The SAS System 13:37 Monday, March 2, 1992 1 General Linear Models Procedure Class Levels Values T 2 1 2 B 3 1 2 3 Number of observations in data set = 16 Dependent Variable: Y Sum of Mean Source DF Squares Square F Value Pr > F Model 1 76.5625000 76.5625000 5.88 0.0295 Error 14 182.3750000 13.0267857 Corrected Total 15 258.9375000 R-Square C.V. Root MSE Y Mean 0.295679 14.47324 3.60926 24.9375 Source DF Type I SS Mean Square F Value Pr > F T 1 76.5625000 76.5625000 5.88 0.0295 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 27.12500000 B 21.26 0.0001 1.27606748 T 1 -4.37500000 B -2.42 0.0295 1.80463194 2 0.00000000 B . . . -------------------------------------------------------------------------------- NOTE: The X'X matrix has been found to be singular and a generalized inverse was used to solve the normal equations. Estimates followed by the letter 'B' are biased, and are not unique estimators of the parameters. -------------------------------------------------------------------------------- Sum of Mean Source DF Squares Square F Value Pr > F Model 2 94.9375000 47.4687500 3.76 0.0514 Error 13 164.0000000 12.6153846 Corrected Total 15 258.9375000 R-Square C.V. Root MSE Y Mean 0.366643 14.24286 3.55181 24.9375 Source DF Type I SS Mean Square F Value Pr > F B 2 94.9375000 47.4687500 3.76 0.0514 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 28.00000000 B 19.31 0.0001 1.45002210 B 1 -5.60000000 B -2.60 0.0218 2.15073035 2 -4.20000000 B -1.95 0.0727 2.15073035 3 0.00000000 B . . . -------------------------------------------------------------------------------- Sum of Mean Source DF Squares Square F Value Pr > F Model 3 167.306731 55.768910 7.30 0.0048 Error 12 91.630769 7.635897 Corrected Total 15 258.937500 R-Square C.V. Root MSE Y Mean 0.646128 11.08095 2.76331 24.9375 Source DF Type I SS Mean Square F Value Pr > F T 1 76.5625000 76.5625000 10.03 0.0081 B 2 90.7442308 45.3721154 5.94 0.0161 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 30.15384615 B 22.72 0.0001 1.32745251 T 1 -4.30769231 B -3.08 0.0096 1.39925781 2 0.00000000 B . . . B 1 -5.16923077 B -3.08 0.0096 1.67910937 2 -4.63076923 B -2.76 0.0173 1.67910937 3 0.00000000 B . . . -------------------------------------------------------------------------------- Sum of Mean Source DF Squares Square F Value Pr > F Model 5 238.937500 47.787500 23.89 0.0001 Error 10 20.000000 2.000000 Corrected Total 15 258.937500 R-Square C.V. Root MSE Y Mean 0.922761 5.671032 1.41421 24.9375 Source DF Type I SS Mean Square F Value Pr > F T 1 76.5625000 76.5625000 38.28 0.0001 B 2 90.7442308 45.3721154 22.69 0.0002 T*B 2 71.6307692 35.8153846 17.91 0.0005 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 32.00000000 B 39.19 0.0001 0.81649658 T 1 -8.00000000 B -6.93 0.0001 1.15470054 2 0.00000000 B . . . B 1 -6.00000000 B -4.65 0.0009 1.29099445 2 -9.00000000 B -7.79 0.0001 1.15470054 3 0.00000000 B . . . T*B 1 1 2.00000000 B 1.15 0.2751 1.73205081 1 2 10.00000000 B 5.77 0.0002 1.73205081 1 3 0.00000000 B . . . 2 1 0.00000000 B . . . 2 2 0.00000000 B . . . 2 3 0.00000000 B . . . -------------------------------------------------------------------------------- Type I Estimable Functions for: T Effect Coefficients INTERCEPT 0 T 1 L2 2 -L2 B 1 0.125*L2 2 -0.125*L2 3 0 T*B 1 1 0.375*L2 1 2 0.25*L2 1 3 0.375*L2 2 1 -0.25*L2 2 2 -0.375*L2 2 3 -0.375*L2 Type I Estimable Functions for: B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 L4 2 L5 3 -L4-L5 T*B 1 1 0.5692*L4+0.0308*L5 1 2 -0.0308*L4+0.4308*L5 1 3 -0.5385*L4-0.4615*L5 2 1 0.4308*L4-0.0308*L5 2 2 0.0308*L4+0.5692*L5 2 3 -0.4615*L4-0.5385*L5 Type I Estimable Functions for: T*B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 0 2 0 3 0 T*B 1 1 L7 1 2 L8 1 3 -L7-L8 2 1 -L7 2 2 -L8 2 3 L7+L8 -------------------------------------------------------------------------------- Type II Estimable Functions for: T Effect Coefficients INTERCEPT 0 T 1 L2 2 -L2 B 1 0 2 0 3 0 T*B 1 1 0.3077*L2 1 2 0.3077*L2 1 3 0.3846*L2 2 1 -0.3077*L2 2 2 -0.3077*L2 2 3 -0.3846*L2 Type II Estimable Functions for: B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 L4 2 L5 3 -L4-L5 T*B 1 1 0.5692*L4+0.0308*L5 1 2 -0.0308*L4+0.4308*L5 1 3 -0.5385*L4-0.4615*L5 2 1 0.4308*L4-0.0308*L5 2 2 0.0308*L4+0.5692*L5 2 3 -0.4615*L4-0.5385*L5 Type II Estimable Functions for: T*B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 0 2 0 3 0 T*B 1 1 L7 1 2 L8 1 3 -L7-L8 2 1 -L7 2 2 -L8 2 3 L7+L8 -------------------------------------------------------------------------------- Type III Estimable Functions for: T Effect Coefficients INTERCEPT 0 T 1 L2 2 -L2 B 1 0 2 0 3 0 T*B 1 1 0.3333*L2 1 2 0.3333*L2 1 3 0.3333*L2 2 1 -0.3333*L2 2 2 -0.3333*L2 2 3 -0.3333*L2 Type III Estimable Functions for: B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 L4 2 L5 3 -L4-L5 T*B 1 1 0.5*L4 1 2 0.5*L5 1 3 -0.5*L4-0.5*L5 2 1 0.5*L4 2 2 0.5*L5 2 3 -0.5*L4-0.5*L5 Type III Estimable Functions for: T*B Effect Coefficients INTERCEPT 0 T 1 0 2 0 B 1 0 2 0 3 0 T*B 1 1 L7 1 2 L8 1 3 -L7-L8 2 1 -L7 2 2 -L8 2 3 L7+L8 -------------------------------------------------------------------------------- Sum of Mean Source DF Squares Square F Value Pr > F Model 5 238.937500 47.787500 23.89 0.0001 Error 10 20.000000 2.000000 Corrected Total 15 258.937500 R-Square C.V. Root MSE Y Mean 0.922761 5.671032 1.41421 24.9375 Source DF Type I SS Mean Square F Value Pr > F T 1 76.5625000 76.5625000 38.28 0.0001 B 2 90.7442308 45.3721154 22.69 0.0002 T*B 2 71.6307692 35.8153846 17.91 0.0005 Source DF Type II SS Mean Square F Value Pr > F T 1 72.3692308 72.3692308 36.18 0.0001 B 2 90.7442308 45.3721154 22.69 0.0002 T*B 2 71.6307692 35.8153846 17.91 0.0005 Source DF Type III SS Mean Square F Value Pr > F T 1 61.7142857 61.7142857 30.86 0.0002 B 2 77.1692308 38.5846154 19.29 0.0004 T*B 2 71.6307692 35.8153846 17.91 0.0005 -------------------------------------------------------------------------------- Least Squares Means T Y Std Err Pr > |T| Pr > |T| H0: LSMEAN LSMEAN H0:LSMEAN=0 LSMEAN1=LSMEAN2 1 23.0000000 0.5091751 0.0001 0.0002 2 27.0000000 0.5091751 0.0001 B Y Std Err Pr > |T| LSMEAN LSMEAN LSMEAN H0:LSMEAN=0 Number 1 23.0000000 0.6454972 0.0001 1 2 24.0000000 0.6454972 0.0001 2 3 28.0000000 0.5773503 0.0001 3 Pr > |T| H0: LSMEAN(i)=LSMEAN(j) i/j 1 2 3 1 . 0.2990 0.0002 2 0.2990 . 0.0010 3 0.0002 0.0010 . T B Y Std Err Pr > |T| LSMEAN LSMEAN LSMEAN H0:LSMEAN=0 Number 1 1 20.0000000 0.8164966 0.0001 1 1 2 25.0000000 1.0000000 0.0001 2 1 3 24.0000000 0.8164966 0.0001 3 2 1 26.0000000 1.0000000 0.0001 4 2 2 23.0000000 0.8164966 0.0001 5 2 3 32.0000000 0.8164966 0.0001 6 Pr > |T| H0: LSMEAN(i)=LSMEAN(j) i/j 1 2 3 4 5 6 1 . 0.0031 0.0061 0.0009 0.0266 0.0001 2 0.0031 . 0.4565 0.4956 0.1524 0.0003 3 0.0061 0.4565 . 0.1524 0.4068 0.0001 4 0.0009 0.4956 0.1524 . 0.0425 0.0009 5 0.0266 0.1524 0.4068 0.0425 . 0.0001 6 0.0001 0.0003 0.0001 0.0009 0.0001 . NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. -------------------------------------------------------------------------------- T tests (LSD) for variable: Y NOTE: This test controls the type I comparisonwise error rate not the experimentwise error rate. Alpha= 0.05 df= 10 MSE= 2 Critical Value of T= 2.23 Least Significant Difference= 1.5755 Means with the same letter are not significantly different. T Grouping Mean N T A 27.1250 8 2 B 22.7500 8 1 Least Significant Difference= 1.9368 WARNING: Cell sizes are not equal. Harmonic Mean of cell sizes= 5.294118 T Grouping Mean N B A 28.0000 6 3 B 23.8000 5 2 B 22.4000 5 1 Level of Level of --------------Y-------------- T B N Mean SD 1 1 3 20.0000000 1.00000000 1 2 2 25.0000000 1.41421356 1 3 3 24.0000000 1.73205081 2 1 2 26.0000000 1.41421356 2 2 3 23.0000000 1.73205081 2 3 3 32.0000000 1.00000000