The GAMPL Procedure

Dispersion Parameter

Some distribution families (Gaussian, gamma, inverse Gaussian, and negative binomial) have a dispersion parameter that you can specify in the DISPERSION= option in the MODEL statement or you can estimate from the data. The following three suboptions for the SCALE= option in the MODEL statement correspond to three ways to estimate the dispersion parameter:

DEVIANCE

estimates the dispersion parameter by the deviance, given the regression parameter estimates:

\[ \hat{\phi } = \frac{\sum _ i D_ i(y_ i,\mu _ i)}{n-\mathrm{df}} \]
MLE

estimates the dispersion parameter by maximizing the penalized likelihood, given the regression parameter estimates:

\[ \hat{\phi } = \underset {\phi }{\operatorname {argmax}}~ \ell _ p(\hat{\bbeta },\phi ) \]

The MLE option is the only option that you can use to estimate the dispersion parameter for the negative binomial distribution.

PEARSON

estimates the dispersion parameter by Pearson’s statistic, given the regression parameter estimates:

\[ \hat{\phi } = \frac{\sum _ i \omega _ i(y_ i-\mu _ i)^2/\nu _ i}{n-\mathrm{df}} \]

If the dispersion parameter is estimated, it contributes one additional degree of freedom to the fitted model.