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

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

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

4. The problem of model choice
Model Selection II: several explanatory variables
The problem of model choice
With 5 x-variables, there are 25=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 + 5x x32768 = models.
With multiple models, there are many p-values and possible “right-leg/left-leg” and “poets’ dates” effects.

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

6. The problem of model choice
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. Economy of variables Model Selection II: several explanatory variables
all variables increase R2
F<1 - adding the variable decreased R2 adj
F>1 - adding the variable increased R2 adj

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

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

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

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

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

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

17. Multiplicity of p-values
Model Selection II: several explanatory variables
Multiplicity of p-values
DF SeqSS
1 42.7
1 14.7
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
LET K2=1-K1

18. Model Selection II: several explanatory variables
Stepwise regression

19. Model Selection II: several explanatory variables
Stepwise regression

20. Stepwise regression Model Selection II: several explanatory variables
General Linear Model: LRGWHAL versus
Source DF Seq SS Adj SS Adj MS F P
VIS
Error
Total
Term Coef SE Coef T P
Constant
VIS

21. Stepwise regression Model Selection II: several explanatory variables
General Linear Model: LRGWHAL versus
Source DF Seq SS Adj SS Adj MS F P
VIS
Error
Total
Term Coef SE Coef T P
Constant
VIS

22. Stepwise regression Model Selection II: several explanatory variables
General Linear Model: LRGWHAL versus
Source DF Seq SS Adj SS Adj MS F P
VIS
Error
Total
Term Coef SE Coef T P
Constant
VIS

23. Stepwise regression Model Selection II: several explanatory variables
General Linear Model: LRGWHAL versus
Source DF Seq SS Adj SS Adj MS F P
VIS
Error
Total
Term Coef SE Coef T P
Constant
VIS

24. Model Selection II: several explanatory variables
Stepwise regression

25. Stepwise regression Model Selection II: several explanatory variables
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. Random Effects Read Chapter 12 Last words…
Model Selection II: several explanatory variables
Last words…
Economy of variables: prediction, adjusted R2
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