Multilevel Quantile Function Modeling with Application to Birth Outcomes

Abstract

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.

Probability Distributions of Crop Yields: A Bayesian Spatial Quantile Regression Approach

“…Several extensions of the model have since been implemented. These include Reich (2012), who accounted for residual correlation in predictors, Reich and Smith (2013), who considered censored data, and Smith et al (2015), who developed a hierarchical variant of the model.…”

“…To obtain estimates of the quantile process, it is necessary to identify a suitable basis that permits the monotonicity constraint on the quantile process. Selection can be made among a number of different basis functions, including parametric quantile functions (Reich 2012), bernstein polynomials (Reich, Fuentes, and Dunson 2011), or splines (Smith et al 2015). The R package BSquare implements non‐spatial versions of Bayesian simultaneous quantile regression (Reich 2013).…”

Smooth Density Spatial Quantile Regression

“…The final approach, which we will adopt and extend, is to model the entire quantile function jointly using basis functions. This is the approach taken by Reich et al (2012) and others (Reich, 2012;Smith et al, 2015) and is more naturally implemented using a Bayesian framework. Regardless of the approach taken, * hlbrantl@ncsu.edu ensuring monotonicity requires either some form of distributional assumption, or constraints on the quantile regression coefficients and the parameter space of the predictors.…”

Composite marginal quantile regression analysis for longitudinal adolescent body mass index data

“…The Frumento and Bottai method [5] concentrated on quantile regression with univariate outcome. Quantile regression coefficients have also been specified as nonparametric functions: [6][7][8][9] adopted piecewise Gaussian basis functions,…”