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

$\backslash operatorname\{logit\}(p)=\backslash ln\backslash left(\backslash frac\{p\}\{1-p\}\backslash right)\; =\; \backslash alpha\; +\; \backslash beta\_1\; x\_\{1,i\}\; +\; \backslash cdots\; +\; \backslash beta\_k\; x\_\{k,i\},$

$i\; =\; 1,\; \backslash dots,\; n,\backslash ,$

where

$p\; =\; \backslash Pr(Y\_i\; =\; 1).\backslash ,$

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

$p\; =\; \backslash Pr(Y\_i\; =\; 1|X)\; =\; \backslash frac\{e^\{\backslash alpha\; +\; \backslash beta\_1\; x\_\{1,i\}\; +\; \backslash cdots\; +\; \backslash beta\_k\; x\_\{k,i\}\}\}\{1+e^\{\backslash alpha\; +\; \backslash beta\_1\; x\_\{1,i\}\; +\; \backslash cdots\; +\; \backslash beta\_k\; x\_\{k,i\}\}\}.$

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

The parameters $\backslash alpha,\; \backslash beta\_1,\; ...,\; \backslash beta\_k$ 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.

statistics

Logistische Regression | Logit模型