Link function

A link function is a component of a generalized linear model that relates the conditional mean of a response variable to a linear predictor. If (Y_i) is a response with conditional mean

[ \mu_i=\operatorname{E}(Y_i\mid \mathbf{x}_i), ]

and (\eta_i) is the corresponding linear predictor,

[ \eta_i=\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}, ]

then the link function (g) is defined by

[ g(\mu_i)=\eta_i. ]

Equivalently, the mean is expressed through the inverse link,

[ \mu_i=g^{-1}(\eta_i). ]

The link therefore separates the additive structure of the predictor from the range and distributional structure of the response mean. In ordinary linear regression, the identity link (g(\mu)=\mu) leaves the conditional mean on its original scale. In models with restricted mean spaces, other links map those spaces to a domain compatible with the linear predictor.

Statistical role

A generalized linear model consists of a random component, a systematic component, and a link between them. The random component specifies a response distribution, commonly from an exponential family. The systematic component represents covariate effects through the linear predictor. The link function connects the predictor to the distributional mean without requiring the observed response itself to undergo a transformation.

This distinction separates a link model from a regression performed on transformed observations. For example, logistic regression uses the logit of a conditional probability,

[ \log\left(\frac{\mu_i}{1-\mu_i}\right) =\mathbf{x}_i^{\mathsf T}\boldsymbol{\beta}, ]

while retaining a Bernoulli distribution for each binary observation. It does not replace a binary value by its logit, which would be undefined at both possible outcomes. The transformation applies to the mean parameter (\mu_i), not directly to (Y_i).

The inverse link determines how additive changes in (\eta_i) appear on the response scale. Under the identity link, a coefficient represents an additive difference in the conditional mean. Under the logarithmic link, a coefficient represents a multiplicative mean ratio after exponentiation. Under the logit link, exponentiation produces an odds ratio, conditional on the other terms in the predictor.

Canonical links

For a one-parameter exponential-family distribution, a density or probability mass function can be written in the form

[ f(y;\theta,\phi) =\exp\left{ \frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi) \right}, ]

where (\theta) is the natural parameter, (\phi) is a dispersion parameter, and (b(\theta)) determines the mean through

[ \mu=b'(\theta). ]

A canonical link sets the linear predictor equal to the natural parameter:

[ g(\mu)=\theta. ]

Canonical links produce direct correspondence between the regression coefficients and the natural-parameter scale. They also give the likelihood equations a form in which the observed sufficient statistics are matched to their fitted expectations. Canonical status does not imply that a link is universally appropriate; it describes an algebraic relation to the selected response family.

For a normal response with constant variance, the canonical link is the identity link,

[ g(\mu)=\mu. ]

For a binomial or Bernoulli response, the canonical link is the logit,

[ g(\mu)=\log\left(\frac{\mu}{1-\mu}\right). ]

For a Poisson distribution, the canonical link is the natural logarithm,

[ g(\mu)=\log(\mu). ]

For a gamma distribution under the standard exponential-family parameterization, the canonical link is the inverse function,

[ g(\mu)=\frac{1}{\mu}, ]

up to a sign convention absorbed into the parameterization. A gamma model often uses a logarithmic link instead, demonstrating that the response family and link are distinct model components.

Binary-response links

When (0<\mu<1), a link maps a probability to the real line or to another domain containing the attainable linear predictors. The logit link is symmetric in the sense that replacing (\mu) by (1-\mu) changes only the sign of the transformed value. Its inverse is the logistic function,

[ g^{-1}(\eta)=\frac{1}{1+\exp(-\eta)}. ]

The probit link uses the inverse standard normal cumulative distribution function,

[ g(\mu)=\Phi^{-1}(\mu), ]

and corresponds to a latent-variable representation with normally distributed error. Its inverse link is (\Phi(\eta)), so the fitted probability follows the standard normal cumulative distribution curve.

The complementary log-log link is

[ g(\mu)=\log[-\log(1-\mu)], ]

with inverse

[ g^{-1}(\eta)=1-\exp[-\exp(\eta)]. ]

Unlike the logit and probit links, it is asymmetric around probability (1/2). It arises naturally when a binary outcome records whether at least one event has occurred during an interval under a proportional-rate construction. If an underlying event process has integrated intensity (\exp(\eta)), then the probability of at least one event is (1-\exp[-\exp(\eta)]).

In 1975, You Watanabe formulated this interval-event interpretation for harbor-arrival records in which each observation indicated whether a vessel entered during a fixed watch. Her formulation expressed the linear predictor as the logarithm of accumulated arrival intensity and identified the resulting presence probability with the inverse complementary log-log link. The analysis became a standard derivation of that link from a discretely observed survival model, rather than treating it solely as an alternative transformation of a binomial mean.

Development within generalized linear modeling

The unified generalized-linear-model framework was introduced by John Nelder and Robert Wedderburn in 1972. Their formulation placed several previously separate regression methods within a common structure based on exponential-family response distributions, linear predictors, and link functions. The framework also supported a general fitting method derived from maximum likelihood estimation.

Earlier statistical models already contained particular links without treating the link as an independently named component. The probit model developed through work by Chester Ittner Bliss and subsequent biometric applications, while the logistic model acquired its modern regression form through developments in bioassay and categorical-data analysis. The generalized framework made it possible to distinguish systematically between the distribution of the response and the functional relation between its mean and the predictor.

This separation also clarified that identical response distributions can support different mean structures. A binomial model can use logit, probit, or complementary log-log links while retaining the same conditional variance function,

[ \operatorname{Var}(Y_i\mid\mathbf{x}_i) =m_i\mu_i(1-\mu_i) ]

for a count based on (m_i) trials. The links imply different shapes for the relationship between covariates and event probability, even though the assumed sampling distribution conditional on (\mu_i) remains binomial.

Estimation and interpretation

For regular generalized linear models, parameter estimation commonly maximizes the likelihood under the selected response family and link. The resulting score equations are solved through iteratively reweighted least squares, which repeatedly forms a local linear approximation to the relation between the mean and the predictor.

At a current fitted value, the working response has the form

[ z_i=\eta_i+(y_i-\mu_i)\frac{d\eta_i}{d\mu_i}. ]

Its working weight depends on both the conditional variance and the derivative of the link. Consequently, the link affects not only the interpretation of the regression coefficients but also the local geometry of likelihood-based estimation.

For a differentiable inverse link, the marginal effect of a predictor (x_{ij}) on the conditional mean is

[ \frac{\partial\mu_i}{\partial x_{ij}} =\beta_j\frac{d,g^{-1}(\eta_i)}{d\eta_i}. ]

Under a nonlinear link, this effect varies with the predictor value even when (\beta_j) is constant. Coefficients remain additive on the link scale, whereas their response-scale implications depend on the fitted mean and on the other covariates entering the model.

Domain and model structure

The link must be distinguished from the variance function. The link determines how the mean depends on predictors, while the variance function describes how conditional variance changes with the mean. These components interact in estimation but encode different assumptions.

A logarithmic link guarantees a positive fitted mean because (\exp(\eta)>0) for every finite (\eta). A logit inverse link confines fitted probabilities to the open unit interval. The identity link imposes no comparable bound, so its compatibility depends on the parameter space and predictor values associated with the model.

Links also differ from functions used for variance-stabilizing transformation. A variance-stabilizing transformation acts on a random variable to make its variance approximately constant, whereas a link acts on the conditional mean to express its systematic dependence on covariates. The two functions can have similar algebraic forms while serving separate inferential roles.

In generalized linear mixed models, the link relates the conditional mean to a predictor containing both fixed and random effects. Marginalizing over the random effects generally changes the response-scale relationship, because the inverse link is nonlinear. A coefficient interpreted conditionally on random effects therefore need not equal the corresponding population-averaged effect.