The odds ratio is the ratio of two event odds. For rare events it is close to the corresponding risk ratio, because when is small.
The log odds ratio is the natural logarithm of an odds ratio. For two independent binomial groups with positive event and non-event counts and , its estimate is , with first-order variance by the delta method. Zero cells need a method that handles boundary data; the displayed normal approximation is not valid there.
For two independent binomial distributions with event and nonevent counts and , the odds ratio is . The delta method applied to , whose derivative is , gives a group variance . Substitution of the fitted proportion gives in the first group and in the second. Their independence gives the displayed sum. A normal approximation then gives a confidence interval on the log odds ratio scale; exponentiation produces an odds-ratio interval. Zero cells require another approximation or model.
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The odds ratio (OR) is a statistic that quantifies the strength of the association between two events, commonly used in epidemiology and various fields of research. It compares the odds of an event occurring in one group to the odds of it occurring in another group. Here's how it works: 1. **Definition of Odds**: The odds of an event is the ratio of the probability that the event occurs to the probability that it does not occur.