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Register a new userJoint modelling of longitudinal measurements and survival times via a copula approach
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Zili Zhang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Joint modelling of longitudinal and time-to-event data is usually described by a random effect joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the latent effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas, to introduce non-linear correlation between the marginal distributions of the longitudinal and time-to-event processes. A Gaussian copula joint model has been proposed in the literature to fit joint data by applying a Monte Carlo expectation-maximisation algorithm. Enlightening as it is, its original estimation procedure comes with some limitations. In the original approach, the log-likelihood function can not be derived analytically thus requires a Monte Carlo integration, which not only comes with intensive computation but also introduces extra variation/noise into the estimation. The combination with the EM algorithm slows down the computation further and convergence to the maximum likelihood estimators can not be always guaranteed. In addition, the assumption that the length of planned measurements is uniform and balanced across all subjects is not suitable when subjects have varying number of observations. In this paper, we relax this restriction and propose an exact likelihood estimation approach to replace the more computationally expensive Monte Carlo expectation-maximisation algorithm. We also provide a straightforward way to compute dynamic predictions of survival probabilities, showing that our proposed model is comparable in prediction performance to the shared random effects joint model.
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We introduce a numerically tractable formulation of Bayesian joint models for longitudinal and survival data. The longitudinal process is modelled using generalised linear mixed models, while the survival process is modelled using a parametric general hazard structure. The two processes are linked by sharing fixed and random effects, separating the effects that play a role at the time scale from those that affect the hazard scale. This strategy allows for the inclusion of non-linear and time-dependent effects while avoiding the need for numerical integration, which facilitates the implementation of the proposed joint model. We explore the use of flexible parametric distributions for modelling the baseline hazard function which can capture the basic shapes of interest in practice. We discuss prior elicitation based on the interpretation of the parameters. We present an extensive simulation study, where we analyse the inferential properties of the proposed models, and illustrate the trade-off between flexibility, sample size, and censoring. We also apply our proposal to two real data applications in order to demonstrate the adaptability of our formulation both in univariate time-to-event data and in a competing risks framework. The methodology is implemented in rstan.
A new class of copulas, termed the MGL copula class, is introduced. The new copula originates from extracting the dependence function of the multivariate generalized log-Moyal-gamma distribution whose marginals follow the univariate generalized log-Moyal-gamma (GLMGA) distribution as introduced in citet{li2019jan}. The MGL copula can capture nonelliptical, exchangeable, and asymmetric dependencies among marginal coordinates and provides a simple formulation for regression applications. We discuss the probabilistic characteristics of MGL copula and obtain the corresponding extreme-value copula, named the MGL-EV copula. While the survival MGL copula can be also regarded as a special case of the MGB2 copula from citet{yang2011generalized}, we show that the proposed model is effective in regression modelling of dependence structures. Next to a simulation study, we propose two applications illustrating the usefulness of the proposed model. This method is also implemented in a user-friendly R package: texttt{rMGLReg}.
In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution $F_{true}$. First, the Dirichlet process is constructed on the unknown marginal distributions of $F_{true}$. Then a Gaussian copula model is utilized to capture the dependence structure of $F_{true}$. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Several interesting theoretical results of the approach are derived. Finally, through several simulated examples and a real data set, the proposed approach reveals excellent performance.
This paper introduces a general framework for survival analysis based on ordinary differential equations (ODE). Specifically, this framework unifies many existing survival models, including proportional hazards models, linear transformation models, accelerated failure time models, and time-varying coefficient models as special cases. Such a unified framework provides a novel perspective on modeling censored data and offers opportunities for designing new and more flexible survival model structures. Further, the aforementioned existing survival models are traditionally estimated by procedures that suffer from lack of scalability, statistical inefficiency, or implementation difficulty. Based on well-established numerical solvers and sensitivity analysis tools for ODEs, we propose a novel, scalable, and easy-to-implement general estimation procedure that is applicable to a wide range of models. In particular, we develop a sieve maximum likelihood estimator for a general semi-parametric class of ODE models as an illustrative example. We also establish a general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semi-parametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example.
The article develops marginal models for multivariate longitudinal responses. Overall, the model consists of five regression submodels, one for the mean and four for the covariance matrix, with the latter resulting by considering various matrix decompositions. The decompositions that we employ are intuitive, easy to understand, and they do not rely on any assumptions such as the presence of an ordering among the multivariate responses. The regression submodels are semiparametric, with unknown functions represented by basis function expansions. We use spike-slap priors for the regression coefficients to achieve variable selection and function regularization, and to obtain parameter estimates that account for model uncertainty. An efficient Markov chain Monte Carlo algorithm for posterior sampling is developed. The simulation studies presented investigate the effects of priors on posteriors, the gains that one may have when considering multivariate longitudinal analyses instead of univariate ones, and whether these gains can counteract the negative effects of missing data. We apply the methods on a highly unbalanced longitudinal dataset with four responses observed over of period of 20 years
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