Logistic regression

From Free net encyclopedia

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

<math>\operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right) = \alpha + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i}</math>

i, = 1, ..., n, where

<math>p = \Pr(Y_i = 1).\,</math>

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, X₁ to X_k. This can be written equivalently as

<math>p = \Pr(Y_i = 1|X) = \frac{\exp(\alpha + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i})}{1+\exp(\alpha + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i})}.</math>

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

The parameters α β₁, ..., β_k are usually estimated by maximum likelihood.

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

See also

• Probit model

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.de:Logistische Regression

zh:Logit模型