MANOVA CRF-32 Example Same data as Assignment 8 data crf32; input a b y1 y2; cards; 1 1 10 21 1 1 12 22 1 1 9 19 1 1 10 21 1 1 14 23 1 2 9 14 1 2 8 15 1 2 11 16 1 2 9 17 1 2 9 17 2 1 11 23 2 1 14 27 2 1 13 24 2 1 15 26 2 1 14 24 2 2 11 15 2 2 12 18 2 2 10 16 2 2 9 17 2 2 9 18 3 1 8 17 3 1 7 15 3 1 10 18 3 1 8 17 3 1 7 19 3 2 9 22 3 2 8 18 3 2 10 17 3 2 9 19 3 2 8 19 ; proc glm; classes a b; model y1 y2 = a b a*b; contrast '1vs2' a 1 -1 0; contrast '1vs3' a 1 0 -1; contrast '2vs3' a 0 1 -1; manova h=a b a*b; run; ---------------------------------------------------------------------------- General Linear Models Procedure Class Level Information Class Levels Values A 3 1 2 3 B 2 1 2 Number of observations in data set = 30 General Linear Models Procedure Dependent Variable: Y1 Sum of Mean Source DF Squares Square F Value Pr > F Model 5 93.1000000 18.6200000 9.80 0.0001 Error 24 45.6000000 1.9000000 Corrected Total 29 138.7000000 R-Square C.V. Root MSE Y1 Mean 0.671233 13.64757 1.37840 10.1000 General Linear Models Procedure Dependent Variable: Y1 Source DF Type I SS Mean Square F Value Pr > F A 2 57.8000000 28.9000000 15.21 0.0001 B 1 14.7000000 14.7000000 7.74 0.0104 A*B 2 20.6000000 10.3000000 5.42 0.0114 Source DF Type III SS Mean Square F Value Pr > F A 2 57.8000000 28.9000000 15.21 0.0001 B 1 14.7000000 14.7000000 7.74 0.0104 A*B 2 20.6000000 10.3000000 5.42 0.0114 Contrast DF Contrast SS Mean Square F Value Pr > F 1vs2 1 14.4500000 14.4500000 7.61 0.0109 1vs3 1 14.4500000 14.4500000 7.61 0.0109 2vs3 1 57.8000000 57.8000000 30.42 0.0001 General Linear Models Procedure Dependent Variable: Y2 Sum of Mean Source DF Squares Square F Value Pr > F Model 5 283.466667 56.693333 24.30 0.0001 Error 24 56.000000 2.333333 Corrected Total 29 339.466667 R-Square C.V. Root MSE Y2 Mean 0.835035 7.983581 1.52753 19.1333 General Linear Models Procedure Dependent Variable: Y2 Source DF Type I SS Mean Square F Value Pr > F A 2 42.466667 21.233333 9.10 0.0011 B 1 112.133333 112.133333 48.06 0.0001 A*B 2 128.866667 64.433333 27.61 0.0001 Source DF Type III SS Mean Square F Value Pr > F A 2 42.466667 21.233333 9.10 0.0011 B 1 112.133333 112.133333 48.06 0.0001 A*B 2 128.866667 64.433333 27.61 0.0001 Contrast DF Contrast SS Mean Square F Value Pr > F 1vs2 1 26.4500000 26.4500000 11.34 0.0026 1vs3 1 0.8000000 0.8000000 0.34 0.5637 2vs3 1 36.4500000 36.4500000 15.62 0.0006 Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SS&CP Matrix for A E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 1.44074791 92.81 0.11533443 0.05243270 0.11166794 7.19 -0.11227282 0.13544994 Multivariate Analysis of Variance Manova Test Criteria and F Approximations for the Hypothesis of no Overall A Effect H = Type III SS&CP Matrix for A E = Error SS&CP Matrix S=2 M=-0.5 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.36855473 7.4429 4 46 0.0001 Pillai's Trace 0.69074032 6.3310 4 48 0.0004 Hotelling-Lawley Trace 1.55241585 8.5383 4 44 0.0001 Roy's Greatest Root 1.44074791 17.2890 2 24 0.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. General Linear Models Procedure Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SS&CP Matrix for B E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 2.00259072 100.00 0.00164735 0.13304117 0.00000000 0.00 0.16094875 -0.05827455 Multivariate Analysis of Variance Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall B Effect H = Type III SS&CP Matrix for B E = Error SS&CP Matrix S=1 M=0 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.33304572 23.0298 2 23 0.0001 Pillai's Trace 0.66695428 23.0298 2 23 0.0001 Hotelling-Lawley Trace 2.00259072 23.0298 2 23 0.0001 Roy's Greatest Root 2.00259072 23.0298 2 23 0.0001 Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SS&CP Matrix for A*B E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 2.30781925 99.80 0.00863494 0.13038512 0.00460326 0.20 0.16072540 -0.06399685 Multivariate Analysis of Variance Manova Test Criteria and F Approximations for the Hypothesis of no Overall A*B Effect H = Type III SS&CP Matrix for A*B E = Error SS&CP Matrix S=2 M=-0.5 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.30092873 9.4636 4 46 0.0001 Pillai's Trace 0.70226819 6.4938 4 48 0.0003 Hotelling-Lawley Trace 2.31242251 12.7183 4 44 0.0001 Roy's Greatest Root 2.30781925 27.6938 2 24 0.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SS&CP Matrix for 1vs2 E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 0.57418716 100.00 0.06779500 0.09722835 0.00000000 0.00 -0.14598306 0.10790052 Multivariate Analysis of Variance Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall 1vs2 Effect H = Contrast SS&CP Matrix for 1vs2 E = Error SS&CP Matrix S=1 M=0 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.63524848 6.6032 2 23 0.0054 Pillai's Trace 0.36475152 6.6032 2 23 0.0054 Hotelling-Lawley Trace 0.57418716 6.6032 2 23 0.0054 Roy's Greatest Root 0.57418716 6.6032 2 23 0.0054 Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SS&CP Matrix for 1vs3 E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 0.32894761 100.00 0.15742332 -0.02781242 0.00000000 0.00 -0.03354270 0.14255646 Multivariate Analysis of Variance Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall 1vs3 Effect H = Contrast SS&CP Matrix for 1vs3 E = Error SS&CP Matrix S=1 M=0 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.75247511 3.7829 2 23 0.0380 Pillai's Trace 0.24752489 3.7829 2 23 0.0380 Hotelling-Lawley Trace 0.32894761 3.7829 2 23 0.0380 Roy's Greatest Root 0.32894761 3.7829 2 23 0.0380 Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Contrast SS&CP Matrix for 2vs3 E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root Y1 Y2 1.42548900 100.00 0.11864955 0.04834705 0.00000000 0.00 -0.10876350 0.13696145 Multivariate Analysis of Variance Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall 2vs3 Effect H = Contrast SS&CP Matrix for 2vs3 E = Error SS&CP Matrix S=1 M=0 N=10.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.41228800 16.3931 2 23 0.0001 Pillai's Trace 0.58771200 16.3931 2 23 0.0001 Hotelling-Lawley Trace 1.42548900 16.3931 2 23 0.0001 Roy's Greatest Root 1.42548900 16.3931 2 23 0.0001