We provide a systematic treatment of $D$-optimal design for binary regression and quantal response models in toxicology studies. For the two-parameter case, we provide an analytical equation (WC equation) for computing the $D$-optimal design quickly and when analytical solution is not available, we apply particle swarm optimization to solve for the $D$-optimal design. Examples with various link functions are given as well as the sensitivity functions. We extend the two-parameter case to three-parameter case by providing a neat formula for the determinant of the information matrix. We also suggest practitioners to work with the neat formula to derive optimal designs for three-parameter binary regression models.

翻译：我们系统地处理毒理学研究中的二元回归和四元反应模型的美元-最佳设计。关于两参数案例,我们提供分析方程式(WC方程),用于快速计算美元-最佳设计,当没有分析解决方案时,我们应用粒子群优化来解决美元-最佳设计。我们提供了各种关联功能的实例以及敏感功能。我们通过为信息矩阵的决定因素提供一个整洁的公式,将两参数案例扩大到三参数案例。我们还建议从业人员与整洁的公式合作,为三参数二元回归模型得出最佳设计。