Posterior propriety for Bayesian binomial regression models with a parametric family of link functions

We consider a Bayesian analysis of Binomial response data with covariates. To describe the problem under investigation, suppose we have n independent binomial observations Y1, . . . , Yn where Yi ∼ Bin(mi, θi) and let xi be p-dimensional covariate vector associated with Yi for i = 1, . . . , n. Binomial observations can be analyzed through a generalized linear model (GLM) where we assume θi = F (xi β) for some known distribution function F (·) and β is the vector of unknown regression coefficients. In this paper, we state necessary and sufficient conditions for propriety of the posterior distribution of β if an improper uniform prior is used on β. 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 β and the link function parameters. There are a number of parametric family of link functions available in the literature. As a specific example, we consider Pregibon’s (1980) 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 (1980) link. We show that Pregibon’s (1980) 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 (1980) link can be easily checked and are milder than those required by the flexible GEV link of Wang and Dey (2010). As an illustration, we analyze a data set from Ramsey and Schafer (2002) 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 (1980) skewed link yields a slightly better fit than the GEV link.