Relative Risk vs. Odds Ratio

Same data, two different numbers

Risk is simply the probability of an outcome happening in a group, the number of people who had the outcome divided by the total number of people in that group. Relative risk compares that probability between two groups, the risk in one group divided by the risk in the other.

Odds measure something related but distinct, the probability of an event happening divided by the probability of it not happening. An odds ratio compares odds between two groups the same way relative risk compares risks.

Consider a trial where 30 of 100 treated patients have an outcome, compared with 20 of 100 control patients. The risk in the treated group is 30 percent, the risk in the control group is 20 percent, so the relative risk is 30 divided by 20, or 1.5. The odds in the treated group are 30 to 70, about 0.43, and the odds in the control group are 20 to 80, or 0.25, making the odds ratio about 1.71. Same data, same trial, two different numbers describing the same comparison.

The rare disease assumption

Notice that 1.71 is further from 1 than 1.5 is. That gap is not a coincidence. As an outcome becomes less common, the denominator in an odds calculation, the probability of the event not happening, gets closer and closer to 1, which means odds get closer and closer to risk itself. Once odds and risk nearly match in both groups, the odds ratio and the relative risk nearly match too.

Run the same style of comparison on a rare outcome, say 3 events per 1,000 treated patients versus 2 per 1,000 control patients. The relative risk is 1.5, and the odds ratio comes out to about 1.50 as well, barely distinguishable. This is called the rare disease assumption: when an outcome is uncommon, generally under about 10 percent, the odds ratio is a reasonable stand in for the relative risk. When the outcome is common, as in the first example, the odds ratio consistently overstates the size of the effect compared with the relative risk.

Why case-control studies can only report an odds ratio

This matters beyond arithmetic because of how certain studies are built. In a case-control study, researchers do not sample a population and watch what happens to it. They start by deliberately selecting a group who already has the outcome, the cases, and a comparison group who does not, the controls, then look backward at prior exposures. Because the number of cases and controls was chosen by the researchers rather than occurring naturally in a sampled population, there is no way to calculate a true incidence, the actual risk, in either group.

Without a true risk, relative risk cannot be calculated directly. An odds ratio can be, because odds can be computed from the exposure pattern within cases and controls regardless of how they were sampled. This turns out to be a convenient match rather than a limitation in practice, since case-control studies are typically used to study rare outcomes in the first place, exactly the setting where the rare disease assumption makes the odds ratio a trustworthy approximation of the relative risk anyway.

When doubling your risk means less than it sounds like

A relative risk of 2 sounds the same no matter where it comes from, but what it means for an actual patient depends entirely on the baseline. If a rare side effect affects 1 in 100,000 people and a medication doubles that to 2 in 100,000, the relative risk is 2, and the absolute increase is a single additional case per 100,000 people, a change most patients would reasonably shrug off.

If a common complication affects 10 percent of patients and something doubles that to 20 percent, the relative risk is also exactly 2, but the absolute increase is 10 percentage points, a difference large enough to change how a clinician and patient weigh a decision together. A headline reporting only the relative risk cannot distinguish between these two very different realities. The baseline, the absolute risk before the change, is what turns a relative risk into something meaningful.

The same underlying result can be reported as a relative risk, an odds ratio, or an absolute risk difference, and which one gets chosen can make an identical effect sound alarming or unremarkable.

Key Takeaways

• Relative risk and odds ratio are genuinely different quantities, even when calculated from the exact same data.

• The two converge under the rare disease assumption, when an outcome is uncommon, but the odds ratio overstates the effect compared with relative risk when the outcome is common.

• Case-control studies cannot compute a true incidence, so they can only report an odds ratio, not a relative risk.

• A relative risk on its own says nothing about the baseline absolute risk it was calculated from.

• Any reported effect measure should be read alongside the baseline rate before deciding how much it actually matters.

Up Next:

Next, we will turn an absolute risk difference into a single practical number with number needed to treat and number needed to harm, the figures that translate a study result into how many patients actually need to be treated for one of them to benefit, or to be harmed.