Optimal designs for bivariate binary random variables in a bivariate treatment space

We consider the design of an experiment involving two drugs; associated with each drug is an identifiable type of toxicity. For example, in cancer chemotherapy, cyclophosphamide is toxic to the heart and busulfan is toxic to the liver. Therefore, four responses are possible: no toxicity, toxicity to the heart, toxicity to the liver and toxicity to both organs. Consequently, responses are bivariate binary random variables that depend on a bivariate treatment space. Assuming a bivariate probit response function with $0\le\rho\le1,$ we have shown that the D-optimal design consists of 4 points with equal allocation to each. These 4 points are not unique. Indeed, given a standardized design space, any four equidistant points inscribed in a circle whose radius is determined by the correlation coefficient are optimal. In addition, the optimal design approaches one point when $\rho$ converges to 1. We also introduce Quantal Contour Optimality for use when one is interested in any drug combination that produces a specified level of toxicity.