Maximum Likelihood Estimation under the Emax Model: Existence, Geometry and Efficiency

TL;DR

This paper investigates the existence and properties of maximum likelihood estimates in the Emax dose-response model, providing new theoretical insights, explicit solutions for certain designs, and strategies to address estimation failures.

Contribution

It offers a comprehensive analysis of MLE existence in the Emax model, derives explicit MLEs for three-point designs, and proposes methods to overcome estimation issues.

Findings

Exact MLE derived for three-point design

Identification of scenarios where MLE fails

Firth's modification improves estimation in problematic cases

Abstract

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.