# Definition of Link function

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**Link function**Definition from Science & Technology Dictionaries & Glossaries

Electronic Statistics Textbook

The link function in generalized linear 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, Possion, 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, and Poisson distributions:

Identity link:

Log link:

Power link:

Binomial , and Ordinal Multinomial distributions:

Logit link:

Probit link:

Complementary log-log link:

Loglog link:

Multinomial distribution:

Generalized logit link:

For discussion of the role of link functions, see the Generalized Linear Models chapter.

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, and Poisson distributions:

Identity link:

**f(z) = z**Log link:

**f(z) = log(z)**Power link:

**f(z) = z**, for a given^{a}**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 Generalized Linear Models chapter.

Common Concepts in Statistics

A particular function of the expected value of the response variable used in modelling as a linear combination of the explanatory variables. For example, logistic regression uses a

*logit*link function rather than the raw expected values of the response variable; and Poisson regression uses the*log*link function. The response variable values are predicted from a linear combination of explanatory variables, which are connected to the response variable via one of these link functions. In the case of the general linear model for a single response variable (a special case of the generalized linear model), the response variable has a normal distribution and the link function is a simple identity function (i.e., the linear combination of values for the predictor variable(s) is not transformed).