The BGLIMM Procedure

Treatment of Subjects in the RANDOM Statement

PROC BGLIMM supports the syntax of specifying random effects without specifying a SUBJECT= option, but this is different from specifying an intercept random effect and using the same random effects in the SUBJECT= option.

For example, suppose you have a nested effect A(B), where both A and B are categorical variables. You can specify the following statement: 

random A(B);

In the traditional mixed modeling procedures, such as PROC MIXED and PROC GLIMMIX, this specification is the same as 

random int / subject = A(B);

But this is not the case in PROC BGLIMM. The difference is as follows:

  • When you specify the SUBJECT= option in PROC BGLIMM, the procedure fits a normal prior (with default TYPE=VC),

    pi left-parenthesis gamma Subscript j Baseline right-parenthesis tilde normal left-parenthesis 0 comma sigma Subscript gamma Superscript 2 Baseline right-parenthesis comma j equals 1 comma ellipsis comma upper J

    where gamma Subscript j is the intercept from cluster j and J is the total number of unique clusters in effect A(B). This is the expected way of specifying a prior on intercept random effects.

  • When you omit the SUBJECT= option, PROC BGLIMM assumes that there is only one cluster (instead of J clusters). This cluster, which represents the entire data set, has nested (categorical) effects of A(B).

    This is equivalent to treating A(B) as categorical fixed effects in the regression model (in the MODEL statement): 

    model resp = A(B);

    The exception is that, instead of a noninformative or flat prior on the regression coefficients, a hierarchical normal prior is assumed (again, following the specification of the TYPE= covariance),

    beta Subscript j Baseline tilde normal left-parenthesis 0 comma sigma Subscript beta Superscript 2 Baseline right-parenthesis comma j equals 1 comma ellipsis comma upper K

    where beta Subscript j (used here instead of gamma Subscript j to emphasize the different treatment in modeling by PROC BGLIMM) is a categorical coefficient for the jth level and K is less or equal to J, depending on the rank of the regression model.

    In other words, after you specify a random effect without using the SUBJECT= option, the effect becomes equivalent to a fixed effect (but with a shrinkage prior). The number of estimable parameters from that effect depends on the rank of the model, so you will often see that some of the cluster parameters are inestimable (hence, their value is displayed as zero).

In PROC MIXED and PROC GLIMMIX, the two specifications lead to identical numerical estimates, because neither procedure requires the design matrix (of the regression model) to be nonsingular. But PROC BGLIMM has this requirement, and this results in different posterior estimates.

If you want to specify a random-effects model in the strict Bayesian sense, use the SUBJECT= option in the RANDOM statement.

Last updated: December 09, 2022