StatReason
What this answers
This page answers "why do the odds ratio and risk ratio give different numbers for the same data, and which one should I read?" using a fixed 2x2 comparison you can adjust by baseline risk and treatment effect.
Three ways to summarize the same table
Risk difference is the treatment group's event risk minus the control group's event risk, expressed in the outcome's own probability units. Risk ratio is the treatment group's risk divided by the control group's risk. Odds ratio is the treatment group's odds of the event (risk divided by 1 minus risk) divided by the control group's odds. All three are computed from the identical event counts; they are different lenses on the same table, not competing estimates.
Why odds ratio and risk ratio diverge
When an event is rare in both groups, odds and risk are numerically close, so odds ratio and risk ratio nearly agree. As baseline risk rises, odds grow faster than risk (odds has no upper bound while risk is capped at 1), which pushes the odds ratio further from the risk ratio, usually making the odds ratio look like a more extreme effect than the risk ratio for the same underlying data. Move the baseline risk slider above from a small value toward a large one to see this gap widen.
Worked example
With a 10 percent baseline risk and a true risk ratio of .70, the treatment group's risk is 7 percent, giving a risk difference of 3 percentage points and a risk ratio of .70. The corresponding odds ratio is close to but not exactly .70, since 10 percent is still a fairly low baseline. Raise the baseline risk toward 50 percent with the same relative effect and the odds ratio noticeably overstates the risk ratio's more direct message.
Which one to report
Risk difference answers an absolute question: how many more or fewer events per group. Risk ratio and risk difference are usually the most directly interpretable for a reader outside the study. Odds ratio is the natural output of logistic regression and case-control designs where risk itself cannot be estimated directly, and is not wrong, but it should not be read as if it were a risk ratio when baseline risk is not small.
What none of these measures decide
None of the three measures establishes that the treatment caused the difference; that depends on the study design, not the arithmetic. None of them tells you whether the difference is clinically or practically meaningful; a small risk difference can matter enormously at a large population scale, and a large relative effect on a rare event can be a small absolute effect.
Source
This comparison follows the measures-of-effect guidance in the Cochrane Handbook and the shared statistical reasoning contract every StatReason engine is built against.
Limitations
This page uses a fixed hypothetical baseline and treatment effect to illustrate the relationship between the three measures; it does not compute a confidence interval or test a hypothesis. Use the linked engines below for a confidence interval or hypothesis test on your own event counts.