Predictive Variable Selection in Generalized Linear Models

Abstract

Here we extend predictive method for model selection of Laud and Ibrahim to the generalized linear model. This prescription avoids the need to directly specify prior probabilities of models and prior densities for the parameters. Instead, a prior prediction for the response induces the required priors. We propose normal and conjugate priors for generalized linear models, each using a single prior prediction for the mean response to induce suitable priors for each variable-subset model. In this way, an informative prior is used to select a subset of variables. In addition to producing a ranking of models by size of the predictive criterion, the standard deviation of the criterion is used as a calibration number to produce a set of equally good models. A straightforward Markov chain Monte Carlo algorithm is used to accomplish the necessary computations. We illustrate this method with real and simulated datasets and compare results with the Bayes factors and the Akaike information and Bayes information model selection criteria. The simulation results confirm the efficacy of the method, because the correct model is known. An illustrative application demonstrates selection of important predictors of success in identifying the sentinel lymph node during surgical treatment of breast cancer. A forward selection procedure is described to avoid a full search over the 218 possible models in this case.