Clustering for multivariate continuous and discrete longitudinal data

Abstract

Multiple outcomes, both continuous and discrete, are routinely gathered on subjects in longitudinal studies and during routine clinical follow-up in general. To motivate our work, we consider a longitudinal study on patients with primary biliary cirrhosis (PBC) with a continuous bilirubin level, a discrete platelet count and a dichotomous indication of blood vessel malformations as examples of such longitudinal outcomes. An apparent requirement is to use all the outcome values to classify the subjects into groups (e.g., groups of subjects with a similar prognosis in a clinical setting). In recent years, numerous approaches have been suggested for classification based on longitudinal (or otherwise correlated) outcomes, targeting not only traditional areas like biostatistics, but also rapidly evolving bioinformatics and many others. However, most available approaches consider only continuous outcomes as a basis for classification, or if noncontinuous outcomes are considered, then not in combination with other outcomes of a different nature. Here, we propose a statistical method for clustering (classification) of subjects into a prespecified number of groups with a priori unknown characteristics on the basis of repeated measurements of several longitudinal outcomes of a different nature. This method relies on a multivariate extension of the classical generalized linear mixed model where a mixture distribution is additionally assumed for random effects. We base the inference on a Bayesian specification of the model and simulation-based Markov chain Monte Carlo methodology. To apply the method in practice, we have prepared ready-to-use software for use in R (http://www.R-project.org). We also discuss evaluation of uncertainty in the classification and also discuss usage of a recently proposed methodology for model comparison - the selection of a number of clusters in our case - based on the penalized posterior deviance proposed by Plummer [Biostatistics 9 (2008) 523-539].Comment: Published in at http://dx.doi.org/10.1214/12-AOAS580 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Trajectories of Glycemic Change in a National Cohort of Adults With Previously Controlled Type 2 Diabetes

“…19 The MGLMM specifies that there is unobserved clustering of random effect parameters across an unknown number of population subgroups (trajectories) with different weights. Model parameters of MGLMM were estimated using the Markov Chain Monte Carlo (MCMC) method and implemented using the R package mixAK, 19 which has the advantage that if the computation fails (i.e. does not converge), then model assumptions are violated 28 indicating that the proposed model is not appropriate for the data and should be rejected.…”

“…However, model selection based on a single criteria such as the PED ignores uncertainty in the comparison, which increases with increasing number of model parameters. 19, 31 An alternative approach is to assess the full posterior distribution of the deviances of the models. Thus, suppose Diff is the difference in deviance between a large and a small model, if the posterior probability P( Diff < − 2log(9) = −4.39) is ≥0.9, then there is strong evidence in favor of the larger model.…”

“…Specifically, we employ a Bayesian estimation of the generalized linear mixed model (GLMM), with the random effect component replaced by a mixture of normal distributions. 19 This procedure enables data-driven discovery of homogenous longitudinal trajectories characterized by time-dependent covariates. We then apply the model to identify trajectories of HbA 1c change among adults with stable controlled type 2 diabetes and predict who will lose glycemic control.…”

Dynamic classification using credible intervals in longitudinal discriminant analysis

“…Our proposal to classify a new patient on the basis of credible intervals for the individual group probabilities (explained in Section 3) starts from considering the following procedure developed by Hughes et al based on the multivariate generalised linear mixed model (MGLMM) proposed by Komárek and Komárková . It is assumed that given U = g (given the allocation of a particular patient into group g , g =0,…, G −1), the values of the longitudinal markers Y 1 ,…, Y R are generated by the group specific MGLMM.…”

“…To the best of our knowledge, this is the first time credible (or confidence) intervals for the group membership probabilities have been used in dynamic LoDA, although we are not the first to consider credible intervals in a classification scheme. Komárek and Komárková do so in the context of longitudinal cluster analysis (where the classification is not “dynamic” and clustering [ie, unsupervised classification] is performed only once using all available longitudinal data) whilst Gugliemli et al do so in the context of survival analysis (although we note that this is not in a longitudinal context, and there is no sequential updating). Leaving a group as unclassified is similar to the neutral zone classifiers proposed by Zhang et al, although their unclassified group is based upon an analysis of the misclassification costs.…”