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Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing

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  • Stephen G. Walker
  • Bani K. Mallick

Abstract

This paper proposes Bayesian nonparametric mixing for some well‐known and popular models. The distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term, a function of the observed covariates, and an additive or multiplicative random effects term. Typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family. This assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified. Another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution. As a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately. This paper therefore proposes a Bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates’ effect on the distribution of the mixed effects.

Suggested Citation

  • Stephen G. Walker & Bani K. Mallick, 1997. "Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(4), pages 845-860.
    • Handle: RePEc:bla:jorssb:v:59:y:1997:i:4:p:845-860
    DOI: 10.1111/1467-9868.00101
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    Cited by:

    1. Bhattacharjee, Arnab & Bhattacharjee, Madhuchhanda, 2007. "Bayesian Analysis of Hazard Regression Models under Order Restrictions on Covariate Effects and Ageing," MPRA Paper 3938, University Library of Munich, Germany.
    2. Ernesto San Martín & Alejandro Jara & Jean-Marie Rolin & Michel Mouchart, 2011. "On the Bayesian Nonparametric Generalization of IRT-Type Models," Psychometrika, Springer;The Psychometric Society, vol. 76(3), pages 385-409, July.
    3. Kumar Prabhash & Vijay M Patil & Vanita Noronha & Amit Joshi & Atanu Bhattacharjee, 2016. "Bayesian Accelerated Failure Time And Its Application In Chemotherapy Drug Treatment Trial," Statistics in Transition New Series, Polish Statistical Association, vol. 17(4), pages 671-690, December.
    4. Nalini Ravishanker & Dipak K. Dey, 2000. "Multivariate Survival Models with a Mixture of Positive Stable Frailties," Methodology and Computing in Applied Probability, Springer, vol. 2(3), pages 293-308, September.
    5. Lau, John W., 2006. "Bayesian semi-parametric modeling for mixed proportional hazard models with right censoring," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 719-728, April.
    6. Antonio Lijoi & Igor Prunster, 2009. "Models beyond the Dirichlet process," Quaderni di Dipartimento 103, University of Pavia, Department of Economics and Quantitative Methods.
    7. Nieto-Barajas, Luis E. & Walker, Stephen G., 2007. "A Bayesian semi-parametric bivariate failure time model," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6102-6113, August.
    8. Levine, Richard A. & Fan, Juanjuan & Strickland, Pamela Ohman & Demirel, Shaban, 2012. "Frailty modeling via the empirical Bayes–Hastings sampler," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1303-1318.
    9. repec:jss:jstsof:40:i05 is not listed on IDEAS
    10. Jianjun Zhang & Lei Yang & Xianyi Wu, 2019. "Polya tree priors and their estimation with multi-group data," Statistical Papers, Springer, vol. 60(3), pages 849-875, June.
    11. Chen, Yuhui & Hanson, Timothy E., 2014. "Bayesian nonparametric k-sample tests for censored and uncensored data," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 335-346.
    12. Antonio Lijoi & Igor Pruenster, 2009. "Models beyond the Dirichlet process," ICER Working Papers - Applied Mathematics Series 23-2009, ICER - International Centre for Economic Research.
    13. Chiara Gigliarano & Pietro Muliere, 2013. "Estimating the Lorenz curve and Gini index with right censored data: a Polya tree approach," METRON, Springer;Sapienza Università di Roma, vol. 71(2), pages 105-122, September.
    14. Prabhash Kumar & Patil Vijay M & Noronha Vanita & Joshi Amit & Bhattacharjee Atanu, 2016. "Bayesian Accelerated Failure Time and its Application in Chemotherapy Drug Treatment Trial," Statistics in Transition New Series, Polish Statistical Association, vol. 17(4), pages 671-690, December.
    15. Zhuang, Haoxin & Diao, Liqun & Yi, Grace Y., 2023. "Polya tree Monte Carlo method," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    16. Thomas A. Murray & Peter F. Thall & Ying Yuan & Sarah McAvoy & Daniel R. Gomez, 2017. "Robust Treatment Comparison Based on Utilities of Semi-Competing Risks in Non-Small-Cell Lung Cancer," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 11-23, January.
    17. Komárek, Arnost & Lesaffre, Emmanuel, 2008. "Generalized linear mixed model with a penalized Gaussian mixture as a random effects distribution," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3441-3458, March.

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