The BGLIMM Procedure

PROC BGLIMM is a simulation-based procedure that uses a variety of sampling algorithms to draw samples from the joint posterior distribution of parameters from a generalized linear mixed model (GLMM). Sampling methods include the conjugate sampler, direct sampler, Gamerman algorithm (a variation of the Metropolis-Hastings algorithm that is tailored to generalized linear models), and No-U-Turn Sampler (NUTS, a self-tuning variation of the Hamiltonian Monte Carlo (HMC) method).

For situations in which the conjugate samplers are used, see the section Conjugate Sampling. The direct sampling method is used for missing values, where the sampling distribution is known. The Gamerman algorithm is used for both the fixed-effects and random-effects parameters in nonnormal models. The NUTS algorithm is used for covariance parameters when conjugacy is not available.

PROC BGLIMM updates parameters conditionally, through Gibbs sampling. The fixed-effects parameters are drawn jointly at each iteration. The random-effect parameters (in a RANDOM statement) are updated by clusters, unless the SUBJECT= option is not specified. In that situation, the random-effects parameters from the same RANDOM statement are updated jointly (for more information about how the random-effects parameters can be parameterized differently with or without the presence of the SUBJECT= option, see the section Treatment of Subjects in the RANDOM Statement). Missing data values are updated in sequence, and the G-side and the R-side covariance parameters are updated separately, in their full posterior conditionals.

The rest of this section describes how PROC BGLIMM computes the full conditional distributions in the Gibbs updating. Let , the collection of all fixed-effects parameters and the covariance matrices; let denote random-effects parameters and denote the random-effects parameters from cluster j. For simplicity, it is assumed that there is only one random effect, thus omitting an extra subindex for . The treatment of random effects is identical for effects in multiple RANDOM statements.