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.

$$g(\mu)=\frac{\mu}{1-\mu}$$

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

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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