Quantitative Methods Model Selection II datasets with several

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

Model Selection II: several explanatory variables The problem of model choice With 5 x-variables,

Model Selection II: several explanatory variables The problem of model choice • Economy of

Model Selection II: several explanatory variables Economy of variables

Model Selection II: several explanatory variables Economy of variables all variables increase R 2

Model Selection II: several explanatory variables Economy of variables continuous

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

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

Model Selection II: several explanatory variables Multiplicity of p-values Focus, don’t fish - reduce

Model Selection II: several explanatory variables Multiplicity of p-values DF 1 1 1 3

Model Selection II: several explanatory variables Stepwise regression

Model Selection II: several explanatory variables Stepwise regression General Linear Model: LRGWHAL versus Source

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

Model Selection II: several explanatory variables Last words… • Economy of variables: prediction, adjusted

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Quantitative Methods Model Selection II: datasets with several explanatory variables

Model Selection II: several explanatory variables 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 1 x 1 + 5 x 1 + 10 x 2 + 10 x 8 + 5 x 64 + 1 x 1024 = 1450 models If we do allow squared terms, there are 1 x 1 + 5 x 2 + 10 x 8 + 10 x 64 + 5 x 1024 + 1 x 32768 = 38619 models. With multiple models, there are many p-values and possible “right-leg/left-leg” and “poets’ dates” effects.

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

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

Model Selection II: several explanatory variables Economy of variables

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

Model Selection II: several explanatory variables Economy of variables continuous

Model Selection II: several explanatory variables Economy of variables

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

Model Selection II: several explanatory variables 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

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

Model Selection II: several explanatory variables Multiplicity of p-values DF 1 1 1 3 Seq. SS 366. 9 42. 7 14. 7 424. 3 MS=424. 3/3=141. 4 F = 141. 4/108. 9 = 1. 30 on 3 and 30 DF CDF 1. 30 K 1; F 3 30. LET K 2=1 -K 1 Single p-value from Minitab using CDF: p=0. 293

Model Selection II: several explanatory variables Stepwise regression

Model Selection II: several explanatory variables Stepwise regression General Linear Model: LRGWHAL versus Source DF Seq SS VIS 1 61. 166 Error 230 72. 759 Total 231 133. 925 Adj SS 61. 166 72. 759 Term Constant VIS SE Coef 0. 06116 0. 009005 Coef -4. 52464 0. 125222 Adj MS 61. 166 0. 316 F 193. 35 T -73. 98 13. 91 P 0. 000

Model Selection II: several explanatory variables Stepwise regression

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

Model Selection II: several explanatory variables Stepwise regression

Model Selection II: several explanatory variables 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