Abstract
BackgroundJoint modelling of longitudinal and time-to-event outcomes has received considerable attention over recent years. Commensurate with this has been a rise in statistical software options for fitting these models. However, these tools have generally been limited to a single longitudinal outcome. Here, we describe the classical joint model to the case of multiple longitudinal outcomes, propose a practical algorithm for fitting the models, and demonstrate how to fit the models using a new package for the statistical software platform R, joineRML.ResultsA multivariate linear mixed sub-model is specified for the longitudinal outcomes, and a Cox proportional hazards regression model with time-varying covariates is specified for the event time sub-model. The association between models is captured through a zero-mean multivariate latent Gaussian process. The models are fitted using a Monte Carlo Expectation-Maximisation algorithm, and inferences are based on approximate standard errors from the empirical profile information matrix, which are contrasted to an alternative bootstrap estimation approach. We illustrate the model and software on a real data example for patients with primary biliary cirrhosis with three repeatedly measured biomarkers.ConclusionsAn open-source software package capable of fitting multivariate joint models is available. The underlying algorithm and source code makes use of several methods to increase computational speed.
Highlights
Joint modelling of longitudinal and time-to-event outcomes has received considerable attention over recent years
Longitudinal data were simulated according to a follow-up schedule of 6 time points, with each model including subject-and-outcome-specific random-intercepts and random-slopes: bi = (b0i1, b1i1, b0i2, b1i2), Correlation was induced between the 2 outcomes by assuming correlation of − 0.5 between the random intercepts for each outcome
Event times were simulated from a Gompertz distribution with shape θ1 = −3.5 and scale exp(θ0) = exp(0.25) ≈ 1.28, following the methodology described by Austin [39]
Summary
Joint modelling of longitudinal and time-to-event outcomes has received considerable attention over recent years. Commensurate with this has been a rise in statistical software options for fitting these models. These tools have generally been limited to a single longitudinal outcome. It has been repeatedly shown elsewhere that if the longitudinal and event-time outcomes are correlated, modelling the two outcome processes separately, for example using linear mixed models and Cox regression models, can lead to biased effect size estimates [1]. Joint modelling has until recently been predominated by modelling a single longitudinal outcome together with a solitary event time outcome; referred to as univariate joint modelling. Recent innovations in the field of joint models have included the incorporation of multivariate longitudinal data [8], competing risks data
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