A shared parameter model of longitudinal measurements and survival time with heterogeneous random-effects distribution

Joint modelling of longitudinal measurements and survival times via a multivariate copula approach

Joint modelling of longitudinal and time-to-event data is usually described by a 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 random 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. The multivariate Gaussian copula joint model has been proposed in the literature to fit joint data by applying a Monte Carlo expectation-maximisation algorithm. In this paper, we propose an exact likelihood estimation approach to replace the more computationally expensive Monte Carlo expectation-maximisation algorithm and we consider results based on using both the multivariate Gaussian and t copula functions. 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.

Approximate Bayesian inference for joint linear and partially linear modeling of longitudinal zero-inflated count and time to event data

Joint modeling of zero-inflated count and time-to-event data is usually performed by applying the shared random effect model. This kind of joint modeling can be considered as a latent Gaussian model. In this paper, the approach of integrated nested Laplace approximation (INLA) is used to perform approximate Bayesian approach for the joint modeling. We propose a zero-inflated hurdle model under Poisson or negative binomial distributional assumption as sub-model for count data. Also, a Weibull model is used as survival time sub-model. In addition to the usual joint linear model, a joint partially linear model is also considered to take into account the non-linear effect of time on the longitudinal count response. The performance of the method is investigated using some simulation studies and its achievement is compared with the usual approach via the Bayesian paradigm of Monte Carlo Markov Chain (MCMC). Also, we apply the proposed method to analyze two real data sets. The first one is the data about a longitudinal study of pregnancy and the second one is a data set obtained of a HIV study.

Exploratory assessment of treatment-dependent random-effects distribution using gradient functions

In analyzing repeated measurements from randomized controlled trials with mixed-effects models, it is important to carefully examine the conventional normality assumption regarding the random-effects distribution and its dependence on treatment allocation in order to avoid biased estimation and correctly interpret the estimated random-effects distribution. In this article, we propose the use of a gradient function method in modeling with the different random-effects distributions depending on the treatment allocation. This method can be effective for considering in advance whether a proper fit requires a model that allows dependence of the random-effects distribution on covariates, or for finding the subpopulations in the random effects.

Bayesian joint modelling of longitudinal and time to event data: a methodological review

BACKGROUND

In clinical research, there is an increasing interest in joint modelling of longitudinal and time-to-event data, since it reduces bias in parameter estimation and increases the efficiency of statistical inference. Inference and prediction from frequentist approaches of joint models have been extensively reviewed, and due to the recent popularity of data-driven Bayesian approaches, a review on current Bayesian estimation of joint model is useful to draw recommendations for future researches.

METHODS

We have undertaken a comprehensive review on Bayesian univariate and multivariate joint models. We focused on type of outcomes, model assumptions, association structure, estimation algorithm, dynamic prediction and software implementation.

RESULTS

A total of 89 articles have been identified, consisting of 75 methodological and 14 applied articles. The most common approach to model the longitudinal and time-to-event outcomes jointly included linear mixed effect models with proportional hazards. A random effect association structure was generally used for linking the two sub-models. Markov Chain Monte Carlo (MCMC) algorithms were commonly used (93% articles) to estimate the model parameters. Only six articles were primarily focused on dynamic predictions for longitudinal or event-time outcomes.

CONCLUSION

Methodologies for a wide variety of data types have been proposed; however the research is limited if the association between the two outcomes changes over time, and there is also lack of methods to determine the association structure in the absence of clinical background knowledge. Joint modelling has been proved to be beneficial in producing more accurate dynamic prediction; however, there is a lack of sufficient tools to validate the prediction.

Sensitivity analysis for the generalized shared-parameter model framework

In this paper, we assess the effect of tuberculosis pericarditis treatment (prednisolone) on CD4 count changes over time and draw inferences in the presence of missing data. We accounted for the missing data and performed sensitivity analyses to assess robustness of inferences, from a model that assumes that the data are missing at random, to models that assume that the data are not missing at random. Our sensitivity approaches are within the shared-parameter model framework. We implemented the approach by Creemers and colleagues to the CD4 count data and performed simulation studies to evaluate the performance of this approach. We also assessed the influence of potentially influential subjects, on parameter estimates, via the global influence approach. Our results revealed that inferences from missing at random analysis model are robust to not missing at random models and influential subjects did not overturn the study conclusions about prednisolone effect and missing data mechanism. Prednisolone was found to have no significant effect on CD4 count changes over time and also did not interact with anti-retroviral therapy. The simulation studies produced unbiased estimates of prednisolone effect with lower mean square errors and coverage probabilities approximately equal the nominal coverage probability.

Joint modeling of survival time and longitudinal outcomes with flexible random effects

Joint models with shared Gaussian random effects have been conventionally used in analysis of longitudinal outcome and survival endpoint in biomedical or public health research. However, misspecifying the normality assumption of random effects can lead to serious bias in parameter estimation and future prediction. In this paper, we study joint models of general longitudinal outcomes and survival endpoint but allow the underlying distribution of shared random effect to be completely unknown. For inference, we propose to use a mixture of Gaussian distributions as an approximation to this unknown distribution and adopt an Expectation-Maximization (EM) algorithm for computation. Either AIC and BIC criteria are adopted for selecting the number of mixtures. We demonstrate the proposed method via a number of simulation studies. We illustrate our approach with the data from the Carolina Head and Neck Cancer Study (CHANCE).