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NMST412 Generalized Linear Models

Link functions for binary data


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The logistic link (logit)

Based on the distribution function of the standard logistic distribution.

g(μ)=μ1μ

g1(η)=exp(η)1+exp(η)

Plot of the inverse logistic link function:

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The probit link

Based on the distribution function of the standard normal distribution.

g(μ)=Φ1(μ)

g1(η)=Φ(η)

Plot of the inverse probit link function:

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The cauchit link

Based on the distribution function of the standard Cauchy distribution.

g(μ)=1πarctan(μ)+12

g1(η)=tan[π(η12)]

Plot of the inverse cauchit link function:

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The complementary log-log link

Related to the distribution function of the extreme value (Gumbel) distribution. The link is not symmetric.

g(μ)=log(log(1μ))

g1(η)=1exp(exp(η))

Plot of the inverse complementary log-log link function:

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Comparison of inverse link functions

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