Link Function

Fit Analyses

Link Function

The link function links the response mean $\mu$ to the linear predictor ${\eta}$ .SAS/INSIGHT software provides six types of link functions:

Identity: $g(\mu) = \mu$
Log: $g(\mu) = \log(\mu)$
Logit: $g(\mu) = \log(\frac{\mu}{1-\mu})$
Probit: $g(\mu) = \Phi^{-1}(\mu)$
Comp. Log-log: $g(\mu) = \log(-\log(1-\mu))$
Power: $g(\mu) = \mu^{\lambda}$ where $\lambda$ is the value in the Power entry field.

For each response distribution in the exponential family, there exists a special link function, the canonical link, for which ${\theta}$ = ${\eta}$ .The canonical links expressed in terms of the mean parameter $\mu$ are

Normal: $g(\mu) = \mu$
Inverse Gaussian: $g(\mu) = \mu^{-2}$
Gamma: $g(\mu) = \mu^{-1}$
Poisson: $g(\mu) = \log(\mu)$
Binomial: $g(\mu) = \log(\frac{\mu}{1-\mu})$

Note Some links are not appropriate for all distributions. For example, logit, probit, and complementary log-log links are only appropriate for the binomial distribution.

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