Binary regression

In statistics, specifically regression analysis, a binary regression estimates a relationship between one or more explanatory variables and a single output binary variable. Generally the probability of the two alternatives is modeled, instead of simply outputting a single value, as in linear regression.

Binary regression is usually analyzed as a special case of binomial regression, with a single outcome (

n

=

1

{\displaystyle n=1}

), and one of the two alternatives considered as "success" and coded as 1: the value is the count of successes in 1 trial, either 0 or 1. The most common binary regression models are the logit model (logistic regression) and the probit model (probit regression).

Applications

Binary regression is principally applied either for prediction (binary classification), or for estimating the association between the explanatory variables and the output. In economics, binary regressions are used to model binary choice.

Interpretations

Binary regression models can be interpreted as latent variable models, together with a measurement model; or as probabilistic models, directly modeling the probability.

Latent variable model

The latent variable interpretation has traditionally been used in bioassay, yielding the probit model, where normal variance and a cutoff are assumed. The latent variable interpretation is also used in item response theory (IRT).

Formally, the latent variable interpretation posits that the outcome y is related to a vector of explanatory variables x by

y

=

1

[

y

∗

>

0

]

{\displaystyle y=1[y^{*}>0]}

where

y

∗

=

x

β

+

ε

{\displaystyle y^{*}=x\beta +\varepsilon }

and

ε

∣

x

∼

G

{\displaystyle \varepsilon \mid x\sim G}

, β is a vector of parameters and G is a probability distribution.