A note on R² measures for Poisson and logistic regression models when both models are applicable

Abstract 

The aim of many epidemiological studies is the regression of a dichotomous outcome (e.g., death or affection by a certain disease) on prognostic covariables. Thereby the Poisson regression model is often used alternatively to the logistic regression model. Modelling the number of events and individual outcomes, respectively, both models lead to nearly the same results concerning the parameter estimates and their significances. However, when calculating the proportion of explained variation, quantified by an R² measure, a large difference between both models usually occurs. We illustrate this difference by an example and explain it with theoretical arguments. We conclude, the R² measure of the Poisson regression quantifies the predictability of event rates, but it is not adequate to quantify the predictability of the outcome of individual observations.

Keywords:  R² measure, Poisson regression, Logistic regression, Approximation, Deviance, Sums-of-squares