1. Quantitative Methods Model Selection II: datasets with several explanatory variables

2. Model Selection II: several explanatory variables The problem of model choice

3. Model Selection II: several explanatory variables The problem of model choice

4. Model Selection II: several explanatory variables The problem of model choice With 5 x-variables, there are 2 5 =32 possible models, not including interactions. If we include two-way interactions without squared terms, there are 1x1 + 5x1 + 10x2 + 10x8 + 5x64 + 1x1024 = 1450 models If we do allow squared terms, there are 1x1 + 5x2 + 10x8 + 10x64 + 5x1024 + 1x32768 = 38619 models. With multiple models, there are many p-values and possible “right-leg/left-leg” and “poets’ dates” effects.

5. Model Selection II: several explanatory variables The problem of model choice Economy of variables Multiplicity of p-values Marginality

6. Model Selection II: several explanatory variables The problem of model choice

7. Model Selection II: several explanatory variables Economy of variables

8. Model Selection II: several explanatory variables Economy of variables

9. Model Selection II: several explanatory variables Economy of variables all variables increase R 2 F<1 - adding the variable decreased R 2 adj F>1 - adding the variable increased R 2 adj

10. Model Selection II: several explanatory variables Economy of variables continuous

11. Model Selection II: several explanatory variables Economy of variables

12. Model Selection II: several explanatory variables Economy of variables (Predictions for datapoint 39)

13. Model Selection II: several explanatory variables Multiplicity of p-values

14. Model Selection II: several explanatory variables Multiplicity of p-values

15. Model Selection II: several explanatory variables Multiplicity of p-values Focus, don’t fish - reduce number of X-variables - use outside information to decide on inclusion - use outside information to decide on exclusion Stringency - reduce nominal p-value Combine model terms - for once, reverse the usual splitting

16. Model Selection II: several explanatory variables Multiplicity of p-values

17. Model Selection II: several explanatory variables Multiplicity of p-values DFSeqSS 1366.9 142.7 114.7 3424.3 MS=424.3/3=141.4 F = 141.4/108.9 = 1.30 on 3 and 30 DF Single p-value from Minitab using CDF: p=0.293 CDF 1.30 K1; F 3 30. LET K2=1-K1

18. Model Selection II: several explanatory variables Stepwise regression

19. Model Selection II: several explanatory variables Stepwise regression

20. Model Selection II: several explanatory variables Stepwise regression General Linear Model: LRGWHAL versus Source DF Seq SS Adj SS Adj MS F P VIS 1 61.166 61.166 61.166 193.35 0.000 Error 230 72.759 72.759 0.316 Total 231 133.925 Term Coef SE Coef T P Constant -4.52464 0.06116 -73.98 0.000 VIS 0.125222 0.009005 13.91 0.000

21. Model Selection II: several explanatory variables Stepwise regression General Linear Model: LRGWHAL versus Source DF Seq SS Adj SS Adj MS F P VIS 1 61.166 61.166 61.166 193.35 0.000 Error 230 72.759 72.759 0.316 Total 231 133.925 Term Coef SE Coef T P Constant -4.52464 0.06116 -73.98 0.000 VIS 0.125222 0.009005 13.91 0.000

22. Model Selection II: several explanatory variables Stepwise regression General Linear Model: LRGWHAL versus Source DF Seq SS Adj SS Adj MS F P VIS 1 61.166 61.166 61.166 193.35 0.000 Error 230 72.759 72.759 0.316 Total 231 133.925 Term Coef SE Coef T P Constant -4.52464 0.06116 -73.98 0.000 VIS 0.125222 0.009005 13.91 0.000

23. General Linear Model: LRGWHAL versus Source DF Seq SS Adj SS Adj MS F P VIS 1 61.166 61.166 61.166 193.35 0.000 Error 230 72.759 72.759 0.316 Total 231 133.925 Term Coef SE Coef T P Constant -4.52464 0.06116 -73.98 0.000 VIS 0.125222 0.009005 13.91 0.000 Model Selection II: several explanatory variables Stepwise regression

24. Model Selection II: several explanatory variables Stepwise regression

25. Model Selection II: several explanatory variables Stepwise regression Forward = Backward Forward ≠ Backward

26. Model Selection II: several explanatory variables Stepwise regression

27. Model Selection II: several explanatory variables Stepwise regression

28. Model Selection II: several explanatory variables Stepwise regression

29. Model Selection II: several explanatory variables Stepwise regression

30. Model Selection II: several explanatory variables Stepwise regression

31. Last words… Economy of variables: prediction, adjusted R 2 Multiplicity: outside information, focussing, stringency, combining model terms Stepwise regressions not usually suitable -- but are for initial sifting of a large number of potential predictors in a preliminary study Random Effects Read Chapter 12 Model Selection II: several explanatory variables