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
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.
Binary regression models can be interpreted as latent variable models, together with a measurement model; or as probabilistic models, directly modeling the probability.
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]
where
y*=x\beta+\varepsilon
\varepsilon\midx\simG
This model can be applied in many economic contexts. For instance, the outcome can be the decision of a manager whether invest to a program,
y*
\varepsilon
The simplest direct probabilistic model is the logit model, which models the log-odds as a linear function of the explanatory variable or variables. The logit model is "simplest" in the sense of generalized linear models (GLIM): the log-odds are the natural parameter for the exponential family of the Bernoulli distribution, and thus it is the simplest to use for computations.
Another direct probabilistic model is the linear probability model, which models the probability itself as a linear function of the explanatory variables. A drawback of the linear probability model is that, for some values of the explanatory variables, the model will predict probabilities less than zero or greater than one.