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Properties of R 2 statistics for logistic regression
Author(s) -
Hu Bo,
Palta Mari,
Shao Jun
Publication year - 2006
Publication title -
statistics in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.996
H-Index - 183
eISSN - 1097-0258
pISSN - 0277-6715
DOI - 10.1002/sim.2300
Subject(s) - statistics , logistic regression , cross sectional regression , medical statistics , econometrics , regression analysis , regression , computer science , mathematics , polynomial regression
Various R 2 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 R 2 statistics. We find that two variance‐based R 2 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.