The BGLIMM Procedure

Multinomial Models

The BGLIMM procedure fits two kinds of models to multinomial data: one that has a cumulative link that applies to ordinal data, and one that has a generalized logit link that applies to nominal data.

PROC BGLIMM models the proportional-odds cumulative probabilities for ordinal data. This model uses cumulative probabilities up to a threshold, thereby making the whole range of ordinal categories binary at that threshold. If the nominal response has J categories and the ordering is natural, the associated probabilities are pi 1 comma pi 2 comma ellipsis comma pi Subscript upper J Baseline, and a cumulative probability of a response less than equal to j is

log left-brace StartFraction normal upper P normal r left-parenthesis upper Y less-than-or-equal-to j right-parenthesis Over normal upper P normal r left-parenthesis upper Y greater-than j right-parenthesis EndFraction right-brace equals log left-brace StartFraction normal upper P normal r left-parenthesis upper Y less-than-or-equal-to j right-parenthesis Over 1 minus normal upper P normal r left-parenthesis upper Y less-than-or-equal-to j right-parenthesis EndFraction right-brace equals log left-brace StartFraction pi 1 plus pi 2 plus midline-horizontal-ellipsis plus pi Subscript j Baseline Over pi Subscript j plus 1 Baseline plus midline-horizontal-ellipsis plus pi Subscript upper J Baseline EndFraction right-brace

This describes the log odds of two cumulative probabilities: a less-than type and a greater-than type. This measures how likely the response is to be in category j or below, or how likely it is to be in a category higher than j.

The linear predictor depends on the response category only through the intercepts (cutoffs) alpha 1 comma ellipsis comma alpha Subscript upper J minus 1 Baseline,

StartLayout 1st Row 1st Column eta 1 2nd Column equals alpha 1 plus bold x prime bold-italic beta plus bold z prime bold-italic gamma 2nd Row 1st Column eta 2 2nd Column equals alpha 2 plus bold x prime bold-italic beta plus bold z prime bold-italic gamma 3rd Row 1st Column Blank 2nd Column vertical-ellipsis 4th Row 1st Column eta Subscript upper J minus 1 2nd Column equals alpha Subscript upper J minus 1 Baseline plus bold x prime bold-italic beta plus bold z prime bold-italic gamma 5th Row 1st Column eta Subscript upper J 2nd Column equals 0 comma EndLayout

where alpha 1 less-than-or-equal-to alpha 2 midline-horizontal-ellipsis less-than-or-equal-to alpha Subscript upper J minus 1 to ensure the cumulative property and left-parenthesis bold-italic beta comma bold-italic gamma right-parenthesis remains the same across all levels of the response variable. The cumulative logits are formed as

log left-brace StartFraction normal upper P normal r left-parenthesis upper Y less-than-or-equal-to j right-parenthesis Over normal upper P normal r left-parenthesis upper Y greater-than j right-parenthesis EndFraction right-brace equals eta Subscript j Baseline equals alpha Subscript j Baseline plus bold x prime bold-italic beta plus bold z prime bold-italic gamma equals alpha Subscript j Baseline plus eta overTilde

The odds ratio that compares two conditions represented by the linear predictors eta Subscript j Baseline 1 and eta Subscript j Baseline 0 is then

StartLayout 1st Row 1st Column psi left-parenthesis eta Subscript j Baseline 1 Baseline comma eta Subscript j Baseline 0 Baseline right-parenthesis 2nd Column equals exp left-parenthesis eta Subscript j Baseline 1 Baseline minus eta Subscript j Baseline 0 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column equals exp left-parenthesis eta overTilde Subscript 1 Baseline minus eta overTilde Subscript 0 Baseline right-parenthesis EndLayout

and is independent of category. You might think of this as a set of parallel lines (or hyperplanes) with different intercepts. The proportional-odds condition forces the lines that correspond to each cumulative logit to be parallel.

In the generalized logit model, you model baseline-category logits. By default, the BGLIMM procedure chooses the last category as the baseline category or reference, but you can use another category as the reference. If the nominal response has J categories, both the fixed effects and the random effects in the linear predictor depend on the response category:

StartLayout 1st Row 1st Column eta 1 2nd Column equals bold x prime bold-italic beta 1 plus bold z prime bold-italic gamma 1 2nd Row 1st Column eta 2 2nd Column equals bold x prime bold-italic beta 2 plus bold z prime bold-italic gamma 2 3rd Row 1st Column Blank 2nd Column vertical-ellipsis 4th Row 1st Column eta Subscript upper J minus 1 2nd Column equals bold x prime bold-italic beta Subscript upper J minus 1 Baseline plus bold z prime bold-italic gamma Subscript upper J minus 1 Baseline 5th Row 1st Column eta Subscript upper J 2nd Column equals 0 EndLayout

The last category defaults to zero in order to ensure the estimability property. The baseline-category logit for category j is

log left-parenthesis pi Subscript j Baseline slash pi Subscript upper J Baseline right-parenthesis equals eta Subscript j Baseline equals bold x prime bold-italic beta Subscript j Baseline plus bold z prime bold-italic gamma Subscript j

and

StartLayout 1st Row 1st Column pi Subscript j 2nd Column equals StartFraction exp left-parenthesis eta Subscript j Baseline right-parenthesis Over sigma-summation Underscript k equals 1 Overscript upper J Endscripts exp left-parenthesis eta Subscript k Baseline right-parenthesis EndFraction EndLayout

Suppose that the two conditions to be compared are identified using subscripts 1 and 0. The log odds ratio of outcome j versus J for the two conditions is then

StartLayout 1st Row 1st Column log left-brace psi left-parenthesis eta Subscript j Baseline 1 Baseline comma eta Subscript j Baseline 0 Baseline right-parenthesis right-brace 2nd Column equals log left-brace StartFraction pi Subscript j Baseline 1 Baseline slash pi Subscript upper J Baseline 1 Baseline Over pi Subscript j Baseline 0 Baseline slash pi Subscript upper J Baseline 0 Baseline EndFraction right-brace equals log left-brace StartFraction exp left-brace eta Subscript j Baseline 1 Baseline right-brace Over exp left-brace eta Subscript j Baseline 0 Baseline right-brace EndFraction right-brace 2nd Row 1st Column Blank 2nd Column equals eta Subscript j Baseline 1 Baseline minus eta Subscript j Baseline 0 EndLayout

Note that the log odds ratios are again differences on the scale of the linear predictor, but they depend on the response category. The BGLIMM procedure determines the estimable functions whose differences represent log odds ratios, but it produces separate estimates for each nonreference response category.

Last updated: December 09, 2022