 0 1); */ MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall GROUP Effect on the Variables Defined by the M Matrix Transformation H = Type III SSCP Matrix for GROUP E = Error SSCP Matrix S=1 M=0 N=14 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.53598494 12.99 2 30 <.0001 Pillai's Trace 0.46401506 12.99 2 30 <.0001 Hotelling-Lawley Trace 0.86572405 12.99 2 30 <.0001 Roy's Greatest Root 0.86572405 12.99 2 30 <.0001

  25. Summary of the Example There is a lot of variation in the write-ups of multivariate analysis of variance. The write-up below is fairly minimal, more detail may be required for most instances. The multivariate test of differences between groups using the Wilks Lambda criteria was statistically significant (F(6, 56) = 3.54; p=0.0049). Follow-up multivariate comparisons showed that the treatment group was significantly different from the average of control 1 and control 2 (F(3,28) = 8.31; p=0.0004). Further, it was determined that control 1 and control 2 were not significant different (F(3,28) = 0.06; p=0.9785). Each of the F-ratio transformations of the Wilks criteria were exact. None of the separate univariate anovas were statistically significant. In particular, the univariate test for difficulty has an F less than 1, so the multivariate test was rerun using the combination of useful and importance, which was statistically significant (F(2,30) = 12.99; p<0.0001).

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