# NMST432 Advanced Regression Models

## Link functions for binary data

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## The logistic link (logit)

Based on the distribution function of the standard logistic distribution.

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

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

#### Plot of the inverse logistic link function:

## The probit link

Based on the distribution function of the standard normal distribution.

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

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

#### Plot of the inverse probit link function:

## The cauchit link

Based on the distribution function of the standard Cauchy distribution.

\[g(\mu)=\frac{1}{\pi}\arctan(\mu)+\frac{1}{2}\]

\[g^{-1}(\eta)=\tan[\pi(\eta-\frac{1}{2})]\]

#### Plot of the inverse cauchit link function:

## The complementary log-log link

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

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

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

#### Plot of the inverse complementary log-log link function:

## Comparison of inverse link functions

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