OPTIONS LS=88 PS=55 NODATE; FILENAME surgical URL "http://www.stat.wmich.edu/naranjo/stat6620/surgicalunitdata.txt"; DATA one; INFILE surgical; INPUT x1 x2 x3 x4 x5 x6 x7 x8 y; logy=log(y); PROC REG DATA=one; MODEL logy = x1-x5 / SELECTION=RSQUARE SSE ADJRSQ CP AIC PRESS; RUN; PROC REG DATA=one; MODEL logy = x1-x8 / SELECTION=FORWARD; MODEL logy = x1-x8 / SELECTION=BACKWARD; MODEL logy = x1-x8 / SELECTION=STEPWISE; RUN; **************************************************************************************** The SAS System 1 The REG Procedure Dependent Variable: logy R-Square Selection Method Number of Observations Read 54 Number of Observations Used 54 Number in Adjusted Model R-Square R-Square C(p) AIC SSE Variables in Model 1 0.4273 0.4162 69.0323 -103.8110 7.33373 x3 1 0.4215 0.4103 70.2356 -103.2679 7.40786 x4 1 0.2210 0.2061 111.8887 -87.2053 9.97416 x2 1 0.0607 0.0426 145.2169 -77.0963 12.02755 x1 1 0.0210 0.0022 153.4549 -74.8644 12.53510 x5 ------------------------------------------------------------------------------------ 2 0.6632 0.6500 22.0014 -130.4785 4.31287 x2 x3 2 0.5992 0.5835 35.2956 -121.0890 5.13194 x3 x4 2 0.5484 0.5307 45.8554 -114.6436 5.78254 x1 x3 2 0.4830 0.4627 59.4441 -107.3420 6.61976 x2 x4 2 0.4459 0.4242 67.1527 -103.6003 7.09470 x3 x5 2 0.4300 0.4076 70.4635 -102.0697 7.29868 x1 x4 2 0.4216 0.3989 72.2120 -101.2785 7.40641 x4 x5 2 0.2630 0.2341 105.1724 -88.1939 9.43713 x1 x2 2 0.2361 0.2062 110.7557 -86.2606 9.78113 x2 x5 2 0.0803 0.0442 143.1444 -76.2348 11.77663 x1 x5 ------------------------------------------------------------------------------------ 3 0.7572 0.7427 4.4537 -146.1614 3.10851 x1 x2 x3 3 0.7177 0.7007 12.6736 -138.0109 3.61495 x2 x3 x4 3 0.6760 0.6566 21.3264 -130.5825 4.14806 x2 x3 x5 3 0.6119 0.5886 34.6639 -120.8226 4.96980 x1 x3 x4 3 0.6013 0.5774 36.8629 -119.3702 5.10528 x3 x4 x5 3 0.5650 0.5389 44.4130 -114.6614 5.57046 x1 x3 x5 3 0.4870 0.4563 60.6095 -105.7631 6.56834 x1 x2 x4 3 0.4834 0.4524 61.3726 -105.3779 6.61536 x2 x4 x5 3 0.4300 0.3958 72.4630 -100.0699 7.29865 x1 x4 x5 3 0.2772 0.2339 104.2084 -87.2491 9.25452 x1 x2 x5 ------------------------------------------------------------------------------------ 4 0.7688 0.7500 4.0424 -146.8058 2.95995 x1 x2 x3 x5 4 0.7591 0.7395 6.0573 -144.5872 3.08409 x1 x2 x3 x4 4 0.7217 0.6989 13.8477 -136.7764 3.56407 x2 x3 x4 x5 4 0.6159 0.5846 35.8206 -119.3901 4.91785 x1 x3 x4 x5 4 0.4872 0.4453 62.5792 -103.7784 6.56648 x1 x2 x4 x5 ------------------------------------------------------------------------------------ 5 0.7690 0.7450 6.0000 -144.8535 2.95734 x1 x2 x3 x4 x5 The SAS System 2 The REG Procedure Model: MODEL1 Dependent Variable: logy Number of Observations Read 54 Number of Observations Used 54 Forward Selection: Step 1 Variable x3 Entered: R-Square = 0.4273 and C(p) = 117.4783 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 5.47078 5.47078 38.79 <.0001 Error 52 7.33373 0.14103 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 5.26489 0.19401 103.86327 736.45 <.0001 x3 0.01512 0.00243 5.47078 38.79 <.0001 Bounds on condition number: 1, 1 ---------------------------------------------------------------------------------------- Forward Selection: Step 2 Variable x2 Entered: R-Square = 0.6632 and C(p) = 50.4918 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 8.49164 4.24582 50.21 <.0001 Error 51 4.31287 0.08457 Corrected Total 53 12.80451 The SAS System 3 The REG Procedure Model: MODEL1 Dependent Variable: logy Forward Selection: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.35094 0.21437 34.83799 411.96 <.0001 x2 0.01413 0.00236 3.02086 35.72 <.0001 x3 0.01538 0.00188 5.66129 66.95 <.0001 Bounds on condition number: 1.0006, 4.0022 ---------------------------------------------------------------------------------------- Forward Selection: Step 3 Variable x8 Entered: R-Square = 0.7780 and C(p) = 18.9015 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 9.96253 3.32084 58.42 <.0001 Error 50 2.84198 0.05684 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.29102 0.17614 33.73340 593.48 <.0001 x2 0.01494 0.00194 3.35315 58.99 <.0001 x3 0.01447 0.00155 4.94031 86.92 <.0001 x8 0.42927 0.08438 1.47089 25.88 <.0001 Bounds on condition number: 1.0208, 9.1267 ---------------------------------------------------------------------------------------- Forward Selection: Step 4 Variable x1 Entered: R-Square = 0.8299 and C(p) = 5.7340 The SAS System 4 The REG Procedure Model: MODEL1 Dependent Variable: logy Forward Selection: Step 4 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 10.62670 2.65668 59.77 <.0001 Error 49 2.17781 0.04445 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.85273 0.19265 17.77526 399.94 <.0001 x1 0.07333 0.01897 0.66417 14.94 0.0003 x2 0.01419 0.00173 2.98915 67.25 <.0001 x3 0.01545 0.00140 5.44592 122.53 <.0001 x8 0.35315 0.07717 0.93070 20.94 <.0001 Bounds on condition number: 1.1026, 17.052 ---------------------------------------------------------------------------------------- Forward Selection: Step 5 Variable x6 Entered: R-Square = 0.8375 and C(p) = 5.5282 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 10.72329 2.14466 49.46 <.0001 Error 48 2.08122 0.04336 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.86739 0.19054 17.86320 411.99 <.0001 x1 0.07125 0.01879 0.62359 14.38 0.0004 x2 0.01390 0.00172 2.82872 65.24 <.0001 x3 0.01511 0.00140 5.07420 117.03 <.0001 x6 0.08682 0.05817 0.09659 2.23 0.1421 x8 0.36285 0.07650 0.97544 22.50 <.0001 The SAS System 5 The REG Procedure Model: MODEL1 Dependent Variable: logy Forward Selection: Step 5 Bounds on condition number: 1.1087, 26.833 ---------------------------------------------------------------------------------------- Forward Selection: Step 6 Variable x5 Entered: R-Square = 0.8435 and C(p) = 5.7725 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 6 10.80017 1.80003 42.21 <.0001 Error 47 2.00434 0.04265 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.05439 0.23474 12.72167 298.31 <.0001 x1 0.07152 0.01863 0.62835 14.73 0.0004 x2 0.01376 0.00171 2.76443 64.82 <.0001 x3 0.01511 0.00139 5.07517 119.01 <.0001 x5 -0.00345 0.00257 0.07688 1.80 0.1858 x6 0.08723 0.05769 0.09750 2.29 0.1372 x8 0.35107 0.07637 0.90108 21.13 <.0001 Bounds on condition number: 1.1145, 38.409 ---------------------------------------------------------------------------------------- Forward Selection: Step 7 Variable x7 Entered: R-Square = 0.8460 and C(p) = 7.0288 The SAS System 6 The REG Procedure Model: MODEL1 Dependent Variable: logy Forward Selection: Step 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 10.83274 1.54753 36.10 <.0001 Error 46 1.97177 0.04286 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.03736 0.23615 12.52870 292.29 <.0001 x1 0.07144 0.01868 0.62689 14.62 0.0004 x2 0.01361 0.00172 2.67549 62.42 <.0001 x3 0.01514 0.00139 5.09344 118.83 <.0001 x5 -0.00371 0.00259 0.08758 2.04 0.1597 x6 0.08695 0.05784 0.09689 2.26 0.1396 x7 0.05806 0.06662 0.03256 0.76 0.3880 x8 0.38781 0.08741 0.84378 19.68 <.0001 Bounds on condition number: 1.4524, 57.081 ---------------------------------------------------------------------------------------- No other variable met the 0.5000 significance level for entry into the model. Summary of Forward Selection Variable Number Partial Model Step Entered Vars In R-Square R-Square C(p) F Value Pr > F 1 x3 1 0.4273 0.4273 117.478 38.79 <.0001 2 x2 2 0.2359 0.6632 50.4918 35.72 <.0001 3 x8 3 0.1149 0.7780 18.9015 25.88 <.0001 4 x1 4 0.0519 0.8299 5.7340 14.94 0.0003 5 x6 5 0.0075 0.8375 5.5282 2.23 0.1421 6 x5 6 0.0060 0.8435 5.7725 1.80 0.1858 7 x7 7 0.0025 0.8460 7.0288 0.76 0.3880 The SAS System 7 The REG Procedure Model: MODEL2 Dependent Variable: logy Number of Observations Read 54 Number of Observations Used 54 Backward Elimination: Step 0 All Variables Entered: R-Square = 0.8461 and C(p) = 9.0000 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 8 10.83400 1.35425 30.93 <.0001 Error 45 1.97051 0.04379 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.05095 0.25174 11.33894 258.94 <.0001 x1 0.06855 0.02542 0.31845 7.27 0.0098 x2 0.01346 0.00195 2.09285 47.79 <.0001 x3 0.01495 0.00181 2.98863 68.25 <.0001 x4 0.00793 0.04671 0.00126 0.03 0.8659 x5 -0.00357 0.00275 0.07359 1.68 0.2014 x6 0.08415 0.06075 0.08403 1.92 0.1728 x7 0.05731 0.06748 0.03159 0.72 0.4002 x8 0.38819 0.08837 0.84489 19.29 <.0001 Bounds on condition number: 3.0249, 106 ---------------------------------------------------------------------------------------- Backward Elimination: Step 1 Variable x4 Removed: R-Square = 0.8460 and C(p) = 7.0288 The SAS System 8 The REG Procedure Model: MODEL2 Dependent Variable: logy Backward Elimination: Step 1 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 10.83274 1.54753 36.10 <.0001 Error 46 1.97177 0.04286 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.03736 0.23615 12.52870 292.29 <.0001 x1 0.07144 0.01868 0.62689 14.62 0.0004 x2 0.01361 0.00172 2.67549 62.42 <.0001 x3 0.01514 0.00139 5.09344 118.83 <.0001 x5 -0.00371 0.00259 0.08758 2.04 0.1597 x6 0.08695 0.05784 0.09689 2.26 0.1396 x7 0.05806 0.06662 0.03256 0.76 0.3880 x8 0.38781 0.08741 0.84378 19.68 <.0001 Bounds on condition number: 1.4524, 57.081 ---------------------------------------------------------------------------------------- Backward Elimination: Step 2 Variable x7 Removed: R-Square = 0.8435 and C(p) = 5.7725 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 6 10.80017 1.80003 42.21 <.0001 Error 47 2.00434 0.04265 Corrected Total 53 12.80451 The SAS System 9 The REG Procedure Model: MODEL2 Dependent Variable: logy Backward Elimination: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.05439 0.23474 12.72167 298.31 <.0001 x1 0.07152 0.01863 0.62835 14.73 0.0004 x2 0.01376 0.00171 2.76443 64.82 <.0001 x3 0.01511 0.00139 5.07517 119.01 <.0001 x5 -0.00345 0.00257 0.07688 1.80 0.1858 x6 0.08723 0.05769 0.09750 2.29 0.1372 x8 0.35107 0.07637 0.90108 21.13 <.0001 Bounds on condition number: 1.1145, 38.409 ---------------------------------------------------------------------------------------- Backward Elimination: Step 3 Variable x5 Removed: R-Square = 0.8375 and C(p) = 5.5282 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 10.72329 2.14466 49.46 <.0001 Error 48 2.08122 0.04336 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.86739 0.19054 17.86320 411.99 <.0001 x1 0.07125 0.01879 0.62359 14.38 0.0004 x2 0.01390 0.00172 2.82872 65.24 <.0001 x3 0.01511 0.00140 5.07420 117.03 <.0001 x6 0.08682 0.05817 0.09659 2.23 0.1421 x8 0.36285 0.07650 0.97544 22.50 <.0001 Bounds on condition number: 1.1087, 26.833 ---------------------------------------------------------------------------------------- Backward Elimination: Step 4 The SAS System 10 The REG Procedure Model: MODEL2 Dependent Variable: logy Backward Elimination: Step 4 Variable x6 Removed: R-Square = 0.8299 and C(p) = 5.7340 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 10.62670 2.65668 59.77 <.0001 Error 49 2.17781 0.04445 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.85273 0.19265 17.77526 399.94 <.0001 x1 0.07333 0.01897 0.66417 14.94 0.0003 x2 0.01419 0.00173 2.98915 67.25 <.0001 x3 0.01545 0.00140 5.44592 122.53 <.0001 x8 0.35315 0.07717 0.93070 20.94 <.0001 Bounds on condition number: 1.1026, 17.052 ---------------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1000 level. Summary of Backward Elimination Variable Number Partial Model Step Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 x4 7 0.0001 0.8460 7.0288 0.03 0.8659 2 x7 6 0.0025 0.8435 5.7725 0.76 0.3880 3 x5 5 0.0060 0.8375 5.5282 1.80 0.1858 4 x6 4 0.0075 0.8299 5.7340 2.23 0.1421 The SAS System 11 The REG Procedure Model: MODEL3 Dependent Variable: logy Number of Observations Read 54 Number of Observations Used 54 Stepwise Selection: Step 1 Variable x3 Entered: R-Square = 0.4273 and C(p) = 117.4783 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 5.47078 5.47078 38.79 <.0001 Error 52 7.33373 0.14103 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 5.26489 0.19401 103.86327 736.45 <.0001 x3 0.01512 0.00243 5.47078 38.79 <.0001 Bounds on condition number: 1, 1 yy---------------------------------------------------------------------------------------- Stepwise Selection: Step 2 Variable x2 Entered: R-Square = 0.6632 and C(p) = 50.4918 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 8.49164 4.24582 50.21 <.0001 Error 51 4.31287 0.08457 Corrected Total 53 12.80451 The SAS System 12 The REG Procedure Model: MODEL3 Dependent Variable: logy Stepwise Selection: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.35094 0.21437 34.83799 411.96 <.0001 x2 0.01413 0.00236 3.02086 35.72 <.0001 x3 0.01538 0.00188 5.66129 66.95 <.0001 Bounds on condition number: 1.0006, 4.0022 ---------------------------------------------------------------------------------------- Stepwise Selection: Step 3 Variable x8 Entered: R-Square = 0.7780 and C(p) = 18.9015 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 9.96253 3.32084 58.42 <.0001 Error 50 2.84198 0.05684 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 4.29102 0.17614 33.73340 593.48 <.0001 x2 0.01494 0.00194 3.35315 58.99 <.0001 x3 0.01447 0.00155 4.94031 86.92 <.0001 x8 0.42927 0.08438 1.47089 25.88 <.0001 Bounds on condition number: 1.0208, 9.1267 ---------------------------------------------------------------------------------------- Stepwise Selection: Step 4 Variable x1 Entered: R-Square = 0.8299 and C(p) = 5.7340 The SAS System 13 The REG Procedure Model: MODEL3 Dependent Variable: logy Stepwise Selection: Step 4 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 10.62670 2.65668 59.77 <.0001 Error 49 2.17781 0.04445 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.85273 0.19265 17.77526 399.94 <.0001 x1 0.07333 0.01897 0.66417 14.94 0.0003 x2 0.01419 0.00173 2.98915 67.25 <.0001 x3 0.01545 0.00140 5.44592 122.53 <.0001 x8 0.35315 0.07717 0.93070 20.94 <.0001 Bounds on condition number: 1.1026, 17.052 ---------------------------------------------------------------------------------------- Stepwise Selection: Step 5 Variable x6 Entered: R-Square = 0.8375 and C(p) = 5.5282 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 10.72329 2.14466 49.46 <.0001 Error 48 2.08122 0.04336 Corrected Total 53 12.80451 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 3.86739 0.19054 17.86320 411.99 <.0001 x1 0.07125 0.01879 0.62359 14.38 0.0004 x2 0.01390 0.00172 2.82872 65.24 <.0001 x3 0.01511 0.00140 5.07420 117.03 <.0001 x6 0.08682 0.05817 0.09659 2.23 0.1421 x8 0.36285 0.07650 0.97544 22.50 <.0001 The SAS System 14 The REG Procedure Model: MODEL3 Dependent Variable: logy Stepwise Selection: Step 5 Bounds on condition number: 1.1087, 26.833 ---------------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1500 level. No other variable met the 0.1500 significance level for entry into the model. Summary of Stepwise Selection Variable Variable Number Partial Model Step Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 x3 1 0.4273 0.4273 117.478 38.79 <.0001 2 x2 2 0.2359 0.6632 50.4918 35.72 <.0001 3 x8 3 0.1149 0.7780 18.9015 25.88 <.0001 4 x1 4 0.0519 0.8299 5.7340 14.94 0.0003 5 x6 5 0.007