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Posterior Predictive "p"-values in Bayesian Hierarchical Models

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  • GUNNHILDUR HÖGNADÃTTIR STEINBAKK
  • GEIR OLVE STORVIK

Abstract

The present work focuses on extensions of the posterior predictive "p"-value (ppp-value) for models with hierarchical structure, designed for testing assumptions made on underlying processes. The ppp-values are popular as tools for model criticism, yet their lack of a common interpretation limit their practical use. We discuss different extensions of ppp-values to hierarchical models, allowing for discrepancy measures that can be used for checking properties of the model at all stages. Through analytical derivations and simulation studies on simple models, we show that similar to the standard ppp-values, these extensions are typically far from uniformly distributed under the model assumptions and can give poor power in a hypothesis testing framework. We propose a calibration of the "p"-values, making the resulting calibrated "p"-values uniformly distributed under the model conditions. Illustrations are made through a real example of multinomial regression to age distributions of fish. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.

Suggested Citation

  • Gunnhildur Högnadãttir Steinbakk & Geir Olve Storvik, 2009. "Posterior Predictive "p"-values in Bayesian Hierarchical Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 320-336.
  • Handle: RePEc:bla:scjsta:v:36:y:2009:i:2:p:320-336
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9469.2008.00630.x
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    References listed on IDEAS

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    1. David Hirst & Sondre Aanes & Geir Storvik & Ragnar Bang Huseby & Ingunn Fride Tvete, 2004. "Estimating catch at age from market sampling data by using a Bayesian hierarchical model," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 53(1), pages 1-14.
    2. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392.
    3. Fredrik A. Dahl & Jørund Gåsemyr & Bent Natvig, 2007. "A Robust Conflict Measure of Inconsistencies in Bayesian Hierarchical Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 816-828.
    4. Andrew Gelman & Iven Van Mechelen & Geert Verbeke & Daniel F. Heitjan & Michel Meulders, 2005. "Multiple Imputation for Model Checking: Completed-Data Plots with Missing and Latent Data," Biometrics, The International Biometric Society, vol. 61(1), pages 74-85, March.
    5. Dipak Dey & Alan Gelfand & Tim Swartz & Pantelis Vlachos, 1998. "A simulation-intensive approach for checking hierarchical models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 325-346, December.
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