Maximum likelihood estimation under the Emax model: existence, geometry and efficiency

This study focuses on the estimation of the Emax dose-response model, a widely utilized framework in clinical trials, experiments in pharmacology, agriculture, environmental science, and more. 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 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 identify the two scenarios where the MLE fails to exist. To address these challenges, we propose utilizing Firth's modified score, which we express analytically as a function of the experimental design. Through a simulation study, we demonstrate that the Firth modification yields a finite estimate in one of the problematic scenarios. For the remaining case, we introduce a design-augmentation strategy akin to a hypothesis test.