Even though the formulas presented in the GLIMMIX chapter of the SAS/STAT User's Guide, Version 9 are correct, the variance for the Beta and Negative Binomial distribution is incorrectly computed in the GLIMMIX procedure.

In general the GLIMMIX procedure computes Var(Y) using the formula,

   Var(Y)=scale*vf, 

where vf is the variance function.

For example, the variance of the Gamma distribution is

   Var(Y)=Φu²,Φ > 0. 

Here the scale parameter = Φ, and the vf=u².

The GLIMMIX procedure does not handle the Beta and Negative Binomial distributions differently and computes the Var(Y) = scale*vf as usual and is incorrect. Since Var(Y) is the denominator in the PEARSON residual, the PEARSON residual is also incorrectly computed.

For the Beta and Negative Binomial distribution there is an extra parameter in the variance function that is treated as a “scale” parameter by GLIMMIX. The variance function of the Beta distribution is defined as follows:

   Var(Y) = μ(1-μ)/(1 + Φ), Φ > 0

where Φ is treated as a scale parameter by the GLIMMIX procedure for the Beta distribution.

The variance function of the Negative Binomial distribution is defined as follows:

   Var(Y) = μ + ku², k > 0

where k is treated as scale parameter by the GLIMMIX procedure for the Negative Binomial distribution.

Note that Φ and k are not “scale” parameters in the context of the Var(Y)computation because they cannot be factored out in the formula of Var(Y). In other words, Var(Y)=vf is the correct way to compute the variance for the Beta and Negative Binomial distribution.

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