What is the mathematical formula for odds?

Odds = Probability / (1-probability). Odds ratio (OR) = ratio of odds of event occurring in exposed vs. unexposed group.

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Odds: the ratio of the probability that an event will occur versus the probability that the event will not occur, or probability / (1-probability). For example, if you are normally on call 2 out of 7 days in a week, then the odds of you being on call on a certain day of the week is [(2/7)/(5/7)] = 0.40. Note that this differs from risk (or probability): the risk of being on call is equal to (# of call days )/ (total # of days in a week) = 2/7 = 0.285. Odds ratio: a ratio of odds; in general they refer to the ratio of the odds of an event occurring in the exposed group versus the unexposed group. For example, lets say you want to compare the differences between PONV in women undergoing total abdominal hysterectomy receiving Drug X and those who do not, controlling for all other variables. You compare 100 different cases. The odds of PONV having received Drug X is 20/80 or 0.25. The odds of PONV without Drug X is 40/60 or 0.67. Therefore, the odds ratio for PONV with Drug X vs. PONV without Drug X is 0.25/0.67 or 0.37. The probability of PONV having received Drug X is 20/100 or 0.20. The probability of PONV with no Drug X is 40/100 or 0.40. Therefore, the relative risk for PONV with Drug X vs. PONV without Drug X is 0.20/0.40 = 0.5. Odds ratios are used instead of relative risk for case-control studies. To be able to calculate relative risk, we compare the risks of outcome in different groups. In case-control studies, we already know what the outcome is and we separate groups into those with the outcome vs. controls. Our objective in such studies is to try to identify risk factors that are more strongly associated with one group than the other; thus, risk and therefore relative risk cannot be calculated from these studies. We use odds ratios instead, which can give us a measure of how strongly the risk factor is associated with the outcome. For example, if we suspect that Drug X is associated with less PONV, then we could take 100 patients with out PONV to 100 patients with PONV and see how many in each group received Drug X. Since we select the outcome in both groups, we cannot calculate the relative chance (risk) of less PONV in the Drug X group because we do not know the chance of no PONV in the general population (who are not Drug X users), and therefore we have no comparison group. However, we can compare the odds of the use of Drug X in those who had no PONV vs. those who had PONV by calculating the odds ratio. So for example:

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You can say that the odds of use of Drug X were 2.7 times greater in non PONV patients vs. PONV patients in this study. This implies an association between use of Drug X and preventing PONV. However, many other things could have contributed to this apparent association: chance alone could have accounted for this difference (helpful to know the 95% CI for the OR); the sample selected for both groups could have been skewed to favor Drug X use in the non-PONV group.

Summary

Odds = Probability / (1-probability).

Odds ratio (OR) = ratio of odds of event occurring in exposed vs. unexposed group. Odds ratio are used to estimate how strongly a variable is associated with the outcome of interest; in prospective trials, it is simply a different way of expressing this association than relative risk. In case-control studies, we separate groups by their outcomes and retrospectively try to identify variables that appear to be more associated with one outcome than another. Therefore, we cannot deduce a calculable risk because the outcome has already been predetermined. We therefore use odds ratios instead to estimate the strength of association of the variable with the outcome of interest. In a prospective study, either a Randomized Clinical Trial or a Cohort study, use Relative Risk.

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