Binary regression: Difference between revisions

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In [[statistics]], specifically [[regression analysis]], a '''binary regression''' estimates a relationship between one or more [[explanatory variable]]s 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 ({{tmath|<math>n = 1}}</math>), and one of the two alternatives considered as "success" and coded as 1: the value is the [[Count data|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==

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Binary regression models can be interpreted as [[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]], 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

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 model]]s (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.

: <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 [[statistical parameter|parameters]] and ''G'' is a [[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]] 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 [[errors and residuals|error term]] <math>\varepsilon</math> is assumed to follow a [[normal distribution]] conditional on the explanatory variables ''x''. This generates the standard [[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]]], 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 model]]s (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.

==See also ==

*{{sectionlink|Generalized linear model#Binary data}}

*[[Fractional model]]

==References==

{{reflist}}

{{refbegin}}

* {{cite chapterbook

|title=Regression Models for Categorical Dependent Variables Using Stata, Second Edition

|chapter=4. Models for binary outcomes: 4.1 The statistical model

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|year=2006

|isbn=978-1-59718011-5

|ref=harv

}}

* {{cite chapterbook

|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

|edition=2nd

|pages=[https://archive.org/details/introductiontoca00agre/page/n88 68]–73

|pages=68–73

|ref=harv

}}

{{refend}}

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[[Category:GeneralizedRegression linear modelsanalysis]]