Many longitudinal clinical studies involve the repeated periodic measurement of continuous responses over a period to detect any changes that might occur in the condition of the individual, such as events of interest like survival after 90 days. In such investigations, the measurements might concern several biomarkers. A current approach to modeling such data is the use of joint modeling. A joint model is typically composed of two or more submodels related through a common set of latent variables, reflecting their interconnection by joint likelihoods.
The expectation-maximization (EM) algorithm has been a standard approach for calculating joint likelihoods since the late 90s, despite being computationally costly in the estimation step. There have been many attempts to cope with this drawback. For example, Bernhardt, Zhang, and Wang, in a study with a binary outcome (survival after 90 days), used a normal density to approximate the distribution of random effects . The present work is an extension of Bernhardt et al., applied to a survival outcome model (length of time the patients diagnosed with the disease are still alive) instead of a binary one, but keeping the normality assumption for multivariate normal distribution of random and fixed effects.
The first section summarizes the state of the art in this domain. Section 2 presents the details of this procedure, especially the expectation step of the EM algorithm, detailing the initialization phase and convergence issues; this part is supplemented by additional material. Section 3 shows the results of two comprehensive simulations.
The application part describes the data analyses performed on two dedicated datasets: the first is a set of primary biliary cirrhosis (PBC) data collected between 1974 and 1984, and the second takes advantage of the Alzheimer’s disease neuroimaging initiative (ADNI) study and is focused on mild cognitive impairment (MCI) as the baseline. In both cases, trivariate joint models are generated on three longitudinal biomarkers. The analyses are complemented with results and graphic representation that confirm the accomplishment of the primary goal: to achieve satisfactory performance in a multivariate setup and survival model, in a reasonable convergence time.
The work provides relevant and useful information for scientists working in biomedical or medical fields, but also in other domains where longitudinal studies are performed.