Relative Risk Calculator

Relative risk is the most intuitive effect measure for prospective designs because it is a ratio of probabilities, not a ratio of odds. In a cohort study or a randomized controlled trial you can directly estimate the probability of the outcome in each group, so the question becomes how the risk in the exposed or treated group compares to the risk in the control group. Cummings (2009) and Greenland (1987) argue that the relative risk should be the default effect measure for these designs because it maps cleanly onto how clinicians and patients reason about treatment effects, while the odds ratio is more useful for case-control studies where direct risk estimation is impossible.

The arithmetic is straightforward. For a 2x2 table with cells a, b, c, d the risk in the exposed group is a divided by (a plus b) and the risk in the unexposed group is c divided by (c plus d); the relative risk is the first divided by the second. Katz et al. (1978) derived the closed-form log-Wald confidence interval that this calculator uses, which the Cochrane Handbook (Higgins et al. 2024) treats as the default for individual studies. For sparse data, alternative methods such as the inverse hyperbolic sine, Koopman score, or Miettinen score intervals can be more accurate, but they require iterative computation and are reserved for situations where Wald intervals misbehave.

Conversion between OR and RR is a common need because much of the published literature reports odds ratios from logistic regression even when the design is a cohort study or trial. Zhang and Yu (1998) gave the standard correction formula: RR equals OR divided by (1 minus p0 plus p0 multiplied by OR), where p0 is the baseline risk in the control group. This calculator implements that conversion in a dedicated tab and transforms the confidence interval bounds through the same function so the corrected RR has a usable interval. The conversion is most useful when p0 exceeds roughly 10 percent, because below that threshold the OR and RR are numerically close.

Three reporting reminders matter. First, always pair the RR with the absolute risks in each group so readers can judge clinical magnitude. A small RR can correspond to a large absolute benefit when baseline risk is high, and a large RR can be clinically trivial when baseline risk is low. Second, log(RR) and its standard error are the values you feed into a forest plot or meta-regression because they are approximately normally distributed across studies. Third, when several 2x2 tables share an exposure-outcome contrast, the Mantel-Haenszel calculator pools them while preserving the conditional structure that fixed-effect meta-analysis requires.

For clinical communication the most useful companions to the RR are the absolute risk reduction and the number needed to treat. The calculator reports both with their confidence intervals. For full systematic reviews and meta-analyses of binary outcomes, our meta-analysis service handles effect size extraction, pooling, heterogeneity, subgroup analysis, and publication-bias assessment to Cochrane and PRISMA 2020 standards, with reproducible code and a journal-ready manuscript delivered by a PhD statistician.