A comparison of the Hosmer-Lemeshow, Pigeon-Heyse, and Tsiatis goodness-of-fit tests for binary logistic regression under two grouping methods.

Algebraic relationships between Hosmer-Lemeshow (HL), Pigeon-Heyse (J(2)), and Tsiatis (T) goodness-of-fit statistics for binary logistic regression models with continuous covariates were investigated, and their distributional properties and performances studied using simulations. Groups were formed under deciles-of-risk (DOR) and partition-covariate-space (PCS) methods. Under DOR, HL and T followed reported null distributions, while J(2) did not. Under PCS, only T followed its reported null distribution, with HL and J(2) dependent on model covariate number and partitioning. Generally, all had similar power. Of the three, T performed best, maintaining Type-I error rates and having a distribution invariant to covariate characteristics, number, and partitioning.