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

Area of Science:

• Pharmacometrics and Biostatistics
• Experimental Design and Statistical Modeling

Background:

• The Emax dose-response model is crucial in various scientific fields, including clinical trials and pharmacology.
• Estimating model parameters using maximum likelihood estimation (MLE) faces challenges not due to computation, but due to the non-existence of MLE in certain scenarios.

Purpose of the Study:

• To provide a comprehensive understanding and control over experimental situations encountered during Emax model parameter estimation.
• To identify specific conditions where maximum likelihood estimates (MLE) for the Emax model do not exist.

Main Methods:

• Derivation of exact maximum likelihood estimates (MLE) for a three-point experimental design.
• Identification of two distinct scenarios where MLE fails to exist.
• Application of Firth's modified score, expressed analytically as a function of the experimental design, to address non-existence of MLE.

Main Results:

• The study analytically derives the exact MLE for a three-point design.
• Firth's modified score successfully yields a finite estimate in one of the identified problematic scenarios.
• A design-augmentation strategy, analogous to a hypothesis test, is proposed for the remaining challenging scenario.

Conclusions:

• The non-existence of MLE in Emax model estimation is an inherent property of specific experimental designs, not a computational limitation.
• Firth's modification and design-augmentation offer practical solutions for robust parameter estimation in challenging experimental designs.
• This work enhances the reliability and applicability of the Emax dose-response model across scientific disciplines.