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 a MLE. Our contribution provides a new understanding and control of all the experimental situations that practitioners 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-augmentation strategy akin to a hypothesis test.

翻译：本研究聚焦于Emax剂量-反应模型的参数估计问题，该模型在临床试验、农业及环境实验中应用广泛。现有研究中获取模型参数最大似然估计（MLE）的挑战常被归因于计算问题，实则源于MLE的不存在性。本研究的贡献在于对实践者可能面临的所有实验情境提供了全新的理解与控制方法，从而指导其估计过程。我们推导出三点实验设计下的精确MLE，并识别出MLE失效的两种情形。针对这些挑战，我们提出采用Firth修正评分函数，并给出其作为实验设计函数的解析表达式。通过模拟研究，我们证明在其中一个问题情形下，Firth修正能够产生有限估计值。对于剩余情形，我们提出了一种类似于假设检验的设计增强策略。