Brazilian Journal of Probability and Statistics

A semiparametric Bayesian model for multiple monotonically increasing count sequences

Valeria Leiva-Yamaguchi and Fernando A. Quintana

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Abstract

In longitudinal clinical trials, subjects may be evaluated many times over the course of the study. This article is motivated by a medical study conducted in the U.S. Veterans Administration Cooperative Urological Research Group to assess the effectiveness of a treatment in preventing recurrence on subjects affected by bladder cancer. The data consist of the accumulated tumor counts over a sequence of regular checkups, with many missing observations. We propose a hierarchical nonparametric Bayesian model for sequences of monotonically increasing counts. Unlike some of the previous analyses for these data, we avoid interpolation by explicitly incorporating the missing observations under the assumption of these being missing completely at random. Our formulation involves a generalized linear mixed effects model, using a dependent Dirichlet process prior for the random effects, with an autoregressive component to include serial correlation along patients. This provides great flexibility in the desired inference, that is, assessing the treatment effect. We discuss posterior computations and the corresponding results obtained for the motivating dataset, including a comparison with parametric alternatives.

Article information

Source
Braz. J. Probab. Stat., Volume 30, Number 2 (2016), 155-170.

Dates
Received: July 2014
Accepted: November 2014
First available in Project Euclid: 31 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1459429708

Digital Object Identifier
doi:10.1214/14-BJPS268

Mathematical Reviews number (MathSciNet)
MR3481099

Zentralblatt MATH identifier
1381.62273

Keywords
Autoregressive model Dirichlet process generalized linear mixed model hierarchical model

Citation

Leiva-Yamaguchi, Valeria; Quintana, Fernando A. A semiparametric Bayesian model for multiple monotonically increasing count sequences. Braz. J. Probab. Stat. 30 (2016), no. 2, 155--170. doi:10.1214/14-BJPS268. https://projecteuclid.org/euclid.bjps/1459429708


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References

  • Aldous, D. J. (1985). Exchangeability and Related Topics, École d’été de probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math. 1117, 1–198. Berlin: Springer.
  • Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics 2, 1152–1174.
  • Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Annals of Statistics 1, 353–355.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized in linear mixed models. Journal of the American Statistical Association 88, 9–25.
  • Bush, C. A. and MacEachern, S. N. (1996). A semiparametric Bayesian model for randomised block designs. Biometrika 83, 275–285.
  • Byar, D., Blackard, C. and Urological Research Group (1977). Comparisons of placebo, pyridoxine, and topical thiotepa in preventing recurrence of stage I bladder cancer. Urology 10, 556–561.
  • Celeux, G., Forbes, F., Robert, C. P. and Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis 1, 651–673 (electronic).
  • Davis, C. S. and Wei, L. J. (1988). Nonparametric methods for analyzing incomplete nondecreasing repeated measurements. Biometrics 44, 1005–1018.
  • Di Lucca, M., Gugliemi, A., Müller, P. and Quintana, F. A. (2013). A simple class of Bayesian nonparametric autoregression models. Bayesian Analysis 8, 63–88.
  • Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association 89, 268–277.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics 1, 209–230.
  • Geisser, S. and Eddy, W. F. (1979). A predictive approach to model selection. Journal of the American Statistical Association 74, 153–160.
  • Gelfand, A. E. and Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society. Series B, Methodological 56, 501–514.
  • Giardina, F., Guglielmi, A., Ruggeri, F. and Quintana, F. A. (2011). Bayesian first order autoregressive latent variable models for multiple binary sequences. Statistical Modelling International Journal 11, 471–488.
  • Hjort, N. L., Holmes, C., Müller, P. and Walker, S. (2010). Bayesian Nonparametrics. Cambridge, UK: Cambridge University Press.
  • Ibrahim, J. G. and Kleinman, K. P. (1998). Semiparametric Bayesian methods for random effects models. In Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133, 89–114. New York: Springer.
  • Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96, 161–173.
  • MacEachern, S. N. and Müller, P. (1998). Estimating mixture of Dirichlet process models. Journal of Computational and Graphical Statistics 7, 223–338.
  • Müller, P. and Mitra, R. (2013). Bayesian nonparametric inference—Why and how. Bayesian Analysis 8, 269–302.
  • Müller, P. and Quintana, F. A. (2004). Nonparametric Bayesian data analysis. Statistical Science 19, 95–110.
  • Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics 9, 249–265.
  • Plummer, M., Best, N., Cowles, K. and Vines, K. (2006). Coda: Convergence diagnosis and output analysis for MCMC. R News 6, 7–11.
  • Roberts, G. O. and Rosenthal, J. S. (2009). Examples of adaptive MCMC. Journal of Computational and Graphical Statistics 18, 349–367.
  • Rolin, J.-M. (1992). Some useful properties of the Dirichlet process. Technical Report 9207, Center for Operations Research & Econometrics, Univ. Catholique de Louvain.
  • Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 4, 639–650.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society. Series B, Statistical Methodology 64, 583–639.
  • Zeger, S. L. and Karim, M. R. (1991). Generalized linear models with random effects; a Gibbs sampling approach. Journal of the American Statistical Association 86, 79–86.