An approach to model clustered survival data with dependent censoring.
TL;DR: This study introduces a likelihood-based method, via the Weibull and piecewise exponential distributions, capable of accommodating the dependence between failure and censoring times and devise a Monte Carlo EM algorithm to carry out inferences.
Abstract: In this study we introduce a likelihood-based method, via the Weibull and piecewise exponential distributions, capable of accommodating the dependence between failure and censoring times. The methodology is developed for the analysis of clustered survival data and it assumes that failure and censoring times are mutually independent conditional on a latent frailty. The dependent censoring mechanism is accounted through the frailty effect and this is accomplished by means of a key parameter accommodating the correlation between failure and censored observations. The full specification of the likelihood in our work simplifies the inference procedures with respect to Huang and Wolfe since it reduces the computation burden of working with the profile likelihood. In addition, the assumptions made for the baseline distributions lead to models with continuous survival functions. In order to carry out inferences, we devise a Monte Carlo EM algorithm. The performance of the proposed models is investigated through a simulation study. Finally, we explore a real application involving patients from the Dialysis Outcomes and Practice Patterns Study observed between 1996 and 2015.
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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.
16 citations
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
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
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
TL;DR: A penalized Cox regression method using five M-spline basis functions is introduced for analysis of right-censored data and is more flexible than the existing parametric models as it produces the increasing, decreasing, convex, concave, and constant hazard functions.
Abstract: Hazard models with cubic spline functions have a number of advantages to the existing regression models For analysis of right-censored data, we introduce a penalized Cox regression method using fi
7 citations
Cites background from "An approach to model clustered surv..."
...Bryson and Johnson (1981) and Wichitsa-Nguan, L€auter, and Liero (2016) investigated the case where the partial likelihood does not have a maximum....
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References
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Journal Article•
TL;DR: Copyright (©) 1999–2012 R Foundation for Statistical Computing; permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and permission notice are preserved on all copies.
Abstract: Copyright (©) 1999–2012 R Foundation for Statistical Computing. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.
272,030 citations
Additional excerpts
...R Core Team (2017)....
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...All models were fitted in R (R Core Team, 2017)....
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49,597 citations
Journal Article•
4,506 citations
TL;DR: A procedure is derived for extracting the observed information matrix when the EM algorithm is used to find maximum likelihood estimates in incomplete data problems and a method useful in speeding up the convergence of the EM algorithms is developed.
Abstract: A procedure is derived for extracting the observed information matrix when the EM algorithm is used to find maximum likelihood estimates in incomplete data problems. The technique requires computation of a complete-data gradient vector or second derivative matrix, but not those associated with the incomplete data likelihood. In addition, a method useful in speeding up the convergence of the EM algorithm is developed. Two examples are presented.
2,145 citations
TL;DR: The use of regression models for making covariance adjustments in the comparsion of survival curves is illustrated by application to a clinical trial of maintenance therapy for childhood leukemia.
Abstract: The use of regression models for making covariance adjustments in the comparsion of survival curves is illustrated by application to a clinical trial of maintenance therapy for childhood leukemia. Three models are considered: the log linear exponential (Glasser [1967]); Cox's [1972] nonparametric generalization of this; and the linear exponential (Feigl and Zelen [1965]). Age and white blood count at diagnosis are both shown to be important for prognosis; adjustment for the latter variable has marked effects on the treatment comparisons. Both advantages and disadvantages with the regression approach are noted.
1,728 citations
"An approach to model clustered surv..." refers methods in this paper
...…alternatives; (d) the PE distribution provides a flexible framework for the baseline hazard function, since no constraints are imposed on its shape; in addition, according to Breslow (1974) and Collett (2015), the PE model with constant hazards between event times is equivalent to the Cox model....
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...In order to specify the time grid of the PE model, we have followed the approach suggested by Breslow (1974) and Demarqui, Loschi, Dey, and Colosimo (2012); Demarqui, Dey, Loschi, and Colosimo (2014), which consists in using the distinct observed failure times as end-points....
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