Two-dimensional toxic dose and multivariate logistic regression, with application to decompression sickness

In toxicological experiments with laboratory animals, there are usually many response variables of interest. When the response variables are continuous, parametric or nonparametric multivariate analysis of variance techniques can be applied to analyze the data. However, multivariate methods for dichotomous response variables are less developed in the statistical literature. An example of the need for such a method is a decompression sickness (DCS) study in which each animal subject is examined for the presence of multiple types of DCS. Two risk factors related to the outcomes are studied. Finding the range of these 2 risk factors for a fixed probability to develop DCS translates to the statistical question of estimating a 2D toxic dose corresponding to a fixed risk for multiple dichotomous outcomes. We propose a Gumbel-type generalization of logistic regression and use maximum likelihood estimation method to fit the model. The estimation of 2D toxic dose can then be based on the fitted model. We illustrate our methods with the Wisconsin sheep data.