Complementary log-log link
= Complementary log-log link
{title2=$\operatorname{cloglog}(p)=\ln[-\ln(1-p)]$}
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 $\operatorname{cloglog}(p)=\ln\lambda+\ln e$. Thus a <binomial regression> with a log-exposure <offset> estimates the log hazard as its intercept.