Stochastically ordered multiple regression

In various application areas, prior information is available about the direction of the effects of multiple predictors on the conditional response distribution. For example, in epidemiology studies of potentially adverse exposures and continuous health responses, one can typically assume a priori that increasing the level of an exposure does not lead to an improvement in the health response. Such an assumption can be formalized through a stochastic ordering assumption in each of the exposures, leading to a potentially large improvement in efficiency in nonparametric modeling of the conditional response distribution. This article proposes a Bayesian nonparametric approach to this problem based on characterizing the conditional response density as a Gaussian mixture, with the locations of the Gaussian means varying flexibly with predictors subject to minimal constraints to ensure stochastic ordering. Theoretical properties are considered and Markov chain Monte Carlo methods are developed for posterior computation. The methods are illustrated using simulation examples and a reproductive epidemiology application.