Completing the model specification #

Supported Families #

The package currently supports the following families and link functions, which are specified using the standard R families in the stats package.

The Beta family is provided by the package function Beta(), which generates a barebones list specifying the family and link. We use a mean-variance parameterisation of the Beta family. The likelihood is:

$$ f(y_i | \mu_i, \phi) = \frac{y_i^{\mu_i\phi - 1}(1-y_i)^{(1-\mu_i)\phi - 1}}{B(\mu_i\phi, (1-\mu_i)\phi)} $$

where $B()$ is the Beta function, and we use logit link

$$ \log\left( \frac{\mu_i}{1-\mu_i} \right) = \mathbf{x}_i\beta + \mathbf{z}_i \mathbf{u} $$

We similarly use a mean-variance parameterisation for the Gamma regression function:

$$ f(y_i | \mu_i, \nu) = \frac{1}{\Gamma(\nu)}\left( \frac{\nu y_i}{\mu_i} \right)^\nu \exp \left( -\frac{\nu y_i}{\mu_i} \right) \frac{1}{y} $$

where we also provide logit, inverse, and identity link functions for the specification of $\mu_i$.