Properties ofR2 statistics for logistic regression

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Abstract
Various R2 statistics have been proposed for logistic regression to quantify the extent to which the binary response can be predicted by a given logistic regression model and covariates. We study the asymptotic properties of three popular variance-based R2 statistics. We find that two variance-based R2 statistics, the sum of squares and the squared Pearson correlation, have identical asymptotic distribution whereas the third one, Gini's concentration measure, has a different asymptotic behaviour and may overstate the predictivity of the model and covariates when the model is misspecified. Our result not only provides a theoretical basis for the findings in previous empirical and numerical work, but also leads to asymptotic confidence intervals. Statistical variability can then be taken into account when assessing the predictive value of a logistic regression model. Copyright © 2005 John Wiley & Sons, Ltd.