Owen Yang
Odds ratio is something that is used commonly in epidemiology to describe an association between two factors.
The odds is most commonly (is it? I have no idea actually) used in betting scenarios, i.e. gambling. If the odds of England football winning is 9 to 1, it means for every 9 winning games we expect to see 1 non-winning game (could be draw or lose). The odds here is 9 to 1, which is basically 9 (9 divided by 1). However, the winning rate is actually 90% because for every 10 games we expect to see 9 winning games.
Odds of A at two conditions
The way we using odds ratios to describe the association between A and B is to describe that the odds of A is changed when the status of B changed. So if the winning odds is 9 to 1 during non-summer seasons (odds=9) but 9 to 2 during summer seasons (odds=4.5), here B being summer vs non-summer, then the odds has halves from non-summer to summer seasons. The odds ratio is 4.5/9=0.5. In other words, being in summer seasons halved the odds of winning.
The word ‘odds ratio’ is therefore confusing. Because odds ratio of A vs B is not the odds of A vs. the odds of B. It is the odds of A when B is positive vs the odds of A when B is negative. Is it the ratio of two conditional odds of A.
The odds ratio can be a little ‘misleading.’ Because in this case, the winning rate is 90% during non-summer seasons (9/10) and 81% during summer seasons (9/11), so the ‘rate ratio’ of the winning rate is 0.81/0.90=0.9. It is not as drastic as the impression ‘0.5’ or ‘halves the odds.’
Why do we use odds ratios?
There are two features that are sometimes useful when we use odds. The first is that odds ratio can be ‘legit’ transformed, and so that logistic models can be used, and logistic models are a easy, simple and elegant model.
The second is that odds ratio tend to be identical when you look at it both ways. So if you swap A and B, the odds ratio will be exactly the same. This gives us an easy interpretation of findings. So if you look across all the games, the odds of games that had taken place in the summer among winning games vs among non-winning games should also be 0.5, but the rate ratio would not be the same (0.3/0.5=0.6?).
Although the odds ratio itself is difficult to interpret, in some topics where the events are rare, such as some cancer, the odds ratio is nearly identical to rate ratio. This helps us to be able to use the elegant logistic regression, but still interpret the result of odds ratio as if we are looking at the rate ratio. Say if the cancer rate is not 1/10, but 1/10000. In some scenarios it will be even lower! usually the incidence is calculate with the unit of 1/100,000 person-year, but the life-time risk would be higher. Say the rate in men is 2/10000 and in women is 1/10000, then the rate ratio for male vs female (or sex vs. cancer) is 2. The odds ratio would be 2/9998 vs 1/9999, i.e. 2.0002. Very little difference.
