How should you interpret an odds ratio differently from a relative risk?

Why

Risk is events divided by everyone at risk, while odds are events divided by non-events. Davies, Crombie and Tavakoli explain that when risks in both groups are small, say below 20 percent, the odds approximate the risks and the odds ratio approximates the relative risk. As the outcome becomes common they diverge, and the OR always lies further from 1.

Suppose 40 percent of controls and 60 percent of treated patients respond. The relative risk is 1.5. The odds are 0.67 and 1.5, so the odds ratio is 2.25.

Reporting that treatment more than doubles response would be wrong, even though 2.25 is the correct odds ratio. The authors offer a rule of thumb: if the initial risk times the odds ratio is below 100 percent, the OR overstates the relative risk by less than twofold. Odds ratios exist for good reasons.

Logistic regression estimates them naturally, and in a case-control study they are the measure you can estimate, because the sampling fixes the number of cases. In cohort studies and trials with common outcomes, a risk ratio from a log-binomial or Poisson model with robust errors, or a risk difference, is easier to read correctly.