Abstract
ABSTRACTGeneralized linear latent variable models (GLLVMs) are a powerful class of models for understanding the relationships among multiple, correlated responses. Estimation, however, presents a major challenge, as the marginal likelihood does not possess a closed form for nonnormal responses. We propose a variational approximation (VA) method for estimating GLLVMs. For the common cases of binary, ordinal, and overdispersed count data, we derive fully closed-form approximations to the marginal log-likelihood function in each case. Compared to other methods such as the expectation-maximization algorithm, estimation using VA is fast and straightforward to implement. Predictions of the latent variables and associated uncertainty estimates are also obtained as part of the estimation process. Simulations show that VA estimation performs similar to or better than some currently available methods, both at predicting the latent variables and estimating their corresponding coefficients. They also show that VA estimation offers dramatic reductions in computation time particularly if the number of correlated responses is large relative to the number of observational units. We apply the variational approach to two datasets, estimating GLLVMs to understanding the patterns of variation in youth gratitude and for constructing ordination plots in bird abundance data. R code for performing VA estimation of GLLVMs is available online. Supplementary materials for this article are available online.
Highlights
In many areas of applied science, data on multiple, correlated responses are often collected, with one of the primary aims being to understand the latent variables driving these correlations
We have proposed a variational approximation method for estimating Generalized Linear Latent Variable Models (GLLVMs), deriving fully closed form approximations to the log-likelihood for the common cases of binary, ordinal, and overdispersed count data
Simulations showed that the variational approximation (VA) approach performs similar to or better than some of popular methods used for fitting GLLVMs, with potentially significant reductions in computation time
Summary
In many areas of applied science, data on multiple, correlated responses are often collected, with one of the primary aims being to understand the latent variables driving these correlations. In psychometrics, subjects are given a series of questions that all relate to some latent trait/s such as gratitude Another example is in ecology, where the abundances of many, interacting species are collected at each site, and ordination is commonly applied to visualize patterns between sites on a latent species composition space (Hui et al, 2015; Warton et al, 2015). Generalized linear latent variable models (GLLVMs, Moustaki and Knott, 2000) offer a general framework for analyzing multiple, correlated responses. Variational methods have become increasingly popular for approximating posterior distributions in Bayesian modeling (e.g. Bishop et al, 2006) By contrast, their use in maximum likelihood estimation for dealing with intractable likelihoods has received little attention. We apply the proposed VA method to datasets in psychometrics and ecology, demonstrating in both examples how GLLVMs offer a model-based framework to understanding the major patterns of variation behind the correlated data on a latent space
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