Selective inference in regression models with groups of variables

We provide a general mathematical framework for selective inference with supervised model selection procedures characterized by quadratic forms in the outcome variable. Forward stepwise with groups of variables is an important special case as it allows models with categorical variables or factors. Models can be chosen by AIC, BIC, or a fixed number of steps. We provide an exact significance test for each group of variables in the selected model based on an appropriately truncated $\chi$ or $F$ distribution for the cases of known and unknown $\sigma^2$ respectively. An efficient software implementation is available as a package in the R statistical programming language.