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.
翻译:对纵向和时间对活动数据的联合建模通常用随机效果联合模型来描述,该模型使用共享或相关的潜在影响来捕捉两个进程之间的关联。在这个框架内,两个进程的联合分布可以通过假定有条件的独立性来直截了当地地得出,因为其潜在影响是有条件的。文献中也考虑了促使次级模型相互依存的替代方法,其中一个这样的方法是使用Copula,在纵向和时间对活动进程的边际分布之间引入非线性的相关性。文献中提出了一个高斯比亚相交联合模型,用一个共享或相关的潜在影响来捕获两个进程之间的关联。在这个框架内,两个进程的联合分布可以通过假定有条件的独立独立来直接得出。在最初的方法中,无法从分析的角度导引出对子模型相互依存的功能,因此,蒙特卡洛的整合不仅需要密集的计算,而且还给估算带来额外的变化/噪音。与EM算法的结合进一步减慢了计算速度,而且与最大可能性估计值的趋同性联合模型在应用蒙特卡洛期望值算法时并非总能保证采用联合的准确性估算。此外,在规划的估测算中,我们所规划的精确度的估定的尺度的尺度是整个测算的准确度的尺度的尺度的尺度的尺度长度长度长度,因此,我们提出一个适当的估测算是整个测算的尺度的尺度是整个的尺度的尺度的尺度的尺度的尺度的尺度。