Binomial regression


Binomial regression

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 one of two possible disjoint outcomes (traditionally denoted "success" or 1, and "failure" or 0). The results are assumed to be binomially distributed and are often fit as a generalized linear model whose predicted values μ are the probabilities that any individual event will result in a success. The likelihood of the predictions is then given by

:L(Y|oldsymbol{mu})=prod_{i=1}^n 1_{y_i=1}(mu_i) + 1_{y_i=0} (1-mu_i), ,!

where 1A is the indicator function which takes on the value one when the event "A" occurs, and zero otherwise. This likelihood is usually maximized over the μs.

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.

Models based on a probability distribution

Many models can be fit into the form

:oldsymbol{mu} = g^{-1}(oldsymbol{eta})

where "g" is the cumulative distribution function of some probability distribution. This form can be arrived at by using the formula

:Y^*= eta^T x_i + epsilon ,!

where epsilon is taken from the probability distribution in question with mean zero and dispersion or variance of one.

Logit model

Here the model is based on a logistic regression.

Probit

In the probit model the probability distribution in question is the normal distribution.

Linear probability model

Here the probability distribution in question is the uniform distribution and the resulting model is referred to as the linear probability model.


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