A nonparametric method for analyzing quantal response data from an indirect bioassay experiment is proposed. Kernel estimates of the dose-response curve are used to develop approximate confidence intervals for (i) the optimal combination dose of a drug with therapeutic effects at low doses and toxic effects at high doses, and (ii) the lethal dose levels of a toxic chemical. This nonparametric procedure was implemented on real and simulated data. The confidence interval for problem (i) has high coverage probabilities when the dose-response curve is symmetric about the optima. However, the coverage probabilities are adversely affected by asymmetry about the optima and consequently are not reliable unless the sample sizes are large. The use of kernel estimators with higher-order kernels may alleviate this sensitivity to asymmetry. The confidence interval for problem (ii) has high coverage probabilities robust with respect to the shape or symmetry of the underlying dose-response curve.