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

ABSTRACT

Algebraic relationships between Hosmer–Lemeshow (HL), Pigeon–Heyse (J²), 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 J² did not. Under PCS, only T followed its reported null distribution, with HL and J² 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.

KEYWORDS:

• Binary logistic regression
• Deciles-of-risk
• Goodness-of-fit
• Hosmer–Lemeshow
• Partition the covariate space
• Pigeon–Heyse
• Tsiatis

MATHEMATICAL SUBJECT CLASSIFICATION:

• 62H10

Acknowledgments

This research was funded by an Australian Postgraduate Award to the first author, as well as a grant from the Australian National Health and Medical Research Council (NHMRC #490000) to the second author.

Conflict of interest

The authors have declared no conflict of interest.

Notes

1 The Stata command fracpoly performs a fractional polynomial regression, fitting fractional polynomials in the specified covariates to the dependent variable. (StataCorp, 2007).