Complementary log-log link ($\mathrm{cloglog}(p)=\ln [-\ln (1-p)]$)

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 $\mathrm{cloglog}(p)=\ln \lambda +\ln e$. Thus a binomial regression with a log-exposure offset estimates the log hazard as its intercept.