In drug-testing experiments the primary responses of interest are efficacy and toxicity. These can be modeled as a bivariate quantal response using the Gumbel model for bivariate logistic regression. D-optimal and Q-optimal experimental designs are developed for this model The Q-optimal design minimizes the average asymptotic prediction variance of p(l,O;d), the probability of efficacy without toxicity at dose d, over a desired range of doses. In addition, a new optimality criterion, T -optimality, is developed which minimizes the asymptotic variance of the estimate of the therapeutic index. Most experimenters will be less familiar with the Gumbel bivariate logistic regression model than with the univariate logistic regression models which comprise its marginals. Therefore, the optimal designs based on the Gumbel model are evaluated based on univariate logistic regression D-efficiencies; conversely, designs derived from the univariate logistic regression model are evaluated with respect to the Gumbel optimality criteria. Further practical considerations motivate an exploration of designs providing a maximum compromise between the three Gumbel-based criteria D, Q and T. Finally, 5-point designs which can be generated by fitted equations are proposed as a practical option for experimental use. / Ph. D.