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 (<i>HL</i>), Pigeon–Heyse (<i>J²</i>), and Tsiatis (<i>T</i>) 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, <i>HL</i> and <i>T</i> followed reported null distributions, while <i>J²</i> did not. Under PCS, only <i>T</i> followed its reported null distribution, with <i>HL</i> and <i>J²</i> dependent on model covariate number and partitioning. Generally, all had similar power. Of the three, <i>T</i> performed best, maintaining Type-I error rates and having a distribution invariant to covariate characteristics, number, and partitioning.