Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty–copula model

Bo Hong Wu, Hirofumi Michimae, Takeshi Emura*

*Corresponding author for this work

Research output: Contribution to journalJournal Article peer-review

13 Scopus citations


In meta-analysis of individual patient data with semi-competing risks, the joint frailty–copula model has been proposed, where frailty terms account for the between-study heterogeneity and copulas account for dependence between terminal and nonterminal event times. In the previous works, the baseline hazard functions in the joint frailty–copula model are estimated by the nonparametric model or the penalized spline model, which requires complex maximization schemes and resampling-based interval estimation. In this article, we propose the Weibull distribution for the baseline hazard functions under the joint frailty–copula model. We show that the Weibull model constitutes a conjugate model for the gamma frailty, leading to explicit expressions for the moments, survival functions, hazard functions, quantiles, and mean residual lifetimes. These results facilitate the parameter interpretation of prognostic inference. We propose a maximum likelihood estimation method and make our computer programs available in the R package, joint.Cox. We also show that the delta method is feasible to calculate interval estimates, which is a useful alternative to the resampling-based method. We conduct simulation studies to examine the accuracy of the proposed methods. Finally, we use the data on ovarian cancer patients to illustrate the proposed method.

Original languageEnglish
Pages (from-to)1525-1552
Number of pages28
JournalComputational Statistics
Issue number4
StatePublished - 01 12 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.


  • Clustered survival data
  • Gamma frailty
  • Hierarchical model
  • Mean residual life
  • Random effects
  • Survival analysis


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