 Brian J. Reich
Assistant Professor
Department of Statistics
North Carolina State University

Research 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. We have recently developed an R package entitled BSquare to impliment a basic version of this model-based approach. Here is a description and sample code for four examples. Continuous data Survival data Discrete data Spline-based approach Selected papers

## Reich, Smith (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 (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. ## Bondell, Reich, Wang (2010). Non-crossing quantile regression curve estimation. Biometrika.

Since quantile regression curves are estimated individually, the quantile curves can cross, leading to an invalid distribution for the response. A simple constrained version of quantile regression is proposed to avoid the crossing problem for both linear and nonparametric quantile curves. A simulation study and a reanalysis of tropical cyclone intensity data shows the usefulness of the procedure. Asymptotic properties of the estimator are equivalent to the typical approach under standard conditions, and the proposed estimator reduces to the classical one if there is no crossing. The performance of the constrained estimator has shown significant improvement by adding smoothing and stability across the quantile levels. ## Reich, Bondell, Wang (2010). Flexible Bayesian quantile regression for independent and clustered data. Biostatistics.

Quantile regression has emerged as a useful supplement to ordinary mean regression. Traditional frequentist quantile regression makes very minimal assumptions on the form of the error distribution, and thus is able to accommodate non-normal errors which are common in many applications. However, inference for these models is challenging, particularly for clustered or censored data. A Bayesian approach enables exact inference and is well-suited to incorporate clustered, missing, or censored data. In this paper, we propose a flexible Bayesian quantile regression model. We assume that the error distribution is an infinite mixture of Gaussian densities subject to a stochastic constraint which enables inference on the quantile of interest. This method outperforms the traditional frequentist method under a wide array of simulated data models. We extend the proposed approach to analyze clustered data. Here we differentiate between and develop conditional and marginal models for clustered data. We apply our methods to analyze a multi-patient apnea duration data set. 