Why is the odds ratio 2.25 when the relative risk is 1.5 for an event rate rising from 40% to 60%?
answer
- odds are not probabilities
- p over one-minus-p, versus p
- the gap grows as events get common
- rare-outcome approximation fails at 40%
- always further from 1 than the risk ratio
basics
~20 sOdds and probabilities diverge once events are common. Risks of 40% and 60% give a relative risk of 1.5, but the matching odds of 0.667 and 1.5 give an odds ratio of 2.25. An odds ratio always sits further from 1.
solid answer
~40 sOdds are `p / (1 - p)`, not `p`. At 40% the odds are `0.4 / 0.6 = 0.667`; at 60% they are `0.6 / 0.4 = 1.5`. The relative risk divides the probabilities, `0.6 / 0.4 = 1.5`; the odds ratio divides the odds, `1.5 / 0.667 = 2.25`. The two agree only when the outcome is rare, because then `1 - p` is close to 1 and odds are close to the probability itself. At a 40% baseline that approximation is badly wrong, and the odds ratio is pulled further from 1 than the relative risk in the same direction. The practical hazard is that audiences hear "2.25" as "risk more than doubled", when risk rose by half. For decisions I quote the absolute difference too: 20 percentage points.
go deeper
Be ready to compute odds from a probability and a ratio from a 2x2 table, and to state that odds equal p divided by one minus p rather than p itself.
Expect to show algebraically why the odds ratio sits further from 1 than the relative risk, and to say at what baseline rate the rare-outcome approximation stops being usable.
Demonstrate that you choose the measure by design and audience: an absolute risk difference for the decision, a ratio for describing relative change, an odds ratio when the sampling design forces it.
Own how ratios are communicated outside the analytics team — the standard that a baseline rate and an absolute difference accompany every ratio, so leadership never prices a decision off an exaggerated multiplier.
## Odds are not probabilities A probability `p` is the share of cases in which an event happens. The corresponding **odds** are the ratio of happening to not happening: `odds = p / (1 - p)` Probabilities live in [0, 1]; odds live in [0, infinity). For rare events the two nearly coincide, because `1 - p` is close to 1: a probability of 0.01 gives odds of 0.0101. For common events they separate sharply: a probability of 0.5 gives odds of 1, and a probability of 0.9 gives odds of 9. ## Working the example Take a 2x2 table where the event rate rises from 40% in one group to 60% in the other. - Group A: `p = 0.40`, odds `= 0.40 / 0.60 = 0.667` - Group B: `p = 0.60`, odds `= 0.60 / 0.40 = 1.50` Three effect sizes describe this one table: - **Risk difference** (absolute): `0.60 - 0.40 = 0.20`, i.e. 20 percentage points. - **Relative risk (risk ratio)**: `0.60 / 0.40 = 1.5`. The event is 1.5 times as likely — 50% more likely. - **Odds ratio**: `1.50 / 0.667 = 2.25`. The odds are 2.25 times as large. All three are correct summaries of the same data. They differ because they answer different questions, and the ratio measures answer questions about different quantities — one about probabilities, the other about odds. ## Why the odds ratio is always further from 1 Write the odds ratio in terms of the two risks: `OR = [p1 / (1 - p1)] / [p0 / (1 - p0)] = RR * (1 - p0) / (1 - p1)` When the event is more common in group 1, `1 - p1 < 1 - p0`, so the multiplier `(1 - p0) / (1 - p1)` exceeds 1 and the odds ratio exceeds the relative risk. Here that multiplier is `0.60 / 0.40 = 1.5`, and indeed `1.5 * 1.5 = 2.25`. As the baseline rate approaches zero, the multiplier approaches 1 and the two measures converge — which is exactly the **rare-outcome approximation**. A common working rule is that the approximation is tolerable when the event rate stays under roughly 10% in both groups; at a 40% baseline it fails badly, as this example shows. Note also the direction: when the event is *less* common in the exposed group, the odds ratio sits further *below* 1 than the relative risk. "Further from 1" holds in both directions; the odds ratio exaggerates in whichever direction the effect runs. ## Why the odds ratio is used at all Given that it is harder to interpret, why report it? Two reasons. **Study design.** In a case-control design, cases and controls are sampled separately, so the observed event rates are an artefact of the sampling and the relative risk cannot be estimated from the table. The odds ratio is invariant to that sampling and can be. **Symmetry.** The odds ratio is symmetric under swapping the roles of outcome and exposure, and under redefining the event as its complement — flipping "success" to "failure" inverts the odds ratio exactly, whereas the relative risk for the complementary event bears no such simple relation in general. That symmetry makes it mathematically convenient. ## Reporting it responsibly The recurring communication failure is that an odds ratio is read as a risk ratio. "2.25 times the odds" becomes "more than twice the risk" in a summary, and the overstatement is large exactly when the outcome is common — which is when readers care most. Three habits guard against it: 1. **Always give the baseline rate.** A ratio without the level it multiplies is uninterpretable. 2.25 on a 40% baseline and 2.25 on a 0.4% baseline describe wildly different situations. 2. **Report the absolute difference alongside the ratio.** Here, 20 percentage points. This is the number the decision consumes. 3. **Translate for the audience.** The reciprocal of the risk difference, `1 / 0.20 = 5`, is the number of people who must be exposed for one additional event to occur. That framing — number needed to treat when the event is good, number needed to harm when it is bad — is far harder to misread than any ratio.
- How would you turn this same table into a decision-facing number?Use the absolute risk difference: 60% minus 40% is 20 percentage points. Its reciprocal, 1 / 0.20 = 5, is the number of people who must be exposed for one extra event — number needed to treat if the event is beneficial, number needed to harm if not. That single integer is much harder to misread than a ratio.
- When is an odds ratio a reasonable stand-in for a relative risk?When the outcome is rare in both groups, conventionally under about 10%. Then one minus the probability is close to one in both arms, so odds are nearly equal to probabilities and the two ratios nearly coincide. The further the event rates rise above that, the more the odds ratio exaggerates the risk ratio.
- Why is the odds ratio preferred in a case-control study?Because cases and controls are sampled separately, the event rates in the table reflect the sampling design rather than the population, so a relative risk computed from them is meaningless. The odds ratio is invariant to that separate sampling, so it estimates the same quantity it would in the population.
saying these in an interview costs you the question
- Reads an odds ratio of 2.25 as risk more than doubled
- Says odds ratio and relative risk are interchangeable
- Computes odds as events divided by the total
- Assumes the rare-outcome approximation holds at any baseline
- Quotes a ratio without giving the baseline rate