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Regularized bidimensional estimation of the hazard rate

  • Vivien Goepp ORCID logo EMAIL logo , Jean-Christophe Thalabard , Grégory Nuel and Olivier Bouaziz


In epidemiological or demographic studies, with variable age at onset, a typical quantity of interest is the incidence of a disease (for example the cancer incidence). In these studies, the individuals are usually highly heterogeneous in terms of dates of birth (the cohort) and with respect to the calendar time (the period) and appropriate estimation methods are needed. In this article a new estimation method is presented which extends classical age-period-cohort analysis by allowing interactions between age, period and cohort effects. We introduce a bidimensional regularized estimate of the hazard rate where a penalty is introduced on the likelihood of the model. This penalty can be designed either to smooth the hazard rate or to enforce consecutive values of the hazard to be equal, leading to a parsimonious representation of the hazard rate. In the latter case, we make use of an iterative penalized likelihood scheme to approximate the L0 norm, which makes the computation tractable. The method is evaluated on simulated data and applied on breast cancer survival data from the SEER program.

Corresponding author: Vivien Goepp, MAP5, CNRS UMR 8145, 45, rue des Saints-Pères, 75006, Paris, France; MINES ParisTech, CBIO–Centre for Computational Biology, PSL Research University, 75006, Paris, France; Institut Curie, PSL Research University, 75005, Paris, France; and Inserm, U900, Paris, France, E-mail: .


The authors are thankful to the National Cancer Institute for providing U.S. mortality data on cancer. We also thank an anonymous reviewer for comments on the adaptive ridge and its link to related methods which significantly improved the quality of this paper.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of Interest statement: The authors have declared no conflict of interest.


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Supplementary Material

The online version of this article offers supplementary material (

Received: 2019-01-07
Revised: 2020-12-03
Accepted: 2021-02-26
Published Online: 2021-03-24

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