Fast approximate inference for multivariate longitudinal data

Abstract

Collecting information on multiple longitudinal outcomes is increasingly common in many clinical settings. In many cases, it is desirable to model these outcomes jointly. However, in large data sets, with many outcomes, computational burden often prevents the simultaneous modeling of multiple outcomes within a single model. We develop a mean field variational Bayes algorithm, to jointly model multiple Gaussian, Poisson, or binary longitudinal markers within a multivariate generalized linear mixed model. Through simulation studies and clinical applications (in the fields of sight threatening diabetic retinopathy and primary biliary cirrhosis), we demonstrate substantial computational savings of our approximate approach when compared to a standard Markov Chain Monte Carlo, while maintaining good levels of accuracy of model parameters.

Keywords: Bayesian computing; Generalized linear mixed model; Markov chain Monte Carlo; Mean field variational Bayes; Multivariate mixed models; Repeated measurements.

Fig. 1

Accuracy scores for mean field variational Bayes compared to MCMC for simulated data sets with three continuous longitudinal markers (top panel) and three types of markers (bottom panel) in the simulation with individuals.

Fig. 2

Model results for a 12 marker multivariate generalized linear mixed model in the diabetic retinopathy. Panel (a) shows the accuracy heat maps of the MFVB fixed effects estimates and residual standard deviations (compared to the MCMC estimates), (b) shows the accuracy of the MFVB random effects covariance matrix (compared to the MCMC estimates), and (c) shows the implied matrix of correlations between the 12 longitudinal markers calculated using MFVB.

Fig. 3

Fitted longitudinal markers for mean field variational Bayes (dashed lines) compared to MCMC (solid lines) for the 12 markers in the diabetic retinopathy data, for three patients. The orange stars, green dots, and blue triangles show the observed values for three different patients, with the respectively colored lines showing the fitted models for each individual. All continuous values, including time, have been scaled prior to analysis and the results plotted here are in terms of the scaled variables. The y-axis of each plot shows the scale version of the variable noted in the title of each panel. The original units for each variable can be found in the description at the start of Section 5.