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

Author

Listed:
  • Giacomo Aletti

    (Università degli Studi di Milano)

  • Nancy Flournoy

    (University of Missouri)

  • Caterina May

    (Università del Piemonte Orientale
    King’s College London)

  • Chiara Tommasi

    (Università degli Studi di Milano)

Abstract

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.

Suggested Citation

  • Giacomo Aletti & Nancy Flournoy & Caterina May & Chiara Tommasi, 2025. "Maximum likelihood estimation under the Emax model: existence, geometry and efficiency," Statistical Papers, Springer, vol. 66(5), pages 1-28, August.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:5:d:10.1007_s00362-025-01673-2
    DOI: 10.1007/s00362-025-01673-2
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    References listed on IDEAS

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    1. Ioannis Kosmidis & David Firth, 2011. "Multinomial logit bias reduction via the Poisson log-linear model," Biometrika, Biometrika Trust, vol. 98(3), pages 755-759.
    2. Ioannis Kosmidis & David Firth, 2021. "Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models," Biometrika, Biometrika Trust, vol. 108(1), pages 71-82.
    3. H. Dette & C. Kiss & M. Bevanda & F. Bretz, 2010. "Optimal designs for the emax, log-linear and exponential models," Biometrika, Biometrika Trust, vol. 97(2), pages 513-518.
    4. Younan Chen & Michael Fries & Sergei Leonov, 2023. "Longitudinal model for a dose-finding study for a rare disease treatment," Statistical Papers, Springer, vol. 64(4), pages 1343-1360, August.
    5. Nancy Flournoy & José Moler & Fernando Plo, 2020. "Performance Measures in Dose‐Finding Experiments," International Statistical Review, International Statistical Institute, vol. 88(3), pages 728-751, December.
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