Dynamic predictions in Bayesian functional joint models for longitudinal and time-to-event data: An application to Alzheimer’s disease

Abstract

In the study of Alzheimer’s disease (AD), researchers often collect repeated measurements of clinical variables, event history, and functional data. If the health measurements deteriorate rapidly, patients may reach a level of cognitive impairment and are diagnosed as having dementia. An accurate prediction of the time to dementia based on the information collected is helpful for physicians to monitor patients’ disease progression and to make early informed medical decisions. In this article, we first propose a functional joint model (FJM) to account for functional predictors in both longitudinal and survival submodels in the joint modeling framework. We then develop a Bayesian approach for parameter estimation and a dynamic prediction framework for predicting the subjects’ future outcome trajectories and risk of dementia, based on their scalar and functional measurements. The proposed Bayesian FJM provides a flexible framework to incorporate many features both in joint modeling of longitudinal and survival data and in functional data analysis. Our proposed model is evaluated by a simulation study and is applied to the motivating Alzheimer’s Disease Neuroimaging Initiative (ADNI) study.

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

“…We have found the joint modelling methods developed under the four categories: single outcome for both of the longitudinal and event-time data (39/75, 52%); single longitudinal outcome and multiple event-time outcomes (13/ 75, 17.3%); multiple longitudinal outcomes and single event-time outcome (15/75, 20%); both outcomes are multiple (8/75, 10.7%). The majority of the articles were based on shared random effect joint models , whereas several articles explored joint models in terms of latent classes [42,54,58,[67][68][69][70], additive model [71,72] and functional model [73,74]. We reviewed the methodology for each sub-model and association structure.…”

“…The functional joint model approach involves modelling the longitudinal outcome, event-time outcome and exposure variables that include both scalar predictors and functional predictors. The functional predictors consist of a sample of functions that have information about curves, surfaces, or other geometric features that are varying over time [73]. These types of function are defined on a one-dimensional time domain, e.g.…”

Bayesian functional joint models for multivariate longitudinal and time-to-event data

“…As AD progresses and the hippocampus atrophies, the radial distance of some subfields shrinks. It has been shown that the baseline vertex-based HRD is predictive of time of MCI-to-AD as a functional predictor [10]. In this paper, we propose a Bayesian personalized prediction model based on a multivariate functional joint model (MFJM) of longitudinal ADAS-Cog 11 score as a scalar predictor, longitudinal vertex-based HRD as a functional predictor, and the time to AD diagnosis.…”

“…This gives the survival submodel in JM as $$h_i(t)=h_0(t)\exp \{{\gamma }_1{\mathrm{italicgender}}_i+{\gamma }_{2}bAg{e}_i+{\gamma }_{3}Ed{u}_i+{\gamma }_{4}APOE-\epsilon 4+{\gamma }_{5}bH{V}_i+{\alpha }^{\ast }{m}_i^{\ast }(t)\}.$$ The second model is a function joint model (refer to as model FJM ) which includes baseline hippocampal radial distance bHRD(s) , instead of hippocampal volume, as a time-independent functional predictor in the survival submodel. The model was proposed and applied to ADNI study in the previous work [10], in which the survival submodel is defined as $$\begin{array}{cc}right{h}_{i}false(tfalse)=& left{h}_{0}false(tfalse)\exp false\{{\gamma }_1{\mathit{gender}}_i+{\gamma }_{2}bAg{e}_i+{\gamma }_{3}Ed{u}_i+{\gamma }_{4}APOE-\epsilon 4\\ right& left+{\int }_{s}bHR{D}_{i}false(sfalse)B_{bHRD}false(sfalse)ds+{\alpha }^{\ast }{m}_i^{\ast }false(tfalse)false\}.\end{array}$$ The third model is a multivariate functional joint model (refer to as model MFJM ) that accounts for the longitudinal hippocampal radial distance lHRD ( s , t ) in the survival submodel, where lHRD ( s , t ) is modeled as $$\begin{array}{cc}right{\mathrm{italiclHRD}}_{i}false(s,t_{\mathit{ij}}false)=& left{m}_{i}false(s,t_{\mathit{ij}}false)+{\epsilon }_{\mathit{ij}}false(sfalse)\\ right{m}_{i}false(s,t_{\mathit{ij}}false)=& left{B}_{0}false(sfalse)+B_{1}false(sfalse)normal\mathit{APOE}-\epsilon 4_i+B_{2}false(...\end{array}$$…”

“…Li and Luo [9] proposed a functional joint model (FJM) that incorporates surface-based hippocampus measure as a functional predictor in the joint model of longitudinal and survival framework. They developed a dynamic prediction method and demonstrated that using such a functional predictor, in addition to other scalar markers, improves predictive performance of the progression of MCI to AD [10]. However, the proposed FJM only accommodates baseline imaging marker as a time-invariant function predictor.…”