linkfcn maps the mean of the response variable mu to
the linear predictor z. linkinv is its inverse.
Note that the logit link for the
binomial family is defined as the quantile of the logistic
distribution with scale 0.6458.
For the Gaussian family, if the link parameter is positive, then
the extended link is used, defined by $$z =
\frac{sign(\mu)|\mu|^\nu - 1}{\nu}$$ In the other case, the link function is the same as for the
Poisson and gamma families.
For the Poisson and gamma families, the Box-Cox transformation is
used, defined by $$z = \frac{\mu^\nu - 1}{\nu}$$
For the GEV binomial family, the link function is defined by
$$\mu = 1 - \exp\{-\max(0, 1 + \nu z)^{\frac{1}{\nu}}\}$$ for any real \(\nu\). At
\(\nu = 0\) it reduces to the complementary log-log
link.
The Wallace binomial family is a fast approximation to the robit
family. It is defined as $$\mu =
\Phi(\mbox{sign}(z) c(\nu) \sqrt{\nu \log(1 + z^2/\nu)})$$
where \(c(\nu) = (8\nu+1)/(8\nu+3)\)