Explained Variance in Logistic Regression

R² is widely relied on in linear regression to index a model's discriminatory power. Many counterparts have been proposed for use in logistic regression, but no single measure is consistently used. Two potential criterion values are relevant: the explained variance in the latent scale underlying the binary indicator of event occurrence and the explained risk of the event itself. In this study, Monte Carlo methods were used to examine the performance, with respect to fixed theoretical levels of explained variance and explained risk, of eight R² analogues. The McKelvey-Zavoina measure appears to be best at estimating explained variance and either the sample-estimated explained risk or the ordinary least squares R² to be best at indexing explained risk. Other measures appear to be poor choices, primarily because asymptotic trends suggest they may be inconsistent estimators of the relevant criterion.