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.
Family | Link functions | code |
---|---|---|
Gaussian | Identity, log | gaussian() |
Binomial | Logit, log, identity | binomial() |
Poisson | Log, Identity | poisson() |
Gamma | Log, Inverse, Identity | gamma() |
Beta | Logit | Beta() |
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 $\mu_i$.