Link Function
and Distribution Function
The link function in generalized
linear/nonlinear models specifies a nonlinear transformation
of the predicted values so that the distribution of predicted values is
one of several special members of the exponential family of distributions
(e.g., gamma, Poisson,
binomial, etc.). The link
function is therefore used to model responses when a dependent
variable is assumed to be nonlinearly related to the predictors.
Various link functions
(see McCullagh and Nelder, 1989) are commonly used, depending on the assumed
distribution of the dependent variable (y)
values:
Normal, Gamma,
Inverse normal, Tweedie, Poisson, and Negative
Binomial distributions:
Identity link: 
f(z) = z 
Log link: 
f(z) = log(z) 
Power link: 
f(z) = za, for a given a 
Binomial, and
Ordinal Multinomial
distributions:
Logit link: 
f(z) = log(z/(1z)) 
Probit
link:

f(z) = invnorm(z),
where invnorm is
the inverse of the standard normal cumulative distribution function. 
Complementary
loglog link: 
f(z) = log(log(1z)) 
Loglog
link: 
f(z) = log(log(z)) 
Multinomial
distribution:
Generalized
logit link: 
f(z1z2,...,zc)
= log(x1/(1z1...zc)), where
the model has c+1
categories. 
For discussion of the role of link functions,
see the see the Introductory
Overview for the Generalized
Linear/Nonlinear Models (GLZ) method of analysis.