1. Model selection Stepwise regression

2. Statement of problem A common problem is that there is a large set of candidate predictor variables. (Note: The examples herein are really not that large.) Goal is to choose a small subset from the larger set so that the resulting regression model is simple, yet have good predictive ability.

3. Example: Cement data Response y: heat evolved in calories during hardening of cement on a per gram basis Predictor x 1 : % of tricalcium aluminate Predictor x 2 : % of tricalcium silicate Predictor x 3 : % of tetracalcium alumino ferrite Predictor x 4 : % of dicalcium silicate

4. Example: Cement data

5. Two basic methods of selecting predictors Stepwise regression: Enter and remove predictors, in a stepwise manner, until there is no justifiable reason to enter or remove more. Best subsets regression: Select the subset of predictors that do the best at meeting some well-defined objective criterion.

6. Stepwise regression: the idea Start with no predictors in the “stepwise model.” At each step, enter or remove a predictor based on partial F-tests (that is, the t-tests). Stop when no more predictors can be justifiably entered or removed from the stepwise model.

7. Stepwise regression: Preliminary steps 1.Specify an Alpha-to-Enter (α E = 0.15) significance level. 2.Specify an Alpha-to-Remove (α R = 0.15) significance level.

8. Stepwise regression: Step #1 1.Fit each of the one-predictor models, that is, regress y on x 1, regress y on x 2, … regress y on x p-1. 2.The first predictor put in the stepwise model is the predictor that has the smallest t-test P-value (below α E = 0.15). 3.If no P-value < 0.15, stop.

9. Stepwise regression: Step #2 1.Suppose x 1 was the “best” one predictor. 2.Fit each of the two-predictor models with x 1 in the model, that is, regress y on (x 1, x 2 ), regress y on (x 1, x 3 ), …, and y on (x 1, x p-1 ). 3.The second predictor put in stepwise model is the predictor that has the smallest t-test P-value (below α E = 0.15). 4.If no P-value < 0.15, stop.

10. Stepwise regression: Step #2 (continued) 1.Suppose x 2 was the “best” second predictor. 2.Step back and check P-value for β 1 = 0. If the P-value for β 1 = 0 has become not significant (above α R = 0.15), remove x 1 from the stepwise model.

11. Stepwise regression: Step #3 1.Suppose both x 1 and x 2 made it into the two-predictor stepwise model. 2.Fit each of the three-predictor models with x 1 and x 2 in the model, that is, regress y on (x 1, x 2, x 3 ), regress y on (x 1, x 2, x 4 ), …, and regress y on (x 1, x 2, x p-1 ).

12. Stepwise regression: Step #3 (continued) 1.The third predictor put in stepwise model is the predictor that has the smallest t-test P-value (below α E = 0.15). 2.If no P-value < 0.15, stop. 3.Step back and check P-values for β 1 = 0 and β 2 = 0. If either P-value has become not significant (above α R = 0.15), remove the predictor from the stepwise model.

13. Stepwise regression: Stopping the procedure The procedure is stopped when adding an additional predictor does not yield a t-test P-value below α E = 0.15.

14. Example: Cement data

15. Predictor Coef SE Coef T P Constant 81.479 4.927 16.54 0.000 x1 1.8687 0.5264 3.55 0.005 Predictor Coef SE Coef T P Constant 57.424 8.491 6.76 0.000 x2 0.7891 0.1684 4.69 0.001 Predictor Coef SE Coef T P Constant 110.203 7.948 13.87 0.000 x3 -1.2558 0.5984 -2.10 0.060 Predictor Coef SE Coef T P Constant 117.568 5.262 22.34 0.000 x4 -0.7382 0.1546 -4.77 0.001

16. Predictor Coef SE Coef T P Constant 103.097 2.124 48.54 0.000 x4 -0.61395 0.04864 -12.62 0.000 x1 1.4400 0.1384 10.40 0.000 Predictor Coef SE Coef T P Constant 94.16 56.63 1.66 0.127 x4 -0.4569 0.6960 -0.66 0.526 x2 0.3109 0.7486 0.42 0.687 Predictor Coef SE Coef T P Constant 131.282 3.275 40.09 0.000 x4 -0.72460 0.07233 -10.02 0.000 x3 -1.1999 0.1890 -6.35 0.000

17. Predictor Coef SE Coef T P Constant 71.65 14.14 5.07 0.001 x4 -0.2365 0.1733 -1.37 0.205 x1 1.4519 0.1170 12.41 0.000 x2 0.4161 0.1856 2.24 0.052 Predictor Coef SE Coef T P Constant 111.684 4.562 24.48 0.000 x4 -0.64280 0.04454 -14.43 0.000 x1 1.0519 0.2237 4.70 0.001 x3 -0.4100 0.1992 -2.06 0.070

18. Predictor Coef SE Coef T P Constant 52.577 2.286 23.00 0.000 x1 1.4683 0.1213 12.10 0.000 x2 0.66225 0.04585 14.44 0.000

19. Predictor Coef SE Coef T P Constant 71.65 14.14 5.07 0.001 x1 1.4519 0.1170 12.41 0.000 x2 0.4161 0.1856 2.24 0.052 x4 -0.2365 0.1733 -1.37 0.205 Predictor Coef SE Coef T P Constant 48.194 3.913 12.32 0.000 x1 1.6959 0.2046 8.29 0.000 x2 0.65691 0.04423 14.85 0.000 x3 0.2500 0.1847 1.35 0.209

20. Predictor Coef SE Coef T P Constant 52.577 2.286 23.00 0.000 x1 1.4683 0.1213 12.10 0.000 x2 0.66225 0.04585 14.44 0.000

21. Stepwise Regression: y versus x1, x2, x3, x4 Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15 Response is y on 4 predictors, with N = 13 Step 1 2 3 4 Constant 117.57 103.10 71.65 52.58 x4 -0.738 -0.614 -0.237 T-Value -4.77 -12.62 -1.37 P-Value 0.001 0.000 0.205 x1 1.44 1.45 1.47 T-Value 10.40 12.41 12.10 P-Value 0.000 0.000 0.000 x2 0.416 0.662 T-Value 2.24 14.44 P-Value 0.052 0.000 S 8.96 2.73 2.31 2.41 R-Sq 67.45 97.25 98.23 97.87 R-Sq(adj) 64.50 96.70 97.64 97.44 C-p 138.7 5.5 3.0 2.7

22. Caution about stepwise regression! Do not jump to the conclusion … –that all the important predictor variables for predicting y have been identified, or –that all the unimportant predictor variables have been eliminated.

23. Caution about stepwise regression! Many t-tests for β k = 0 are conducted in a stepwise regression procedure. The probability is high … –that we included some unimportant predictors –that we excluded some important predictors

24. Drawbacks of stepwise regression The final model is not guaranteed to be optimal in any specified sense. The procedure yields a single final model, although in practice there are often several equally good models. It doesn’t take into account a researcher’s knowledge about the predictors.

25. Example: Modeling PIQ

26. Stepwise Regression: PIQ versus MRI, Height, Weight Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15 Response is PIQ on 3 predictors, with N = 38 Step 1 2 Constant 4.652 111.276 MRI 1.18 2.06 T-Value 2.45 3.77 P-Value 0.019 0.001 Height -2.73 T-Value -2.75 P-Value 0.009 S 21.2 19.5 R-Sq 14.27 29.49 R-Sq(adj) 11.89 25.46 C-p 7.3 2.0

27. The regression equation is PIQ = 111 + 2.06 MRI - 2.73 Height Predictor Coef SE Coef T P Constant 111.28 55.87 1.99 0.054 MRI 2.0606 0.5466 3.77 0.001 Height -2.7299 0.9932 -2.75 0.009 S = 19.51 R-Sq = 29.5% R-Sq(adj) = 25.5% Analysis of Variance Source DF SS MS F P Regression 2 5572.7 2786.4 7.32 0.002 Error 35 13321.8 380.6 Total 37 18894.6 Source DF Seq SS MRI 1 2697.1 Height 1 2875.6

28. Example: Modeling BP

29. Stepwise Regression: BP versus Age, Weight, BSA, Duration, Pulse, Stress Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15 Response is BP on 6 predictors, with N = 20 Step 1 2 3 Constant 2.205 -16.579 -13.667 Weight 1.201 1.033 0.906 T-Value 12.92 33.15 18.49 P-Value 0.000 0.000 0.000 Age 0.708 0.702 T-Value 13.23 15.96 P-Value 0.000 0.000 BSA 4.6 T-Value 3.04 P-Value 0.008 S 1.74 0.533 0.437 R-Sq 90.26 99.14 99.45 R-Sq(adj) 89.72 99.04 99.35 C-p 312.8 15.1 6.4

30. The regression equation is BP = - 13.7 + 0.702 Age + 0.906 Weight + 4.63 BSA Predictor Coef SE Coef T P Constant -13.667 2.647 -5.16 0.000 Age 0.70162 0.04396 15.96 0.000 Weight 0.90582 0.04899 18.49 0.000 BSA 4.627 1.521 3.04 0.008 S = 0.4370 R-Sq = 99.5% R-Sq(adj) = 99.4% Analysis of Variance Source DF SS MS F P Regression 3 556.94 185.65 971.93 0.000 Error 16 3.06 0.19 Total 19 560.00 Source DF Seq SS Age 1 243.27 Weight 1 311.91 BSA 1 1.77

31. Stepwise regression in Minitab Stat >> Regression >> Stepwise … Specify response and all possible predictors. If desired, specify predictors that must be included in every model. (This is where researcher’s knowledge helps!!) Select OK. Results appear in session window.