Logistic regression: Difference between revisions

Revision as of 15:41, 15 November 2006

Revision as of 16:21, 16 November 2006

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:<math>\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},</math>

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

where

:<math>p_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, ''X<mathsub>X_1i</math> to <math>X_k</mathsub>''. This can be written equivalently as

:<math>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}}}.</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 sexgender, <math>e^\beta</math> exp(the antilog of <math>\''&beta</math>;'') is the estimate of the [[odds-ratio]] of having the outcome for, say, males compared with females.

The parameters <math>\''&alpha;'', \beta_1''&beta;''₁, ..., \beta_k''&beta;<sub>k</mathsub>'' are usually estimated by [[maximum likelihood]].

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