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 binomially distributed errors.

The model takes the following form:

${\displaystyle \log({\frac {p}{1-p}})=\alpha +\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}$

The log 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 equvalently as:

${\displaystyle p=P(Y=1|X)={\frac {e^{\alpha +\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}{1+e^{\alpha +\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}}}$

The interpretation of the ${\displaystyle \beta }$ parameter estimates is as an additive effect on the log of the odds. 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.

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.