Convex mixture regression for quantitative risk assessment

Abstract

There is wide interest in studying how the distribution of a continuous response changes with a predictor. We are motivated by environmental applications in which the predictor is the dose of an exposure and the response is a health outcome. A main focus in these studies is inference on dose levels associated with a given increase in risk relative to a baseline. In addressing this goal, popular methods either dichotomize the continuous response or focus on modeling changes with the dose in the expectation of the outcome. Such choices may lead to information loss and provide inaccurate inference on dose-response relationships. We instead propose a Bayesian convex mixture regression model that allows the entire distribution of the health outcome to be unknown and changing with the dose. To balance flexibility and parsimony, we rely on a mixture model for the density at the extreme doses, and express the conditional density at each intermediate dose via a convex combination of these extremal densities. This representation generalizes classical dose-response models for quantitative outcomes, and provides a more parsimonious, but still powerful, formulation compared to nonparametric methods, thereby improving interpretability and efficiency in inference on risk functions. A Markov chain Monte Carlo algorithm for posterior inference is developed, and the benefits of our methods are outlined in simulations, along with a study on the impact of dde exposure on gestational age.

Keywords: Additional risk; Benchmark dose; Conditional density estimation; Convex density regression; Dose-response; Nonparametric density regression.

Figure 1

Histograms of the observed gestational age at delivery for selected dose intervals, along with the pointwise posterior mean (continuous blue lines), and 95% credible intervals (shaded blue areas) for $f\left(y|x\right)$ calculated in the central points (7.5, 22.5, 37.5, 52.5, 67.5, 125) of each dose interval, under the proposed convex mixture regression model. The figure appears in color in the electronic version of this article.

Figure 2

Goodness-of-fit assessments for n = 500 in Scenario 1 (a), 2 (b), and 3 (c). Smoothed empirical estimate of $F_x\left(37\right)=\text{pr}\left(y\le 37|x\right)$ computed from the observed data (black line), and from 50 data sets simulated from the posterior predictive distribution induced by CoMiRe (grey lines). In the x axis we also report the simulated dose exposures.

Figure 3

Inference on the additional risk function for n = 500 in the three scenarios. The dashed lines represent the true additional risk function R_A(x, 37), whereas the red, green, and blue continuous lines denote the posterior mean of R_A(x, 37) under anova–ddp, f–dmix, and CoMiRe, respectively. The shaded areas represent the pointwise 95% posterior credible bands. In the x axis we report the simulated dose exposures. Lower panels provide a zoom on the range of the additional risk typically considered in benchmark dose analysis.

Figure 4

Goodness-of-fit assessment in the application. Smoothed empirical estimate of $F_x\left(37\right)=\text{pr}\left(y\le 37|x\right)$ computed from the observed data (black line), and from 50 data sets simulated from the posterior predictive distribution induced by CoMiRe (grey lines). In the x axis we report the observed exposures.

Figure 5

Posterior mean (solid lines) and pointwise 95% credible bands (shaded areas) for (a) β(x), (b) R_A(x, 37) and (c) the related BMD_q. In the x axis in (a) and (b) we report the observed exposures.