The SAS System Obs trt unit day y 1 1 1 2 0.22 2 1 1 4 0.56 3 1 1 6 0.66 4 1 1 8 0.89 5 1 2 2 0.68 6 1 2 4 0.91 7 1 2 6 1.06 8 1 2 8 0.80 9 1 3 2 0.68 10 1 3 4 0.45 11 1 3 6 0.72 12 1 3 8 0.89 13 2 4 2 2.53 14 2 4 4 2.70 15 2 4 6 2.10 16 2 4 8 1.50 17 2 5 2 2.59 18 2 5 4 1.43 19 2 5 6 1.35 20 2 5 8 0.74 21 2 6 2 0.56 22 2 6 4 1.37 23 2 6 6 1.87 24 2 6 8 1.21 25 3 7 2 0.22 26 3 7 4 0.22 27 3 7 6 0.20 28 3 7 8 0.11 29 3 8 2 0.45 30 3 8 4 0.28 31 3 8 6 1.24 32 3 8 8 0.86 33 3 9 2 0.22 34 3 9 4 0.33 35 3 9 6 0.34 36 3 9 8 0.20 37 4 10 2 0.22 38 4 10 4 0.80 39 4 10 6 0.80 40 4 10 8 0.37 41 4 11 2 0.22 42 4 11 4 0.62 43 4 11 6 0.89 44 4 11 8 0.95 45 4 12 2 0.22 46 4 12 4 0.56 47 4 12 6 0.69 48 4 12 8 0.63 The SAS System The GLM Procedure Class Level Information Class Levels Values trt 4 1 2 3 4 unit 12 1 2 3 4 5 6 7 8 9 10 11 12 day 4 2 4 6 8 Number of Observations Read 48 Number of Observations Used 48 The SAS System The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 23 16.36531458 0.71153542 5.32 <.0001 Error 24 3.21006667 0.13375278 Corrected Total 47 19.57538125 R-Square Coeff Var Root MSE y Mean 0.836015 43.76631 0.365722 0.835625 Source DF Type I SS Mean Square F Value Pr > F trt 3 11.56497292 3.85499097 28.82 <.0001 unit(trt) 8 2.76513333 0.34564167 2.58 0.0343 day 3 0.48957292 0.16319097 1.22 0.3239 trt*day 9 1.54563542 0.17173727 1.28 0.2955 Source DF Type III SS Mean Square F Value Pr > F trt 3 11.56497292 3.85499097 28.82 <.0001 unit(trt) 8 2.76513333 0.34564167 2.58 0.0343 day 3 0.48957292 0.16319097 1.22 0.3239 trt*day 9 1.54563542 0.17173727 1.28 0.2955 Tests of Hypotheses Using the Type III MS for unit(trt) as an Error Term Source DF Type III SS Mean Square F Value Pr > F trt 3 11.56497292 3.85499097 11.15 0.0031 The SAS System Obs trt unit y2 y4 y6 y8 1 1 1 0.22 0.56 0.66 0.89 2 1 2 0.68 0.91 1.06 0.80 3 1 3 0.68 0.45 0.72 0.89 4 2 4 2.53 2.70 2.10 1.50 5 2 5 2.59 1.43 1.35 0.74 6 2 6 0.56 1.37 1.87 1.21 7 3 7 0.22 0.22 0.20 0.11 8 3 8 0.45 0.28 1.24 0.86 9 3 9 0.22 0.33 0.34 0.20 10 4 10 0.22 0.80 0.80 0.37 11 4 11 0.22 0.62 0.89 0.95 12 4 12 0.22 0.56 0.69 0.63 The SAS System The GLM Procedure Class Level Information Class Levels Values trt 4 1 2 3 4 unit 12 1 2 3 4 5 6 7 8 9 10 11 12 Number of Observations Read 12 Number of Observations Used 12 The SAS System The GLM Procedure Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable y2 y4 y6 y8 Level of time 2 4 6 8 Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r| DF = 8 y2 y4 y6 y8 y2 1.000000 0.485121 0.001991 -0.032073 0.1856 0.9959 0.9347 y4 0.485121 1.000000 0.452771 0.369243 0.1856 0.2210 0.3281 y6 0.001991 0.452771 1.000000 0.832221 0.9959 0.2210 0.0054 y8 -0.032073 0.369243 0.832221 1.000000 0.9347 0.3281 0.0054 E = Error SSCP Matrix time_N represents the nth degree polynomial contrast for time time_1 time_2 time_3 time_1 2.14272 -0.78776 -0.15431 time_2 -0.78776 0.69797 -0.15122 time_3 -0.15431 -0.15122 0.36938 Partial Correlation Coefficients from the Error SSCP Matrix of the Variables Defined by the Specified Transformation / Prob > |r| DF = 8 time_1 time_2 time_3 time_1 1.000000 -0.644160 -0.173450 0.0611 0.6554 time_2 -0.644160 1.000000 -0.297817 0.0611 0.4364 time_3 -0.173450 -0.297817 1.000000 0.6554 0.4364 The SAS System The GLM Procedure Repeated Measures Analysis of Variance Sphericity Tests Mauchly's Variables DF Criterion Chi-Square Pr > ChiSq Transformed Variates 5 0.1802439 11.518153 0.0420 Orthogonal Components 5 0.1802439 11.518153 0.0420 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time Effect H = Type III SSCP Matrix for time E = Error SSCP Matrix S=1 M=0.5 N=2 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.33247637 4.02 3 6 0.0696 Pillai's Trace 0.66752363 4.02 3 6 0.0696 Hotelling-Lawley Trace 2.00773252 4.02 3 6 0.0696 Roy's Greatest Root 2.00773252 4.02 3 6 0.0696 MANOVA Test Criteria and F Approximations for the Hypothesis of no time*trt Effect H = Type III SSCP Matrix for time*trt E = Error SSCP Matrix S=3 M=-0.5 N=2 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.21379182 1.45 9 14.753 0.2530 Pillai's Trace 0.94322114 1.22 9 24 0.3272 Hotelling-Lawley Trace 2.96982620 1.87 9 6.8205 0.2143 Roy's Greatest Root 2.72345425 7.26 3 8 0.0113 NOTE: F Statistic for Roy's Greatest Root is an upper bound. The SAS System The GLM Procedure Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects Source DF Type III SS Mean Square F Value Pr > F trt 3 11.56497292 3.85499097 11.15 0.0031 Error 8 2.76513333 0.34564167 The SAS System The GLM Procedure Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Source DF Type III SS Mean Square F Value Pr > F time 3 0.48957292 0.16319097 1.22 0.3239 time*trt 9 1.54563542 0.17173727 1.28 0.2955 Error(time) 24 3.21006667 0.13375278 Adj Pr > F Source G - G H - F time 0.3165 0.3238 time*trt 0.3298 0.3040 Error(time) Greenhouse-Geisser Epsilon 0.5245 Huynh-Feldt Epsilon 0.8755 The SAS System The GLM Procedure Repeated Measures Analysis of Variance Analysis of Variance of Contrast Variables time_N represents the nth degree polynomial contrast for time Contrast Variable: time_1 Source DF Type III SS Mean Square F Value Pr > F Mean 1 0.03060042 0.03060042 0.11 0.7441 trt 3 1.31980458 0.43993486 1.64 0.2553 Error 8 2.14272000 0.26784000 Contrast Variable: time_2 Source DF Type III SS Mean Square F Value Pr > F Mean 1 0.36575208 0.36575208 4.19 0.0748 trt 3 0.15600625 0.05200208 0.60 0.6351 Error 8 0.69796667 0.08724583 Contrast Variable: time_3 Source DF Type III SS Mean Square F Value Pr > F Mean 1 0.09322042 0.09322042 2.02 0.1931 trt 3 0.06982458 0.02327486 0.50 0.6901 Error 8 0.36938000 0.04617250