The BGLIMM Procedure

Response Probability Distributions

Probability distributions of the response y in generalized linear models are usually parameterized in terms of the mean mu and dispersion parameter phi instead of the natural parameter theta. The probability distributions that are available in the BGLIMM procedure are shown in the following list. The PROC BGLIMM scale parameter and the variance of y are also shown.

  • Normal:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction 1 Over StartRoot 2 pi EndRoot sigma EndFraction exp left-bracket minus one-half left-parenthesis StartFraction y minus mu Over sigma EndFraction right-parenthesis squared right-bracket for negative normal infinity less-than y less-than normal infinity 2nd Row 1st Column phi 2nd Column equals 3rd Column sigma squared 3rd Row 1st Column normal s normal c normal a normal l normal e 2nd Column equals 3rd Column sigma squared 4th Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column sigma squared EndLayout

  • Inverse Gaussian:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction 1 Over StartRoot 2 pi y cubed EndRoot sigma EndFraction exp left-bracket minus StartFraction 1 Over 2 y EndFraction left-parenthesis StartFraction y minus mu Over mu sigma EndFraction right-parenthesis squared right-bracket for 0 less-than y less-than normal infinity 2nd Row 1st Column phi 2nd Column equals 3rd Column sigma squared 3rd Row 1st Column normal s normal c normal a normal l normal e 2nd Column equals 3rd Column sigma squared 4th Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column sigma squared mu cubed EndLayout

  • Gamma:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction 1 Over normal upper Gamma left-parenthesis nu right-parenthesis y EndFraction left-parenthesis StartFraction y nu Over mu EndFraction right-parenthesis Superscript nu Baseline exp left-parenthesis minus StartFraction y nu Over mu EndFraction right-parenthesis for 0 less-than y less-than normal infinity 2nd Row 1st Column phi 2nd Column equals 3rd Column nu Superscript negative 1 3rd Row 1st Column normal s normal c normal a normal l normal e 2nd Column equals 3rd Column nu 4th Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column StartFraction mu squared Over nu EndFraction EndLayout

  • Geometric: This is a special case of the negative binomial where k = 1.

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction left-parenthesis mu right-parenthesis Superscript y Baseline Over left-parenthesis 1 plus mu right-parenthesis Superscript y plus 1 Baseline EndFraction for y equals 0 comma 1 comma 2 comma ellipsis 2nd Row 1st Column phi 2nd Column equals 3rd Column 1 3rd Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column mu left-parenthesis 1 plus mu right-parenthesis EndLayout

  • Negative binomial:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction normal upper Gamma left-parenthesis y plus 1 slash k right-parenthesis Over normal upper Gamma left-parenthesis y plus 1 right-parenthesis normal upper Gamma left-parenthesis 1 slash k right-parenthesis EndFraction StartFraction left-parenthesis k mu right-parenthesis Superscript y Baseline Over left-parenthesis 1 plus k mu right-parenthesis Superscript y plus 1 slash k Baseline EndFraction for y equals 0 comma 1 comma 2 comma ellipsis 2nd Row 1st Column phi 2nd Column equals 3rd Column 1 3rd Row 1st Column normal d normal i normal s normal p normal e normal r normal s normal i normal o normal n 2nd Column equals 3rd Column k 4th Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column mu plus k mu squared EndLayout

  • Poisson:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartFraction mu Superscript y Baseline normal e Superscript negative mu Baseline Over y factorial EndFraction for y equals 0 comma 1 comma 2 comma ellipsis 2nd Row 1st Column phi 2nd Column equals 3rd Column 1 3rd Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column mu EndLayout

  • Binomial:

    StartLayout 1st Row 1st Column f left-parenthesis y right-parenthesis 2nd Column equals 3rd Column StartBinomialOrMatrix n Choose r EndBinomialOrMatrix mu Superscript r Baseline left-parenthesis 1 minus mu right-parenthesis Superscript n minus r Baseline for y equals StartFraction r Over n EndFraction comma r equals 0 comma 1 comma 2 comma ellipsis comma n 2nd Row 1st Column phi 2nd Column equals 3rd Column 1 3rd Row 1st Column normal upper V normal a normal r left-parenthesis upper Y right-parenthesis 2nd Column equals 3rd Column StartFraction mu left-parenthesis 1 minus mu right-parenthesis Over n EndFraction EndLayout

The negative binomial and zero-inflated negative binomial distributions contain a parameter k, called the negative binomial dispersion parameter. This is not the same as the generalized linear model dispersion phi; rather, it is an additional distribution parameter that must be estimated or set to a fixed value.

For the binomial distribution, the response is the binomial proportion y equals e v e n t s slash t r i a l s. The variance function is upper V left-parenthesis mu right-parenthesis equals mu left-parenthesis 1 minus mu right-parenthesis, and the binomial trials parameter n is regarded as a weight w.

Last updated: December 09, 2022