Likelihood Models for Clustered Binary and Continuous Out comes: Application to Developmental Toxicology

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Abstract
Summary. In developmental toxicology, methods based on dose‐response modeling and quantitative risk assessment are being actively pursued. Among live fetuses, the presence of malformations and reduction in fetal weight are of primary interest, but ordinarily, the dose‐response relationships are characterized in each of the outcomes separately while appropriately accounting for clustering within litters. Jointly modeling the outcomes, allowing different relationships with dose while incorporating the correlation between the fetuses and the outcomes, may be more appropriate. We propose a likelihood‐based model that is an extension of a correlated probit model to incorporate continuous outcomes. Our model maintains a marginal dose–response interpretation for the individual outcomes while taking into account both the correlations between outcomes on an individual fetus and those due to clustering. The joint risk of malformation and low birth weight can then be estimated directly. This approach is particularly well suited to estimating safe dose levels as part of quantitative risk assessment.