Binomial Regression - Specification of Model

Specification of Model

The results are assumed to be binomially distributed. They are often fitted as a generalised linear model where the predicted values μ are the probabilities that any individual event will result in a success. The likelihood of the predictions is then given by

where 1_A is the indicator function which takes on the value one when the event A occurs, and zero otherwise: in this formulation, for any given observation y_i, only one of the two terms inside the product contributes, according to whether y_i=0 or 1. The likelihood function is more fully specified by defining the formal parameters μ_i as parameterised functions of the explanatory variables: this defines the likelihood in terms of a much reduced number of parameters. Fitting of the model is usually achieved by employing the method of maximum likelihood to determine these parameters. In practice, the use of a formulation as a generalised linear model allows advantage to be taken of certain algorithmic ideas which are applicable across the whole class of more general models but which do not apply to all maximum likelihood problems.

Models used in binomial regression can often be extended to multinomial data.

There are many methods of generating the values of μ in systematic ways that allow for interpretation of the model; they are discussed below.