RPD - Example 7.1 - Linthurst Data 1 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio R-Square Selection Method Number of Observations Read 45 Number of Observations Used 45 Number in Adjusted Model R-Square R-Square C(p) AIC MSE SBC 1 0.5994 0.5900 7.4206 546.1394 178618 549.75274 1 0.3899 0.3757 32.7381 565.0662 272011 568.67948 1 0.0740 0.0525 70.9131 583.8403 412835 587.45364 1 0.0419 0.0196 74.7978 585.3758 427165 588.98916 1 0.0106 -.0124 78.5729 586.8195 441091 590.43282 ----------------------------------------------------------------------------- 2 0.6584 0.6422 2.2816 540.9615 155909 546.38153 2 0.6476 0.6308 3.5937 542.3696 160865 547.78961 2 0.6083 0.5896 8.3430 547.1266 178801 552.54660 2 0.6034 0.5845 8.9331 547.6841 181030 553.10405 2 0.5526 0.5313 15.0704 553.1057 204209 558.52571 2 0.4300 0.4029 29.8862 564.0033 260164 569.42329 2 0.4152 0.3874 31.6782 565.1590 266932 570.57897 2 0.0776 0.0337 72.4807 585.6661 421031 591.08610 2 0.0743 0.0303 72.8748 585.8249 422520 591.24491 2 0.0534 0.0083 75.4034 586.8307 432069 592.25065 ----------------------------------------------------------------------------- 3 0.6625 0.6378 3.7960 542.4291 157833 549.65573 3 0.6604 0.6355 4.0487 542.7070 158811 549.93363 3 0.6590 0.6341 4.2073 542.8805 159424 550.10718 3 0.6522 0.6267 5.0374 543.7780 162636 551.00468 3 0.6521 0.6266 5.0482 543.7895 162677 551.01615 3 0.6366 0.6101 6.9149 545.7442 169900 552.97089 3 0.5765 0.5455 14.1850 552.6379 198026 559.86457 3 0.5644 0.5326 15.6428 553.9016 203666 561.12824 3 0.4300 0.3883 31.8861 566.0032 266509 573.22987 3 0.0776 0.0101 74.4796 587.6657 431296 594.89230 ----------------------------------------------------------------------------- 4 0.6749 0.6423 4.2964 542.7437 155832 551.77705 4 0.6718 0.6389 4.6696 543.1691 157312 552.20242 4 0.6642 0.6306 5.5882 544.1993 160955 553.23257 4 0.6617 0.6279 5.8863 544.5286 162137 553.56189 4 0.5773 0.5350 16.0873 554.5519 202589 563.58518 ----------------------------------------------------------------------------- 5 0.6773 0.6359 6.0000 544.4031 158622 555.24304 RPD - Example 7.1 - Linthurst Data 2 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio R-Square Selection Method Number in Model R-Square Variables in Model 1 0.5994 ph 1 0.3899 zn 1 0.0740 na 1 0.0419 k 1 0.0106 sal --------------------------------------- 2 0.6584 ph na 2 0.6476 ph k 2 0.6083 ph zn 2 0.6034 sal ph 2 0.5526 sal zn 2 0.4300 na zn 2 0.4152 k zn 2 0.0776 sal na 2 0.0743 k na 2 0.0534 sal k --------------------------------------- 3 0.6625 ph na zn 3 0.6604 ph k na 3 0.6590 sal ph na 3 0.6522 sal ph k 3 0.6521 ph k zn 3 0.6366 sal ph zn 3 0.5765 sal k zn 3 0.5644 sal na zn 3 0.4300 k na zn 3 0.0776 sal k na --------------------------------------- 4 0.6749 sal ph k zn 4 0.6718 sal ph na zn 4 0.6642 ph k na zn 4 0.6617 sal ph k na 4 0.5773 sal k na zn --------------------------------------- 5 0.6773 sal ph k na zn RPD - Example 7.1 - Linthurst Data 3 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Number of Observations Read 45 Number of Observations Used 45 Forward Selection: Step 1 Variable ph Entered: R-Square = 0.5994 and C(p) = 7.4206 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 11490388 11490388 64.33 <.0001 Error 43 7680575 178618 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -885.21051 243.44073 2361739 13.22 0.0007 ph 409.80431 51.09422 11490388 64.33 <.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Forward Selection: Step 2 Variable na Entered: R-Square = 0.6584 and C(p) = 2.2816 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 12622789 6311394 40.48 <.0001 Error 42 6548174 155909 Corrected Total 44 19170963 RPD - Example 7.1 - Linthurst Data 4 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Forward Selection: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -475.72558 273.52269 471625 3.03 0.0893 ph 404.94819 47.76984 11203720 71.86 <.0001 na -0.02333 0.00866 1132401 7.26 0.0101 Bounds on condition number: 1.0014, 4.0057 -------------------------------------------------------------------------------- Forward Selection: Step 3 Variable zn Entered: R-Square = 0.6625 and C(p) = 3.7960 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 12699815 4233272 26.82 <.0001 Error 41 6471149 157833 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -195.42611 486.54995 25463 0.16 0.6900 ph 369.78511 69.59675 4455726 28.23 <.0001 na -0.02253 0.00878 1038493 6.58 0.0141 zn -7.36781 10.54677 77026 0.49 0.4888 Bounds on condition number: 2.1259, 15.733 -------------------------------------------------------------------------------- Forward Selection: Step 4 RPD - Example 7.1 - Linthurst Data 5 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Forward Selection: Step 4 Variable sal Entered: R-Square = 0.6718 and C(p) = 4.6696 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 12878489 3219622 20.47 <.0001 Error 40 6292475 157312 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 936.78645 1168.15684 101167 0.64 0.4273 sal -24.29157 22.79321 178674 1.14 0.2929 ph 314.20826 86.87467 2057834 13.08 0.0008 na -0.01924 0.00930 673405 4.28 0.0451 zn -18.32554 14.71677 243922 1.55 0.2203 Bounds on condition number: 4.1529, 42.365 -------------------------------------------------------------------------------- Forward Selection: Step 5 Variable k Entered: R-Square = 0.6773 and C(p) = 6.0000 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 12984700 2596940 16.37 <.0001 Error 39 6186263 158622 Corrected Total 44 19170963 RPD - Example 7.1 - Linthurst Data 6 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Forward Selection: Step 5 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 1252.27938 1234.75049 163158 1.03 0.3167 sal -30.28508 24.03135 251921 1.59 0.2151 ph 305.52544 87.87869 1917306 12.09 0.0013 k -0.28512 0.34843 106211 0.67 0.4182 na -0.00867 0.01593 47011 0.30 0.5893 zn -20.67639 15.05460 299209 1.89 0.1775 Bounds on condition number: 4.3099, 80.873 -------------------------------------------------------------------------------- All variables have been entered 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 ph 1 0.5994 0.5994 7.4206 64.33 <.0001 2 na 2 0.0591 0.6584 2.2816 7.26 0.0101 3 zn 3 0.0040 0.6625 3.7960 0.49 0.4888 4 sal 4 0.0093 0.6718 4.6696 1.14 0.2929 5 k 5 0.0055 0.6773 6.0000 0.67 0.4182 RPD - Example 7.1 - Linthurst Data 7 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Number of Observations Read 45 Number of Observations Used 45 Backward Elimination: Step 0 All Variables Entered: R-Square = 0.6773 and C(p) = 6.0000 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 12984700 2596940 16.37 <.0001 Error 39 6186263 158622 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 1252.27938 1234.75049 163158 1.03 0.3167 sal -30.28508 24.03135 251921 1.59 0.2151 ph 305.52544 87.87869 1917306 12.09 0.0013 k -0.28512 0.34843 106211 0.67 0.4182 na -0.00867 0.01593 47011 0.30 0.5893 zn -20.67639 15.05460 299209 1.89 0.1775 Bounds on condition number: 4.3099, 80.873 -------------------------------------------------------------------------------- Backward Elimination: Step 1 Variable na Removed: R-Square = 0.6749 and C(p) = 4.2964 RPD - Example 7.1 - Linthurst Data 8 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Backward Elimination: Step 1 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 12937689 3234422 20.76 <.0001 Error 40 6233274 155832 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept 1505.48820 1133.69403 274802 1.76 0.1917 sal -35.94326 21.47611 436496 2.80 0.1020 ph 293.86111 84.47376 1885805 12.10 0.0012 k -0.43882 0.20238 732606 4.70 0.0361 zn -23.45191 14.03989 434796 2.79 0.1027 Bounds on condition number: 3.8156, 39.099 -------------------------------------------------------------------------------- Backward Elimination: Step 2 Variable zn Removed: R-Square = 0.6522 and C(p) = 5.0374 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 12502893 4167631 25.63 <.0001 Error 41 6668070 162636 Corrected Total 44 19170963 RPD - Example 7.1 - Linthurst Data 9 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Backward Elimination: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -131.24547 582.52494 8255.73731 0.05 0.8229 sal -12.05733 16.36921 88239 0.54 0.4656 ph 410.20688 48.82724 11478835 70.58 <.0001 k -0.49006 0.20437 935178 5.75 0.0211 Bounds on condition number: 1.003, 9.0203 -------------------------------------------------------------------------------- Backward Elimination: Step 3 Variable sal Removed: R-Square = 0.6476 and C(p) = 3.5937 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 12414654 6207327 38.59 <.0001 Error 42 6756309 160865 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -506.97735 279.77138 528239 3.28 0.0771 ph 412.03955 48.49753 11611782 72.18 <.0001 k -0.48710 0.20321 924266 5.75 0.0211 Bounds on condition number: 1.0004, 4.0015 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1000 level. RPD - Example 7.1 - Linthurst Data 10 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Summary of Backward Elimination Variable Number Partial Model Step Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 na 4 0.0025 0.6749 4.2964 0.30 0.5893 2 zn 3 0.0227 0.6522 5.0374 2.79 0.1027 3 sal 2 0.0046 0.6476 3.5937 0.54 0.4656 RPD - Example 7.1 - Linthurst Data 11 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Number of Observations Read 45 Number of Observations Used 45 Stepwise Selection: Step 1 Variable ph Entered: R-Square = 0.5994 and C(p) = 7.4206 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 11490388 11490388 64.33 <.0001 Error 43 7680575 178618 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -885.21051 243.44073 2361739 13.22 0.0007 ph 409.80431 51.09422 11490388 64.33 <.0001 Bounds on condition number: 1, 1 -------------------------------------------------------------------------------- Stepwise Selection: Step 2 Variable na Entered: R-Square = 0.6584 and C(p) = 2.2816 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 12622789 6311394 40.48 <.0001 Error 42 6548174 155909 Corrected Total 44 19170963 RPD - Example 7.1 - Linthurst Data 12 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Stepwise Selection: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -475.72558 273.52269 471625 3.03 0.0893 ph 404.94819 47.76984 11203720 71.86 <.0001 na -0.02333 0.00866 1132401 7.26 0.0101 Bounds on condition number: 1.0014, 4.0057 -------------------------------------------------------------------------------- Stepwise Selection: Step 3 Variable zn Entered: R-Square = 0.6625 and C(p) = 3.7960 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 12699815 4233272 26.82 <.0001 Error 41 6471149 157833 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -195.42611 486.54995 25463 0.16 0.6900 ph 369.78511 69.59675 4455726 28.23 <.0001 na -0.02253 0.00878 1038493 6.58 0.0141 zn -7.36781 10.54677 77026 0.49 0.4888 Bounds on condition number: 2.1259, 15.733 -------------------------------------------------------------------------------- Stepwise Selection: Step 4 RPD - Example 7.1 - Linthurst Data 13 10:26 Monday, July 27, 2009 The REG Procedure Model: MODEL1 Dependent Variable: bio Stepwise Selection: Step 4 Variable zn Removed: R-Square = 0.6584 and C(p) = 2.2816 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 12622789 6311394 40.48 <.0001 Error 42 6548174 155909 Corrected Total 44 19170963 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -475.72558 273.52269 471625 3.03 0.0893 ph 404.94819 47.76984 11203720 71.86 <.0001 na -0.02333 0.00866 1132401 7.26 0.0101 Bounds on condition number: 1.0014, 4.0057 -------------------------------------------------------------------------------- All variables left in the model are significant at the 0.1500 level. The stepwise method terminated because the next variable to be entered was just removed. 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 ph 1 0.5994 0.5994 7.4206 64.33 <.0001 2 na 2 0.0591 0.6584 2.2816 7.26 0.0101 3 zn 3 0.0040 0.6625 3.7960 0.49 0.4888 4 zn 2 0.0040 0.6584 2.2816 0.49 0.4888