Complementary log-log link
Topic of Introduction to generalised linear models, part of Generalised linear models
Complementary log-log link
| Type | Topic |
| Taught in | Introduction to generalised linear models |
| Part | Generalised linear models |
g(p)=\log\{-\log(1-p)\}. Unlike the logit and probit it is not symmetric about p=1/2. It arises when a binary response records whether a Poisson count is positive, since then \log\{-\log(1-p)\}=\log\lambda, and in ordinal regression it gives a proportional-hazards model.
In Introduction to generalised linear models
The complementary log-log link is \log\bigl(-\log(1-p_i)\bigr)={\boldsymbol{X}}_i^\top\boldsymbol{\beta} \qquad\text{equivalently}\qquad p_i=1-\exp\bigl\{-\exp({\boldsymbol{X}}_i^\top\boldsymbol{\beta})\bigr\}. The exercise asks you to check that the right-hand side lies in (0,1) for every real linear predictor. This link is the one that does not treat the two tails symmetrically. Logit and probit both approach 0 and 1 in the same fashion, one a mirror of the other. Complementary log-log can stay near zero for longer and then rise more sharply towards one, or the reverse, depending on the sign of the slope. Use it when that asymmetry is the scientific point. Do not use it because you want a sentence about odds. The coefficient is a log-hazard, an instantaneous risk, and a hazard is much harder to explain to a medical collaborator than an odds ratio.
Used in
See also
Logit link · Probit link · Identity link · Log link · Inverse link · Canonical link · Binary responses in a GLM · Ordinal responses · Proportional-hazards model · Exponential family