Binomial regression

Regression analysis technique

In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is the number of successes in a series of n independent Bernoulli trials, where each trial has probability of success p. the details were as follows. The observed outcome variable was whether or not a fault occurred in an industrial process. There were two explanatory variables: the first was a simple two case factor representing whether or not a modified version of the process was used and the second was an ordinary quantitative variable measuring the purity of the material being supplied for the process.

The response variable Y is assumed to be binomially distributed conditional on the explanatory variables X. The number of trials n is known, and the probability of success for each trial p is specified as a function θ(X). This implies that the conditional expectation and conditional variance of the observed fraction of successes, Y/n, are

The goal of binomial regression is to estimate the function θ(X). Typically the statistician assumes , for a known function m, and estimates β. Common choices for m include the logistic function. The data 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 1A 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 yi, only one of the two terms inside the product contributes, according to whether yi=0 or 1. The likelihood function is more fully specified by defining the formal parameters μi 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.

There is a requirement that the modelling linking the probabilities μ to the explanatory variables should be of a form which only produces values in the range 0 to 1. Many models can be fitted into the form

Here η is an intermediate variable representing a linear combination, containing the regression parameters, of the explanatory variables. The function
g is the cumulative distribution function (cdf) of some probability distribution. Usually this probability distribution has a support from minus infinity to plus infinity so that any finite value of η is transformed by the function g to a value inside the range 0 to 1. In the case of logistic regression, the link function is the log of the odds ratio or logistic function. In the case of probit, the link is the cdf of the normal distribution. The linear probability model is not a proper binomial regression specification because predictions need not be in the range of zero to one; it is sometimes used for this type of data when the probability space is where interpretation occurs or when the analyst lacks sufficient sophistication to fit or calculate approximate linearizations of probabilities for interpretation.

Binomial regression is closely connected with binary regression. If the response is a binary variable (two possible outcomes), then these alternatives can be coded as 0 or 1 by considering one of the outcomes as "success" and the other as "failure" and considering these as count data: "success" is 1 success out of 1 trial, while "failure" is 0 successes out of 1 trial. This can now be considered a binomial distribution with trial, so a binary regression is a special case of a binomial regression. If these data are grouped (by adding counts), they are no longer binary data, but are count data for each group, and can still be modeled by a binomial regression; the individual binary outcomes are then referred to as "ungrouped data". An advantage of working with grouped data is that one can test the goodness of fit of the model; for example, grouped data may exhibit overdispersion relative to the variance estimated from the ungrouped data.