Quantitative risk assessment for multivariate continuous outcomes with application to neurotoxicology: the bivariate case

The neurotoxic effects of chemical agents are often investigated in controlled studies on rodents, with multiple binary and continuous endpoints routinely collected. One goal is to conduct quantitative risk assessment to determine safe dose levels. Such studies face two major challenges for continuous outcomes. First, characterizing risk and defining a benchmark dose are difficult. Usually associated with an adverse binary event, risk is clearly definable in quantal settings as presence or absence of an event; finding a similar probability scale for continuous outcomes is less clear. Often, an adverse event is defined for continuous outcomes as any value below a specified cutoff level in a distribution assumed normal or log normal. Second, while continuous outcomes are traditionally analyzed separately for such studies, recent literature advocates also using multiple outcomes to assess risk. We propose a method for modeling and quantitative risk assessment for bivariate continuous outcomes that address both difficulties by extending existing percentile regression methods. The model is likelihood based; it allows separate dose-response models for each outcome while accounting for the bivariate correlation and overall characterization of risk. The approach to estimation of a benchmark dose is analogous to that for quantal data without the need to specify arbitrary cutoff values. We illustrate our methods with data from a neurotoxicity study of triethyl tin exposure in rats.