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OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex67; INPUT strength length height @@;

6-3 Multiple Regression. OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex67; INPUT strength length height @@; label strength='Pull Strength' length='Wire length' height='Die Height'; CARDS; 9.95 2 50 24.45 8 110 31.75 11 120 35 10 550

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OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex67; INPUT strength length height @@;

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  1. 6-3 Multiple Regression OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex67; INPUT strength length height @@; label strength='Pull Strength' length='Wire length' height='Die Height'; CARDS; 9.95 2 50 24.45 8 110 31.75 11 120 35 10 550 25.02 8 295 16.86 4 200 14.38 2 375 9.6 2 52 24.35 9 100 27.5 8 300 17.08 4 412 37 11 400 41.95 12 500 11.66 2 360 21.65 4 205 17.89 4 400 69 20 600 10.3 1 585 34.93 10 540 46.59 15 250 44.88 15 290 54.12 16 510 56.63 17 590 22.13 6 100 21.15 5 400 PROC SGSCATTER data=ex67; MATRIX STRENGTH LENGTH HEIGHT; TITLE 'Scatter Plot Matrix for Wire Bond Data'; PROC REG data=ex67; MODEL strength=length height/xpx r CLB CLM CLI influence; TITLE 'Multiple Regression'; ods graphics on; procreg data=ex67 plots(label)=(CooksDRStudentByLeverage DFFITS DFBETAS); model strength=length height; run; ods graphics off; DATA EX67N; INPUT LENGTH HEIGHT @@; DATALINES; 11 35 5 20 DATA EX67N1; SET EX67 EX67N; PROC REG DATA=EX67N1; MODEL STRENGTH=LENGTH HEIGHT/CLM CLI; TITLE 'CIs FOR MEAN RESPONSE AND FUTURE OBSERVATION'; RUN; QUIT; Example 6-7 • PLOT npp.*Residual.; /* Normal Probability Plot */ • PLOT RESIDual.*Pred.; /* Residual Plot */ • PLOT Residual.*length; • PLOT Residual.*height;

  2. 6-3 Multiple Regression

  3. 6-3 Multiple Regression • The REG Procedure • Model: MODEL1 • Model Crossproducts X'X X'Y Y'Y • Variable Label Intercept length height strength • Intercept Intercept 25 206 8294 725.82 • length Wire length 206 2396 77177 8008.47 • height Die Height 8294 77177 3531848 274816.71 • strength 725.82 8008.47 274816.71 27178.5316 • ------------------------------------------------------------------------------------------------------------------------------------------------------- • The REG Procedure • Model: MODEL1 • Dependent Variable: strength • Number of Observations Read 25 • Number of Observations Used 25 • Analysis of Variance • Sum of Mean • Source DF Squares Square F Value Pr > F • Model 2 5990.77122 2995.38561 572.17 <.0001 • Error 22 115.17348 5.23516 • Corrected Total 24 6105.94470 • Root MSE 2.28805 R-Square 0.9811 • Dependent Mean 29.03280 Adj R-Sq 0.9794 • CoeffVar 7.88090 • Parameter Estimates • Parameter Standard • Variable Label DF Estimate Error t Value Pr > |t| 95% Confidence Limits • Intercept Intercept 1 2.26379 1.06007 2.14 0.0441 0.06535 4.46223 • length Wire length 1 2.74427 0.09352 29.34 <.0001 2.55031 2.93823 • height Die Height 1 0.01253 0.00280 4.48 0.0002 0.00672 0.01833

  4. 6-3 Multiple Regression Multiple Regression The REG Procedure Model: MODEL1 Dependent Variable: strength Output Statistics Dependent Predicted Std Error Std Error Student Cook's Obs Variable Value Mean Predict 95% CL Mean 95% CL Predict Residual ResidualResidual -2-1 0 1 2 D 1 9.9500 8.3787 0.9074 6.4968 10.2606 3.2740 13.4834 1.5713 2.100 0.748 | |* | 0.035 2 24.4500 25.5960 0.7645 24.0105 27.1815 20.5930 30.5990 -1.1460 2.157 -0.531 | *| | 0.012 3 31.7500 33.9541 0.8620 32.1665 35.7417 28.8834 39.0248 -2.2041 2.119 -1.040 | **| | 0.060 4 35.0000 36.5968 0.7303 35.0821 38.1114 31.6158 41.5778 -1.5968 2.168 -0.736 | *| | 0.021 5 25.0200 27.9137 0.4677 26.9437 28.8836 23.0704 32.7569 -2.8937 2.240 -1.292 | **| | 0.024 6 16.8600 15.7464 0.6261 14.4481 17.0448 10.8269 20.6660 1.1136 2.201 0.506 | |* | 0.007 7 14.3800 12.4503 0.7862 10.8198 14.0807 7.4328 17.4677 1.9297 2.149 0.898 | |* | 0.036 8 9.6000 8.4038 0.9039 6.5291 10.2784 3.3018 13.5058 1.1962 2.102 0.569 | |* | 0.020 9 24.3500 28.2150 0.8185 26.5175 29.9125 23.1754 33.2546 -3.8650 2.137 -1.809 | ***| | 0.160 10 27.5000 27.9763 0.4651 27.0118 28.9408 23.1341 32.8184 -0.4763 2.240 -0.213 | | | 0.001 11 17.0800 18.4023 0.6960 16.9588 19.8458 13.4425 23.3621 -1.3223 2.180 -0.607 | *| | 0.013 12 37.0000 37.4619 0.5246 36.3739 38.5498 32.5936 42.3301 -0.4619 2.227 -0.207 | | | 0.001 13 41.9500 41.4589 0.6553 40.0999 42.8179 36.5230 46.3948 0.4911 2.192 0.224 | | | 0.001 14 11.6600 12.2623 0.7689 10.6678 13.8568 7.2565 17.2682 -0.6023 2.155 -0.280 | | | 0.003 15 21.6500 15.8091 0.6213 14.5206 17.0976 10.8921 20.7260 5.8409 2.202 2.652 | |***** | 0.187 16 17.8900 18.2520 0.6785 16.8448 19.6592 13.3026 23.2014 -0.3620 2.185 -0.166 | | | 0.001 17 69.0000 64.6659 1.1652 62.2494 67.0824 59.3409 69.9909 4.3341 1.969 2.201 | |**** | 0.565 18 10.3000 12.3368 1.2383 9.7689 14.9048 6.9414 17.7323 -2.0368 1.924 -1.059 | **| | 0.155 19 34.9300 36.4715 0.7096 34.9999 37.9431 31.5034 41.4396 -1.5415 2.175 -0.709 | *| | 0.018 20 46.5900 46.5598 0.8780 44.7389 48.3807 41.4773 51.6423 0.0302 2.113 0.0143 | | | 0.000 21 44.8800 47.0609 0.8238 45.3524 48.7694 42.0176 52.1042 -2.1809 2.135 -1.022 | **| | 0.052 22 54.1200 52.5613 0.8432 50.8127 54.3099 47.5042 57.6183 1.5587 2.127 0.733 | |* | 0.028 23 56.6300 56.3078 0.9771 54.2814 58.3342 51.1481 61.4675 0.3222 2.069 0.156 | | | 0.002 24 22.1300 19.9822 0.7557 18.4149 21.5494 14.9850 24.9794 2.1478 2.160 0.995 | |* | 0.040 25 21.1500 20.9963 0.6176 19.7153 22.2772 16.0813 25.9112 0.1537 2.203 0.0698 | | | 0.000

  5. 6-3 Multiple Regression Multiple Regression The REG Procedure Model: MODEL1 Dependent Variable: strength Output Statistics Hat DiagCov -------------DFBETAS------------- ObsRStudent H Ratio DFFITS Intercept length height 1 0.7404 0.1573 1.2629 0.3199 0.3179 -0.1005 -0.2001 2 -0.5226 0.1116 1.2451 -0.1853 -0.1403 -0.0515 0.1483 3 -1.0419 0.1419 1.1519 -0.4237 -0.2219 -0.2371 0.3393 4 -0.7285 0.1019 1.1879 -0.2454 0.0788 0.0223 -0.1843 5 -1.3131 0.0418 0.9470 -0.2742 -0.1572 -0.0097 0.0553 6 0.4973 0.0749 1.1999 0.1415 0.1301 -0.0581 -0.0494 7 0.8940 0.1181 1.1655 0.3271 0.1480 -0.2618 0.1422 8 0.5602 0.1561 1.3031 0.2409 0.2395 -0.0766 -0.1498 9 -1.9155 0.1280 0.8133 -0.7338 -0.5012 -0.2837 0.6056 10 -0.2079 0.0413 1.1919 -0.0432 -0.0241 -0.0010 0.0075 11 -0.5978 0.0925 1.2045 -0.1909 -0.0602 0.1321 -0.1027 12 -0.2028 0.0526 1.2065 -0.0478 0.0018 -0.0169 -0.0085 13 0.2191 0.0820 1.2440 0.0655 -0.0224 0.0173 0.0338 14 -0.2736 0.1129 1.2824 -0.0976 -0.0484 0.0779 -0.0381 15 3.1422 0.0737 0.3906 0.8866 0.8097 -0.3743 -0.2920 16 -0.1620 0.0879 1.2559 -0.0503 -0.0178 0.0347 -0.0253 17 2.4352 0.2593 0.7361 1.4410 -0.8514 1.0089 0.4136 18 -1.0617 0.2929 1.3899 -0.6833 -0.0219 0.5216 -0.5324 19 -0.7004 0.0962 1.1870 -0.2285 0.0701 0.0180 -0.1676 20 0.0140 0.1473 1.3483 0.0058 0.0006 0.0048 -0.0031 21 -1.0228 0.1296 1.1418 -0.3947 -0.0085 -0.3241 0.1706 22 0.7249 0.1358 1.2354 0.2873 -0.1326 0.1830 0.0764 23 0.1522 0.1824 1.4016 0.0719 -0.0413 0.0402 0.0304 24 0.9943 0.1091 1.1242 0.3479 0.3085 0.0165 -0.2621 25 0.0682 0.0729 1.2393 0.0191 0.0063 -0.0116 0.0095 Sum of Residuals 0 Sum of Squared Residuals 115.17348 Predicted Residual SS (PRESS) 156.16295

  6. 6-3 Multiple Regression Multiple Regression The REG Procedure Model: MODEL1 Dependent Variable: strength Pull Strength Number of Observations Read 25 Number of Observations Used 25 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 5990.77122 2995.38561 572.17 <.0001 Error 22 115.17348 5.23516 Corrected Total 24 6105.94470 Root MSE 2.28805 R-Square 0.9811 Dependent Mean 29.03280 Adj R-Sq 0.9794 CoeffVar 7.88090 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 2.26379 1.06007 2.14 0.0441 length Wire length 1 2.74427 0.09352 29.34 <.0001 height Die Height 1 0.01253 0.00280 4.48 0.0002

  7. 6-3 Multiple Regression

  8. 6-3 Multiple Regression

  9. 6-3 Multiple Regression

  10. 6-3 Multiple Regression

  11. 6-3 Multiple Regression

  12. 6-3 Multiple Regression

  13. 6-3 Multiple Regression CIs FOR MEAN RESPONSE AND FUTURE OBSERVATION The REG Procedure Model: MODEL1 Dependent Variable: strength Pull Strength Number of Observations Read 27 Number of Observations Used 25 Number of Observations with Missing Values 2 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 5990.77122 2995.38561 572.17 <.0001 Error 22 115.17348 5.23516 Corrected Total 24 6105.94470 Root MSE 2.28805 R-Square 0.9811 Dependent Mean 29.03280 Adj R-Sq 0.9794 CoeffVar 7.88090 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 2.26379 1.06007 2.14 0.0441 length Wire length 1 2.74427 0.09352 29.34 <.0001 height Die Height 1 0.01253 0.00280 4.48 0.0002

  14. 6-3 Multiple Regression CIs FOR MEAN RESPONSE AND FUTURE OBSERVATION The REG Procedure Model: MODEL1 Dependent Variable: strength Pull Strength Output Statistics Dependent Predicted Std Error Obs Variable Value Mean Predict 95% CL Mean 95% CL Predict Residual 1 9.9500 8.3787 0.9074 6.4968 10.2606 3.2740 13.4834 1.5713 2 24.4500 25.5960 0.7645 24.0105 27.1815 20.5930 30.5990 -1.1460 3 31.7500 33.9541 0.8620 32.1665 35.7417 28.8834 39.0248 -2.2041 4 35.0000 36.5968 0.7303 35.0821 38.1114 31.6158 41.5778 -1.5968 5 25.0200 27.9137 0.4677 26.9437 28.8836 23.0704 32.7569 -2.8937 6 16.8600 15.7464 0.6261 14.4481 17.0448 10.8269 20.6660 1.1136 7 14.3800 12.4503 0.7862 10.8198 14.0807 7.4328 17.4677 1.9297 8 9.6000 8.4038 0.9039 6.5291 10.2784 3.3018 13.5058 1.1962 9 24.3500 28.2150 0.8185 26.5175 29.9125 23.1754 33.2546 -3.8650 10 27.5000 27.9763 0.4651 27.0118 28.9408 23.1341 32.8184 -0.4763 11 17.0800 18.4023 0.6960 16.9588 19.8458 13.4425 23.3621 -1.3223 12 37.0000 37.4619 0.5246 36.3739 38.5498 32.5936 42.3301 -0.4619 13 41.9500 41.4589 0.6553 40.0999 42.8179 36.5230 46.3948 0.4911 14 11.6600 12.2623 0.7689 10.6678 13.8568 7.2565 17.2682 -0.6023 15 21.6500 15.8091 0.6213 14.5206 17.0976 10.8921 20.7260 5.8409 16 17.8900 18.2520 0.6785 16.8448 19.6592 13.3026 23.2014 -0.3620 17 69.0000 64.6659 1.1652 62.2494 67.0824 59.3409 69.9909 4.3341 18 10.3000 12.3368 1.2383 9.7689 14.9048 6.9414 17.7323 -2.0368 19 34.9300 36.4715 0.7096 34.9999 37.9431 31.5034 41.4396 -1.5415 20 46.5900 46.5598 0.8780 44.7389 48.3807 41.4773 51.6423 0.0302 21 44.8800 47.0609 0.8238 45.3524 48.7694 42.0176 52.1042 -2.1809 22 54.1200 52.5613 0.8432 50.8127 54.3099 47.5042 57.6183 1.5587 23 56.6300 56.3078 0.9771 54.2814 58.3342 51.1481 61.4675 0.3222 24 22.1300 19.9822 0.7557 18.4149 21.5494 14.9850 24.9794 2.1478 25 21.1500 20.9963 0.6176 19.7153 22.2772 16.0813 25.9112 0.1537 26 . 32.8892 1.0620 30.6867 35.0918 27.6579 38.1206 . 27 . 16.2357 0.9286 14.3099 18.1615 11.1147 21.3567 . Sum of Residuals 0 Sum of Squared Residuals 115.17348 Predicted Residual SS (PRESS) 156.16295

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