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

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) te...

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Bibliographic Details
Published in:Test (Madrid, Spain) Spain), 2024, Vol.33 (2), p.589-608
Main Authors: Surjanovic, Nikola, Lockhart, Richard A., Loughin, Thomas M.
Format: Article
Language:English
Subjects:
Asymptotic methods
Comparative analysis
Economics
Finance
Generalized linear models
Goodness of fit
Insurance
Management
Mathematics and Statistics
Original Paper
Regression analysis
Sampling techniques
Simulation
Statistical tests
Statistical Theory and Methods
Statistics
Statistics for Business
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Summary: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.
ISSN:1133-0686
1863-8260
DOI:10.1007/s11749-023-00912-8