Goodness‐of‐Fit Testing for the Logistic Regression Model when the Estimated Probabilities are Small

The distribution of the Hosmer‐Lemeshow chi‐square type goodness‐of‐fit tests ( Č g , Ȟ g ) for the logistic regression model are examined via simulations designed to examine their behavior when most of the estimated probabilities are small or are expected to fall in a few deciles. The results of the simulations show statistic Č g should be used when the two outcome groups ( y = 0, 1) are not well separated, Δ≤2, where Δ 2 is the Mahalanobis distance. Statistic Ȟ g should be used when Δ ≥ 8. Either statistic may be used when 2 ≦ Δ ≦ 8. All tests should be used with caution when the proportion in the sample with y = 1 is less than 0.1.