Link function

A link function connects the response mean to the linear predictor through $g({\mu }_i)={\eta }_i$.

The complementary log-log link maps $p\in (0,1)$ to $\ln [-\ln (1-p)]$, with inverse $p=1-\exp (-e^{\eta })$. For constant infection hazard $\lambda$ and exposure $e$, $p=1-e^{-\lambda e}$ implies $\mathrm{cloglog}(p)=\ln \lambda +\ln e$. Thus a binomial regression with a log-exposure offset estimates the log hazard as its intercept.

The identity link function equates the response mean with the linear predictor. In a generalized linear model it gives ${\mu }_i=x_i^T\beta$. It is the canonical link function for the normal distribution. For response families whose means must remain positive or between zero and one, using the identity link requires corresponding restrictions on the linear predictor.

The logit link maps a mean probability to its log odds: $g(p)=\log [p/(1-p)]$, $0<p<1$. Its inverse is $p=(1+e^{-\eta }{)}^{-1}$, used in logistic regression.

The logarithmic link maps a positive mean to $\log \mu$. It is the Poisson canonical link.