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I'm trying to estimate the predicted probabilities of an observation being a particular integer, $y$, after a negative binomial regression model. Long's Regression models for categorical and limited dependent variables gives this predicted probability as (pg.237):

$$ \hat{\text{Pr}}(y \mid x) = \frac{ \Gamma(y + \hat{a}^{-1}) }{ y!\Gamma(\hat{a}^{-1}) } \left( \frac{\hat{a}^{-1}}{\hat{a}^{-1}+\hat{\mu}}\right)^{\hat{a}^{-1}} \left( \frac{\hat{\mu}}{\hat{a}^{-1}+\hat{\mu}} \right) $$

Where $\hat{\mu}$ is the predicted mean of the variable, $\hat{a}$ is the dispersion estimate, and $\Gamma$ is the Gamma function. Now, my question is the statistical software I use takes both a shape and a scale parameter for the $\Gamma$ distribution, so I am confused as to how to actually estimate the predicted probabilities for any particular integer $y$.

In the above equation, what does Long expect me to supply as the shape and the scale for the $\Gamma$ function?

I'm trying to estimate the predicted probabilities of an observation being a particular integer, $y$, after a negative binomial regression model. Long's Regression models for categorical and limited dependent variables gives this predicted probability as (pg.237):

$$ \hat{\text{Pr}}(y \mid x) = \frac{ \Gamma(y + \hat{a}^{-1}) }{ y!\Gamma(\hat{a}^{-1}) } \left( \frac{\hat{a}^{-1}}{\hat{a}^{-1}+\hat{\mu}}\right)^{\hat{a}^{-1}} \left( \frac{\hat{\mu}}{\hat{a}^{-1}+\hat{\mu}} \right)^y $$

Where $\hat{\mu}$ is the predicted mean of the variable, $\hat{a}$ is the dispersion estimate, and $\Gamma$ is the Gamma function. Now, my question is the statistical software I use takes both a shape and a scale parameter for the $\Gamma$ distribution, so I am confused as to how to actually estimate the predicted probabilities for any particular integer $y$.

In the above equation, what does Long expect me to supply as the shape and the scale for the $\Gamma$ function?

I'm trying to estimate the predicted probabilities of an observation being a particular integer, $x$, after a negative binomial regression model. Long's Regression models for categorical and limited dependent variables gives this predicted probability as (pg.237):

$$ \hat{\text{Pr}}(y \mid x) = \frac{ \Gamma(y + \hat{a}^{-1}) }{ y!\Gamma(\hat{a}^{-1}) } \left( \frac{\hat{a}^{-1}}{\hat{a}^{-1}+\hat{\mu}}\right)^{\hat{a}^{-1}} \left( \frac{\hat{\mu}}{\hat{a}^{-1}+\hat{\mu}} \right) $$

Where $\hat{\mu}$ is the predicted mean of the variable, $\hat{a}$ is the dispersion estimate, and $\Gamma$ is the Gamma function. Now, my question is the statistical software I use takes both a shape and a scale parameter for the $\Gamma$ distribution, so I am confused as to how to actually estimate the predicted probabilities for any particular integer $x$.

In the above equation, what does Long expect me to supply as the shape and the scale for the $\Gamma$ function?

I'm trying to estimate the predicted probabilities of an observation being a particular integer, $y$, after a negative binomial regression model. Long's Regression models for categorical and limited dependent variables gives this predicted probability as (pg.237):

$$ \hat{\text{Pr}}(y \mid x) = \frac{ \Gamma(y + \hat{a}^{-1}) }{ y!\Gamma(\hat{a}^{-1}) } \left( \frac{\hat{a}^{-1}}{\hat{a}^{-1}+\hat{\mu}}\right)^{\hat{a}^{-1}} \left( \frac{\hat{\mu}}{\hat{a}^{-1}+\hat{\mu}} \right) $$

Where $\hat{\mu}$ is the predicted mean of the variable, $\hat{a}$ is the dispersion estimate, and $\Gamma$ is the Gamma function. Now, my question is the statistical software I use takes both a shape and a scale parameter for the $\Gamma$ distribution, so I am confused as to how to actually estimate the predicted probabilities for any particular integer $y$.

In the above equation, what does Long expect me to supply as the shape and the scale for the $\Gamma$ function?