Logistic regression: Difference between revisions

Revision as of 11:31, 18 August 2006

Revision as of 15:02, 13 November 2006

Line 3:

The model takes the form

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

:<math>i = 1, \dots, n,\,</math>

Line 9:

where

:<math>pp_i = \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, <math>X_1</math> to <math>X_k</math>. This can be written equivalently as

:<math>pp_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}}}.</math>

The interpretation of the <math>\beta</math> parameter estimates is as a multiplicative effect on 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.