The BGLIMM Procedure

In a random-effects model, the conditional distribution of is similar to that of the fixed-effects-only model,

where the log-likelihood function now includes the random effects . This construction reflects two PROC BGLIMM modeling settings: all random-effects parameters enter the likelihood function (linearly at the mean level), and the fixed-effects parameters cannot be hyperparameters of (hence no terms).

The conditional distribution of again mirrors that of :

For , the following conditional is used:

In this computation, only subjects from the jth cluster are used. This reflects the conditional independence assumption that the RANDOM statement makes. This simplification in the calculation makes updating the random-effects parameters computationally efficient and enables the procedure to handle random effects that contain large number of clusters just as easily.

The G-side covariance matrix depends only on the random effects and not on the data or other parameters, or ,

where is the prior distribution of .