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/(1-z))

Probit link:

 

f(z) = invnorm(z), where invnorm is the inverse of the standard normal cumulative distribution function.

Complementary log-log link:

f(z) = log(-log(1-z))

Loglog link:

f(z) = -log(-log(z))

Multinomial distribution:

Generalized logit link:

f(z1|z2,...,zc) = log(x1/(1-z1-...-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.