A Comparison of Parametric and Coarsened Bayesian Interval Estimation in the Presence of a Known Mean-Variance Relationship

While the use of Bayesian methods of analysis have become increasingly common, classical frequentist hypothesis testing still holds sway in medical research especially clinical trials. One major difference between a standard frequentist approach and the most common Bayesian approaches is that even when a frequentist hypothesis test is derived from parametric models, the interpretation and operating characteristics of the test may be considered in a distribution-free manner. Bayesian inference, on the other hand, is often conducted in a parametric setting where the interpretation of the results is dependent on the parametric model. Here we consider a Bayesian counterpart to the most standard frequentist approach to inference. Instead of specifying a sampling distribution for the data we specify an approximate distribution of a summary statistic, thereby resulting in a “coarsening” of the data. This approach is robust in that it provides some protection against model misspecification and allows one to account for the possibility of a specified mean-variance relationship. Notably, the method also allows one to place prior mass directly on the quantity of interest or, alternatively, to employ a noninformative prior a counterpart to the standard frequentist approach. We explore interval estimation of a population location parameter in the presence of a mean-variance relationship a problem that is not well addressed by standard nonparametric frequentist methods. We find that the method has performance comparable to the correct parametric model, and performs notably better than some plausible yet incorrect models. Finally, we apply the method to a real data set and compare ours to previously reported results.