The Relative Risk Ratio and Odds Ratio are both used to measure the medical effect of a treatment or variable to which people are exposed. The effect could be beneficial (from a therapy) or harmful (from a hazard).  Risk is the number of those having the outcome of interest (death, infection, illness, etc.) divided by the total number exposed to the treatment. Odds is the number having the outcome divided by the number not having the outcome.  The risk or odds ratio is the risk or odds in the exposed group divided by the risk or odds in the control group.  A risk or odds ratio = 1 indicates no difference between the groups.  A risk or odds ratio > 1 indicates a heightened probability of the outcome in the treatment group.

The two metrics track each other, but are not equal.  An example with a control group and a therapy treatment group:

Treatment group:   5 deaths, 95 survive:   Risk = 5/100 = 0.05, Odds = 5/95 = 0.053

Control group:  8 deaths, 92 survive:  Risk = 8/100 = 0.08, Odds = 8/92 = 0.087

Risk ratio = 0.05/0.08 = 0.625    

Odds ratio =  0.053/0.087 = 0.609

So we can say that the treatment reduces the risk of the outcome to 62.5% of what it would otherwise have been.  The odds of the outcome would have been reduced to 60.9%.  So why use odds?  While the general public’s literacy rate with probability (risk) is not super high, its familiarity with odds is even more tenuous.  Except for gamblers.

For the answer, consider a retrospective, or case-control study.  Let’s use a concrete example: Are blood thinners useful in reducing the risk of stroke with Covid patients?  From a research perspective, we’d like to set up a study, and treat some patients with blood thinners and some without. Then wait and see how many in each group had strokes. But in the fast-moving Covid environment, a leisurely, well-designed study like that is not feasible.  So we take a reverse approach:  look at the patients who did have strokes, and see if their treatment protocols were less likely to include blood thinners.  

But… less likely than what?  For that calculation, we find a group of patients who are as similar to the stroke group as possible (“case-controls”), but who did not get blood thinners; in this “control” group we count the number of strokes.  So now we have two groups; one with all-strokes and the other, the case-control group, with strokes and no strokes.  We still can calculate odds for each group, and we calculate it as “odds of having received blood thinners.”  It is a useful statistical and medical calculation for purposes of evaluating treatments, but it does not relate directly to the “risk of having a stroke.”  The latter calculation would be impossible since we lack any information on the denominator; the size of the group at risk for a stroke.

One other area where we deal with odds, rather than probabilities, where the latter would seem more natural, is logistic regression.  In logistic regression, it turns out that we can fit a standard linear model to the log of the odds of an outcome; we can’t do that for the probability of an outcome. However, after the model is fit, it is possible to convert from odds to probabilities.