An R-squared measure of goodness of fit for some common nonlinear regression models

For regression models other than the linear model, R-squared type goodness-of-fit summary statistics have been constructed for particular models using a variety of methods. We propose an R-squared measure of goodness of fit for the class of exponential family regression models, which includes legit, probit, Poisson, geometric, gamma, and exponential. This R-squared is defined as the proportionate reduction in uncertainty, measured by Kullback-Leibler divergence, due to the inclusion of regressors. Under further conditions concerning the conditional mean function it can also be interpreted as the fraction of uncertainty explained by the fitted model.