Logistic regression

Logistic regression is a statistical regression model for binomially distributed dependent variables. It is a generalized linear model that utilizes the logit as its link function.

The model takes the form

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

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

where

${\displaystyle p_{i}=\Pr(Y_{i}=1).\,\!}$

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_i. This can be written equivalently as

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

The interpretation of the β parameter estimates is as a multiplicative effect on the odds ratio. In the case of a dichotomous explanatory variable, for instance gender, exp(β) 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

• Data mining
• Linear discriminant analysis
• Perceptron
• Probit model

References

• Alan Agresti, Categorical Data Analysis. New York: Wiley, 1990.
• Amemiya, T., Advanced Econometrics, Harvard University Press, 1985.
• David W. Hosmer and Stanley Lemeshow, Applied logistic regression., 2nd ed., New York; Chichester, Wiley, 2000, ISBN 0-471-35632-8