Binomial regression

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 {displaystyle n} independent Bernoulli trials, where each trial has probability of success p {displaystyle p} . In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables. 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 {displaystyle n} independent Bernoulli trials, where each trial has probability of success p {displaystyle p} . In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables. Binomial regression is closely related to binary regression: if the response is a binary variable (two possible outcomes), then it can be considered as a binomial distribution with n = 1 {displaystyle n=1} trial by considering one of the outcomes as 'success' and the other as 'failure', counting the outcomes as either 1 or 0: counting a success as 1 success out of 1 trial, and counting a failure as 0 successes out of 1 trial. Binomial regression models are essentially the same as binary choice models, one type of discrete choice model. The primary difference is in the theoretical motivation: Discrete choice models are motivated using utility theory so as to handle various types of correlated and uncorrelated choices, while binomial regression models are generally described in terms of the generalized linear model, an attempt to generalize various types of linear regression models. As a result, discrete choice models are usually described primarily with a latent variable indicating the 'utility' of making a choice, and with randomness introduced through an error variable distributed according to a specific probability distribution. Note that the latent variable itself is not observed, only the actual choice, which is assumed to have been made if the net utility was greater than 0. Binary regression models, however, dispense with both the latent and error variable and assume that the choice itself is a random variable, with a link function that transforms the expected value of the choice variable into a value that is then predicted by the linear predictor. It can be shown that the two are equivalent, at least in the case of binary choice models: the link function corresponds to the quantile function of the distribution of the error variable, and the inverse link function to the cumulative distribution function (CDF) of the error variable. The latent variable has an equivalent if one imagines generating a uniformly distributed number between 0 and 1, subtracting from it the mean (in the form of the linear predictor transformed by the inverse link function), and inverting the sign. One then has a number whose probability of being greater than 0 is the same as the probability of success in the choice variable, and can be thought of as a latent variable indicating whether a 0 or 1 was chosen. In machine learning, binomial regression is considered a special case of probabilistic classification, and thus a generalization of binary classification. In one published example of an application of binomial regression, 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 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 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.