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.

Cardiometabolic diseases have substantially increased in China in the past 20 years and blood pressure is a primary modifiable risk factor. Using data from the China Health and Nutrition Survey we examine blood pressure trends in China from 1991 to 2009, with a concentration on age cohorts and urbanicity. Very large values of blood pressure are of interest, so we model the conditional quantile functions of systolic and diastolic blood pressure. This allows the covariate effects in the middle of the distribution to vary from those in the upper tail, the focal point of our analysis. We join the distributions of systolic and diastolic blood pressure using a copula, which permits the relationships between the covariates and the two responses to share information and enables probabilistic statements about systolic and diastolic blood pressure jointly. Our copula maintains the marginal distributions of the group quantile effects while accounting for within-subject dependence, enabling inference at the population and subject levels. Our population level regression effects change across quantile level, year, and blood pressure type, providing a rich environment for inference. To our knowledge, this is the first quantile function model to explicitly model within-subject autocorrelation and is the first quantile function approach that simultaneously models multivariate conditional response. We find that the association between high blood pressure and living in an urban area has evolved from positive to negative, with the strongest changes occurring in the upper tail. The increase in urbanization over the last twenty years coupled with the transition from the positive association between urbanization and blood pressure in earlier years to a more uniform association with urbanization suggests increasing blood pressure over time throughout China, even in less urbanized areas. Our methods are available in the R package BSquare.

Summary: Infants born preterm or small for gestational age have elevated rates of morbidity and mortality. Using birth certificate records in Texas from [2002][2003][2004] and Environmental Protection Agency air pollution estimates, we relate the quantile functions of birth weight and gestational age to ozone exposure and multiple predictors, including parental age, race, and education level. We introduce a semi-parametric Bayesian quantile approach that models the full quantile function rather than just a few quantile levels. Our multilevel quantile function model establishes relationships between birth weight and the predictors separately for each week of gestational age and between gestational age and the predictors separately across Texas Public Health Regions. We permit these relationships to vary nonlinearly across gestational age, spatial domain and quantile level and we unite them in a hierarchical model via a basis expansion on the regression coefficients that preserves interpretability. Very low birth weight is a primary concern, so we leverage extreme value theory to supplement our model in the tail of the distribution. Gestational ages are rounded into weekly values, so we present methodology for modeling quantile functions of discrete response data. In a simulation study we show that pooling information across gestational age and quantile level substantially reduces MSE of predictor effects relative to standard frequentist quantile regression. We find that ozone is negatively associated with the lower tail of gestational age in south Texas and across the distribution of birth weight for high gestational ages. Our methods are available in the R package BSquare.

I thank Yang et al. (2015) (henceforth Yang) for a well-written paper that explores the asymptotic validity of the asymmetric Laplace (AL) distribution in Bayesian quantile regression. Asymptotically, the mode of the posterior distribution of the regression parameters is the maximum likelihood estimator of the AL distribution. Yang use this fact to inform a simple correction to the estimate of the posterior variance that dramatically improves coverage probabilities. Yang convincingly present asymptotic and numerical arguments illustrating the improvement.The proposed approach consists of first employing standard frequentist quantile regression of the median of the response on the predictors to estimate the variance of the errors. Then, the parameters are estimated using a Bayesian model with the asymmetric Laplace working likelihood that incorporates a point mass prior for the error variance. Finally, the covariance of the final estimates is used as the 'meat' in a 'sandwich' estimator that is designed to mimic the asymptotic sampling distribution of the regression parameters. The approach borrows asymptotic validity and flexibility from the frequentist domain and the ease of approximating covariance matrices from the Bayesian domain to greatly improve coverage probabilities.The simulation study in the paper introducing the AL in Bayesian quantile regression (Yu & Moyeed, 2001) did not include predictors and assumed standard normal errors. The improvement shown in coverage probabilities in a more challenging simulation setting is quite an achievement. I tested the improvement in another simulation setting, with the assistance of the code from Yang. Define the functions a. / D sign.0:5 / log.