A Bayesian hierarchical model for quantitative and qualitative responses

摘要

In many science and engineering systems both quantitative and qualitative output observations are collected. If modeled separately the important relationship between the two types of responses is ignored. In this article, we propose a Bayesian hierarchical modeling framework to jointly model a continuous and a binary response. Compared with the existing methods, the Bayesian method overcomes two restrictions. First, it solves the problem in which the model size (specifically, the number of parameters to be estimated) exceeds the number of observations for the continuous response. We use one example to show how such a problem can easily occur if the design of the experiment is not proper; all the frequentist approaches would fail in this case. Second, the Bayesian model can provide statistical inference on the estimated parameters and predictions, whereas it is not clear how to obtain inference using the latest method proposed by Deng and Jin (2015), which jointly models the two responses via constrained likelihood. We also develop a Gibbs sampling scheme to generate accurate estimation and prediction for the Bayesian hierarchical model. Both the simulation and the real case study are shown to illustrate the proposed method.