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Maximum likelihood estimation in the additive hazards model

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  • Chengyuan Lu
  • Jelle Goeman
  • Hein Putter

Abstract

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non‐parametric model, additive hazards offer a very flexible way of modeling time‐varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non‐negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log‐likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non‐negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least‐squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least‐squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.

Suggested Citation

  • Chengyuan Lu & Jelle Goeman & Hein Putter, 2023. "Maximum likelihood estimation in the additive hazards model," Biometrics, The International Biometric Society, vol. 79(3), pages 1646-1656, September.
    • Handle: RePEc:bla:biomet:v:79:y:2023:i:3:p:1646-1656
    DOI: 10.1111/biom.13764
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    References listed on IDEAS

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    1. Marschner, Ian C. & Gillett, Alexandra C. & O’Connell, Rachel L., 2012. "Stratified additive Poisson models: Computational methods and applications in clinical epidemiology," Computational Statistics & Data Analysis, Elsevier, vol. 56(5), pages 1115-1130.
    2. Sheila M. Gore & Stuart J. Pocock & Gillian R. Kerr, 1984. "Regression Models and Non‐Proportional Hazards in the Analysis of Breast Cancer Survival," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(2), pages 176-195, June.
    3. Torben Martinussen & Stijn Vansteelandt & Per Kragh Andersen, 2020. "Subtleties in the interpretation of hazard contrasts," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(4), pages 833-855, October.
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