Risk Assessment Via a Robust Probit Model, with Application to Toxicology

Article excerpt

Various frameworks have been suggested for assessing the risk associated with continuous toxicity outcomes. The first formulates the effect of exposure on the adverse effect via a simple normal model and then computes the risk function using tail probabilities from the standard normal distribution. Because this risk function depends heavily on the assumed model, it may be sensitive to model misspecification. Recently, a semiparametric approach that utilizes an alternative definition of excess risk has been studied. Unfortunately, it is not yet clear how the two approaches relate to one another. In this article, we investigate a semiparametric normal model in which an unknown transformation of the adverse response satisfies the linear model. We demonstrate that this formulation unifies the two existing approaches and allows for a coherent risk analysis of dose--response data. In addition, estimation and inference procedures for the unknown transformation in the semiparametric model for the continuous response are developed. These are incorporated in novel model-checking procedures, including a formal sup-norm test of the simple normal model. A well-known toxicological study of aconiazide, a drug under investigation for treatment of tuberculosis, serves as a case study for the risk assessment methodology.

KEY WORDS: Dose-response curve; H-H plot; Kolmogorov-Smirnov test; Risk function; Semiparametric transformation model; Toxicity study.

1. OVERVIEW OF EXISTING METHODS

Until recently, the literature on risk assessment focused almost exclusively on the relationship of a binary response to the dose of a potentially toxic substance. That is, interest centered on the occurrence or nonoccurrence of some adverse effect, such as tumor presence. In many cases, however, the dichotomous outcomes arise by splitting a continuous response. Thus the analysis may be highly dependent on the way in which the Bernoulli response is constructed; that is, sensitive to the method used to select the cutoff.

Instead of grouping the continuous response, one can model the raw outcome directly and then determine the probability of certain extreme events using the model. This was the approach taken by Chen and Gaylor (1992), Gaylor and Slikker (1990, 1992), and Kodell and West (1993). First, a linear normal model is fit to the data. Next, in accordance with Kodell and West (1993), an adverse event is defined to be a realization of the continuous response that falls in the "extreme" tail. Kodell and West (1993) characterize "extreme" by distance in standard deviations from the mean of the control (dose = 0) group. The evaluation of the dose effect is then based on the difference in this probability of a tail event compared to control. Unfortunately, the procedure may be sensitive to skewed and nonnormal errors. To address this issue, Grump (1995) considered a parametric model with an arbitrary error distribution. Even with this generalization, the key difficulty is that the tail probabilities used to compute the risk function may be highly dependent on the choice of error distribution. In particular, there may be no control group observations having the tail event.

A novel approach put forth by Bosch, Wypij, and Ryan (1995, 1996) uses a different measure of risk. Specifically, the authors attempt to answer the question "What is the probability that the response of a control subject (dose = 0) is greater than the response of a subject with exposure d (dose = d)?" When the number of dose groups is small, these probabilities can be estimated nonparametrically by looking at all possible pairings of subjects in dose group d with subjects in the control group. However, this procedure can lead to loss of information, especially in small laboratory studies in which there may only be a few animals in each dose group. To increase efficiency, the authors use the popular probit model to formulate the effect of dose on the underlying probabilities. …