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Odds ratios and interpretations
What is an odds ratio, and how do we estimate its uncertainty?
Research papers may include an odds ratio. This ratio measures the association between an intervention or exposure, and an outcome.
What are the odds?
Odds are different to probabilities. In classical statistics, probabilities are the long-run frequency of an event. In Bayesian statistics, probabilities represent a state of knowledge or reflect personal belief.
Through probabilities come odds. Divide the probability an event will occur by the probability it will not. In a full pack of 52 playing cards, the odds of drawing a Diamond card is one-to-three (or 1/3). There are three times as many cards in other suits than Diamond cards. An event with 90% probability would have odds of nine-to-one (or nine).
We also move from odds to probabilities. That probability is the odds number, divided by the odds number with one added on. If the odds number were four, then the probability is four divided by five: 80%.
What is an odds ratio?
Whilst odds are themselves a ratio, odds ratio divide one set of odds by another. Relative odds compares odds in one group — like those receiving a new treatment — to another.
Suppose we had 30 patients each in two groups. Our concern was the development of infectious complications. Half of patients got an experimental prophylaxis, whilst the others did not.
- Experimental group: the adverse event occurs in one patient (labelled ‘a’), and does not happen in 29 others (‘b’).
- Control group: the adverse event happens in nine patients (‘c’), and not in the other 21 (‘d’).
The odds are in the respective groups are then 1/29 (a/b) and 9/21 (c/d). The odds ratio is (1/29)/(9/21), equal to 0.08. Odds of the adverse event are lower in the experimental group than control patients.
