The BGLIMM Procedure

Scale and Dispersion Parameters

The parameter phi in the log likelihood functions is a scale parameter. McCullagh and Nelder (1989, p. 29) refer to it as the dispersion parameter. With the exception of the normal distribution, phi does not correspond to the variance of an observation; the variance of an observation in a generalized linear model is a function of phi and mu. In a generalized linear model, the BGLIMM procedure displays the estimate of phi as "Scale" in the "Posterior Summaries and Intervals" table. Note that the scale parameter is the same as that reported by the GLIMMIX procedure for almost all distributions, but it is different from that reported by the GENMOD procedure for some distributions of this scale (see the "Parameter Estimates" table in PROC GLIMMIX and PROC GENMOD). The scale that is reported by PROC GENMOD is sometimes a transformation of the dispersion parameter in the log-likelihood function. Table 13 displays the "Scale" entries that are reported by the three procedures in terms of the phi (or k) parameter.

Table 13: Scale Reported in Output Table

Distribution PROC BGLIMM PROC GLIMMIX PROC GENMOD
Gamma ModifyingAbove phi With caret 1 slash ModifyingAbove phi With caret 1 slash ModifyingAbove phi With caret
Inverse Gaussian ModifyingAbove phi With caret ModifyingAbove phi With caret StartRoot ModifyingAbove phi With caret EndRoot
Negative binomial ModifyingAbove k With caret ModifyingAbove k With caret ModifyingAbove k With caret
Normal ModifyingAbove phi With caret equals ModifyingAbove normal upper V normal a normal r With caret left-bracket y right-bracket ModifyingAbove phi With caret equals ModifyingAbove normal upper V normal a normal r With caret left-bracket y right-bracket StartRoot ModifyingAbove phi With caret EndRoot


Last updated: December 09, 2022