Brian Reich

Selected Papers on Quantile Regression


Quantile regression provides a comprehensive analysis of the relationship between covariates and a response. In quantile regression, by specifying different covariate effects at different quantile levels we allow covariates to affect not only the center of the distribution, but also its spread and the magnitude of extreme events. My primary focus in quantile regession research is to develop model-based approaches which specify the response distribution in a way that has the appropriate quantile function. This is conducive to MCMC, borrows stength across quantile levels, and permits the user to center the prior on a parametric model.


Reich BJ, Smith LB (2013). Bayesian quantile regression for censored data. Biometrics.

In this paper we propose a semiparametric quantile regression model for censored survival data. Quantile regression permits covariates to affect survival differently at different stages in the follow-up period, thus providing a comprehensive study of the survival distribution. We take a semiparametric approach, representing the quantile process as a linear combination of basis functions. The basis functions are chosen so that the prior for the quantile process is centered on a simple location-scale model, but flexible enough to accommodate a wide range of quantile processes. We show in a simulation study that this approach is competitive with existing methods. The method is illustrated using data from a drug treatment study, where we find that the Bayesian model often gives smaller measures of uncertainty than its competitors, and thus identifies more significant effects.


Reich BJ (2012). Spatiotemporal quantile regression for detecting distributional changes in environmental processes. JRSS-C.

Climate change may lead to changes in several aspects of the distribution of climate variables, including changes in the mean, increased variability, and severity of extreme events. In this paper, we propose using spatiotemporal quantile regression as a flexible and interpretable method for simultaneously detecting changes in several features of the distribution of climate variables. The spatiotemporal quantile regression model assumes that each quantile level changes linearly in time, permitting straight-forward inference on the time trend for each quantile level. Unlike classical quantile regression which uses model-free methods to analyze a single quantile or several quantiles separately, we take a model-based approach which jointly models all quantiles, and thus the entire response distribution. In the spatiotemporal quantile regression model, each spatial location has its own quantile function that evolves over time, and the quantile functions are smoothed spatially using Gaussian process priors. We propose a basis expansion for the quantile function that permits a closed-form for the likelihood, and allows for residual correlation modeling via a Gaussian spatial copula. We illustrate the methods using temperature data for the southeast US from the years 1931-2009. For these data, borrowing information across space identifies more significant time trends than classical non-spatial quantile regression. We find a decreasing time trend for much of the spatial domain for monthly mean and maximum temperatures. For the lower quantiles of monthly minimum temperature, we find a decrease in Georgia and Florida, and an increase in Virginia and the Carolinas.


Reich, Fuentes, Dunson (2011). Bayesian spatial quantile regression. JASA.

Tropospheric ozone is one of the six criteria pollutants regulated by the US EPA under the Clean Air Act and has been linked with several adverse health effects, including mortality. Due to the strong dependence on weather conditions, ozone may be sensitive to climate change and there is great interest in studying the potential effect of climate change on ozone, and how this change may affect public health. In this paper we develop a Bayesian spatial model to predict ozone under different meteorological conditions, and use this model to study spatial and temporal trends and to forecast ozone concentrations under different climate scenarios. We develop a spatial quantile regression model that does not assume normality and allows the covariates to affect the entire conditional distribution, rather than just the mean. The conditional distribution is allowed to vary from site-to-site and is smoothed with a spatial prior. For extremely large data sets our model is computationally infeasible, and we develop an approximate method. We apply the approximate version of our model to summer ozone from 1997-2005 in the Eastern US, and use deterministic climate models to project ozone under future climate conditions. Our analysis suggests that holding all other factors fixed, an increase in daily average temperature will lead to the largest increase in ozone in the Industrial Midwest and Northeast.