General Design Bayesian Generalized Linear Mixed Models

Abstract

Linear mixed models are able to handle an extraordinary range of complications in regression-type analyses. Their most common use is to account for within-subject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when treated in full generality, mixed models can also handle spline-type smoothing and closely approximate kriging. This allows for nonparametric regression models (e.g., additive models and varying coefficient models) to be handled within the mixed model framework. The key is to allow the random effects design matrix to have general structure; hence our label general design. For continuous response data, particularly when Gaussianity of the response is reasonably assumed, computation is now quite mature and supported by the R, SAS and S-PLUS packages. Such is not the case for binary and count responses, where generalized linear mixed models (GLMMs) are required, but are hindered by the presence of intractable multivariate integrals. Software known to us supports special cases of the GLMM (e.g., PROC NLMIXED in SAS or glmmML in R) or relies on the sometimes crude Laplace-type approximation of integrals (e.g., the SAS macro glimmix or glmmPQL in R). This paper describes the fitting of general design generalized linear mixed models. A Bayesian approach is taken and Markov chain Monte Carlo (MCMC) is used for estimation and inference. In this generalized setting, MCMC requires sampling from nonstandard distributions. In this article, we demonstrate that the MCMC package WinBUGS facilitates sound fitting of general design Bayesian generalized linear mixed models in practice.Comment: Published at http://dx.doi.org/10.1214/088342306000000015 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Improved models of harvest-induced bark damage

“…This provides an alternative strategy for marginalising the random effects and may be more robust (Zhao et al 2006;Brown and Draper 2006).…”

“…Zeger and Karim 1991;Damien et al 1999;Sorensen and Gianola 2002;Zhao et al 2006), and several software packages are now available which implement such techniques. For the purposes of our study, we used R's MCMCglmm package (Hadfield 2010).…”

Variables associated with detection probability, detection latency, and behavioral responses of Golden-winged Warblers (Vermivora chrysoptera)

“…Although techniques that approximate these integrals (Breslow and Clayton 1993) are now popular, Markov chain Monte Carlo (MCMC) methods provide an alternative strategy for marginalizing the random effects that may be more robust (Zhao, Staudenmayer, Coull, and Wand 2006;Browne and Draper 2006). Developing MCMC methods for generalized linear mixed models (GLMM) is an active area of research (e.g., Zeger and Karim 1991;Damien, Wakefield, and Walker 1999;Sorensen and Gianola 2002;Zhao et al 2006), and several software packages are now available that implement these techniques (e.g., WinBUGS (Spiegelhalter, Thomas, Best, and Lunn 2003), MLwiN (Rasbash, Steele, Browne, and Prosser 2005), glmmBUGS (Brown 2009), JAGS (Plummer 2003)). However, these methods often require a certain level of expertise on behalf of the user and may take a great deal of computing time.…”

A Review of R-packages for Random-Intercept Probit Regression in Small Clusters

“…MCMC methods simulate the likelihood rather than computing it, by calculating the sample average of independently simulated realizations of the integrand. As such, MCMC is thought to provide a more robust approach to marginalizing the random effects [18,38].…”