Binomial regression

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In statistics, binomial regression is a technique in which the response (often referred to as Y) is the result of a series of Bernoulli trials, or a series of ones and zeros. The results are assumed to be binomially distributed and are often fit with a generalized linear model that predicts values ( $μ$ ) that are regarded as the probability that any individual event will result in a success or a one. The Likelihood of the predictions is then given by

$L(Y|\boldsymbol{\mu})=\prod_{i=1}^n 1_{y_i=1} \mu_i + 1_{y_i=0} (1-\mu_i)$

Where $1 A$ is the indicator function which takes on the value one when the event $A$ occurs. The result is then often solved with maximum likelihood.

It should be noted that data with two states that are not ones and zero can also be fit after assigning one state to one and the other to zero. The state that is assigned to one is often called the upper state and the other the lower state. In addition, multinomial data can be fit under this model after some provisions have been made to allow for this.

There are many methods of generating the values of $μ$ in systematic ways that allow for interpretation of the model; they are discussed below (or at least will be in the future).