Joint latent class models for longitudinal and time-to-event data: a review

Abstract

Most statistical developments in the joint modelling area have focused on the shared random-effect models that include characteristics of the longitudinal marker as predictors in the model for the time-to-event. A less well-known approach is the joint latent class model which consists in assuming that a latent class structure entirely captures the correlation between the longitudinal marker trajectory and the risk of the event. Owing to its flexibility in modelling the dependency between the longitudinal marker and the event time, as well as its ability to include covariates, the joint latent class model may be particularly suited for prediction problems. This article aims at giving an overview of joint latent class modelling, especially in the prediction context. The authors introduce the model, discuss estimation and goodness-of-fit, and compare it with the shared random-effect model. Then, dynamic predictive tools derived from joint latent class models, as well as measures to evaluate their dynamic predictive accuracy, are presented. A detailed illustration of the methods is given in the context of the prediction of prostate cancer recurrence after radiation therapy based on repeated measures of Prostate Specific Antigen.

Keywords: Brier score; joint model; longitudinal data; mixture model; predictive accuracy; prognosis; prostate cancer.

Figure 1

(A) Class-specific predicted mean trajectories and (B) class-specific event-free probabilities from the 4-class JLCM for a subject with Tstage<3, Gleason<7 and iPSA=2 ng/mL. (C) Weighted subject-specific predicted trajectories (pred) and weighted observed trajectories (obs) from the 4-class JLCM. (D) Weighted predicted event-free probabilities (pred) and weighted Kaplan–Meier estimates (obs) from the 4-class JLCM.

Figure 2

Example of individual predicted probability of clinical recurrence within 3 years updated every 6 months from the 4-class JLCM (bold plain line) and the 1-class JLCM (plain line) for a subject who recurred 3.8 years after the end of EBRT and had T-stage=2 Gleason=7, iPSA=9.7 ng/mL and repeated PSA measurements denoted by ×.

Figure 3

(A) CVPOL_a, cross-validated estimate of EPOCE, computed from the SREM (with current PSA level and current PSA slope included as covariates in the survival model), and from JLCM models with G=1 to G=5 latent classes for times at prediction from 1 to 6 years after EBRT, (B) Differences in EPOCE with 95% tracking interval (TI) on UM dataset (N=459).

Figure 4

Predictive accuracy measures at times at prediction from 1 to 6 years after EBRT computed on VANC dataset (N=719) from joint models estimated on UM dataset: (A) EPOCE estimate, (B) difference in EPOCE and 95% tracking interval (TI), (C) data-based estimate of Integrated Brier Score (IBS), and (D) model-based estimate of IBS. The censoring distribution for the IBS data-based estimate is modelled in a semi-parametric proportional hazard model with Gleason, T-stage, iPSA as covariates. SREM refers to the SREM with current PSA level and slope included as covariates in the survival model while G=1 to G=5 refer to JLCM with G=1 to G=5 latent classes.