Logistic regression: Difference between revisions

Revision as of 19:41, 14 March 2006

Revision as of 03:03, 26 March 2006

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The model takes the following form:

:<math>\log(\frac{p}{1-p}) = \alpha + \beta_1 X_1 + \ldotscdots + \beta_k X_k</math>

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 equvalentlyequivalently as:

:<math>p = P(Y=1|X) = \frac{e^{\alpha + \beta_1 X_1 + \ldotscdots + \beta_k X_k}}{1+e^{\alpha + \beta_1 X_1 + \ldotscdots + \beta_k X_k}}.</math>

The interpretation of the <math>\beta</math> parameter estimates is as an additive effect on the log of the odds. 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.