  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.
