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Diagnostic checks for discrete data regression models using posterior predictive simulations

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  • A. Gelman
  • Y. Goegebeur
  • F. Tuerlinckx
  • I. Van Mechelen

Abstract

Model checking with discrete data regressions can be difficult because the usual methods such as residual plots have complicated reference distributions that depend on the parameters in the model. Posterior predictive checks have been proposed as a Bayesian way to average the results of goodness‐of‐fit tests in the presence of uncertainty in estimation of the parameters. We try this approach using a variety of discrepancy variables for generalized linear models fitted to a historical data set on behavioural learning. We then discuss the general applicability of our findings in the context of a recent applied example on which we have worked. We find that the following discrepancy variables work well, in the sense of being easy to interpret and sensitive to important model failures: structured displays of the entire data set, general discrepancy variables based on plots of binned or smoothed residuals versus predictors and specific discrepancy variables created on the basis of the particular concerns arising in an application. Plots of binned residuals are especially easy to use because their predictive distributions under the model are sufficiently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals defined from an underlying continuous model and quantile–quantile plots of these residuals.

Suggested Citation

  • A. Gelman & Y. Goegebeur & F. Tuerlinckx & I. Van Mechelen, 2000. "Diagnostic checks for discrete data regression models using posterior predictive simulations," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 49(2), pages 247-268.
  • Handle: RePEc:bla:jorssc:v:49:y:2000:i:2:p:247-268
    DOI: 10.1111/1467-9876.00190
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    Cited by:

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    2. Clough, Brian J. & Russell, Matthew B. & Domke, Grant M. & Woodall, Christopher W. & Radtke, Philip J., 2016. "Comparing tree foliage biomass models fitted to a multispecies, felled-tree biomass dataset for the United States," Ecological Modelling, Elsevier, vol. 333(C), pages 79-91.
    3. R. B. O'Hara & S. Lampila & M. Orell, 2009. "Estimation of Rates of Births, Deaths, and Immigration from Mark–Recapture Data," Biometrics, The International Biometric Society, vol. 65(1), pages 275-281, March.
    4. Robert Kapłon, 2006. "A retrospective review of categorical data analysis – theory and marketing practice," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 16(1), pages 55-72.
    5. Thomas J. Steenburgh & Andrew Ainslie & Peder Hans Engebretson, 2003. "Massively Categorical Variables: Revealing the Information in Zip Codes," Marketing Science, INFORMS, vol. 22(1), pages 40-57, August.
    6. Meulders, Michel & Boeck, Paul De & Mechelen, Iven Van, 2001. "Probability matrix decomposition models and main-effects generalized linear models for the analysis of replicated binary associations," Computational Statistics & Data Analysis, Elsevier, vol. 38(2), pages 217-233, December.
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    8. Carolina Costa Mota Paraíba & Natalia Bochkina & Carlos Alberto Ribeiro Diniz, 2018. "Bayesian truncated beta nonlinear mixed-effects models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(2), pages 320-346, January.
    9. Silvia Montagna & Vanessa Orani & Raffaele Argiento, 2021. "Bayesian isotonic logistic regression via constrained splines: an application to estimating the serve advantage in professional tennis," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 573-604, June.
    10. Patrizia Ordine & Claudio Lupi, 2009. "Family Income and Students' Mobility," Giornale degli Economisti, GDE (Giornale degli Economisti e Annali di Economia), Bocconi University, vol. 68(1), pages 1-23, April.
    11. Andrew Gelman & Ginger L. Chew & Michael Shnaidman, 2004. "Bayesian Analysis of Serial Dilution Assays," Biometrics, The International Biometric Society, vol. 60(2), pages 407-417, June.
    12. Wagner, Helga & Duller, Christine, 2012. "Bayesian model selection for logistic regression models with random intercept," Computational Statistics & Data Analysis, Elsevier, vol. 56(5), pages 1256-1274.
    13. Andrew Gelman, 2003. "A Bayesian Formulation of Exploratory Data Analysis and Goodness‐of‐fit Testing," International Statistical Review, International Statistical Institute, vol. 71(2), pages 369-382, August.

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