- Binomial regression
In

statistics ,**binomial regression**is a technique in which the response (often referred to as "Y") is the result of a series ofBernoulli trial s, 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 ageneralized linear model whose predicted values μ are the probabilities that any individual event will result in a success. Thelikelihood of the predictions is then given by:$L(Y|\backslash boldsymbol\{mu\})=prod\_\{i=1\}^n\; 1\_\{y\_i=1\}(mu\_i)\; +\; 1\_\{y\_i=0\}\; (1-mu\_i),\; ,!$

where 1

_{A}is theindicator 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

:$\backslash boldsymbol\{mu\}\; =\; g^\{-1\}(\backslash boldsymbol\{eta\})$

where "g" is the

cumulative distribution function of someprobability 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 thenormal distribution .**Linear probability model**Here the probability distribution in question is the

uniform distribution and the resulting model is referred to as thelinear probability model .

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