Test of equality of var-cov matrices across groups via PROC DISCRIM 1 The DISCRIM Procedure Total Sample Size 27 DF Total 26 Variables 5 DF Within Classes 24 Classes 3 DF Between Classes 2 Number of Observations Read 27 Number of Observations Used 27 Class Level Information Variable Prior group Name Frequency Weight Proportion Probability 1 _1 10 10.0000 0.370370 0.333333 2 _2 7 7.0000 0.259259 0.333333 3 _3 10 10.0000 0.370370 0.333333 Within Covariance Matrix Information Natural Log of the Covariance Determinant of the group Matrix Rank Covariance Matrix 1 5 12.77284 2 5 13.87187 3 5 15.56530 Pooled 5 15.43061 Test of equality of var-cov matrices across groups via PROC DISCRIM 2 The DISCRIM Procedure Test of Homogeneity of Within Covariance Matrices Notation: K = Number of Groups P = Number of Variables N = Total Number of Observations - Number of Groups N(i) = Number of Observations in the i'th Group - 1 __ N(i)/2 || |Within SS Matrix(i)| V = ----------------------------------- N/2 |Pooled SS Matrix| _ _ 2 | 1 1 | 2P + 3P - 1 RHO = 1.0 - | SUM ----- - --- | ------------- |_ N(i) N _| 6(P+1)(K-1) DF = .5(K-1)P(P+1) _ _ | PN/2 | | N V | Under the null hypothesis: -2 RHO ln | ------------------ | | __ PN(i)/2 | |_ || N(i) _| is distributed approximately as Chi-Square(DF). Chi-Square DF Pr > ChiSq 22.165042 30 0.8480 Since the Chi-Square value is not significant at the 0.1 level, a pooled covariance matrix will be used in the discriminant function. Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252. Test of equality of var-cov matrices across groups via PROC DISCRIM 3 The DISCRIM Procedure Pairwise Generalized Squared Distances Between Groups 2 _ _ -1 _ _ D (i|j) = (X - X )' COV (X - X ) i j i j Generalized Squared Distance to group From group 1 2 3 1 0 1.85703 7.60821 2 1.85703 0 8.10706 3 7.60821 8.10706 0 Linear Discriminant Function _ -1 _ -1 _ Constant = -.5 X' COV X Coefficient Vector = COV X j j j Linear Discriminant Function for group Variable 1 2 3 Constant -99.32574 -106.82095 -86.91214 w0 3.25494 3.75703 3.55416 w1 -1.16663 -1.52956 -1.49278 w2 0.92676 0.96077 1.19421 w3 -1.01854 -0.90816 -0.83250 w4 0.92616 0.86376 0.55675 Test of equality of var-cov matrices across groups via PROC DISCRIM 4 The DISCRIM Procedure Classification Summary for Calibration Data: WORK.RATS Resubstitution Summary using Linear Discriminant Function Generalized Squared Distance Function 2 _ -1 _ D (X) = (X-X )' COV (X-X ) j j j Posterior Probability of Membership in Each group 2 2 Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X)) j k k Number of Observations and Percent Classified into group From group 1 2 3 Total 1 9 1 0 10 90.00 10.00 0.00 100.00 2 1 6 0 7 14.29 85.71 0.00 100.00 3 1 1 8 10 10.00 10.00 80.00 100.00 Total 11 8 8 27 40.74 29.63 29.63 100.00 Priors 0.33333 0.33333 0.33333 Error Count Estimates for group 1 2 3 Total Rate 0.1000 0.1429 0.2000 0.1476 Priors 0.3333 0.3333 0.3333 This call to PROC GLM tests for parallelism and group effects 5 The GLM Procedure Number of Observations Read 27 Number of Observations Used 27 This call to PROC GLM tests for parallelism and group effects 6 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 1 -1 0 0 0 MVAR2 0 1 -1 0 0 MVAR3 0 0 1 -1 0 MVAR4 0 0 0 1 -1 This call to PROC GLM tests for parallelism and group effects 7 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 1.92715980 91.16 0.01080934 -0.00411714 0.01549824 0.02569399 0.18691743 8.84 0.03363526 -0.01067039 -0.01325003 0.01545930 0.00000000 0.00 0.01595799 -0.05579686 0.03980323 -0.00392112 0.00000000 0.00 0.02082055 0.02394694 0.01132206 -0.02599720 MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Group Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=2 M=0.5 N=9.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.28782801 4.54 8 42 0.0005 Pillai's Trace 0.81585332 3.79 8 44 0.0019 Hotelling-Lawley Trace 2.11407723 5.41 8 27.776 0.0004 Roy's Greatest Root 1.92715980 10.60 4 22 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. This call to PROC GLM tests for parallelism and group effects 8 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 1 0 0 0 0 MVAR2 0 1 0 0 0 MVAR3 0 0 1 0 0 MVAR4 0 0 0 1 0 MVAR5 0 0 0 0 1 This call to PROC GLM tests for parallelism and group effects 9 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 MVAR5 1.93207013 85.85 -0.00661171 0.01304252 -0.01903917 -0.01049910 0.02580900 0.31840406 14.15 -0.07541996 0.05474602 -0.00562348 -0.01683190 0.01006598 0.00000000 0.00 0.02600325 0.01408259 -0.00150257 -0.00435029 0.00266341 0.00000000 0.00 -0.02180838 0.07014773 -0.09612888 0.04691033 -0.00568765 0.00000000 0.00 0.03072397 0.00733224 -0.01294697 -0.04116185 0.02831448 MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Group Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=2 M=1 N=9 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.25868850 3.86 10 40 0.0010 Pillai's Trace 0.90045123 3.44 10 42 0.0023 Hotelling-Lawley Trace 2.25047419 4.37 10 27.385 0.0010 Roy's Greatest Root 1.93207013 8.11 5 21 0.0002 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. This call to PROC GLM tests differences in time 10 The GLM Procedure Number of Observations Read 27 Number of Observations Used 27 This call to PROC GLM tests differences in time 11 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 1 0 0 0 -1 MVAR2 0 1 0 0 -1 MVAR3 0 0 1 0 -1 MVAR4 0 0 0 1 -1 This call to PROC GLM tests differences in time 12 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Time (not parallel) E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 57.4255777 98.39 0.02787583 -0.01334716 -0.01112043 0.01211867 0.8391074 1.44 -0.01868406 -0.00191367 0.04139351 -0.00177496 0.1035346 0.18 0.01637364 -0.05577285 0.03544621 0.03179212 0.0000000 0.00 -0.02328246 0.06365457 -0.08121984 0.05542539 MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Time (not parallel) Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Time (not parallel) E = Error SSCP Matrix S=3 M=0 N=9.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.00843342 23.64 12 55.852 <.0001 Pillai's Trace 1.53296306 6.01 12 69 <.0001 Hotelling-Lawley Trace 58.36821973 98.44 12 32.675 <.0001 Roy's Greatest Root 57.42557773 330.20 4 23 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. This call to PROC GLM tests differences in time for each group separately 13 The GLM Procedure Number of Observations Read 27 Number of Observations Used 27 This call to PROC GLM tests differences in time for each group separately 14 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 1 0 0 0 -1 MVAR2 0 1 0 0 -1 MVAR3 0 0 1 0 -1 MVAR4 0 0 0 1 -1 This call to PROC GLM tests differences in time for each group separately 15 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Time, Group1 E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 26.9209291 100.00 0.02789555 -0.01582346 -0.00811749 0.01342345 0.0000000 0.00 0.01167488 -0.06938002 0.08154151 -0.00001996 0.0000000 0.00 -0.01214472 0.02839495 -0.05454605 0.06365953 0.0000000 0.00 -0.02956968 0.03839377 0.00000000 0.00000000 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Time, Group1 Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Time, Group1 E = Error SSCP Matrix S=1 M=1 N=9.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.03581543 141.33 4 21 <.0001 Pillai's Trace 0.96418457 141.33 4 21 <.0001 Hotelling-Lawley Trace 26.92092908 141.33 4 21 <.0001 Roy's Greatest Root 26.92092908 141.33 4 21 <.0001 Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Time, Group2 E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 17.0767178 100.00 0.02469731 -0.01055311 -0.00820907 0.01016751 0.0000000 0.00 0.01274530 -0.07011163 0.08159323 -0.00000281 This call to PROC GLM tests differences in time for each group separately 16 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Time, Group2 E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 0.0000000 0.00 -0.01011484 0.02651246 -0.05445492 0.06425999 0.0000000 0.00 -0.03258243 0.04017965 0.00000000 0.00000000 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Time, Group2 Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Time, Group2 E = Error SSCP Matrix S=1 M=1 N=9.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.05531978 89.65 4 21 <.0001 Pillai's Trace 0.94468022 89.65 4 21 <.0001 Hotelling-Lawley Trace 17.07671778 89.65 4 21 <.0001 Roy's Greatest Root 17.07671778 89.65 4 21 <.0001 Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Time, Group3 E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 MVAR2 MVAR3 MVAR4 14.3705729 100.00 0.03073952 -0.01153771 -0.01999369 0.01157302 0.0000000 0.00 -0.02370802 0.07590406 -0.09398133 0.04309952 0.0000000 0.00 -0.02049554 0.02374828 0.01347330 -0.00605978 0.0000000 0.00 0.00304462 -0.02976235 0.01662577 0.04695211 This call to PROC GLM tests differences in time for each group separately 17 The GLM Procedure Multivariate Analysis of Variance MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Time, Group3 Effect on the Variables Defined by the M Matrix Transformation H = Contrast SSCP Matrix for Time, Group3 E = Error SSCP Matrix S=1 M=1 N=9.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.06505938 75.45 4 21 <.0001 Pillai's Trace 0.93494062 75.45 4 21 <.0001 Hotelling-Lawley Trace 14.37057287 75.45 4 21 <.0001 Roy's Greatest Root 14.37057287 75.45 4 21 <.0001 This call to PROC GLM tests group effects at each time 18 The GLM Procedure Number of Observations Read 27 Number of Observations Used 27 This call to PROC GLM tests group effects at each time 19 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 1 0 0 0 0 This call to PROC GLM tests group effects at each time 20 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 0.01967059 100.00 0.04394536 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 = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=1 M=0 N=11 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.98070887 0.24 2 24 0.7916 Pillai's Trace 0.01929113 0.24 2 24 0.7916 Hotelling-Lawley Trace 0.01967059 0.24 2 24 0.7916 Roy's Greatest Root 0.01967059 0.24 2 24 0.7916 This call to PROC GLM tests group effects at each time 21 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 0 1 0 0 0 This call to PROC GLM tests group effects at each time 22 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 0.02215448 100.00 0.02462235 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 = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=1 M=0 N=11 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.97832571 0.27 2 24 0.7688 Pillai's Trace 0.02167429 0.27 2 24 0.7688 Hotelling-Lawley Trace 0.02215448 0.27 2 24 0.7688 Roy's Greatest Root 0.02215448 0.27 2 24 0.7688 This call to PROC GLM tests group effects at each time 23 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 0 0 1 0 0 This call to PROC GLM tests group effects at each time 24 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 0.26441242 100.00 0.02096820 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 = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=1 M=0 N=11 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.79088119 3.17 2 24 0.0599 Pillai's Trace 0.20911881 3.17 2 24 0.0599 Hotelling-Lawley Trace 0.26441242 3.17 2 24 0.0599 Roy's Greatest Root 0.26441242 3.17 2 24 0.0599 This call to PROC GLM tests group effects at each time 25 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 0 0 0 1 0 This call to PROC GLM tests group effects at each time 26 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 0.81692992 100.00 0.01574710 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 = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=1 M=0 N=11 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.55037896 9.80 2 24 0.0008 Pillai's Trace 0.44962104 9.80 2 24 0.0008 Hotelling-Lawley Trace 0.81692992 9.80 2 24 0.0008 Roy's Greatest Root 0.81692992 9.80 2 24 0.0008 This call to PROC GLM tests group effects at each time 27 The GLM Procedure Multivariate Analysis of Variance M Matrix Describing Transformed Variables w0 w1 w2 w3 w4 MVAR1 0 0 0 0 1 This call to PROC GLM tests group effects at each time 28 The GLM Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SSCP Matrix for Group E = Error SSCP Matrix Variables have been transformed by the M Matrix Characteristic Characteristic Vector V'EV=1 Root Percent MVAR1 1.45299522 100.00 0.01285345 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 = Contrast SSCP Matrix for Group E = Error SSCP Matrix S=1 M=0 N=11 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.40766488 17.44 2 24 <.0001 Pillai's Trace 0.59233512 17.44 2 24 <.0001 Hotelling-Lawley Trace 1.45299522 17.44 2 24 <.0001 Roy's Greatest Root 1.45299522 17.44 2 24 <.0001 Profile plot of group*week means (group means at each week plotted vs week 29 Plot of grpmn*week. Symbol is value of group. grpmn ‚ 180 ˆ ‚ ‚ ‚ ‚ ‚ 2 160 ˆ 1 ‚ ‚ ‚ ‚ ‚ 140 ˆ ‚ ‚ 2 ‚ 1 ‚ ‚ 3 120 ˆ ‚ ‚ ‚ 3 ‚ 1 ‚ 2 100 ˆ ‚ 3 ‚ ‚ ‚ ‚ 80 ˆ 1 ‚ 2 ‚ ‚ ‚ ‚ 60 ˆ ‚ 2 ‚ 1 ‚ ‚ ‚ 40 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ w0 w1 w2 w3 w4 NAME OF FORMER VARIABLE NOTE: 2 obs hidden.