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On goodness‐of‐fit measures for Poisson regression models

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  • Takeshi Kurosawa
  • Francis K.C. Hui
  • A.H. Welsh
  • Kousuke Shinmura
  • Nobuoki Eshima

Abstract

In this article, we study the statistical properties of the goodness‐of‐fit measure mpp proposed by (Eshima & Tabata 2007, Statistics & Probability Letters 77, 583–593) for generalised linear models. Focusing on the special case of Poisson regression using the canonical log link function, and assuming a random vector X of covariates, we obtain an explicit form for mpp that enables us to study its properties and construct a new estimator for the measure by utilising information about the shape of the covariate distribution. Simulations show that the newly proposed estimator for mpp exhibits better performance in terms of mean squared error than the simple unbiased covariance estimator, especially for larger absolute values of the slope coefficients. In contrast, it may be more unstable when the value of the slope coefficient is close to boundary of the domain of the moment generating function for the corresponding covariate. We illustrate the application of mpp on a data set of counts of complaints against doctors working in an emergency unit in hospital, in particular, showing how our proposed estimator can be efficiently computed across a series of candidate models.

Suggested Citation

  • Takeshi Kurosawa & Francis K.C. Hui & A.H. Welsh & Kousuke Shinmura & Nobuoki Eshima, 2020. "On goodness‐of‐fit measures for Poisson regression models," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 62(3), pages 340-366, September.
    • Handle: RePEc:bla:anzsta:v:62:y:2020:i:3:p:340-366
    DOI: 10.1111/anzs.12303
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    References listed on IDEAS

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    1. Eshima, Nobuoki & Tabata, Minoru, 2011. "Three predictive power measures for generalized linear models: The entropy coefficient of determination, the entropy correlation coefficient and the regression correlation coefficient," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 3049-3058, November.
    2. Ibrahim, Joseph G. & Zhu, Hongtu & Tang, Niansheng, 2008. "Model Selection Criteria for Missing-Data Problems Using the EM Algorithm," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1648-1658.
    3. Francis K.C. Hui & David I. Warton & Scott D. Foster, 2015. "Order selection in finite mixture models: complete or observed likelihood information criteria?," Biometrika, Biometrika Trust, vol. 102(3), pages 724-730.
    4. Eshima, Nobuoki & Tabata, Minoru, 2007. "Entropy correlation coefficient for measuring predictive power of generalized linear models," Statistics & Probability Letters, Elsevier, vol. 77(6), pages 588-593, March.
    5. Takeshi Kurosawa & Nobuhiro Suzuki, 2018. "Asymptotic distribution of regression correlation coefficient for Poisson regression model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(1), pages 166-180, January.
    6. Takahashi, Akihito & Kurosawa, Takeshi, 2016. "Regression correlation coefficient for a Poisson regression model," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 71-78.
    7. John G. Cragg & Russell S. Uhler, 1970. "The Demand for Automobiles," Canadian Journal of Economics, Canadian Economics Association, vol. 3(3), pages 386-406, August.
    8. Eshima, Nobuoki, 2004. "Canonical exponential models for analysis of association between two sets of variables," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 135-144, January.
    9. Eshima, Nobuoki & Tabata, Minoru, 2010. "Entropy coefficient of determination for generalized linear models," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1381-1389, May.
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