# Binary regression

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{{Short description|Statistical estimation method}}
{{regression bar}}
In [statistics](/source/statistics), specifically [regression analysis](/source/regression_analysis), a '''binary regression''' estimates a relationship between one or more [explanatory variable](/source/explanatory_variable)s and a single output [binary variable](/source/binary_variable). Generally the probability of the two alternatives is modeled, instead of simply outputting a single value, as in [linear regression](/source/linear_regression).

Binary regression is usually analyzed as a special case of [binomial regression](/source/binomial_regression), with a single outcome (<math>n = 1</math>), and one of the two alternatives considered as "success" and coded as 1: the value is the [count](/source/Count_data) of successes in 1 trial, either 0 or 1. The most common binary regression models are the [logit model](/source/logit_model) ([logistic regression](/source/logistic_regression)) and the [probit model](/source/probit_model) ([probit regression](/source/probit_regression)).

==Applications==
Binary regression is principally applied either for prediction ([binary classification](/source/binary_classification)), or for estimating the [association](/source/correlation) between the explanatory variables and the output. In economics, binary regressions are used to model [binary choice](/source/binary_choice).

==Interpretations==
Binary regression models can be interpreted as [latent variable model](/source/latent_variable_model)s, 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](/source/bioassay), yielding the [probit model](/source/probit_model), where normal variance and a cutoff are assumed. The latent variable interpretation is also used in [item response theory](/source/item_response_theory) (IRT).

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

: <math>y=1 [y^*>0]</math>

where <math>y^*=x\beta +\varepsilon </math> and <math>\varepsilon \mid x\sim G</math>, {{math|''&beta;''}} is a vector of [parameters](/source/statistical_parameter) and ''G'' is a [probability distribution](/source/probability_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, <math>y^*</math> is the expected net [discounted cash flow](/source/discounted_cash_flow) and ''x'' 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.<ref>For a detailed example, refer to: Tetsuo Yai, Seiji Iwakura, Shigeru Morichi, Multinomial probit with structured covariance for route choice behavior, Transportation Research Part B: Methodological, Volume 31, Issue 3, June 1997, Pages 195–207, ISSN 0191-2615</ref>

Often, the [error term](/source/errors_and_residuals) <math>\varepsilon</math> is assumed to follow a [normal distribution](/source/normal_distribution) conditional on the explanatory variables ''x''. This generates the standard [probit model](/source/probit_model).<ref>Bliss, C. I. (1934). "The Method of Probits". Science 79 (2037): 38–39.</ref>

=== Probabilistic model ===
The simplest direct probabilistic model is the [logit model](/source/logit_model), which models the [log-odds](/source/log-odds) as a linear function of the explanatory variable or variables. The logit model is "simplest" in the sense of [generalized linear model](/source/generalized_linear_model)s (GLIM): the log-odds are the natural parameter for the [exponential family](/source/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](/source/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.

==See also ==
*{{sectionlink|Generalized linear model#Binary data}}
*[Fractional model](/source/Fractional_model)

==References==
{{reflist}}
{{refbegin}}
* {{cite book
|title=Regression Models for Categorical Dependent Variables Using Stata, Second Edition
|chapter=4. Models for binary outcomes: 4.1 The statistical model
|chapter-url=https://books.google.com/books?id=kbrIEvo_zawC&pg=PA131
|pages=131–136
|publisher=Stata Press
|first1=J. Scott |last1=Long
|first2=Jeremy |last2=Freese
|year=2006
|isbn=978-1-59718011-5
}}

* {{cite book
|last=Agresti |first=Alan
|chapter=3.2 Generalized Linear Models for Binary Data
|year=2007
|title=Categorical Data Analysis
|url=https://archive.org/details/introductiontoca00agre |url-access=limited |edition=2nd
|pages=[https://archive.org/details/introductiontoca00agre/page/n88 68]–73
}}
{{refend}}

{{statistics-stub}}

Category:Regression analysis

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Adapted from the Wikipedia article [Binary regression](https://en.wikipedia.org/wiki/Binary_regression) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Binary_regression?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.