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 "" data is: where the dispersion parameter τ is exactly 1 for the binomial distribution. Indeed, the standard binomial likelihood omits τ. 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 τ exceeds 1, the model is said to exhibit overdispersion. Multinomial regression The binomial case may be easily extended to allow for a multinomial distribution as the response (also, a Generalized Linear Model for counts, with a constrained total). There are two ways in which this is usually done: Ordered response If the response variable is ordinal, then one may fit a model function of the form: for m > 2. Different links g lead to ordinal regression models like proportional odds models or ordered probit models. Unordered response If the response variable is a nominal measurement, or the data do not satisfy the assumptions of an ordered model, one may fit a model of the following form: for m > 2. Different links g lead to multinomial logit or multinomial probit models. These are more general than the ordered response models, and more parameters are estimated. Count data Another example of generalized linear models includes Poisson regression which models count data using the Poisson distribution. The link is typically the logarithm, the canonical link. The variance function is proportional to the mean where the dispersion parameter τ is typically fixed at exactly one. When it is not, the resulting quasi-likelihood model is often described as Poisson with overdispersion or quasi-Poisson. Extensions Correlated or clustered data The standard GLM assumes that the observations are uncorrelated. Extensions have been developed to allow for correlation between observations, as occurs for example in longitudinal studies and clustered designs: Generalized estimating equations (GEEs) allow for the correlation between observations without the use of an explicit probability model for the origin of the correlations, so there is no explicit likelihood. They are suitable when the random effects and their variances are not of inherent