Editing Generalized linear model (section)

==== Complementary log-log (cloglog) ====
The complementary log-log function may also be used:
:<math>g(p) = \log(-\log(1-p)).</math>
This link function is asymmetric and will often produce different results from the logit and probit link functions.<ref>{{Cite web|url=http://www.stat.ualberta.ca/~kcarrier/STAT562/comp_log_log.pdf|title=Complementary Log-log Model}}</ref> The cloglog model corresponds to applications where we observe either zero events (e.g., defects) or one or more, where the number of events is assumed to follow the [[Poisson distribution]].<ref>{{Cite web|url=https://bayesium.com/which-link-function-logit-probit-or-cloglog/|title=Which Link Function — Logit, Probit, or Cloglog?|date=2015-08-14|website=Bayesium Analytics|language=en-US|access-date=2019-03-17}}</ref> The Poisson assumption means that

:<math>\Pr(0) = \exp(-\mu),</math>

where ''&mu;'' is a positive number denoting the expected number of events.  If ''p'' represents the proportion of observations with at least one event, its complement

:<math> 1-p = \Pr(0) = \exp(-\mu),</math>

and then

:<math> -\log(1-p) = \mu.</math>

A linear model requires the response variable to take values over the entire real line. Since ''&mu;'' must be positive, we can enforce that by taking the logarithm, and letting log(''&mu;'') be a linear model.  This produces the "cloglog" transformation

:<math>\log(-\log(1-p)) = \log(\mu).</math>