Odds Ratio - Role in Logistic Regression

Role in Logistic Regression

Logistic regression is one way to generalize the odds ratio beyond two binary variables. Suppose we have a binary response variable Y and a binary predictor variable X, and in addition we have other predictor variables Z1, ..., Zp that may or may not be binary. If we use multiple logistic regression to regress Y on X, Z1, ..., Zp, then the estimated coefficient for X is related to a conditional odds ratio. Specifically, at the population level

\exp(\beta_x) = \frac{P(Y=1|X=1, Z_1, \ldots, Z_p)/P(Y=0|X=1, Z_1, \ldots, Z_p)}{P(Y=1|X=0, Z_1, \ldots, Z_p)/P(Y=0|X=0, Z_1, \ldots, Z_p)},

so is an estimate of this conditional odds ratio. The interpretation of is as an estimate of the odds ratio between Y and X when the values of Z1, ..., Zp are held fixed.

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