Latent Class Models for Joint Analysis of Longitudinal Biomarker and Event Process Data

Abstract

A retrospective substudy of the nutritional prevention of cancer (NPC) trials investigated the utility of longitudinally measured prostate-specific antigen (PSA) as a biomarker for subsequent onset of prostate cancer (PCa). Serial PSA levels were determined retrospectively from frozen blood samples that had been collected from all patients at successive clinic visits with the timing and the number of these visits highly variable. Diagnosis dates of all incident cases of PCa were recorded. Heterogeneity in PSA trajectories was observed that could not be fully explained by the usual linear mixed-effects model and measured covariates. Latent class models that incorporate both a longitudinal biomarker process and an event process offer a way to handle additional heterogeneity, to uncover distinct subpopulations, to incorporate correlated nonnormally distributed outcomes, and to classify individuals into risk classes. Our latent class joint model can aid the prediction of PCa probability given the longitudinal biomarker information available on an individual up to any date. The proposed model easily accommodates highly unbalanced longitudinal data and recurrent events. There are two levels of structure in the latent class joint model. First, the uncertainty of latent class membership is specified through a multinomial logistic model. Second, the class-specific marker trajectory and event process are specified parametrically and semiparametrically, under the assumption of conditional independence given the latent class membership. We use a likelihood approach to obtain parameter estimates via the EM algorithm. We fit the latent class joint model to the data from the NPC trials; four distinct subpopulations are identified that differ with regard to their PSA trajectories and risk for prostate cancer. Higher PSA level is significantly associated with increased risk of PCa, but appears to be conditionally independent once the latent classes are taken into account. Among the covariates, selenium supplementation and age at entry are statistically significant for various parts of the model. Assumptions—in particular the conditional independence between the longitudinal PSA biomarker and time to PCa diagnosis—are assessed.

KEY WORDS:

• Biomarker
• Em algorithm
• Joint longitudinal and event process model
• Latent class
• Risk classification
• Semiparametric maximum likelihood estimator