The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Number of Observations Read 197 Number of Observations Used 197 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 102.91160 51.45580 129.22 <.0001 Error 194 77.25265 0.39821 Corrected Total 196 180.16425 Root MSE 0.63104 R-Square 0.5712 Dependent Mean 1.21784 Adj R-Sq 0.5668 Coeff Var 51.81609 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.18759 0.04989 23.80 <.0001 u 1 1.34783 0.08481 15.89 <.0001 v 1 -0.05728 0.10544 -0.54 0.5876 The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Number of Observations Read 197 Number of Observations Used 197 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 116.97791 23.39558 70.72 <.0001 Error 191 63.18634 0.33082 Corrected Total 196 180.16425 Root MSE 0.57517 R-Square 0.6493 Dependent Mean 1.21784 Adj R-Sq 0.6401 Coeff Var 47.22846 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.39016 0.06755 20.58 <.0001 u 1 1.58343 0.10620 14.91 <.0001 v 1 0.36106 0.11940 3.02 0.0028 u2 1 -0.05905 0.15821 -0.37 0.7094 uv 1 -0.75589 0.21164 -3.57 0.0004 v2 1 -1.30256 0.20643 -6.31 <.0001 The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Number of Observations Read 197 Number of Observations Used 197 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 9 143.70487 15.96721 81.90 <.0001 Error 187 36.45938 0.19497 Corrected Total 196 180.16425 Root MSE 0.44155 R-Square 0.7976 Dependent Mean 1.21784 Adj R-Sq 0.7879 Coeff Var 36.25707 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.37008 0.06226 22.01 <.0001 u 1 2.70204 0.13525 19.98 <.0001 v 1 0.51415 0.16191 3.18 0.0017 u2 1 -0.59656 0.20141 -2.96 0.0035 uv 1 1.36021 0.25925 5.25 <.0001 v2 1 -0.44789 0.21326 -2.10 0.0371 u3 1 -2.20518 0.25534 -8.64 <.0001 u2v 1 0.80070 0.35443 2.26 0.0250 uv2 1 -2.54672 0.40381 -6.31 <.0001 v3 1 -1.42896 0.34002 -4.20 <.0001 The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Number of Observations Read 197 Number of Observations Used 197 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 14 149.16473 10.65462 62.55 <.0001 Error 182 30.99952 0.17033 Corrected Total 196 180.16425 Root MSE 0.41271 R-Square 0.8279 Dependent Mean 1.21784 Adj R-Sq 0.8147 Coeff Var 33.88836 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.35939 0.07226 18.81 <.0001 u 1 3.06272 0.20125 15.22 <.0001 v 1 0.28388 0.21075 1.35 0.1797 u2 1 0.02436 0.41245 0.06 0.9530 uv 1 1.30305 0.36818 3.54 0.0005 v2 1 -0.59402 0.43753 -1.36 0.1762 u3 1 -3.90766 0.45717 -8.55 <.0001 u2v 1 2.87411 0.73012 3.94 0.0001 uv2 1 -1.27033 0.71203 -1.78 0.0761 v3 1 -1.14172 0.47944 -2.38 0.0183 u4 1 -1.73995 0.58115 -2.99 0.0031 u3v 1 2.75393 0.77842 3.54 0.0005 u2v2 1 -0.92365 0.84312 -1.10 0.2747 uv3 1 -2.81382 0.91577 -3.07 0.0024 v4 1 -0.21537 0.70274 -0.31 0.7596 The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Number of Observations Read 197 Number of Observations Used 197 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 20 153.94002 7.69700 51.66 <.0001 Error 176 26.22423 0.14900 Corrected Total 196 180.16425 Root MSE 0.38601 R-Square 0.8544 Dependent Mean 1.21784 Adj R-Sq 0.8379 Coeff Var 31.69596 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 1.24292 0.08567 14.51 <.0001 u 1 3.05246 0.23232 13.14 <.0001 v 1 0.81916 0.26555 3.08 0.0024 u2 1 0.64246 0.69942 0.92 0.3596 uv 1 3.51938 0.78136 4.50 <.0001 v2 1 1.46832 0.64701 2.27 0.0245 u3 1 -3.95412 0.94602 -4.18 <.0001 u2v 1 2.13354 1.02941 2.07 0.0397 uv2 1 -3.64357 0.98297 -3.71 0.0003 v3 1 -5.71324 1.15608 -4.94 <.0001 u4 1 -2.71150 1.13794 -2.38 0.0182 u3v 1 0.73790 1.82674 0.40 0.6867 u2v2 1 -3.40152 2.24624 -1.51 0.1317 uv3 1 -9.67354 2.03756 -4.75 <.0001 v4 1 -4.71147 1.14166 -4.13 <.0001 u5 1 -0.05842 0.98487 -0.06 0.9528 u4v 1 1.99197 2.05163 0.97 0.3329 u3v2 1 3.09686 1.90237 1.63 0.1053 u2v3 1 2.41682 2.22373 1.09 0.2786 uv4 1 7.66383 2.33311 3.28 0.0012 v5 1 7.33580 1.48588 4.94 <.0001 The SAS System The Mixed Procedure Model Information Data Set WORK.SO4DATA Dependent Variable y Covariance Structure Diagonal Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Residual Dimensions Covariance Parameters 1 Columns in X 6 Columns in Z 0 Subjects 1 Max Obs Per Subject 197 Number of Observations Number of Observations Read 197 Number of Observations Used 197 Number of Observations Not Used 0 Covariance Parameter Estimates Cov Parm Estimate Residual 0.3308 Fit Statistics -2 Res Log Likelihood 350.4 AIC (smaller is better) 352.4 AICC (smaller is better) 352.4 BIC (smaller is better) 355.6 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 1.3902 0.06755 191 20.58 <.0001 u 1.5834 0.1062 191 14.91 <.0001 v 0.3611 0.1194 191 3.02 0.0028 u2 -0.05905 0.1582 191 -0.37 0.7094 uv -0.7559 0.2116 191 -3.57 0.0004 v2 -1.3026 0.2064 191 -6.31 <.0001 The SAS System The Mixed Procedure Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F u 1 191 222.30 <.0001 v 1 191 9.14 0.0028 u2 1 191 0.14 0.7094 uv 1 191 12.76 0.0004 v2 1 191 39.82 <.0001 The SAS System The Mixed Procedure Model Information Data Set WORK.SO4DATA Dependent Variable y Covariance Structure Spatial Exponential Subject Effect Intercept Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Between-Within Dimensions Covariance Parameters 2 Columns in X 6 Columns in Z 0 Subjects 1 Max Obs Per Subject 197 Number of Observations Number of Observations Read 197 Number of Observations Used 197 Number of Observations Not Used 0 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 350.39394831 1 3 295.96598636 2.32531769 2 2 289.58274489 44.44974317 3 2 279.82137711 0.08600639 4 4 278.39813144 0.00105978 5 1 278.35768273 0.00000709 6 1 278.35742368 0.00000000 Convergence criteria met. Covariance Parameter Estimates Cov Parm Subject Estimate SP(EXP) Intercept 0.09736 Residual 0.3368 Fit Statistics -2 Res Log Likelihood 278.4 AIC (smaller is better) 282.4 AICC (smaller is better) 282.4 BIC (smaller is better) 288.9 The SAS System The Mixed Procedure Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 1 72.04 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 1.3512 0.1393 0 9.70 . u 1.3533 0.1886 191 7.18 <.0001 v 0.3137 0.1860 191 1.69 0.0934 u2 -0.09028 0.2709 191 -0.33 0.7393 uv -0.7035 0.3451 191 -2.04 0.0429 v2 -1.2155 0.3266 191 -3.72 0.0003 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F u 1 191 51.48 <.0001 v 1 191 2.84 0.0934 u2 1 191 0.11 0.7393 uv 1 191 4.15 0.0429 v2 1 191 13.85 0.0003 The SAS System The Mixed Procedure Model Information Data Set WORK.SO4DATA Dependent Variable y Covariance Structure Spatial Exponential Subject Effect Intercept Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Between-Within Dimensions Covariance Parameters 2 Columns in X 21 Columns in Z 0 Subjects 1 Max Obs Per Subject 197 Number of Observations Number of Observations Read 197 Number of Observations Used 197 Number of Observations Not Used 0 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 164.65973120 1 4 164.65973120 0.00000000 Convergence criteria met. Covariance Parameter Estimates Cov Parm Subject Estimate SP(EXP) Intercept 0 Residual 0.1490 Fit Statistics -2 Res Log Likelihood 164.7 AIC (smaller is better) 166.7 AICC (smaller is better) 166.7 BIC (smaller is better) 169.8 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 0 0.00 1.0000 The SAS System The Mixed Procedure Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 1.2429 0.08567 0 14.51 . u 3.0525 0.2323 176 13.14 <.0001 v 0.8192 0.2656 176 3.08 0.0024 u2 0.6425 0.6994 176 0.92 0.3596 uv 3.5194 0.7814 176 4.50 <.0001 v2 1.4683 0.6470 176 2.27 0.0245 u3 -3.9541 0.9460 176 -4.18 <.0001 u2v 2.1335 1.0294 176 2.07 0.0397 uv2 -3.6436 0.9830 176 -3.71 0.0003 v3 -5.7132 1.1561 176 -4.94 <.0001 u4 -2.7115 1.1379 176 -2.38 0.0182 u3v 0.7379 1.8267 176 0.40 0.6867 u2v2 -3.4015 2.2462 176 -1.51 0.1317 uv3 -9.6735 2.0376 176 -4.75 <.0001 v4 -4.7115 1.1417 176 -4.13 <.0001 u5 -0.05842 0.9849 176 -0.06 0.9528 u4v 1.9920 2.0516 176 0.97 0.3329 u3v2 3.0969 1.9024 176 1.63 0.1053 u2v3 2.4168 2.2237 176 1.09 0.2786 uv4 7.6638 2.3331 176 3.28 0.0012 v5 7.3358 1.4859 176 4.94 <.0001 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F u 1 176 172.63 <.0001 v 1 176 9.52 0.0024 u2 1 176 0.84 0.3596 uv 1 176 20.29 <.0001 v2 1 176 5.15 0.0245 u3 1 176 17.47 <.0001 u2v 1 176 4.30 0.0397 uv2 1 176 13.74 0.0003 v3 1 176 24.42 <.0001 u4 1 176 5.68 0.0182 u3v 1 176 0.16 0.6867 u2v2 1 176 2.29 0.1317 uv3 1 176 22.54 <.0001 v4 1 176 17.03 <.0001 u5 1 176 0.00 0.9528 u4v 1 176 0.94 0.3329 u3v2 1 176 2.65 0.1053 u2v3 1 176 1.18 0.2786 uv4 1 176 10.79 0.0012 v5 1 176 24.37 <.0001