Binary data 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 μi is then the probability, p, of Yi 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: 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 , the range of the binomial mean. The normal CDF is a popular choice and yields the probit model. Its link is 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 distributions 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: This link function is asymmetric and will often produce different results from the logit and probit link functions. 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. The Poisson assumption means that where μ is a positive number denoting the expected number of events. If p represents the proportion of observations with at least one event, its complement and then A linear model requires the response variable to take values over the entire real line. Since μ must be positive, we can enforce that by taking the logarithm, and letting log(μ) be a linear model. This produces the "cloglog" transformation 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.