Logistic regression

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Logistic regression is a statistical regression model for binary dependent variables. It can be considered as a generalized linear model that utilizes the logit as its link function, and has binomially distributed errors.

The model takes the form

[\operatorname(p)=\ln\left(\frac\right) = \alpha + \beta_1 x_ + \cdots + \beta_k x_,]

[i = 1, \dots, n,\,]

where

[p = \Pr(Y_i = 1).\,]

The logarithm of the odds (probability divided by one minus the probability) of the outcome is modelled as a linear function of the explanatory variables, [X_1] to [X_k]. This can be written equivalently as

[p = \Pr(Y_i = 1|X) = \frac + \cdots + \beta_k x_}} + \cdots + \beta_k x_}}.]

The interpretation of the [\beta] parameter estimates is as a multiplicative effect on the odds ratio. In the case of a dichotomous explanatory variable, for instance sex, [e^\beta] (the antilog of [\beta]) is the estimate of the odds-ratio of having the outcome for, say, males compared with females.

The parameters [\alpha, \beta_1, ..., \beta_k] are usually estimated by maximum likelihood.

Extensions of the model exist to cope with multi-category dependent variables and ordinal dependent variables.

References

Agresti, Alan: Categorical Data Analysis. New York: Wiley, 1990.
Amemiya, T., 1985, Advanced Econometrics, Harvard University Press.
Hosmer, D. W. and S. Lemeshow: Applied logistic regression. New York; Chichester, Wiley, 2000.

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Logistic regression

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References