Summarizing the predictive power of a generalized linear model

This paper studies summary measures of the predictive power of a generalized linear model, paying special attention to a generalization of the multiple correlation coefficient from ordinary linear regression. The population value is the correlation between the response and its conditional expectation given the predictors, and the sample value is the correlation between the observed response and the model predicted value. We compare four estimators of the measure in terms of bias, mean squared error and behaviour in the presence of overparameterization. The sample estimator and a jack-knife estimator usually behave adequately, but a cross-validation estimator has a large negative bias with large mean squared error. One can use bootstrap methods to construct confidence intervals for the population value of the correlation measure and to estimate the degree to which a model selection procedure may provide an overly optimistic measure of the actual predictive power.