Modelling Sparse Generalized Longitudinal Observations with Latent Gaussian Processes

Abstract

In longitudinal data analysis one frequently encounters non-Gaussian data that are repeatedly collected for a sample of individuals over time. The repeated observations could be binomial, Poisson or of another discrete type or could be continuous. The timings of the repeated measurements are often sparse and irregular. We introduce a latent Gaussian process model for such data, establishing a connection to functional data analysis. The functional methods proposed are non-parametric and computationally straightforward as they do not involve a likelihood. We develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations. This method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model. These scores can then be used for further statistical analysis, such as inference, regression, discriminant analysis or clustering. We illustrate these non-parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models. Copyright (c) 2008 Royal Statistical Society.

Longitudinal Functional Principal Component Analysis

“…Our approach can be seen as an extension of functional principal component analysis for multilevel functional data [7]. Our methods apply to longitudinal data where each observation is functional, and should thus not be confused with nonparametric methods for the longitudinal profiles of scalar variables [17,30,31,37,41,48,50,51]. For good introductions to functional data analysis in general, please see [10,34].…”

Dynamic relations for sparsely sampled Gaussian processes

“…As has been demonstrated in various settings (James et al 2000;Yao et al 2005a;Hall et al 2008), the FDA paradigm to extend classical statistical methodology to the case of functional data can be worked out even when one has available only sparse and irregularly spaced measurements for each subject or item in the sample. A topic that has been of much interest lately is modeling variation in time in addition to variation in amplitude.…”

Rejoinder on: Probability enhanced effective dimension reduction for classifying sparse functional data

“…Nevertheless, based on our experience and the study in Lin and Carroll (2000), it is suggested to ignore the withinsubject correlation in conjunction with cross-validation type search, at least when the measurements are sparsely observed with noise. The issue with the non-negative definiteness of the covariance estimation is less pronounced in our implementation of PACE algorithm, as we followed the suggestion made in Hall et al (2008) to discard the basis associated with negative eigenvalues. It has been shown theoretically that this amendment leads to asymptotically negligible impact on the resulting covariance estimator.…”