## The Gamma Distribution

The gamma distribution can be used for a continuous response variable Y that has positive values (Y > 0), and the distribution function has various forms. Within the context of a GLM, we use (Faraway, 2006)

Before starting to memorise the exact mathematical definition of this density function, let us first look at the mean and variance of a variable Y that is gamma distributed and sketch the density curve for various values of y and v (which is the equivalent of the k in the negative binomial distribution). The mean and variance of Y are y2

The dispersion is determined by v-1; a small value of v (relative to y2) implies that the spread in the data is large. Density curves for difference values of y and v are given in Fig. 8.4. Note the wide range of shapes between these curves. For a large v, the gamma distribution becomes bell shaped and symmetric. In such cases, the Gaussian distribution can be used as well. Faraway (2006) gives an example of a linear regression model and a gamma GLM with a small (0.0045) dispersion parameter v^1; estimated parameters and standard errors obtained by both methods are nearly identical. However, for larger values of v-1, this is not the necessarily the case.

Gamma with mean 2 and v=0.01

Gamma with mean 2 and v=1

Gamma with mean 2 and v=2

4 6 Y values

Gamma with mean 10 and v=0.1