Logistic regression

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

${\displaystyle \operatorname {logit} (p)=\ln \left({\frac {p}{1-p}}\right)=\alpha +\beta _{1}x_{1,i}+\cdots +\beta _{k}x_{k,i}}$,

${\displaystyle i=1,\dots ,n}$,

where

${\displaystyle p=\Pr(Y_{i}=1)}$.

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, ${\displaystyle X_{1}}$ to ${\displaystyle X_{k}}$. This can be written equivalently as

${\displaystyle p=\Pr(Y_{i}=1|X)={\frac {e^{\alpha +\beta _{1}x_{1,i}+\cdots +\beta _{k}x_{k,i}}}{1+e^{\alpha +\beta _{1}x_{1,i}+\cdots +\beta _{k}x_{k,i}}}}.}$

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

The parameters ${\displaystyle \alpha ,\beta _{1},...,\beta _{k}}$ are usually estimated by maximum likelihood.

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

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.