This study focuses on the estimation of the Emax dose-response model, a widely utilized framework in clinical trials, agriculture, and environmental experiments. Existing challenges in obtaining maximum likelihood estimates (MLE) for model parameters are often ascribed to computational issues but, in reality, stem from the absence of MLE. Our contribution provides a new understanding and control of all the experimental situations that pratictioners might face, guiding them in the estimation process. We derive the exact MLE for a three-point experimental design and we identify the two scenarios where the MLE fails. To address these challenges, we propose utilizing Firth's modified score, providing its analytical expression as a function of the experimental design. Through a simulation study, we demonstrate that, in one of the problematic cases, the Firth modification yields a finite estimate. For the remaining case, we introduce a design-correction strategy akin to a hypothesis test.

翻译：本研究聚焦于Emax剂量-响应模型的估计问题，该模型在临床试验、农业及环境实验中应用广泛。现有研究常将模型参数极大似然估计的获取困难归因于计算问题，但实则源于极大似然估计的不存在性。我们的贡献在于：为实践者可能面临的所有实验情形提供全新理解与可控方案，指导其完成估计过程。我们推导了三水平实验设计的精确极大似然估计，并识别出极大似然估计失效的两种场景。针对这些挑战，我们提出采用Firth修正得分函数，并给出其作为实验设计函数的解析表达式。通过模拟研究证明：在其中一个问题场景中，Firth修正能产生有限估计量；对于剩余场景，我们引入类似假设检验的设计修正策略。