[Paper Review] Joint Models with Multiple Longitudinal Outcomes and a Time-to-Event Outcome

Katya Mauff, Ewout W. Steyerberg+3 more|arXiv (Cornell University)|Aug 23, 2018

Statistical Methods and Bayesian Inference19 references3 citations

TL;DR

This paper proposes a computationally efficient two-stage joint modeling approach for multiple longitudinal outcomes and a time-to-event outcome, using a correction factor based on importance sampling to reduce bias. The method achieves near-accurate estimates with significantly faster computation than full joint modeling, even in complex settings.

ABSTRACT

Joint models for longitudinal and survival data have garnered a lot of attention in recent years, with the development of myriad extensions to the basic model, including those which allow for multivariate longitudinal data, competing risks and recurrent events. Several software packages are now also available for their implementation. Although mathematically straightforward, the inclusion of multiple longitudinal outcomes in the joint model remains computationally difficult due to the large number of random effects required, which hampers the practical application of this extension. We present a novel approach that enables the fitting of such models with more realistic computational times. The idea behind the approach is to split the estimation of the joint model in two steps; estimating a multivariate mixed model for the longitudinal outcomes, and then using the output from this model to fit the survival submodel. So called two-stage approaches have previously been proposed, and shown to be biased. Our approach differs from the standard version, in that we additionally propose the application of a correction factor, adjusting the estimates obtained such that they more closely resemble those we would expect to find with the multivariate joint model. This correction is based on importance sampling ideas. Simulation studies show that this corrected-two-stage approach works satisfactorily, eliminating the bias while maintaining substantial improvement in computational time, even in more difficult settings.

Motivation & Objective

• To address the computational burden of fitting joint models with multiple longitudinal outcomes and a time-to-event outcome.
• To reduce the bias inherent in standard two-stage approaches used for such models.
• To develop a method that maintains estimation accuracy while drastically improving computational efficiency.
• To provide a practical alternative to full multivariate joint modeling, which becomes infeasible with many random effects.

Proposed method

• The method splits the joint model estimation into two stages: first fitting a multivariate mixed model to the longitudinal outcomes.
• Second, using the estimated random effects from the first stage as covariates in a survival submodel.
• A correction factor is applied to the survival model estimates using importance sampling to adjust for selection bias due to conditioning on random effects.
• The correction factor accounts for the distributional mismatch between the true joint model and the two-stage approximation.
• The approach leverages the computational simplicity of separate modeling while preserving asymptotic properties close to those of the full joint model.

Experimental results

Research questions

• RQ1 Can a two-stage approach with a correction factor achieve estimation accuracy comparable to full joint modeling for multiple longitudinal and time-to-event outcomes?
• RQ2 How does the proposed method compare to standard two-stage approaches in terms of bias and efficiency?
• RQ3 To what extent does the method reduce computational time without sacrificing estimation accuracy?
• RQ4 Does the importance sampling-based correction effectively mitigate the bias introduced by ignoring the uncertainty in random effects?

Key findings

• The corrected two-stage approach significantly reduces bias compared to standard two-stage methods, approaching the performance of full joint modeling.
• The method maintains substantial computational speedups, even in settings with many random effects or complex correlation structures.
• Simulation studies confirm that the correction factor effectively adjusts for the distortion caused by conditioning on estimated random effects.
• The approach achieves estimation accuracy close to that of the full multivariate joint model, particularly in moderate to large sample sizes.