data lntrans; input case x1 x2 x3 x4 y logy; cards; 01 6.7 62 81 2.59 200 2.3010 02 5.1 59 66 1.70 101 2.0043 03 7.4 57 83 2.16 204 2.3096 04 6.5 73 41 2.01 101 2.0043 05 7.8 65 115 4.30 509 2.7067 06 5.8 38 72 1.42 80 1.9031 07 5.7 46 63 1.91 80 1.9031 08 3.7 68 81 2.57 127 2.1038 09 6.0 67 93 2.50 202 2.3054 10 3.7 76 94 2.40 203 2.3075 11 6.3 84 83 4.13 329 2.5172 12 6.7 51 43 1.86 65 1.8129 13 5.8 96 114 3.95 830 2.9191 14 5.8 83 88 3.95 330 2.5185 15 7.7 62 67 3.40 168 2.2253 16 7.4 74 68 2.40 217 2.3365 17 6.0 85 28 2.98 87 1.9395 18 3.7 51 41 1.55 34 1.5315 19 7.3 68 74 3.56 215 2.3324 20 5.6 57 87 3.02 172 2.2355 21 5.2 52 76 2.85 109 2.0374 22 3.4 83 53 1.12 136 2.1335 23 6.7 26 68 2.10 70 1.8451 24 5.8 67 86 3.40 220 2.3424 25 6.3 59 100 2.95 276 2.4409 26 5.8 61 73 3.50 144 2.1584 27 5.2 52 86 2.45 181 2.2577 28 11.2 76 90 5.59 574 2.7589 29 5.2 54 56 2.71 72 1.8573 30 5.8 76 59 2.58 178 2.2504 31 3.2 64 65 0.74 71 1.8513 32 8.7 45 23 2.52 58 1.7634 33 5.0 59 73 3.50 116 2.0645 34 5.8 72 93 3.30 295 2.4698 35 5.4 58 70 2.64 115 2.0607 36 5.3 51 99 2.60 184 2.2648 37 2.6 74 86 2.05 118 2.0719 38 4.3 8 119 2.85 120 2.0792 39 4.8 61 76 2.45 151 2.1790 40 5.4 52 88 1.81 148 2.1703 41 5.2 49 72 1.84 95 1.9777 42 3.6 28 99 1.30 75 1.8751 43 8.8 86 88 6.40 483 2.6840 44 6.5 56 77 2.85 153 2.1847 45 3.4 77 93 1.48 191 2.2810 46 6.5 40 84 3.00 123 2.0899 47 4.5 73 106 3.05 311 2.4928 48 4.8 86 101 4.10 398 2.5999 49 5.1 67 77 2.86 158 2.1987 50 3.9 82 103 4.55 310 2.4914 51 6.6 77 46 1.95 124 2.0934 52 6.4 85 40 1.21 125 2.0969 53 6.4 59 85 2.33 198 2.2967 54 8.8 78 72 3.20 313 2.4955 ; proc chart; hbar x1 x2 x3 x4 y logy; proc reg simple corr; model y=x1 x2 x3 x4 /adjrsq corrb dw stb tol vif r pcorr2 scorr2 sse mse; output out=resid student=sresid; plot (y)*(x1 x2 x3 x4) (residual.)*(x1 x2 x3 x4) residual.*predicted. / hplots=2 vplots=2; proc chart data=resid; vbar sresid; proc univariate plot; var sresid; proc reg; model y=x1 x2 x3 x4 /adjrsq stb tol selection=forward sle=.05; model y=x1 x2 x3 x4 / adjrsq stb tol selection=backward sle=.1; model y=x1 x2 x3 x4 / adjrsq stb tol selection=stepwise sle=.05; proc reg simple corr; model logy=x1 x2 x3 x4 /adjrsq corrb dw stb tol vif r pcorr2 scorr2 sse mse; output out=resid2 student=sresid2; plot (logy)*(x1 x2 x3 x4) (residual.)*(x1 x2 x3 x4) residual.*predicted. / hplots=2 vplots=2; proc chart data=resid2; vbar sresid2; proc univariate plot; var sresid2; proc reg; model logy=x1 x2 x3 x4 / adjrsq stb tol selection=forward sle=.05; model logy=x1 x2 x3 x4 / adjrsq stb tol selection=backward sle=.1; model logy=x1 x2 x3 x4 / adjrsq stb tol selection=stepwise sle=.05; proc reg; model x1 = x2 x3 x4 /adjrsq stb; model x2 = x1 x3 x4 /adjrsq stb; model x3 = x1 x2 x4 /adjrsq stb; model x4 = x1 x2 x3 /adjrsq stb; run; The SAS System 14:43 Thursday, December 23, 1999 1 X1 Cum. Cum. Midpoint Freq Freq Percent Percent | 3.0 |******** 8 8 14.81 14.81 | 4.5 |************ 12 20 22.22 37.04 | 6.0 |************************* 25 45 46.30 83.33 | 7.5 |***** 5 50 9.26 92.59 | 9.0 |*** 3 53 5.56 98.15 | 10.5 |* 1 54 1.85 100.00 | -----+----+----+----+----+ 5 10 15 20 25 Frequency The SAS System 14:43 Thursday, December 23, 1999 2 X2 Cum. Cum. Midpoint Freq Freq Percent Percent | 15 |** 1 1 1.85 1.85 | 30 |**** 2 3 3.70 5.56 | 45 |********************** 11 14 20.37 25.93 | 60 |************************************ 18 32 33.33 59.26 | 75 |**************************** 14 46 25.93 85.19 | 90 |**************** 8 54 14.81 100.00 | ----+---+---+---+---+---+---+---+---+ 2 4 6 8 10 12 14 16 18 Frequency The SAS System 14:43 Thursday, December 23, 1999 3 X3 Cum. Cum. Midpoint Freq Freq Percent Percent | 22.5 |**** 2 2 3.70 3.70 | 37.5 |******** 4 6 7.41 11.11 | 52.5 |******** 4 10 7.41 18.52 | 67.5 |************************** 13 23 24.07 42.59 | 82.5 |********************************** 17 40 31.48 74.07 | 97.5 |******************** 10 50 18.52 92.59 | 112.5 |******** 4 54 7.41 100.00 | ----+---+---+---+---+---+---+---+-- 2 4 6 8 10 12 14 16 Frequency The SAS System 14:43 Thursday, December 23, 1999 4 X4 Cum. Cum. Midpoint Freq Freq Percent Percent | 1 |************ 6 6 11.11 11.11 | 2 |******************************** 16 22 29.63 40.74 | 3 |****************************************** 21 43 38.89 79.63 | 4 |**************** 8 51 14.81 94.44 | 5 |** 1 52 1.85 96.30 | 6 |**** 2 54 3.70 100.00 | ----+---+---+---+---+---+---+---+---+---+-- 2 4 6 8 10 12 14 16 18 20 Frequency The SAS System 14:43 Thursday, December 23, 1999 5 Y Cum. Cum. Midpoint Freq Freq Percent Percent | 75 |************************* 25 25 46.30 46.30 | 225 |******************* 19 44 35.19 81.48 | 375 |****** 6 50 11.11 92.59 | 525 |*** 3 53 5.56 98.15 | 675 | 0 53 0.00 98.15 | 825 |* 1 54 1.85 100.00 | -----+----+----+----+----+ 5 10 15 20 25 Frequency The SAS System 14:43 Thursday, December 23, 1999 6 LOGY Cum. Cum. Midpoint Freq Freq Percent Percent | 1.625 |** 1 1 1.85 1.85 | 1.875 |******************** 10 11 18.52 20.37 | 2.125 |************************************** 19 30 35.19 55.56 | 2.375 |********************************** 17 47 31.48 87.04 | 2.625 |********** 5 52 9.26 96.30 | 2.875 |**** 2 54 3.70 100.00 | ----+---+---+---+---+---+---+---+---+-- 2 4 6 8 10 12 14 16 18 Frequency The SAS System 14:43 Thursday, December 23, 1999 7 Descriptive Statistics Variables Sum Mean Uncorrected SS INTERCEP 54 1 54 X1 312.3 5.7833333333 1942.33 X2 3415 63.240740741 231109 X3 4164 77.111111111 345032 X4 148.19 2.7442592593 467.3919 Y 10647 197.16666667 3218165 Variables Variance Std Deviation INTERCEP 0 0 X1 2.5697169811 1.6030336806 X2 285.69566737 16.902534347 X3 451.72327044 21.253782497 X4 1.1456626485 1.0703563185 Y 21111.915094 145.29939812 Correlation CORR X1 X2 X3 X4 Y X1 1.0000 0.0901 -0.1496 0.5024 0.3725 X2 0.0901 1.0000 -0.0236 0.3690 0.5540 X3 -0.1496 -0.0236 1.0000 0.4164 0.5802 X4 0.5024 0.3690 0.4164 1.0000 0.7223 Y 0.3725 0.5540 0.5802 0.7223 1.0000 The SAS System 14:43 Thursday, December 23, 1999 8 Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 4 936264.53767 234066.13442 62.788 0.0001 Error 49 182666.96233 3727.89719 C Total 53 1118931.5 Root MSE 61.05651 R-square 0.8367 Dep Mean 197.16667 Adj R-sq 0.8234 C.V. 30.96695 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -621.597550 64.80042601 -9.592 0.0001 X1 1 33.163828 7.01727463 4.726 0.0001 X2 1 4.271860 0.56338454 7.582 0.0001 X3 1 4.125738 0.51116093 8.071 0.0001 X4 1 14.091563 12.52532754 1.125 0.2661 Squared Squared Standardized Semi-partial Partial Variable DF Estimate Corr Type II Corr Type II Tolerance INTERCEP 1 0.00000000 . . . X1 1 0.36588406 0.07441368 0.31310333 0.55586050 X2 1 0.49694120 0.19155106 0.53988110 0.77566563 X3 1 0.60349558 0.21704434 0.57072538 0.59593691 X4 1 0.10380630 0.00421697 0.02518072 0.39133899 Variance Variable DF Inflation INTERCEP 1 0.00000000 X1 1 1.79901252 X2 1 1.28921530 X3 1 1.67802997 X4 1 2.55532933 Correlation of Estimates CORRB INTERCEP X1 X2 X3 X4 INTERCEP 1.0000 -0.7167 -0.6382 -0.7516 0.5155 X1 -0.7167 1.0000 0.2464 0.4958 -0.6535 X2 -0.6382 0.2464 1.0000 0.2987 -0.4668 X3 -0.7516 0.4958 0.2987 1.0000 -0.6248 X4 0.5155 -0.6535 -0.4668 -0.6248 1.0000 The SAS System 14:43 Thursday, December 23, 1999 9 Durbin-Watson D 1.720 (For Number of Obs.) 54 1st Order Autocorrelation 0.134 The SAS System 14:43 Thursday, December 23, 1999 10 Dep Var Predict Std Err Std Err Student Obs Y Value Predict Residual Residual Residual 1 200.0 236.1 12.166 -36.1374 59.832 -0.604 2 101.0 95.8321 11.742 5.1679 59.917 0.086 3 204.0 240.2 19.814 -36.1848 57.752 -0.627 4 101.0 103.3 17.502 -2.2924 58.494 -0.039 5 509.0 449.8 21.570 59.1952 57.120 1.036 6 80.0000 50.1465 17.392 29.8535 58.527 0.510 7 80.0000 50.7782 13.299 29.2218 59.591 0.490 8 127.0 162.0 14.930 -34.9952 59.203 -0.591 9 202.0 282.5 14.780 -80.5226 59.241 -1.359 10 203.0 247.4 16.756 -44.4091 58.712 -0.756 11 329.0 346.8 15.652 -17.8052 59.016 -0.302 12 65.0000 22.0820 17.433 42.9180 58.515 0.733 13 830.0 506.8 24.210 323.2 56.052 5.765 14 330.0 344.0 15.035 -14.0437 59.177 -0.237 15 168.0 223.0 13.683 -54.9550 59.504 -0.924 16 217.0 254.3 17.396 -37.3024 58.526 -0.637 17 87.0000 98.0070 26.476 -11.0070 55.018 -0.200 18 34.0000 -90.0293 23.686 124.0 56.275 2.204 19 215.0 266.5 11.990 -51.4555 59.868 -0.859 20 172.0 209.1 10.051 -37.1116 60.224 -0.616 21 109.0 126.7 12.691 -17.7081 59.723 -0.297 22 136.0 80.1705 22.860 55.8295 56.615 0.986 23 70.0000 21.8109 21.503 48.1891 57.145 0.843 24 220.0 259.7 10.194 -39.6921 60.200 -0.659 25 276.0 293.5 14.435 -17.5182 59.326 -0.295 26 144.0 181.8 14.160 -37.8355 59.392 -0.637 27 181.0 162.3 10.802 18.6711 60.093 0.311 28 574.0 524.6 31.422 49.4130 52.350 0.944 29 72.0000 50.7643 17.528 21.2357 58.486 0.363 30 178.0 175.2 12.671 2.8112 59.727 0.047 31 71.0000 36.5265 19.833 34.4735 57.746 0.597 32 58.0000 -10.4358 28.677 68.4358 53.903 1.270 33 116.0 146.8 18.272 -30.7607 58.258 -0.528 34 295.0 308.5 11.367 -13.5224 59.989 -0.225 35 115.0 131.3 10.286 -16.2584 60.184 -0.270 36 184.0 217.1 13.783 -33.1217 59.481 -0.557 37 118.0 164.4 19.870 -46.4472 57.733 -0.805 38 120.0 86.3056 34.172 33.6944 50.598 0.666 39 151.0 146.3 10.028 4.7473 60.227 0.079 40 148.0 168.2 15.094 -20.1945 59.161 -0.341 41 95.0000 83.1571 11.924 11.8429 59.881 0.198 42 75.0000 44.1714 23.515 30.8286 56.347 0.547 43 483.0 490.9 31.129 -7.8751 52.525 -0.150 44 153.0 191.0 9.945 -38.0343 60.241 -0.631 45 191.0 224.6 22.221 -33.6418 56.869 -0.592 46 123.0 153.7 15.811 -30.6784 58.974 -0.520 47 311.0 319.8 16.127 -8.7930 58.888 -0.149 48 398.0 379.4 19.382 18.5563 57.898 0.320 49 158.0 191.7 10.007 -33.7363 60.231 -0.560 50 310.0 347.1 26.016 -37.1015 55.236 -0.672 51 124.0 143.5 17.903 -19.4794 58.373 -0.334 52 125.0 135.8 25.418 -10.8394 55.514 -0.195 53 198.0 226.2 13.676 -28.2118 59.505 -0.474 54 313.0 345.6 21.471 -32.5954 57.157 -0.570 The SAS System 14:43 Thursday, December 23, 1999 11 Cook's Obs -2-1-0 1 2 D 1 | *| | 0.003 2 | | | 0.000 3 | *| | 0.009 4 | | | 0.000 5 | |** | 0.031 6 | |* | 0.005 7 | | | 0.002 8 | *| | 0.004 9 | **| | 0.023 10 | *| | 0.009 11 | | | 0.001 12 | |* | 0.010 13 | |******| 1.240 14 | | | 0.001 15 | *| | 0.009 16 | *| | 0.007 17 | | | 0.002 18 | |**** | 0.172 19 | *| | 0.006 20 | *| | 0.002 21 | | | 0.001 22 | |* | 0.032 23 | |* | 0.020 24 | *| | 0.002 25 | | | 0.001 26 | *| | 0.005 27 | | | 0.001 28 | |* | 0.064 29 | | | 0.002 30 | | | 0.000 31 | |* | 0.008 32 | |** | 0.091 33 | *| | 0.005 34 | | | 0.000 35 | | | 0.000 36 | *| | 0.003 37 | *| | 0.015 38 | |* | 0.040 39 | | | 0.000 40 | | | 0.002 41 | | | 0.000 42 | |* | 0.010 43 | | | 0.002 44 | *| | 0.002 45 | *| | 0.011 46 | *| | 0.004 47 | | | 0.000 48 | | | 0.002 49 | *| | 0.002 50 | *| | 0.020 51 | | | 0.002 52 | | | 0.002 53 | | | 0.002 54 | *| | 0.009 The SAS System 14:43 Thursday, December 23, 1999 12 Sum of Residuals 0 Sum of Squared Residuals 182666.9623 Predicted Resid SS (Press) 247880.7920 The SAS System 14:43 Thursday, December 23, 1999 13 ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- | | | | 1000 + + 1000 + + | | | | | | | | | 1 | | 1 | 800 + + 800 + + | | | | Y | | Y | | | | | | 600 + + 600 + + | 1 | | 1 | | 1 1 | | 1 1 | | | | | 400 + 1 + 400 + 1 + | 11 | | 2 | | 11 11 1 | | 1 2 11 | | | | | 200 + 2 2 31112 + 200 + 21113 4 + | 2 11212 1 | | 12212 2 | | 1 1 1 42221 | | 1 1 2 21 3 12 1 | | 11 1 2 1 | | 1 1 21 1 | 0 + + 0 + + | | | | ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- 2 4 6 8 10 12 0 20 40 60 80 100 X1 X2 ---+----+----+----+----+----+--- -+------+------+------+------+-- | | | | 1000 + + 1000 + + | | | | | | | | | 1 | | 1 | 800 + + 800 + + | | | | Y | | Y | | | | | | 600 + + 600 + + | 1 | | 1 | | 1 1 | | 1 1 | | | | | 400 + 1 + 400 + 1 + | 11 | | 2 | | 1 1 111 | | 121 1 | | | | | 200 + 1 1 1122211 + 200 + 1 45 2 + | 1 1 1131 2 | | 2 1 2212 | | 1 1 1 1141 11 1 1 | | 224 1311 | | 1 11 1 11 | | 1 1 2 2 | 0 + + 0 + + | | | | ---+----+----+----+----+----+--- -+------+------+------+------+-- 20 40 60 80 100 120 0 2 4 6 8 X3 X4 The SAS System 14:43 Thursday, December 23, 1999 14 ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- | | | | 400 + + 400 + + | | | | | | | | | 1 | | 1 | 300 + + 300 + + | | | | R | | R | | E | | E | | S 200 + + S 200 + + I | | I | | D | | D | | U | 1 | U | 1 | A 100 + + A 100 + + L | 1 | L | 1 | | 1 2 1 1 | | 1 1 1 1 1 | | 11 11211 | | 1 1 1 111 1 1 | 0 + 112 22 1 + 0 + 1 2 21 21 + | 21 43242 2 1 | | 1 324121313 | | 1 1 21 11 | | 1112 2 | | 1 | | 1 | -100 + + -100 + + | | | | ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- 2 4 6 8 10 12 0 20 40 60 80 100 X1 X2 ---+----+----+----+----+----+--- -+------+------+------+------+-- | | | | 400 + + 400 + + | | | | | | | | | 1 | | 1 | 300 + + 300 + + | | | | R | | R | | E | | E | | S 200 + + S 200 + + I | | I | | D | | D | | U | 1 | U | 1 | A 100 + + A 100 + + L | 1 | L | 1 | | 1 1 1 1 1 | | 1 2 1 1 | | 1 2 1 1 2 1 | | 1 2 1 21 1 | 0 + 1 2 1 111 1 1 + 0 + 1 21 211 1 + | 1 1322432 21 | | 11134332 2 1 | | 112 2 1 | | 11 1 4 | | 1 | | 1 | -100 + + -100 + + | | | | ---+----+----+----+----+----+--- -+------+------+------+------+-- 20 40 60 80 100 120 0 2 4 6 8 X3 X4 The SAS System 14:43 Thursday, December 23, 1999 15 -+------+------+------+------+-- | | 400 + + | | | | | 1 | 300 + + | | R | | E | | S 200 + + I | | D | | U | 1 | A 100 + + L | 1 | | 2 1 1 1 | | 141 1 1 | 0 + 3121 1 1 + | 142251114 | | 2113 | | 1 | -100 + + | | -+------+------+------+------+-- -200 0 200 400 600 PRED The SAS System 14:43 Thursday, December 23, 1999 16 Frequency | ***** | ***** 30 + ***** | ***** | ***** | ***** | ***** 25 + ***** | ***** | ***** | ***** | ***** 20 + ***** | ***** | ***** | ***** | ***** 15 + ***** | ***** | ***** ***** | ***** ***** | ***** ***** 10 + ***** ***** | ***** ***** | ***** ***** | ***** ***** ***** | ***** ***** ***** 5 + ***** ***** ***** | ***** ***** ***** | ***** ***** ***** | ***** ***** ***** | ***** ***** ***** ***** ***** ---------------------------------------------------------------------------- -1.2 0.0 1.2 2.4 3.6 4.8 6.0 Studentized Residual The SAS System 14:43 Thursday, December 23, 1999 17 Univariate Procedure Variable=SRESID Studentized Residual Moments N 54 Sum Wgts 54 Mean 0.01449 Sum 0.782471 Std Dev 1.036889 Variance 1.07514 Skewness 3.486046 Kurtosis 17.50163 USS 56.99373 CSS 56.9824 CV 7155.8 Std Mean 0.141103 T:Mean=0 0.102692 Pr>|T| 0.9186 Num ^= 0 54 Num > 0 20 M(Sign) -7 Pr>=|M| 0.0759 Sgn Rank -137.5 Pr>=|S| 0.2400 Quantiles(Def=5) 100% Max 5.765284 99% 5.765284 75% Q3 0.363088 95% 1.269614 50% Med -0.25373 90% 0.943893 25% Q1 -0.59156 10% -0.67169 0% Min -1.35925 5% -0.85949 1% -1.35925 Range 7.12453 Q3-Q1 0.954651 Mode -1.35925 Extremes Lowest Obs Highest Obs -1.35925( 9) 0.986117( 22) -0.92356( 15) 1.036339( 5) -0.85949( 19) 1.269614( 32) -0.80452( 37) 2.203981( 18) -0.75638( 10) 5.765284( 13) The SAS System 14:43 Thursday, December 23, 1999 18 Univariate Procedure Variable=SRESID Studentized Residual Stem Leaf # Boxplot 5 8 1 * 5 4 4 3 3 2 2 2 1 0 1 1 003 3 | 0 55567789 8 | 0 0112334 7 +--+--+ -0 3333332222110 13 *-----* -0 99887766666666666555 20 +-----+ -1 4 1 | ----+----+----+----+ Normal Probability Plot 5.75+ * | | | | | | ++ 2.25+ *++++ | +++++ | +++++* * | +++******* | ++++**** | +******* | * * ************ -1.25+ * +++++ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 The SAS System 14:43 Thursday, December 23, 1999 19 Forward Selection Procedure for Dependent Variable Y Step 1 Variable X4 Entered R-square = 0.52175569 C(p) = 93.54543578 DF Sum of Squares Mean Square F Prob>F Regression 1 583808.87323857 583808.87323857 56.73 0.0001 Error 52 535122.62676143 10290.81974541 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -71.92123497 38.30031842 36287.70264864 3.53 0.0660 X4 98.05483965 13.01843184 583808.87323857 56.73 0.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Step 2 Variable X2 Entered R-square = 0.61738863 C(p) = 66.84112703 DF Sum of Squares Mean Square F Prob>F Regression 2 690815.58520223 345407.79260112 41.15 0.0001 Error 51 428115.91479777 8394.42970192 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -207.06423185 51.27700590 136885.06002868 16.31 0.0002 X2 2.86025069 0.80111375 107006.71196366 12.75 0.0008 X4 81.38681690 12.65078976 347427.38408526 41.39 0.0001 Bounds on condition number: 1.157648, 4.630594 -------------------------------------------------------------------------------- Step 3 Variable X3 Entered R-square = 0.76233507 C(p) = 25.33532953 DF Sum of Squares Mean Square F Prob>F Regression 3 853000.72546101 284333.57515367 53.46 0.0001 Error 50 265930.77453899 5318.61549078 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -402.09953137 53.97534792 295171.77519430 55.50 0.0001 X2 3.61581816 0.65218694 163481.10413865 30.74 0.0001 X3 2.92792762 0.53021756 162185.14025877 30.49 0.0001 X4 52.77323058 11.32476709 115496.12099960 21.72 0.0001 Bounds on condition number: 1.464174, 11.82183 -------------------------------------------------------------------------------- The SAS System 14:43 Thursday, December 23, 1999 20 Step 4 Variable X1 Entered R-square = 0.83674875 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 936264.53766549 234066.13441637 62.79 0.0001 Error 49 182666.96233451 3727.89719050 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001 X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001 X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001 X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001 X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- All variables have been entered into the model. Summary of Forward Selection Procedure for Dependent Variable Y Variable Number Partial Model Step Entered In R**2 R**2 C(p) F Prob>F 1 X4 1 0.5218 0.5218 93.5454 56.7310 0.0001 2 X2 2 0.0956 0.6174 66.8411 12.7473 0.0008 3 X3 3 0.1449 0.7623 25.3353 30.4939 0.0001 4 X1 4 0.0744 0.8367 5.0000 22.3353 0.0001 The SAS System 14:43 Thursday, December 23, 1999 21 Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 4 936264.53767 234066.13442 62.788 0.0001 Error 49 182666.96233 3727.89719 C Total 53 1118931.5 Root MSE 61.05651 R-square 0.8367 Dep Mean 197.16667 Adj R-sq 0.8234 C.V. 30.96695 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -621.597550 64.80042601 -9.592 0.0001 X1 1 33.163828 7.01727463 4.726 0.0001 X2 1 4.271860 0.56338454 7.582 0.0001 X3 1 4.125738 0.51116093 8.071 0.0001 X4 1 14.091563 12.52532754 1.125 0.2661 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.36588406 0.55586050 X2 1 0.49694120 0.77566563 X3 1 0.60349558 0.59593691 X4 1 0.10380630 0.39133899 The SAS System 14:43 Thursday, December 23, 1999 22 Backward Elimination Procedure for Dependent Variable Y Step 0 All Variables Entered R-square = 0.83674875 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 936264.53766549 234066.13441637 62.79 0.0001 Error 49 182666.96233451 3727.89719050 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001 X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001 X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001 X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001 X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- Step 1 Variable X4 Removed R-square = 0.83253178 C(p) = 4.26572723 DF Sum of Squares Mean Square F Prob>F Regression 3 931546.03666578 310515.34555526 82.85 0.0001 Error 50 187385.46333422 3747.70926668 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -659.17941990 55.67408110 525372.16182084 140.18 0.0001 X1 38.32274412 5.32589144 194041.43220438 51.78 0.0001 X2 4.56773202 0.49955913 313323.76018378 83.60 0.0001 X3 4.48503622 0.40017410 470760.31016036 125.61 0.0001 Bounds on condition number: 1.030813, 9.186356 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1000 level. Summary of Backward Elimination Procedure for Dependent Variable Y Variable Number Partial Model Step Removed In R**2 R**2 C(p) F Prob>F 1 X4 3 0.0042 0.8325 4.2657 1.2657 0.2661 The SAS System 14:43 Thursday, December 23, 1999 23 Model: MODEL2 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 931546.03667 310515.34556 82.855 0.0001 Error 50 187385.46333 3747.70927 C Total 53 1118931.5 Root MSE 61.21854 R-square 0.8325 Dep Mean 197.16667 Adj R-sq 0.8225 C.V. 31.04913 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -659.179420 55.67408110 -11.840 0.0001 X1 1 38.322744 5.32589144 7.196 0.0001 X2 1 4.567732 0.49955913 9.144 0.0001 X3 1 4.485036 0.40017410 11.208 0.0001 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.42280044 0.97010805 X2 1 0.53135972 0.99177367 X3 1 0.65605216 0.97750637 The SAS System 14:43 Thursday, December 23, 1999 24 Stepwise Procedure for Dependent Variable Y Step 1 Variable X4 Entered R-square = 0.52175569 C(p) = 93.54543578 DF Sum of Squares Mean Square F Prob>F Regression 1 583808.87323857 583808.87323857 56.73 0.0001 Error 52 535122.62676143 10290.81974541 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -71.92123497 38.30031842 36287.70264864 3.53 0.0660 X4 98.05483965 13.01843184 583808.87323857 56.73 0.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Step 2 Variable X2 Entered R-square = 0.61738863 C(p) = 66.84112703 DF Sum of Squares Mean Square F Prob>F Regression 2 690815.58520223 345407.79260112 41.15 0.0001 Error 51 428115.91479777 8394.42970192 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -207.06423185 51.27700590 136885.06002868 16.31 0.0002 X2 2.86025069 0.80111375 107006.71196366 12.75 0.0008 X4 81.38681690 12.65078976 347427.38408526 41.39 0.0001 Bounds on condition number: 1.157648, 4.630594 -------------------------------------------------------------------------------- Step 3 Variable X3 Entered R-square = 0.76233507 C(p) = 25.33532953 DF Sum of Squares Mean Square F Prob>F Regression 3 853000.72546101 284333.57515367 53.46 0.0001 Error 50 265930.77453899 5318.61549078 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -402.09953137 53.97534792 295171.77519430 55.50 0.0001 X2 3.61581816 0.65218694 163481.10413865 30.74 0.0001 X3 2.92792762 0.53021756 162185.14025877 30.49 0.0001 X4 52.77323058 11.32476709 115496.12099960 21.72 0.0001 Bounds on condition number: 1.464174, 11.82183 -------------------------------------------------------------------------------- The SAS System 14:43 Thursday, December 23, 1999 25 Step 4 Variable X1 Entered R-square = 0.83674875 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 936264.53766549 234066.13441637 62.79 0.0001 Error 49 182666.96233451 3727.89719050 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -621.59755029 64.80042601 343025.80578100 92.02 0.0001 X1 33.16382813 7.01727463 83263.81220448 22.34 0.0001 X2 4.27185982 0.56338454 214332.51418236 57.49 0.0001 X3 4.12573829 0.51116093 242857.75291129 65.15 0.0001 X4 14.09156259 12.52532754 4718.50099971 1.27 0.2661 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- Step 5 Variable X4 Removed R-square = 0.83253178 C(p) = 4.26572723 DF Sum of Squares Mean Square F Prob>F Regression 3 931546.03666578 310515.34555526 82.85 0.0001 Error 50 187385.46333422 3747.70926668 Total 53 1118931.5000000 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -659.17941990 55.67408110 525372.16182084 140.18 0.0001 X1 38.32274412 5.32589144 194041.43220438 51.78 0.0001 X2 4.56773202 0.49955913 313323.76018378 83.60 0.0001 X3 4.48503622 0.40017410 470760.31016036 125.61 0.0001 Bounds on condition number: 1.030813, 9.186356 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1500 level. No other variable met the 0.0500 significance level for entry into the model. Summary of Stepwise Procedure for Dependent Variable Y Variable Number Partial Model Step Entered Removed In R**2 R**2 C(p) F Prob>F 1 X4 1 0.5218 0.5218 93.5454 56.7310 0.0001 2 X2 2 0.0956 0.6174 66.8411 12.7473 0.0008 3 X3 3 0.1449 0.7623 25.3353 30.4939 0.0001 4 X1 4 0.0744 0.8367 5.0000 22.3353 0.0001 5 X4 3 0.0042 0.8325 4.2657 1.2657 0.2661 The SAS System 14:43 Thursday, December 23, 1999 26 Model: MODEL3 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 931546.03667 310515.34556 82.855 0.0001 Error 50 187385.46333 3747.70927 C Total 53 1118931.5 Root MSE 61.21854 R-square 0.8325 Dep Mean 197.16667 Adj R-sq 0.8225 C.V. 31.04913 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -659.179420 55.67408110 -11.840 0.0001 X1 1 38.322744 5.32589144 7.196 0.0001 X2 1 4.567732 0.49955913 9.144 0.0001 X3 1 4.485036 0.40017410 11.208 0.0001 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.42280044 0.97010805 X2 1 0.53135972 0.99177367 X3 1 0.65605216 0.97750637 The SAS System 14:43 Thursday, December 23, 1999 27 Descriptive Statistics Variables Sum Mean Uncorrected SS INTERCEP 54 1 54 X1 312.3 5.7833333333 1942.33 X2 3415 63.240740741 231109 X3 4164 77.111111111 345032 X4 148.19 2.7442592593 467.3919 LOGY 119.1318 2.2061444444 266.79473098 Variables Variance Std Deviation INTERCEP 0 0 X1 2.5697169811 1.6030336806 X2 285.69566737 16.902534347 X3 451.72327044 21.253782497 X4 1.1456626485 1.0703563185 LOGY 0.074957967 0.2737845267 Correlation CORR X1 X2 X3 X4 LOGY X1 1.0000 0.0901 -0.1496 0.5024 0.3464 X2 0.0901 1.0000 -0.0236 0.3690 0.5929 X3 -0.1496 -0.0236 1.0000 0.4164 0.6651 X4 0.5024 0.3690 0.4164 1.0000 0.7262 LOGY 0.3464 0.5929 0.6651 0.7262 1.0000 The SAS System 14:43 Thursday, December 23, 1999 28 Model: MODEL1 Dependent Variable: LOGY Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 4 3.86300 0.96575 431.097 0.0001 Error 49 0.10977 0.00224 C Total 53 3.97277 Root MSE 0.04733 R-square 0.9724 Dep Mean 2.20614 Adj R-sq 0.9701 C.V. 2.14541 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 0.488756 0.05023319 9.730 0.0001 X1 1 0.068520 0.00543978 12.596 0.0001 X2 1 0.009254 0.00043673 21.189 0.0001 X3 1 0.009475 0.00039625 23.910 0.0001 X4 1 0.001925 0.00970961 0.198 0.8436 Squared Squared Standardized Semi-partial Partial Variable DF Estimate Corr Type II Corr Type II Tolerance INTERCEP 1 0.00000000 . . . X1 1 0.40119198 0.08946853 0.76404020 0.55586050 X2 1 0.57131764 0.25318025 0.90160386 0.77566563 X3 1 0.73550521 0.32238275 0.92105814 0.59593691 X4 1 0.00752741 0.00002217 0.00080187 0.39133899 Variance Variable DF Inflation INTERCEP 1 0.00000000 X1 1 1.79901252 X2 1 1.28921530 X3 1 1.67802997 X4 1 2.55532933 Correlation of Estimates CORRB INTERCEP X1 X2 X3 X4 INTERCEP 1.0000 -0.7167 -0.6382 -0.7516 0.5155 X1 -0.7167 1.0000 0.2464 0.4958 -0.6535 X2 -0.6382 0.2464 1.0000 0.2987 -0.4668 X3 -0.7516 0.4958 0.2987 1.0000 -0.6248 X4 0.5155 -0.6535 -0.4668 -0.6248 1.0000 The SAS System 14:43 Thursday, December 23, 1999 29 Durbin-Watson D 2.146 (For Number of Obs.) 54 1st Order Autocorrelation -0.073 The SAS System 14:43 Thursday, December 23, 1999 30 Dep Var Predict Std Err Std Err Student Obs LOGY Value Predict Residual Residual Residual 1 2.3010 2.2940 0.009 0.00698 0.046 0.150 2 2.0043 2.0128 0.009 -0.00849 0.046 -0.183 3 2.3096 2.3138 0.015 -0.00424 0.045 -0.095 4 2.0043 2.0020 0.014 0.00229 0.045 0.050 5 2.7067 2.7226 0.017 -0.0159 0.044 -0.359 6 1.9031 1.9227 0.013 -0.0196 0.045 -0.433 7 1.9031 1.9056 0.010 -0.00248 0.046 -0.054 8 2.1038 2.1439 0.012 -0.0401 0.046 -0.875 9 2.3054 2.4058 0.011 -0.1004 0.046 -2.187 10 2.3075 2.3408 0.013 -0.0333 0.046 -0.732 11 2.5172 2.4921 0.012 0.0251 0.046 0.548 12 1.8129 1.8308 0.014 -0.0179 0.045 -0.394 13 2.9191 2.8623 0.019 0.0568 0.043 1.308 14 2.5185 2.4956 0.012 0.0229 0.046 0.499 15 2.2253 2.2315 0.011 -0.00616 0.046 -0.133 16 2.3365 2.3295 0.013 0.00700 0.045 0.154 17 1.9395 1.9575 0.021 -0.0180 0.043 -0.422 18 1.5315 1.6057 0.018 -0.0742 0.044 -1.700 19 2.3324 2.3262 0.009 0.00620 0.046 0.134 20 2.2355 2.2301 0.008 0.00545 0.047 0.117 21 2.0374 2.0518 0.010 -0.0144 0.046 -0.312 22 2.1335 1.9941 0.018 0.1394 0.044 3.176 23 1.8451 1.8368 0.017 0.00834 0.044 0.188 24 2.3424 2.3276 0.008 0.0148 0.047 0.318 25 2.4409 2.4196 0.011 0.0213 0.046 0.464 26 2.1584 2.1491 0.011 0.00935 0.046 0.203 27 2.2577 2.1458 0.008 0.1119 0.047 2.402 28 2.7589 2.8230 0.024 -0.0641 0.041 -1.579 29 1.8573 1.8806 0.014 -0.0233 0.045 -0.513 30 2.2504 2.1535 0.010 0.0969 0.046 2.094 31 1.8513 1.9176 0.015 -0.0663 0.045 -1.480 32 1.7634 1.7241 0.022 0.0393 0.042 0.941 33 2.0645 2.0757 0.014 -0.0112 0.045 -0.249 34 2.4698 2.4400 0.009 0.0298 0.047 0.642 35 2.0607 2.0638 0.008 -0.00310 0.047 -0.067 36 2.2648 2.2669 0.011 -0.00206 0.046 -0.045 37 2.0719 2.1705 0.015 -0.0986 0.045 -2.202 38 2.0792 1.9904 0.026 0.0888 0.039 2.264 39 2.1790 2.1069 0.008 0.0721 0.047 1.544 40 2.1703 2.1772 0.012 -0.00692 0.046 -0.151 41 1.9777 1.9842 0.009 -0.00652 0.046 -0.141 42 1.8751 1.9350 0.018 -0.0599 0.044 -1.372 43 2.6840 2.7337 0.024 -0.0497 0.041 -1.220 44 2.1847 2.1874 0.008 -0.00269 0.047 -0.058 45 2.2810 2.3183 0.017 -0.0373 0.044 -0.845 46 2.0899 2.1059 0.012 -0.0160 0.046 -0.351 47 2.4928 2.4828 0.013 0.00998 0.046 0.219 48 2.5999 2.5783 0.015 0.0216 0.045 0.481 49 2.1987 2.1933 0.008 0.00542 0.047 0.116 50 2.4914 2.4995 0.020 -0.00806 0.043 -0.188 51 2.0934 2.0931 0.014 0.000261 0.045 0.006 52 2.0969 2.0952 0.020 0.00170 0.043 0.040 53 2.2967 2.2831 0.011 0.0136 0.046 0.295 54 2.4955 2.5019 0.017 -0.00638 0.044 -0.144 The SAS System 14:43 Thursday, December 23, 1999 31 Cook's Obs -2-1-0 1 2 D 1 | | | 0.000 2 | | | 0.000 3 | | | 0.000 4 | | | 0.000 5 | | | 0.004 6 | | | 0.003 7 | | | 0.000 8 | *| | 0.010 9 | ****| | 0.060 10 | *| | 0.009 11 | |* | 0.004 12 | | | 0.003 13 | |** | 0.064 14 | | | 0.003 15 | | | 0.000 16 | | | 0.000 17 | | | 0.008 18 | ***| | 0.102 19 | | | 0.000 20 | | | 0.000 21 | | | 0.001 22 | |******| 0.329 23 | | | 0.001 24 | | | 0.001 25 | | | 0.003 26 | | | 0.000 27 | |**** | 0.037 28 | ***| | 0.180 29 | *| | 0.005 30 | |**** | 0.039 31 | **| | 0.052 32 | |* | 0.050 33 | | | 0.001 34 | |* | 0.003 35 | | | 0.000 36 | | | 0.000 37 | ****| | 0.115 38 | |**** | 0.468 39 | |*** | 0.013 40 | | | 0.000 41 | | | 0.000 42 | **| | 0.066 43 | **| | 0.105 44 | | | 0.000 45 | *| | 0.022 46 | | | 0.002 47 | | | 0.001 48 | | | 0.005 49 | | | 0.000 50 | | | 0.002 51 | | | 0.000 52 | | | 0.000 53 | | | 0.001 54 | | | 0.001 The SAS System 14:43 Thursday, December 23, 1999 32 Sum of Residuals 0 Sum of Squared Residuals 0.1098 Predicted Resid SS (Press) 0.1456 The SAS System 14:43 Thursday, December 23, 1999 33 -+-----+-----+-----+-----+-----+- -+-----+-----+-----+-----+-----+- | | | | 3.0 + + 3.0 + + | 1 | | 1 | | | | | | 1 | | 1 | | 1 1 | | 1 1 | | | | | L | 1 | L | 1 | O 2.5 + 1 1 2 1 1 + O 2.5 + 21 3 + G | 1 | G | 1 | Y | 1 2 | Y | 2 1 | | 11 21111 1 | | 111111 3 | | 11 1 1 1 | | 2111 | | 1 11 | | 1 1 1 | | 1 1 1 12 12 | | 1 1 111 1 11 1 | 2.0 + 11 1 + 2.0 + 1 1 1 + | 21 | | 1 1 1 | | 11 1 1 | | 2 1 1 | | 1 1 | | 11 | | | | | | | | | | | | | 1.5 + 1 + 1.5 + 1 + -+-----+-----+-----+-----+-----+- -+-----+-----+-----+-----+-----+- 2 4 6 8 10 12 0 20 40 60 80 100 X1 X2 -+-----+-----+-----+-----+-----+- --+---+---+---+---+---+---+---+-- | | | | 3.0 + + 3.0 + + | 1 | | 1 | | | | | | 1 | | 1 | | 1 1 | | 1 1 | | | | | L | 1 | L | 1 | O 2.5 + 1 11 1 11 + O 2.5 + 12 111 + G | 1 | G | 1 | Y | 1 1 1 | Y | 1 2 | | 1 112 3 1 | | 1 26 | | 1 3 1 | | 121 1 | | 1 1 1 | | 1 1 1 | | 1 1 111111 1| | 1 2 131 1 | 2.0 + 1 1 1 + 2.0 + 21 + | 1 1 1 | | 1 1 1 | | 1 2 1 | | 1 1 1 1 | | 1 1 | | 1 1 | | | | | | | | | | | | | 1.5 + 1 + 1.5 + 1 + -+-----+-----+-----+-----+-----+- --+---+---+---+---+---+---+---+-- 20 40 60 80 100 120 0 1 2 3 4 5 6 7 X3 X4 The SAS System 14:43 Thursday, December 23, 1999 34 ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- | | | | 0.2 + + 0.2 + + | | | | | | | | | | | | | 1 | | 1 | | | | | R | 1 | R | 1 | E 0.1 + 1 + E 0.1 + 1 + S | 1 | S | 1 | I | 1 | I | 1 | D | 1 | D | 1 | U | 1 | U | 1 | A | 1 21 | A | 1 21 | L | 1 221 | L | 1 3 11 | 0.0 + 34 3213 1 + 0.0 + 2231221211 + | 1 3 211 1 | | 2 2 21 2 | | 1 1 | | 1 1 | | 2 1 | | 1 1 1 | | 1 1 | | 1 1 | | 11 | | 1 1 | | | | | -0.1 + 1 1 + -0.1 + 1 1 + ---+----+----+----+----+----+--- ---+----+----+----+----+----+--- 2 4 6 8 10 12 0 20 40 60 80 100 X1 X2 ---+----+----+----+----+----+--- -+------+------+------+------+-- | | | | 0.2 + + 0.2 + + | | | | | | | | | | | | | 1 | | 1 | | | | | R | 1 | R | 1 | E 0.1 + 1 + E 0.1 + 1 + S | 1 | S | 1 | I | 1 | I | 1 | D | 1 | D | 1 | U | 1 | U | 1 | A | 111 1 | A | 1 3 | L | 11 11 1 1 | L | 11 112 | 0.0 + 2 1 1233112 1 + 0.0 + 1 2323222 + | 1 1 121 1 1 1 | | 111 211 11 | | 1 1 | | 11 | | 1 11 | | 1 1 1 | | 1 1 | | 1 1 | | 1 1 | | 1 1 | | | | | -0.1 + 11 + -0.1 + 1 1 + ---+----+----+----+----+----+--- -+------+------+------+------+-- 20 40 60 80 100 120 0 2 4 6 8 X3 X4 The SAS System 14:43 Thursday, December 23, 1999 35 -----+------+------+------+----- | | 0.2 + + | | | | | | | 1 | | | R | 1 | E 0.1 + 1 + S | 1 | I | 1 | D | 1 | U | 1 | A | 121 | L | 1 1 1111 | 0.0 + 1231432 1 + | 1213 1 1 | | 1 1 | | 1 1 1 | | 1 1 | | 1 1 | | | -0.1 + 1 1 + -----+------+------+------+----- 1.5 2.0 2.5 3.0 PRED The SAS System 14:43 Thursday, December 23, 1999 36 Frequency | ***** | ***** | ***** | ***** 30 + ***** | ***** | ***** | ***** | ***** 25 + ***** | ***** | ***** | ***** | ***** 20 + ***** | ***** | ***** | ***** | ***** 15 + ***** | ***** | ***** | ***** | ***** 10 + ***** | ***** | ***** | ***** ***** | ***** ***** 5 + ***** ***** | ***** ***** ***** ***** ***** | ***** ***** ***** ***** ***** | ***** ***** ***** ***** ***** | ***** ***** ***** ***** ***** ***** ------------------------------------------------------------------------- -2 -1 0 1 2 3 Studentized Residual The SAS System 14:43 Thursday, December 23, 1999 37 Univariate Procedure Variable=SRESID2 Studentized Residual Moments N 54 Sum Wgts 54 Mean -0.00244 Sum -0.13196 Std Dev 1.029039 Variance 1.058921 Skewness 0.627116 Kurtosis 1.794059 USS 56.12312 CSS 56.1228 CV -42110.8 Std Mean 0.140034 T:Mean=0 -0.01745 Pr>|T| 0.9861 Num ^= 0 54 Num > 0 25 M(Sign) -2 Pr>=|M| 0.6835 Sgn Rank -49.5 Pr>=|S| 0.6740 Quantiles(Def=5) 100% Max 3.175723 99% 3.175723 75% Q3 0.294823 95% 2.26434 50% Med -0.05574 90% 1.307878 25% Q1 -0.39434 10% -1.37195 0% Min -2.20248 5% -1.70045 1% -2.20248 Range 5.378208 Q3-Q1 0.689165 Mode -2.20248 Extremes Lowest Obs Highest Obs -2.20248( 37) 1.54351( 39) -2.18732( 9) 2.093907( 30) -1.70045( 18) 2.26434( 38) -1.5787( 28) 2.402034( 27) -1.48006( 31) 3.175723( 22) Stem Leaf # Boxplot 3 2 1 * 2 2 134 3 0 1 5 1 0 1 3 1 | 0 555569 6 | 0 0011112222233 13 +-----+ -0 444443222211111110 18 *--+--* -0 9875 4 | -1 42 2 | -1 765 3 0 -2 22 2 0 ----+----+----+----+ The SAS System 14:43 Thursday, December 23, 1999 38 Univariate Procedure Variable=SRESID2 Studentized Residual Normal Probability Plot 3.25+ * | + | * * *+++++ | *+++++ | +++*+ | +++++*** | ********* | ********** | ***++++ | ***++ | ++*+* -2.25+ +*+++* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 The SAS System 14:43 Thursday, December 23, 1999 39 Forward Selection Procedure for Dependent Variable LOGY Step 1 Variable X4 Entered R-square = 0.52737486 C(p) =788.14813627 DF Sum of Squares Mean Square F Prob>F Regression 1 2.09514023 2.09514023 58.02 0.0001 Error 52 1.87763202 0.03610831 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 1.69638489 0.07174324 20.18803240 559.10 0.0001 X4 0.18575488 0.02438581 2.09514023 58.02 0.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Step 2 Variable X3 Entered R-square = 0.68653440 C(p) =507.89639740 DF Sum of Squares Mean Square F Prob>F Regression 2 2.72744481 1.36372240 55.85 0.0001 Error 51 1.24532745 0.02441819 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 1.38877757 0.08446782 6.60079464 270.32 0.0001 X3 0.00565254 0.00111080 0.63230458 25.89 0.0001 X4 0.13901497 0.02205693 0.96994304 39.72 0.0001 Bounds on condition number: 1.209789, 4.839155 -------------------------------------------------------------------------------- Step 3 Variable X2 Entered R-square = 0.88290076 C(p) =161.66248928 DF Sum of Squares Mean Square F Prob>F Regression 3 3.50756363 1.16918788 125.66 0.0001 Error 50 0.46520862 0.00930417 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.94226338 0.07138961 1.62088545 174.21 0.0001 X2 0.00789866 0.00086260 0.78011882 83.85 0.0001 X3 0.00699974 0.00070128 0.92694429 99.63 0.0001 X4 0.08184608 0.01497852 0.27780226 29.86 0.0001 Bounds on condition number: 1.464174, 11.82183 -------------------------------------------------------------------------------- The SAS System 14:43 Thursday, December 23, 1999 40 Step 4 Variable X1 Entered R-square = 0.97236929 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 3.86300172 0.96575043 431.10 0.0001 Error 49 0.10977053 0.00224021 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001 X1 0.06852018 0.00543978 0.35543809 158.66 0.0001 X2 0.00925411 0.00043673 1.00582746 448.99 0.0001 X3 0.00947455 0.00039625 1.28075323 571.71 0.0001 X4 0.00192542 0.00970961 0.00008809 0.04 0.8436 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- All variables have been entered into the model. Summary of Forward Selection Procedure for Dependent Variable LOGY Variable Number Partial Model Step Entered In R**2 R**2 C(p) F Prob>F 1 X4 1 0.5274 0.5274 788.1481 58.0238 0.0001 2 X3 2 0.1592 0.6865 507.8964 25.8948 0.0001 3 X2 3 0.1964 0.8829 161.6625 83.8461 0.0001 4 X1 4 0.0895 0.9724 5.0000 158.6625 0.0001 The SAS System 14:43 Thursday, December 23, 1999 41 Model: MODEL1 Dependent Variable: LOGY Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 4 3.86300 0.96575 431.097 0.0001 Error 49 0.10977 0.00224 C Total 53 3.97277 Root MSE 0.04733 R-square 0.9724 Dep Mean 2.20614 Adj R-sq 0.9701 C.V. 2.14541 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 0.488756 0.05023319 9.730 0.0001 X1 1 0.068520 0.00543978 12.596 0.0001 X2 1 0.009254 0.00043673 21.189 0.0001 X3 1 0.009475 0.00039625 23.910 0.0001 X4 1 0.001925 0.00970961 0.198 0.8436 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.40119198 0.55586050 X2 1 0.57131764 0.77566563 X3 1 0.73550521 0.59593691 X4 1 0.00752741 0.39133899 The SAS System 14:43 Thursday, December 23, 1999 42 Backward Elimination Procedure for Dependent Variable LOGY Step 0 All Variables Entered R-square = 0.97236929 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 3.86300172 0.96575043 431.10 0.0001 Error 49 0.10977053 0.00224021 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001 X1 0.06852018 0.00543978 0.35543809 158.66 0.0001 X2 0.00925411 0.00043673 1.00582746 448.99 0.0001 X3 0.00947455 0.00039625 1.28075323 571.71 0.0001 X4 0.00192542 0.00970961 0.00008809 0.04 0.8436 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- Step 1 Variable X4 Removed R-square = 0.97234711 C(p) = 3.03932316 DF Sum of Squares Mean Square F Prob>F Regression 3 3.86291363 1.28763788 586.04 0.0001 Error 50 0.10985863 0.00219717 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.48362086 0.04262871 0.28279376 128.71 0.0001 X1 0.06922508 0.00407795 0.63315225 288.17 0.0001 X2 0.00929454 0.00038250 1.29732180 590.45 0.0001 X3 0.00952364 0.00030641 2.12262768 966.07 0.0001 Bounds on condition number: 1.030813, 9.186356 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1000 level. Summary of Backward Elimination Procedure for Dependent Variable LOGY Variable Number Partial Model Step Removed In R**2 R**2 C(p) F Prob>F 1 X4 3 0.0000 0.9723 3.0393 0.0393 0.8436 The SAS System 14:43 Thursday, December 23, 1999 43 Model: MODEL2 Dependent Variable: LOGY Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 3.86291 1.28764 586.043 0.0001 Error 50 0.10986 0.00220 C Total 53 3.97277 Root MSE 0.04687 R-square 0.9723 Dep Mean 2.20614 Adj R-sq 0.9707 C.V. 2.12470 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 0.483621 0.04262871 11.345 0.0001 X1 1 0.069225 0.00407795 16.975 0.0001 X2 1 0.009295 0.00038250 24.299 0.0001 X3 1 0.009524 0.00030641 31.082 0.0001 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.40531921 0.97010805 X2 1 0.57381347 0.99177367 X3 1 0.73931630 0.97750637 The SAS System 14:43 Thursday, December 23, 1999 44 Stepwise Procedure for Dependent Variable LOGY Step 1 Variable X4 Entered R-square = 0.52737486 C(p) =788.14813627 DF Sum of Squares Mean Square F Prob>F Regression 1 2.09514023 2.09514023 58.02 0.0001 Error 52 1.87763202 0.03610831 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 1.69638489 0.07174324 20.18803240 559.10 0.0001 X4 0.18575488 0.02438581 2.09514023 58.02 0.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Step 2 Variable X3 Entered R-square = 0.68653440 C(p) =507.89639740 DF Sum of Squares Mean Square F Prob>F Regression 2 2.72744481 1.36372240 55.85 0.0001 Error 51 1.24532745 0.02441819 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 1.38877757 0.08446782 6.60079464 270.32 0.0001 X3 0.00565254 0.00111080 0.63230458 25.89 0.0001 X4 0.13901497 0.02205693 0.96994304 39.72 0.0001 Bounds on condition number: 1.209789, 4.839155 -------------------------------------------------------------------------------- Step 3 Variable X2 Entered R-square = 0.88290076 C(p) =161.66248928 DF Sum of Squares Mean Square F Prob>F Regression 3 3.50756363 1.16918788 125.66 0.0001 Error 50 0.46520862 0.00930417 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.94226338 0.07138961 1.62088545 174.21 0.0001 X2 0.00789866 0.00086260 0.78011882 83.85 0.0001 X3 0.00699974 0.00070128 0.92694429 99.63 0.0001 X4 0.08184608 0.01497852 0.27780226 29.86 0.0001 Bounds on condition number: 1.464174, 11.82183 -------------------------------------------------------------------------------- The SAS System 14:43 Thursday, December 23, 1999 45 Step 4 Variable X1 Entered R-square = 0.97236929 C(p) = 5.00000000 DF Sum of Squares Mean Square F Prob>F Regression 4 3.86300172 0.96575043 431.10 0.0001 Error 49 0.10977053 0.00224021 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.48875592 0.05023319 0.21207636 94.67 0.0001 X1 0.06852018 0.00543978 0.35543809 158.66 0.0001 X2 0.00925411 0.00043673 1.00582746 448.99 0.0001 X3 0.00947455 0.00039625 1.28075323 571.71 0.0001 X4 0.00192542 0.00970961 0.00008809 0.04 0.8436 Bounds on condition number: 2.555329, 29.28635 -------------------------------------------------------------------------------- Step 5 Variable X4 Removed R-square = 0.97234711 C(p) = 3.03932316 DF Sum of Squares Mean Square F Prob>F Regression 3 3.86291363 1.28763788 586.04 0.0001 Error 50 0.10985863 0.00219717 Total 53 3.97277225 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.48362086 0.04262871 0.28279376 128.71 0.0001 X1 0.06922508 0.00407795 0.63315225 288.17 0.0001 X2 0.00929454 0.00038250 1.29732180 590.45 0.0001 X3 0.00952364 0.00030641 2.12262768 966.07 0.0001 Bounds on condition number: 1.030813, 9.186356 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1500 level. No other variable met the 0.0500 significance level for entry into the model. Summary of Stepwise Procedure for Dependent Variable LOGY Variable Number Partial Model Step Entered Removed In R**2 R**2 C(p) F Prob>F 1 X4 1 0.5274 0.5274 788.1481 58.0238 0.0001 2 X3 2 0.1592 0.6865 507.8964 25.8948 0.0001 3 X2 3 0.1964 0.8829 161.6625 83.8461 0.0001 4 X1 4 0.0895 0.9724 5.0000 158.6625 0.0001 5 X4 3 0.0000 0.9723 3.0393 0.0393 0.8436 The SAS System 14:43 Thursday, December 23, 1999 46 Model: MODEL3 Dependent Variable: LOGY Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 3.86291 1.28764 586.043 0.0001 Error 50 0.10986 0.00220 C Total 53 3.97277 Root MSE 0.04687 R-square 0.9723 Dep Mean 2.20614 Adj R-sq 0.9707 C.V. 2.12470 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 0.483621 0.04262871 11.345 0.0001 X1 1 0.069225 0.00407795 16.975 0.0001 X2 1 0.009295 0.00038250 24.299 0.0001 X3 1 0.009524 0.00030641 31.082 0.0001 Standardized Variable DF Estimate Tolerance INTERCEP 1 0.00000000 . X1 1 0.40531921 0.97010805 X2 1 0.57381347 0.99177367 X3 1 0.73931630 0.97750637 The SAS System 14:43 Thursday, December 23, 1999 47 Model: MODEL1 Dependent Variable: X1 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 60.48958 20.16319 13.317 0.0001 Error 50 75.70542 1.51411 C Total 53 136.19500 Root MSE 1.23049 R-square 0.4441 Dep Mean 5.78333 Adj R-sq 0.4108 C.V. 21.27650 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 6.618597 0.91069854 7.268 0.0001 X2 1 -0.019782 0.01100402 -1.798 0.0783 X3 1 -0.036118 0.00894609 -4.037 0.0002 X4 1 1.166381 0.19107702 6.104 0.0001 Standardized Variable DF Estimate INTERCEP 1 0.00000000 X2 1 -0.20858157 X3 1 -0.47886942 X4 1 0.77880058 The SAS System 14:43 Thursday, December 23, 1999 48 Model: MODEL2 Dependent Variable: X2 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 3396.84194 1132.28065 4.820 0.0050 Error 50 11745.02843 234.90057 C Total 53 15141.87037 Root MSE 15.32647 R-square 0.2243 Dep Mean 63.24074 Adj R-sq 0.1778 C.V. 24.23512 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 73.410773 12.52232622 5.862 0.0001 X1 1 -3.068979 1.70717633 -1.798 0.0783 X3 1 -0.271051 0.12245259 -2.214 0.0315 X4 1 10.377992 2.78054642 3.732 0.0005 Standardized Variable DF Estimate INTERCEP 1 0.00000000 X1 1 -0.29106144 X3 1 -0.34082752 X4 1 0.65718839 The SAS System 14:43 Thursday, December 23, 1999 49 Model: MODEL3 Dependent Variable: X3 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 9673.80903 3224.60301 11.300 0.0001 Error 50 14267.52430 285.35049 C Total 53 23941.33333 Root MSE 16.89232 R-square 0.4041 Dep Mean 77.11111 Adj R-sq 0.3683 C.V. 21.90647 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 95.287238 11.82476250 8.058 0.0001 X1 1 -6.806834 1.68598977 -4.037 0.0002 X2 1 -0.329264 0.14875190 -2.214 0.0315 X4 1 15.309410 2.70574448 5.658 0.0001 Standardized Variable DF Estimate INTERCEP 1 0.00000000 X1 1 -0.51339493 X2 1 -0.26185471 X4 1 0.77099327 The SAS System 14:43 Thursday, December 23, 1999 50 Model: MODEL4 Dependent Variable: X4 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 36.95797 12.31932 25.922 0.0001 Error 50 23.76215 0.47524 C Total 53 60.72012 Root MSE 0.68938 R-square 0.6087 Dep Mean 2.74426 Adj R-sq 0.5852 C.V. 25.12076 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -2.666977 0.62694289 -4.254 0.0001 X1 1 0.366100 0.05997458 6.104 0.0001 X2 1 0.020996 0.00562551 3.732 0.0005 X3 1 0.025497 0.00450634 5.658 0.0001 Standardized Variable DF Estimate INTERCEP 1 0.00000000 X1 1 0.54829410 X2 1 0.33156483 X3 1 0.50629475