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

Author

Listed:
  • Nikola Surjanovic

    (University of British Columbia)

  • Richard A. Lockhart

    (Simon Fraser University)

  • Thomas M. Loughin

    (Simon Fraser University)

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.

Suggested Citation

  • Nikola Surjanovic & Richard A. Lockhart & Thomas M. Loughin, 2024. "A generalized Hosmer–Lemeshow goodness-of-fit test for a family of generalized linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 33(2), pages 589-608, June.
    • Handle: RePEc:spr:testjl:v:33:y:2024:i:2:d:10.1007_s11749-023-00912-8
    DOI: 10.1007/s11749-023-00912-8
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    References listed on IDEAS

    as
    1. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "Rejoinder on: An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 442-447, September.
    2. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    3. D. Y. Lin & L. J. Wei & Z. Ying, 2002. "Model-Checking Techniques Based on Cumulative Residuals," Biometrics, The International Biometric Society, vol. 58(1), pages 1-12, March.
    4. Winfried Stute & Li‐Xing Zhu, 2002. "Model Checks for Generalized Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 535-545, September.
    5. Ronald Christensen & Yong Lin, 2015. "Lack-of-fit Tests Based On Partial Sums of Residuals," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(13), pages 2862-2880, July.
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    Cited by:

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