EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 1 15:05 Tuesday, August 14, 2001 The CORR Procedure 10 Variables: X1 X2 X3 X4 X5 X6 X7 X8 X9 Y Simple Statistics Variable N Mean Std Dev Sum X1 25 5.44240 0.81695 136.06000 X2 25 0.69520 0.12587 17.38000 X3 25 6.35600 1.75406 158.90000 X4 25 30.48000 0.77028 762.00000 X5 25 20.24000 3.01773 506.00000 X6 25 9.16000 10.28219 229.00000 X7 25 52.60000 17.26560 1315 X8 25 43.36400 23.19897 1084 X9 25 4.32000 0.85245 108.00000 Y 25 9.42400 1.63064 235.60000 Simple Statistics Variable Minimum Maximum X1 3.45000 6.72000 X2 0.42000 0.95000 X3 3.70000 10.30000 X4 28.00000 31.00000 X5 11.00000 23.00000 X6 0 28.00000 X7 28.10000 76.70000 X8 13.70000 106.10000 X9 2.00000 6.00000 Y 6.36000 12.51000 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 2 15:05 Tuesday, August 14, 2001 The CORR Procedure Pearson Correlation Coefficients, N = 25 Prob > |r| under H0: Rho=0 X1 X2 X3 X4 X5 X1 1.00000 0.94364 -0.12609 0.38213 0.68509 <.0001 0.5481 0.0594 0.0002 X2 0.94364 1.00000 -0.14367 0.24823 0.76446 <.0001 0.4932 0.2315 <.0001 X3 -0.12609 -0.14367 1.00000 -0.31677 0.23114 0.5481 0.4932 0.1229 0.2663 X4 0.38213 0.24823 -0.31677 1.00000 0.02008 0.0594 0.2315 0.1229 0.9241 X5 0.68509 0.76446 0.23114 0.02008 1.00000 0.0002 <.0001 0.2663 0.9241 X6 -0.19112 -0.22636 0.55810 -0.20475 0.11688 0.3601 0.2766 0.0037 0.3262 0.5779 X7 -0.00188 0.06774 -0.61634 0.07738 -0.20976 0.9929 0.7477 0.0010 0.7131 0.3142 X8 -0.13135 -0.13419 0.98996 -0.32100 0.21248 0.5314 0.5225 <.0001 0.1177 0.3079 X9 0.61630 0.60130 0.07390 -0.05330 0.60059 0.0010 0.0015 0.7255 0.8002 0.0015 Y 0.38318 0.30555 0.47431 0.13674 0.53612 0.0587 0.1375 0.0166 0.5146 0.0057 Pearson Correlation Coefficients, N = 25 Prob > |r| under H0: Rho=0 X6 X7 X8 X9 Y X1 -0.19112 -0.00188 -0.13135 0.61630 0.38318 0.3601 0.9929 0.5314 0.0010 0.0587 X2 -0.22636 0.06774 -0.13419 0.60130 0.30555 0.2766 0.7477 0.5225 0.0015 0.1375 X3 0.55810 -0.61634 0.98996 0.07390 0.47431 0.0037 0.0010 <.0001 0.7255 0.0166 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 3 15:05 Tuesday, August 14, 2001 The CORR Procedure Pearson Correlation Coefficients, N = 25 Prob > |r| under H0: Rho=0 X6 X7 X8 X9 Y X4 -0.20475 0.07738 -0.32100 -0.05330 0.13674 0.3262 0.7131 0.1177 0.8002 0.5146 X5 0.11688 -0.20976 0.21248 0.60059 0.53612 0.5779 0.3142 0.3079 0.0015 0.0057 X6 1.00000 -0.85761 0.49151 0.11751 0.64065 <.0001 0.0126 0.5759 0.0006 X7 -0.85761 1.00000 -0.54142 -0.23695 -0.84524 <.0001 0.0052 0.2541 <.0001 X8 0.49151 -0.54142 1.00000 0.02842 0.39454 0.0126 0.0052 0.8927 0.0510 X9 0.11751 -0.23695 0.02842 1.00000 0.38212 0.5759 0.2541 0.8927 0.0594 Y 0.64065 -0.84524 0.39454 0.38212 1.00000 0.0006 <.0001 0.0510 0.0594 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 4 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 9 58.94665 6.54963 20.18 <.0001 Error 15 4.86915 0.32461 Corrected Total 24 63.81580 Root MSE 0.56975 R-Square 0.9237 Dependent Mean 9.42400 Adj R-Sq 0.8779 Coeff Var 6.04569 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Type I SS Intercept 1 1.89421 6.99636 0.27 0.7903 2220.29440 X1 1 0.70541 0.56490 1.25 0.2309 9.36977 X2 1 -1.89372 4.14629 -0.46 0.6544 1.82950 X3 1 1.13422 0.74609 1.52 0.1493 16.96202 X4 1 0.11876 0.20461 0.58 0.5702 0.72591 X5 1 0.17935 0.08095 2.22 0.0426 5.56947 X6 1 -0.01818 0.02451 -0.74 0.4697 13.20264 X7 1 -0.07742 0.01659 -4.67 0.0003 9.89370 X8 1 -0.08585 0.05200 -1.65 0.1195 0.52327 X9 1 -0.34501 0.21070 -1.64 0.1223 0.87036 Parameter Estimates Variable DF Type II SS Intercept 1 0.02379 X1 1 0.50617 X2 1 0.06771 X3 1 0.75020 X4 1 0.10936 X5 1 1.59339 X6 1 0.17859 X7 1 7.06721 X8 1 0.88475 X9 1 0.87036 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 5 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Dependent Variable: Y Output Statistics Dep Var Predicted Std Error Std Error Student Obs Y Value Mean Predict Residual Residual Residual 1 10.9800 10.8518 0.2492 0.1282 0.512 0.250 2 11.1300 10.4257 0.3279 0.7043 0.466 1.512 3 12.5100 12.2055 0.3045 0.3045 0.482 0.632 4 8.4000 8.2015 0.4677 0.1985 0.325 0.610 5 9.2700 9.3399 0.2964 -0.0699 0.487 -0.144 6 8.7300 8.4601 0.4041 0.2699 0.402 0.672 7 6.3600 5.9452 0.4536 0.4148 0.345 1.203 8 8.5000 8.2329 0.3193 0.2671 0.472 0.566 9 7.8200 8.1715 0.3082 -0.3515 0.479 -0.733 10 9.1400 9.2395 0.3034 -0.0995 0.482 -0.206 11 8.2400 8.6292 0.5253 -0.3892 0.221 -1.764 12 12.1900 11.5147 0.4003 0.6753 0.405 1.666 13 11.8800 11.4877 0.3269 0.3923 0.467 0.841 14 9.5700 9.3254 0.4315 0.2446 0.372 0.657 15 10.9400 10.8647 0.3368 0.0753 0.460 0.164 16 9.5800 9.8470 0.3161 -0.2670 0.474 -0.563 17 10.0900 9.7165 0.3480 0.3735 0.451 0.828 18 8.1100 8.3238 0.3148 -0.2138 0.475 -0.450 19 6.8300 6.8987 0.4704 -0.0687 0.321 -0.214 20 8.8800 8.4303 0.4166 0.4497 0.389 1.157 21 7.6800 7.8379 0.2565 -0.1579 0.509 -0.310 22 8.4700 8.9414 0.4011 -0.4714 0.405 -1.165 23 8.8600 10.0913 0.2535 -1.2313 0.510 -2.413 24 10.3600 11.0558 0.2384 -0.6958 0.517 -1.345 25 11.0800 11.5619 0.3371 -0.4819 0.459 -1.049 Output Statistics Cook's Obs -2-1 0 1 2 D 1 | | | 0.001 2 | |*** | 0.113 3 | |* | 0.016 4 | |* | 0.077 5 | | | 0.001 6 | |* | 0.046 7 | |** | 0.250 8 | |* | 0.015 9 | *| | 0.022 10 | | | 0.002 11 | ***| | 1.763 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 6 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Dependent Variable: Y Output Statistics Cook's Obs -2-1 0 1 2 D 12 | |*** | 0.270 13 | |* | 0.035 14 | |* | 0.058 15 | | | 0.001 16 | *| | 0.014 17 | |* | 0.041 18 | | | 0.009 19 | | | 0.010 20 | |** | 0.154 21 | | | 0.002 22 | **| | 0.133 23 | ****| | 0.144 24 | **| | 0.038 25 | **| | 0.059 Sum of Residuals 0 Sum of Squared Residuals 4.86915 Predicted Residual SS (PRESS) 18.99533 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 7 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Test TEST1 Results for Dependent Variable Y Mean Source DF Square F Value Pr > F Numerator 1 0.50617 1.56 0.2309 Denominator 15 0.32461 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 8 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Test TEST2 Results for Dependent Variable Y Mean Source DF Square F Value Pr > F Numerator 2 0.49518 1.53 0.2494 Denominator 15 0.32461 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 9 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1 Test TEST3 Results for Dependent Variable Y Mean Source DF Square F Value Pr > F Numerator 1 0.10177 0.31 0.5838 Denominator 15 0.32461 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 10 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1.1 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 57.70623 8.24375 22.94 <.0001 Error 17 6.10957 0.35939 Corrected Total 24 63.81580 Root MSE 0.59949 R-Square 0.9043 Dependent Mean 9.42400 Adj R-Sq 0.8648 Coeff Var 6.36130 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 4.88489 6.50830 0.75 0.4632 X1 1 0.35909 0.29357 1.22 0.2379 X4 1 0.17191 0.20370 0.84 0.4104 X5 1 0.17125 0.06617 2.59 0.0191 X6 1 -0.01608 0.02570 -0.63 0.5398 X7 1 -0.08928 0.01588 -5.62 <.0001 X8 1 -0.00573 0.00705 -0.81 0.4275 X9 1 -0.23821 0.21384 -1.11 0.2808 EXAMPLE 4. MULTIPLE LINEAR REGRESSION: STEAM DATA 11 15:05 Tuesday, August 14, 2001 The REG Procedure Model: MODEL1.2 Dependent Variable: Y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 8 58.19645 7.27456 20.71 <.0001 Error 16 5.61935 0.35121 Corrected Total 24 63.81580 Root MSE 0.59263 R-Square 0.9119 Dependent Mean 9.42400 Adj R-Sq 0.8679 Coeff Var 6.28851 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 6.46443 6.57127 0.98 0.3399 X1 1 0.93380 0.56643 1.65 0.1187 X2 1 -4.60134 3.89465 -1.18 0.2547 X4 1 0.09318 0.21211 0.44 0.6663 X5 1 0.22300 0.07872 2.83 0.0120 X6 1 -0.01857 0.02549 -0.73 0.4769 X7 1 -0.08771 0.01575 -5.57 <.0001 X8 1 -0.00749 0.00713 -1.05 0.3091 X9 1 -0.27055 0.21315 -1.27 0.2225