Instatistics, specificallyregression analysis, abinary regression estimates a relationship between one or moreexplanatory variables and a single outputbinary variable. Generally the probability of the two alternatives is modeled, instead of simply outputting a single value, as inlinear regression.

Binary regression is usually analyzed as a special case ofbinomial regression, with a single outcome (${\displaystyle n=1}$), and one of the two alternatives considered as "success" and coded as 1: the value is thecount of successes in 1 trial, either 0 or 1. The most common binary regression models are thelogit model (logistic regression) and theprobit model (probit regression).

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

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

The latent variable interpretation has traditionally been used inbioassay, yielding theprobit model, where normal variance and a cutoff are assumed. The latent variable interpretation is also used initem response theory (IRT).

Formally, the latent variable interpretation posits that the outcomey is related to a vector of explanatory variablesx by

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

where${\displaystyle y^{\ast }=x\beta +\epsilon }$ and${\displaystyle \epsilon \mid x\sim G}$,β is a vector ofparameters andG is aprobability distribution.

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,${\displaystyle y^{\ast }}$ is the expected netdiscounted cash flow andx is a vector of variables which can affect the cash flow of this program. Then the manager will invest only when she expects the net discounted cash flow to be positive.^[1]

Often, theerror term${\displaystyle \epsilon }$ is assumed to follow anormal distribution conditional on the explanatory variablesx. This generates the standardprobit model.^[2]

The simplest direct probabilistic model is thelogit model, which models thelog-odds as a linear function of the explanatory variable or variables. The logit model is "simplest" in the sense ofgeneralized linear models (GLIM): the log-odds are the natural parameter for theexponential family of the Bernoulli distribution, and thus it is the simplest to use for computations.

Another direct probabilistic model is thelinear 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.

• Generalized linear model § Binary data
• Fractional model

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