Criticality of predictors in multiple regression

A new method is proposed for comparing all predictors in a multiple regress ion model. This method generates a measure of predictor criticality, which is distinct from and has several advantages over traditional indices of pre dictor importance. Using the bootstrapping (resampling with replacement) procedure, a large nu mber of samples are obtained from a given data set which contains one respo nse variable and p predictors. For each sample, all 2(p) - 1 subset regress ion models are fitted and the best subset model is selected. Thus, the (mul tinomial) distribution of the probability that each of the 2(p) - 1 subsets is 'the best' model for the data set is obtained. A predictor's criticality is defined as a function of the probabilities ass ociated with the models that include the predictor. That is, a predictor wh ich is included in a large number of probable models is critical to the ide ntification of the best-fitting regression model and, therefore, to the pre diction of the response variable. The procedure can be applied to fixed and random regression models and can use any measure of goodness of fit (e.g., adjusted R-2, C-p, AIC) for ident ifying the best model. Several criticality measures can be defined by using different combinations of the probabilities of the best-fitting models, an d asymptotic confidence intervals for each variable's criticality can be de rived. The procedure is illustrated with several examples.