Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty–copula model
TL;DR: It is shown 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, which facilitate the parameter interpretation of prognostic inference.
Abstract: 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.
Citations
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TL;DR: A frailty‐copula model is proposed, which is a hybrid model including both a frailty term and a copula function for dependence between failure times, and likelihood‐based inference methods based on competing risks data, including accelerated failure time models are developed.
24 citations
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01 Dec 2021
TL;DR: This paper proposes a novel copula-based Markov chain model for describing serial dependence in recurrent event times, and proposes a two-stage estimation method under Weibull distributions for fitting the survival data.
Abstract: Copula modeling for serial dependence has been extensively discussed in a time series context. However, fitting copula-based Markov models for serially dependent survival data is challenging due to the complex censoring mechanisms. The purpose of this paper is to develop likelihood-based methods for fitting a copula-based Markov chain model to serially dependent event times that are dependently censored by a terminal event, such as death. We propose a novel copula-based Markov chain model for describing serial dependence in recurrent event times. We also apply another copula model for handling dependent censoring. Due to the complex likelihood function with the two copulas, we propose a two-stage estimation method under Weibull distributions for fitting the survival data. The asymptotic normality of the proposed estimator is established through the theory of estimating functions. We propose a jackknife method for interval estimates, which is shown to be asymptotically consistent. To select suitable copulas for a given dataset, we propose a model selection method according to the 2nd stage likelihood. We conduct simulation studies to assess the performance of the proposed methods. For illustration, we analyze survival data from colorectal cancer patients. We implement the proposed methods in our original R package “Copula.Markov.survival” that is made available in CRAN (
https://cran.r-project.org/
).
15 citations
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TL;DR: In this paper , the authors comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses, including the exponential, Weibull, Rayleigh, lognormal, log-logistic, gamma, generalized gamma, Pareto (types I, II, and IV), Hjorth, Burr (types III and XII), Dagum, exponential power, Gompertz, Birnbaum-Saunders, exponential-logarithmic, piecewise exponential, generalized exponential, exponentiated Weibell, generalized modified Weibbull, and spline distributions.
Abstract: During its 330 years of history, parametric distributions have been useful for survival and reliability analyses. In this paper, we comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses. We provide encyclopedic coverage of the important parametric distributions, which is more extensive than the existing textbooks on survival and reliability analyses. We also explain how these distributions have been adopted in survival and reliability analyses with original and state-of-the-art references. We cover the exponential, Weibull, Rayleigh, lognormal, log-logistic, gamma, generalized gamma, Pareto (types I, II, and IV), Hjorth, Burr (types III and XII), Dagum, exponential power, Gompertz, Birnbaum-Saunders, exponential-logarithmic, piecewise exponential, generalized exponential, exponentiated Weibull, generalized modified Weibull, and spline distributions. We analyze a real dataset for illustration.
12 citations
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TL;DR: This article provides a tutorial in order to build a web-based application for dynamic risk prediction for cancer patients on the basis of the R packages joint, and demonstrates the proposed methods using a dataset of breast cancer patients from multiple clinical studies.
Abstract: Clinical risk prediction formulas for cancer patients can be improved by dynamically updating the formulas by intermediate events, such as tumor progression. The increased accessibility of individual patient data (IPD) from multiple studies has motivated the development of dynamic prediction formulas accounting for between-study heterogeneity. A joint frailty-copula model for overall survival and time to tumor progression has the potential to develop a dynamic prediction formula of death from heterogenous studies. However, the process of developing, validating, and publishing the prediction formula is complex, which has not been sufficiently described in the literature. In this article, we provide a tutorial in order to build a web-based application for dynamic risk prediction for cancer patients on the basis of the R packages joint.Cox and Shiny. We demonstrate the proposed methods using a dataset of breast cancer patients from multiple clinical studies. Following this tutorial, we demonstrate how one can publish web applications available online, which can be manipulated by any user through a smartphone or personal computer. After learning this tutorial, developers acquire the ability to build an online web application using their own datasets.
11 citations
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TL;DR: In this article, a dynamic prediction method using a bivariate failure time model allows one to build a prediction for the time-to-death for patients, which is one of the most important issues in survival analysis.
Abstract: Predicting time-to-death for patients is one of the most important issues in survival analysis. A dynamic prediction method using a bivariate failure time model allows one to build a prediction for...
9 citations
References
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TL;DR: In this article, the applicability of statistics to a wide field of problems is discussed, and examples of simple and complex distributions are given, as well as a discussion of the application of statistics in a wide range of problems.
Abstract: This paper discusses the applicability of statistics to a wide field of problems. Examples of simple and complex distributions are given.
9,091 citations
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01 Jan 1999
TL;DR: This book discusses the fundamental properties of copulas and some of their primary applications, which include the study of dependence and measures of association, and the construction of families of bivariate distributions.
Abstract: The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. This book is suitable as a text or for self-study.
8,626 citations
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[...]
23 Oct 2007
TL;DR: In this book different methods based on the frailty model are described and it is demonstrated how they can be used to analyze clustered survival data.
Abstract: Readers will find in the pages of this book a treatment of the statistical analysis of clustered survival data. Such data are encountered in many scientific disciplines including human and veterinary medicine, biology, epidemiology, public health and demography. A typical example is the time to death in cancer patients, with patients clustered in hospitals. Frailty models provide a powerful tool to analyze clustered survival data. In this book different methods based on the frailty model are described and it is demonstrated how they can be used to analyze clustered survival data. All programs used for these examples are available on the Springer website.
461 citations
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TL;DR: In this article, the authors considered a variant of the competing risks problem in which a terminal event censors a non-terminal event, but not vice versa, and formulated the joint distribution of the events via a gamma frailty model in the upper wedge where data are observable, with the marginal distributions unspecified.
Abstract: SUMMARY We consider a variation of the competing risks problem in which a terminal event censors a non-terminal event, but not vice versa. The joint distribution of the events is formulated via a gamma frailty model in the upper wedge where data are observable (Day et al., 1997), with the marginal distributions unspecified. An estimator for the association parameter is obtained from a concordance estimating function. A novel plug-in estimator for the marginal distribution of the non-terminal event is shown to be uniformly consistent and to converge weakly to a Gaussian process. The assumptions on the joint distribution outside the upper wedge are weaker than those usually made in competing risks analyses. Simulations demonstrate that the methods work well with practical sample sizes. The proposals are illustrated with data on morbidity and mortality in leukaemia patients.
270 citations