Bivariate binary models of efficacy and toxicity in dose-ranging trials

Methods for the simultaneous analysis of the relationships of binary variables for efficacy and toxicity to dosage of an experimental drug are developed. Properties of two models of ‘within-dose’ dependence of efficacy and toxicity in parallel designs - one a bivariate analogue of the familiar univariate logistic model, and the other an adaptation of a general model developed by D.R. Cox– are explored. The cell probabilities predicted by these models are often quite similar to those predicted by a model of independence of efficacy and toxicity, but large discrepancies can occur when there is approximate equality of the median effective and median toxic doses. Asymptotic variances of estimates of parameters involved in assessing correlation are large when there is little or no dependence in the data, but parameters can be estimated with good precision in at least some cases of moderate to strong dependence between efficacy and toxicity.