The BGLIMM Procedure

Likelihood

The BGLIMM procedure forms the log likelihoods of generalized linear models as

upper L left-parenthesis bold-italic mu comma phi semicolon bold y comma f Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts f Subscript i Baseline l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis

where l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline right-parenthesis is the log-likelihood contribution of the ith observation with weight w Subscript i and f Subscript i is the value of the frequency variable. In the case where observations have weights, the scale parameter is replaced with phi slash w Subscript i in the density, where w Subscript i is the weight associated with the observation y Subscript i. For the determination of w Subscript i and f Subscript i, see the WEIGHT and FREQ statements.

The individual log-likelihood contributions for the various distributions are as follows:

Binary:
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals w Subscript i Baseline left-parenthesis y Subscript i Baseline log left-brace mu Subscript i Baseline right-brace plus left-parenthesis 1 minus y Subscript i Baseline right-parenthesis log left-brace 1 minus mu Subscript i Baseline right-brace right-parenthesis

normal upper V normal a normal r left-bracket upper Y right-bracket equals mu left-parenthesis 1 minus mu right-parenthesis comma phi identical-to 1.

Binomial:
StartLayout 1st Row 1st Column l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis 2nd Column equals w Subscript i Baseline left-parenthesis y Subscript i Baseline log left-brace mu Subscript i Baseline right-brace plus left-parenthesis n Subscript i Baseline minus y Subscript i Baseline right-parenthesis log left-brace 1 minus mu Subscript i Baseline right-brace right-parenthesis 2nd Row 1st Column Blank 2nd Column plus w Subscript i Baseline left-parenthesis log left-brace normal upper Gamma left-parenthesis n Subscript i Baseline plus 1 right-parenthesis right-brace minus log left-brace normal upper Gamma left-parenthesis y Subscript i Baseline plus 1 right-parenthesis right-brace minus log left-brace normal upper Gamma left-parenthesis n Subscript i Baseline minus y Subscript i Baseline plus 1 right-parenthesis right-brace right-parenthesis EndLayout

where y Subscript i and n Subscript i are the events and trials in the events/trials syntax, and 0 less-than mu less-than 1. normal upper V normal a normal r left-bracket upper Y slash n right-bracket equals mu left-parenthesis 1 minus mu right-parenthesis slash n comma phi identical-to 1.

Exponential:
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column minus log left-brace mu Subscript i Baseline right-brace minus y Subscript i Baseline slash mu Subscript i Baseline 2nd Column w Subscript i Baseline equals 1 2nd Row 1st Column w Subscript i Baseline log left-brace StartFraction w Subscript i Baseline y Subscript i Baseline Over mu Subscript i Baseline EndFraction right-brace minus StartFraction w Subscript i Baseline y Subscript i Baseline Over mu Subscript i Baseline EndFraction minus log left-brace y Subscript i Baseline normal upper Gamma left-parenthesis w Subscript i Baseline right-parenthesis right-brace 2nd Column w Subscript i Baseline not-equals 1 EndLayout

normal upper V normal a normal r left-bracket upper Y right-bracket equals mu squared comma phi identical-to 1.

Gamma:
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals w Subscript i Baseline phi log left-brace StartFraction w Subscript i Baseline y Subscript i Baseline phi Over mu Subscript i Baseline EndFraction right-brace minus StartFraction w Subscript i Baseline y Subscript i Baseline phi Over mu Subscript i Baseline EndFraction minus log left-brace y Subscript i Baseline right-brace minus log left-brace normal upper Gamma left-parenthesis w Subscript i Baseline phi right-parenthesis right-brace

normal upper V normal a normal r left-bracket upper Y right-bracket equals phi mu squared comma phi greater-than 0.

Geometric:
StartLayout 1st Row 1st Column l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis 2nd Column equals y Subscript i Baseline log left-brace StartFraction mu Subscript i Baseline Over w Subscript i Baseline EndFraction right-brace minus left-parenthesis y Subscript i Baseline plus w Subscript i Baseline right-parenthesis log left-brace 1 plus StartFraction mu Subscript i Baseline Over w Subscript i Baseline EndFraction right-brace 2nd Row 1st Column Blank 2nd Column plus log left-brace StartFraction normal upper Gamma left-parenthesis y Subscript i Baseline plus w Subscript i Baseline right-parenthesis Over normal upper Gamma left-parenthesis w Subscript i Baseline right-parenthesis normal upper Gamma left-parenthesis y Subscript i Baseline plus 1 right-parenthesis EndFraction right-brace EndLayout

normal upper V normal a normal r left-bracket upper Y right-bracket equals mu plus mu squared comma phi identical-to 1.

Inverse Gaussian:
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals minus one half left-bracket StartFraction w Subscript i Baseline left-parenthesis y Subscript i Baseline minus mu Subscript i Baseline right-parenthesis squared Over y Subscript i Baseline phi mu Subscript i Superscript 2 Baseline EndFraction plus log left-brace StartFraction phi y Subscript i Superscript 3 Baseline Over w Subscript i Baseline EndFraction right-brace plus log left-brace 2 pi right-brace right-bracket

normal upper V normal a normal r left-bracket upper Y right-bracket equals phi mu cubed comma phi greater-than 0.

Multinomial:
l left-parenthesis bold-italic mu Subscript i Baseline comma phi semicolon bold y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals w Subscript i Baseline sigma-summation Underscript j equals 1 Overscript upper J Endscripts y Subscript i j Baseline log left-brace mu Subscript i j Baseline right-brace

phi identical-to 1.

Negative binomial:
StartLayout 1st Row 1st Column l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis 2nd Column equals y Subscript i Baseline log left-brace StartFraction k mu Subscript i Baseline Over w Subscript i Baseline EndFraction right-brace minus left-parenthesis y Subscript i Baseline plus w Subscript i Baseline slash k right-parenthesis log left-brace 1 plus StartFraction k mu Subscript i Baseline Over w Subscript i Baseline EndFraction right-brace 2nd Row 1st Column Blank 2nd Column plus log left-brace StartFraction normal upper Gamma left-parenthesis y Subscript i Baseline plus w Subscript i Baseline slash k right-parenthesis Over normal upper Gamma left-parenthesis w Subscript i Baseline slash k right-parenthesis normal upper Gamma left-parenthesis y Subscript i Baseline plus 1 right-parenthesis EndFraction right-brace EndLayout

normal upper V normal a normal r left-bracket y right-bracket equals mu plus k mu squared comma k greater-than 0 comma phi identical-to 1. For a given k, the negative binomial distribution is a member of the exponential family. The parameter k is related to the scale of the data, because it is part of the variance function. However, it cannot be factored from the variance, as is the case with the phi parameter in many other distributions. The parameter k is designated as "Scale" in the output of PROC BGLIMM.

Normal (Gaussian):
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals minus one half left-bracket StartFraction w Subscript i Baseline left-parenthesis y Subscript i Baseline minus mu Subscript i Baseline right-parenthesis squared Over phi EndFraction plus log left-brace StartFraction phi Over w Subscript i Baseline EndFraction right-brace plus log left-brace 2 pi right-brace right-bracket

normal upper V normal a normal r left-bracket y right-bracket equals phi comma phi greater-than 0.

Poisson:
l left-parenthesis mu Subscript i Baseline comma phi semicolon y Subscript i Baseline comma w Subscript i Baseline right-parenthesis equals w Subscript i Baseline left-parenthesis y Subscript i Baseline log left-brace mu Subscript i Baseline right-brace minus mu Subscript i Baseline minus log left-brace normal upper Gamma left-parenthesis y Subscript i Baseline plus 1 right-parenthesis right-brace right-parenthesis

normal upper V normal a normal r left-bracket y right-bracket equals mu comma phi identical-to 1.

Define the parameter vector for the generalized linear model as bold-italic theta equals bold-italic beta, if phi identical-to 1, and as bold-italic theta equals left-bracket bold-italic beta prime comma phi right-bracket prime otherwise. bold-italic beta denotes the fixed-effects parameters in the linear predictor. For the negative binomial distribution, the relevant parameter vector is bold-italic theta equals left-bracket bold-italic beta prime comma k right-bracket prime. The gradient and Hessian of the negative log likelihood are then

bold g equals minus StartFraction partial-differential upper L left-parenthesis bold-italic theta semicolon bold y right-parenthesis Over partial-differential bold-italic theta EndFraction bold upper H equals minus StartFraction partial-differential squared upper L left-parenthesis bold-italic theta semicolon bold y right-parenthesis Over partial-differential bold-italic theta partial-differential bold-italic theta prime EndFraction
Last updated: December 09, 2022