Best subset selection results in a nested set of best models with different numbers of predictors.

Best subset selection results in a nested set of best models with different numbers of predictors.

• A. True
• B. False
• C. It is always nested
• D. It is never nested

Answer: (B)

The important idea is that nesting is not guaranteed when using best subset selection. Best subset chooses the best model of exactly k predictors for each k, but the set of predictors that makes the best model with k can be completely different from the set that makes the best model with k+1. There’s no requirement that the k-predictor model be contained in the k+1-predictor model. For example, the best single predictor might be X2, while the best two-predictor model could be {X1, X3}, which does not contain X2. So the sequence of “best” models across different numbers of predictors need not be nested. By contrast, forward selection builds models by adding one predictor at a time, guaranteeing nesting because each larger model contains all predictors from the previous one. Since best subset does not enforce such containment, the statement is false.

The important idea is that nesting is not guaranteed when using best subset selection. Best subset chooses the best model of exactly k predictors for each k, but the set of predictors that makes the best model with k can be completely different from the set that makes the best model with k+1. There’s no requirement that the k-predictor model be contained in the k+1-predictor model. For example, the best single predictor might be X2, while the best two-predictor model could be {X1, X3}, which does not contain X2. So the sequence of “best” models across different numbers of predictors need not be nested.

By contrast, forward selection builds models by adding one predictor at a time, guaranteeing nesting because each larger model contains all predictors from the previous one. Since best subset does not enforce such containment, the statement is false.