A Two-Stage Approach for Bayesian Joint Models of Longitudinal and Survival Data: Correcting Bias with Informative Prior

Abstract

Joint models of longitudinal and survival outcomes have gained much popularity in recent years, both in applications and in methodological development. This type of modelling is usually characterised by two submodels, one longitudinal (e.g., mixed-effects model) and one survival (e.g., Cox model), which are connected by some common term. Naturally, sharing information makes the inferential process highly time-consuming. In particular, the Bayesian framework requires even more time for Markov chains to reach stationarity. Hence, in order to reduce the modelling complexity while maintaining the accuracy of the estimates, we propose a two-stage strategy that first fits the longitudinal submodel and then plug the shared information into the survival submodel. Unlike a standard two-stage approach, we apply a correction by incorporating an individual and multiplicative fixed-effect with informative prior into the survival submodel. Based on simulation studies and sensitivity analyses, we empirically compare our proposal with joint specification and standard two-stage approaches. The results show that our methodology is very promising, since it reduces the estimation bias compared to the other two-stage method and requires less processing time than the joint specification approach.

Bridging the gap between two‐stage and joint models: The case of tumor growth inhibition and overall survival models

“…Two‐stage approaches for joint models are usually performed by fitting the longitudinal submodel and then somehow incorporating its information into the survival submodel. Tsiatis' proposal 6 is one of the most popular two‐stage strategies and has its Bayesian version 41 . Specifically, in the first stage, we calculate the maximum a posteriori (MAP) of the random effects $$$ \boldsymbol{b} $$$ and parameters $$$ {\boldsymbol{\theta}}_y $$$ estimated from longitudinal submodel fit separately: $$$$ \left({\hat{\boldsymbol{\theta}}}_y,\hat{\boldsymbol{b}}\right)=\underset{{\boldsymbol{\theta}}_y,\kern0.3em \boldsymbol{b}}{\arg\;\max}\prod \limits_{i=1}^n\prod \limits_{j=1}^{n_i}f\left({y}_{ij}|{\boldsymbol{\theta}}_y,{\boldsymbol{b}}_i\right)\kern0.3em f\left({\boldsymbol{b}}_i|\boldsymbol{\Sigma} \right)\kern0.3em \pi \left({\boldsymbol{\theta}}_y\right), $$$$ where $$$ \pi \left({\boldsymbol{\theta}}_y\right) $$$ is the prior distribution on the parameter vector $$$ {\boldsymbol{\theta}}_y $$$.…”

“…Tsiatis' proposal 6 is one of the most popular two-stage strategies and has its Bayesian version. 41 Specifically, in the first stage, we calculate the maximum a posteriori (MAP) of the random effects b and parameters 𝜽 y estimated from longitudinal submodel fit separately:…”