The GAMPL Procedure
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Overview
- Getting Started
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Syntax
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DetailsMissing ValuesThin-Plate Regression SplinesGeneralized Additive ModelsModel Evaluation CriteriaFitting AlgorithmsDegrees of FreedomModel InferenceDispersion ParameterTests for Smoothing ComponentsComputational Method: MultithreadingChoosing an Optimization TechniqueDisplayed OutputODS Table NamesODS Graphics
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Examples
- References
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
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estimates the dispersion parameter by the deviance, given the regression parameter estimates:
![\[ \hat{\phi } = \frac{\sum _ i D_ i(y_ i,\mu _ i)}{n-\mathrm{df}} \]](images/stathpug_hpgam0203.png)
- MLE
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estimates the dispersion parameter by maximizing the penalized likelihood, given the regression parameter estimates:
![\[ \hat{\phi } = \underset {\phi }{\operatorname {argmax}}~ \ell _ p(\hat{\bbeta },\phi ) \]](images/stathpug_hpgam0204.png)
The MLE option is the only option that you can use to estimate the dispersion parameter for the negative binomial distribution.
- PEARSON
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estimates the dispersion parameter by Pearson’s statistic, given the regression parameter estimates:
![\[ \hat{\phi } = \frac{\sum _ i \omega _ i(y_ i-\mu _ i)^2/\nu _ i}{n-\mathrm{df}} \]](images/stathpug_hpgam0205.png)
If the dispersion parameter is estimated, it contributes one additional degree of freedom to the fitted model.