Abstract

Bayesian quantile regression is an emerging alternative to conventional quantile regression with important computational advantages. However, it has been rarely used in epidemiology research because of the difficulties of doing Bayesian posterior simulation and, additionally, because the method involves an unusual form of model misspecification. In this paper, I investigate Bayesian quantile regression using the Stan programming environment and compare the results with conventional quantile regression. I apply the method in a data example that estimates the effect of chronic medical conditions on depression symptoms in Canadian adolescents. This data is well-suited to demonstrating the properties of quantile regression because it has an unusual outcome variable that is interval scale but taking discrete values on the integers from 0, 1, 2, ..., 27. This work makes new methodological contributions to our understanding of Bayesian quantile regression. First, I develop a novel Bayesian method for assessing the presence of heteroscedastic errors in the outcome variable. Second, I show the surprising result that Bayesian quantile regression may give dramatically different results compared to the conventional quantile regression estimator, even in large samples. This occurs because the point estimator from conventional quantile regression is calculated using the simplex algorithm of Barrodale and Roberts, which is heavily affected by discreteness of the outcome variable. In contrast, Bayesian quantile regression explores a continuous range of values for the unknown model parameters. I illustrate the advantages of inference using the full posterior distribution for inference rather than the conventional quantile regression point estimator by using a logarithmic scoring rule for probabilistic prediction. I demonstrate that inference based on the full posterior distribution for unknown parameters will often yield a better overall fit for the data compared to conventional quantile regression.