Generalized Linear Models with Random Effects; a Gibbs Sampling Approach

Variance components analysis for pedigree-based censored survival data using generalized linear mixed models (GLMMs) and Gibbs sampling in BUGS

“…Reparameterisation (model 5) allows the MCMC process to traverse regions of the parameter space in which σ 2 A and σ 2 CS are negative, thereby avoiding positive bias in their estimators [Zeger and Karim, 1991;Burton et al, 1999]. The modified parameterisation also serves a second purpose.…”

Random variate transformations in the Gibbs sampler: issues of efficiency and convergence

“…Furthermore, if an envelope rejection sampling (Zeger and Karim, 1991) or a ratio of uniforms method (Wakefield et al 1991)are employed, the computationally expensive numerical maximization routines can be used only in the first iteration.…”

“…Generalized linear models are a large family of models where Gibbs sampler requires adaptation of sampling methods dealing with non-conjugacy; see Dellaportas and Smith (1993) and Zeger and Karim (1991). We choose to use the Griddy-Gibbs sampler applying the transformations suggested in Section 2.2, without giving emphasis to inference details but focusing, as in the previous examples, to gain in efficiency.…”

“…We then need to employ sophisticated algorithms for random variate generation, normally requiring the identification of features of the conditional density such as the location of its maximum, its curvature, its logconcavity, or its normalizing constant. Zeger and Karim (1991) and Carlin and Gelfand (1991) suggest the use of rejection sampling algorithms (Devroye 1986), where Wakefield et al (1993 suggest the generalized ratio of uniforms method (Wakefield et al, 1991). They all require location of the maximum of a density, which is normally achieved numerically and it is therefore expensive.…”

Calculation of marginal densities for parameters of multinomial distributions

“…The utility of Gibbs sampling for the reconstruction of marginal posterior densities was first investigated by Gelfand and Smith (1990), and has now been demonstrated for a wide range of applications (see for example Zeger andKarim, 1991, andDellaportas and, for related work on generalized linear models). It is a Monte Carlo technique whereby a sample is generated from the joint posterior density.…”