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Posterior propriety for Bayesian binomial regression models with a parametric family of link functions. (English) Zbl 1365.62098

Summary: We consider a Bayesian analysis of Binomial response data with covariates. To describe the problem under investigation, suppose we have \(n\) independent binomial observations \(Y_1, \ldots, Y_n\) where \(Y_i \sim \text{Bin}(m_i, \theta_i)\) and let \(\mathbf{x}_i\) be \(p\)-dimensional covariate vector associated with \(Y_i\) for \(i = 1, \ldots, n\). Binomial observations can be analyzed through a generalized linear model (GLM) where we assume \(\theta_i = F(\mathbf{x}_i^T \boldsymbol{\beta})\) for some known distribution function \(F(\cdot)\) and \(\boldsymbol{\beta}\) is the vector of unknown regression coefficients. In this paper, we state necessary and sufficient conditions for propriety of the posterior distribution of \(\boldsymbol{\beta}\) if an improper uniform prior is used on \(\boldsymbol{\beta}\). We also consider situations where the link function is not pre-specified but belongs to a parametric family and the link function parameters are estimated along with the regression coefficients. In this case, we investigate the propriety of the joint posterior distributions of \(\boldsymbol{\beta}\) and the link function parameters. There are a number of parametric families of link functions available in the literature. As a specific example, we consider D. Pregibon’s [J. R. Stat. Soc., Ser. C 29, 15–24 (1980; Zbl 0434.62048)] link function and show that our general posterior propriety results can be used to establish propriety of the posterior distributions corresponding to the Pregibon’s [loc. cit.] link. We show that Pregibon’s [loc. cit.] simple one parameter family of link function can be used to fit both positively and negatively skewed response curves. Moreover, the conditions for posterior propriety corresponding to the Pregibon’s [loc. ct.] link can be easily checked and are milder than those required by the flexible GEV link of X. Wang and D. K. Dey [Ann. Appl. Stat. 4, No. 4, 2000–2023 (2010; Zbl 1220.62165)]. As an illustration, we analyze a data set from F. L. Ramsey and D. W. Schafer [The statistical sleuth. A course in methods of data analysis. 3rd edition. Boston, MA: Brooks/Cole; Cengage Learning (2013; Zbl 1329.62005)] regarding the relationship between dose of Aflatoxicol and odds of liver tumor in rainbow trouts. In this example, the symmetric logit link fails to fit the data, whereas Pregibon’s [loc. cit.] skewed link yields a slightly better fit than the GEV link.

MSC:

62F15 Bayesian inference
62G32 Statistics of extreme values; tail inference
62J12 Generalized linear models (logistic models)
62P10 Applications of statistics to biology and medical sciences; meta analysis

Software:

STUKEL; R
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References:

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