Editing Generalized linear model (section)

=== Binary data ===
{{See also|Binary regression}}
When the response data, ''Y'', are binary (taking on only values 0 and 1), the distribution function is generally chosen to be the [[Bernoulli distribution]] and the interpretation of ''μ''<sub>i</sub> is then the probability, ''p'', of ''Y''<sub>i</sub> taking on the value one.

There are several popular link functions for binomial functions.

==== Logit link function ====
The most typical link function is the canonical [[logit]] link:

:<math>g(p) = \operatorname{logit} p = \ln \left( { p \over 1-p } \right).</math>

GLMs with this setup are [[logistic regression]] models (or ''logit models'').

==== Probit link function as popular choice of inverse cumulative distribution function ====
Alternatively, the inverse of any continuous [[cumulative distribution function]] (CDF) can be used for the link since the CDF's range is <math>[0,1]</math>, the range of the binomial mean. The [[Normal distribution#Cumulative distribution function|normal CDF]] <math>\Phi</math> is a popular choice and yields the [[probit model]]. Its link is

:<math>g(p) = \Phi^{-1}(p).\,\!</math>

The reason for the use of the probit model is that a constant scaling of the input variable to a normal CDF (which can be absorbed through equivalent scaling of all of the parameters) yields a function that is practically identical to the logit function, but probit models are more tractable in some situations than logit models. (In a Bayesian setting in which normally distributed [[prior distribution]]s are placed on the parameters, the relationship between the normal priors and the normal CDF link function means that a [[probit model]] can be computed using [[Gibbs sampling]], while a logit model generally cannot.)

==== 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>

==== Identity link ====
The identity link ''g(p) = p'' is also sometimes used for binomial data to yield a [[linear probability model]].  However, the identity link can predict nonsense "probabilities" less than zero or greater than one.  This can be avoided by using a transformation like cloglog, probit or logit (or any inverse cumulative distribution function). A primary merit of the identity link is that it can be estimated using linear math—and other standard link functions are approximately linear matching the identity link near ''p'' = 0.5.

==== Variance function ====
The [[variance function]] for "{{visible anchor|quasibinomial}}" data is:

:<math>\operatorname{Var}(Y_i)= \tau\mu_i (1-\mu_i)\,\!</math>

where the dispersion parameter ''&tau;'' is exactly 1 for the binomial distribution. Indeed, the standard binomial likelihood omits ''&tau;''.  When it is present, the model is called "quasibinomial", and the modified likelihood is called a [[quasi-likelihood]], since it is not generally the likelihood corresponding to any real family of probability distributions.  If ''&tau;'' exceeds 1, the model is said to exhibit [[overdispersion]].