Link function

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link function
The logit is one type of link function.

Link functions in Generalized Linear Models (GLMs) let us model curved relationships with straight lines. More specifically, it maps non-linear relationships to linear ones, which means you can fit a linear model to data. More specifically, link functions allow us to model non-linear relationships between predictors and response variables by transforming the response variable to a linear scale. It specifies the “link” between the random and the systematic components, indicating how the response variable’s expected value relates to the linear combination of explanatory variables [1].

Link function examples

Link functions connect the probability distribution of the response variable to the linear combination of the predictors. For example, in linear regression, the link function is the identity function — mapping every element in a set to itself and directly predicting the outcome. Other examples include the log, logit, reciprocal, and probit. Which link function you use depends upon what analysis you’re performing.

Common link functions include:

Link functionData type
Logit linkBinary data
Log linkCount data
Log log linkData from extreme asymmetric distributions
Identity linkContinuous data
Complementary log-log (cloglog) linkBinary data
Inverse complementary log-log (icloglog) linkBinary data
Negative inverseNon-negative and continuous
Probit linkBinary data
Power linkContinuous and count data

Which link function you use depends on the distribution of the response variable and the desired relationship between the predictors and the response variable.

DistributionSupportLink nameLink function
Bernoulli distributionIntegers: {0, 1}LogitXΒ = ln(μ / 1 – μ)
Binomial distribution integers: {0, 1, …, N}LogitXΒ = ln(μ / n – μ)
Normal Distributionreal: (-∞, ∞)IdentityXΒ = μ
Exponential Distributionreal: (0, ∞)Negative inverseXΒ = -μ-1
Gamma Distributionreal: (0, ∞)Negative inverseXΒ = -μ-1
Poisson Distributionintegers: 0, 1, 2, …LogXΒ = ln(μ)


[1] 6.1 – Introduction to GLMs

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