Summarizing the predictive power of a generalized linear model

This paper studies summary measures of the predictive power of a generalize d linear model, paying special attention to a generalization of the multipl e correlation coefficient from ordinary linear regression. The population v alue is the correlation between the response and its conditional expectatio n given the predictors, and the sample value is the correlation between the observed response and the model predicted value. We compare four estimator s of the measure in terms of bias, mean squared error and behaviour in the presence of overparameterization. The sample estimator and a jack-knife est imator usually behave adequately, but a cross-validation estimator has a la rge negative bias with large mean squared error. One can use bootstrap meth ods to construct confidence intervals for the population value of the corre lation measure and to estimate the degree to which a model selection proced ure may provide an overly optimistic measure of the actual predictive power . Copyright (C) 2000 John Wiley & Sons, Ltd.