A generalized Hosmer–Lemeshow goodness-of-fit test for a family of generalized linear models

Nikola Surjanovic (), Richard A. Lockhart () and Thomas M. Loughin ()
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Nikola Surjanovic: University of British Columbia
Richard A. Lockhart: Simon Fraser University
Thomas M. Loughin: Simon Fraser University

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2024, vol. 33, issue 2, No 10, 589-608

Abstract: Abstract Generalized linear models (GLMs) are very widely used, but formal goodness-of-fit (GOF) tests for the overall fit of the model seem to be in wide use only for certain classes of GLMs. We develop and apply a new goodness-of-fit test, similar to the well-known and commonly used Hosmer–Lemeshow (HL) test, that can be used with a wide variety of GLMs. The test statistic is a variant of the HL statistic, but we rigorously derive an asymptotically correct sampling distribution using methods of Stute and Zhu (Scand J Stat 29(3):535–545, 2002) and demonstrate its consistency. We compare the performance of our new test with other GOF tests for GLMs, including a naive direct application of the HL test to the Poisson problem. Our test provides competitive or comparable power in various simulation settings and we identify a situation where a naive version of the test fails to hold its size. Our generalized HL test is straightforward to implement and interpret and an R package is publicly available.

Keywords: Empirical regression process; Exponential dispersion family; Generalized linear model; Goodness-of-fit test; Hosmer–Lemeshow test; 62J12; 62E20; 62P99 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11749-023-00912-8

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