An augmented probit model for missing predictable covariates in quantal bioassay with small sample size

Quantal bioassay experiments relate the amount or potency of some compound; for example, poison, antibody, or drug to a binary outcome such as death or infection in animals. For infectious diseases, probit regression is commonly used for inference and a key measure of potency is given by the ID(P) , the amount that results in P% of the animals being infected. In some experiments, a validation set may be used where both direct and proxy measures of the dose are available on a subset of animals with the proxy being available on all. The proxy variable can be viewed as a messy reflection of the direct variable, leading to an errors-in-variables problem. We develop a model for the validation set and use a constrained seemingly unrelated regression (SUR) model to obtain the distribution of the direct measure conditional on the proxy. We use the conditional distribution to derive a pseudo-likelihood based on probit regression and use the parametric bootstrap for statistical inference. We re-evaluate an old experiment in 21 monkeys where neutralizing antibodies (nABs) to HIV were measured using an old (proxy) assay in all monkeys and with a new (direct) assay in a validation set of 11 who had sufficient stored plasma. Using our methods, we obtain an estimate of the ID(1) for the new assay, an important target for HIV vaccine candidates. In simulations, we compare the pseudo-likelihood estimates with regression calibration and a full joint likelihood approach.