Quantify association in a 2×2 table — the odds ratio and the relative risk, with confidence intervals, and which to use for cohort vs case-control studies
Learn to compute the odds ratio and the relative risk in R for a 2×2 contingency table — the standard effect sizes in epidemiology and clinical research. Compute both with base R (and their confidence intervals from the log scale), understand when to use each (cohort vs case-control study designs), and why the odds ratio approximates the relative risk for rare outcomes.
A chi-square or Fisher test tells you whether two categorical variables are associated; for a 2×2 table you also want how strongly — and in which direction. The two standard measures are the relative risk and the odds ratio, the workhorses of epidemiology and clinical research.
For a 2×2 table of exposure (yes/no) × outcome (event/no event), label the cells:
| Event | No event | |
|---|---|---|
| Exposed | a | b |
| Unexposed | c | d |
This lesson computes both effect sizes with base R, with confidence intervals, using the famous Physicians’ Health Study (aspirin and heart attack).
In a randomized trial, 104 of 11037 on aspirin had a heart attack (MI), vs 189 of 11034 on placebo:
# a = aspirin & MI, b = aspirin & no MI, c = placebo & MI, d = placebo & no MI
a <- 104; b <- 10933
c <- 189; d <- 10845
tab <- matrix(c(a, b, c, d), nrow = 2, byrow = TRUE,
dimnames = list(group = c("Aspirin", "Placebo"), outcome = c("MI", "no MI")))
tab outcome
group MI no MI
Aspirin 104 10933
Placebo 189 10845The relative risk is the ratio of the probability of the event in the exposed group to that in the unexposed group:
a <- 104; b <- 10933; c <- 189; d <- 10845
risk_exposed <- a / (a + b)
risk_unexposed <- c / (c + d)
RR <- risk_exposed / risk_unexposed
# 95% CI on the log scale
se_logRR <- sqrt(1/a - 1/(a + b) + 1/c - 1/(c + d))
RR_ci <- exp(log(RR) + c(-1, 1) * 1.96 * se_logRR)
c(RR = RR, lower = RR_ci[1], upper = RR_ci[2]) RR lower upper
0.5501150 0.4336712 0.6978247 The relative risk is 0.55 (95% CI 0.43–0.70) — aspirin takers had about half the risk of a heart attack. The CI is entirely below 1, so the protective effect is significant.
The odds ratio is the ratio of the odds of the event — for a 2×2 table this is the cross-product (a × d) / (b × c):
a <- 104; b <- 10933; c <- 189; d <- 10845
OR <- (a * d) / (b * c)
# 95% CI on the log scale
se_logOR <- sqrt(1/a + 1/b + 1/c + 1/d)
OR_ci <- exp(log(OR) + c(-1, 1) * 1.96 * se_logOR)
c(OR = OR, lower = OR_ci[1], upper = OR_ci[2]) OR lower upper
0.5458355 0.4290391 0.6944271 The odds ratio is 0.55 (95% CI 0.43–0.69) — almost identical to the relative risk, because a heart attack is rare here (≈ 1–2%). When the outcome is rare, the OR approximates the RR.
fisher.test() also gives an odds ratio. For a 2×2 table, fisher.test(tab)$estimate returns the conditional maximum-likelihood odds ratio (slightly different from the sample OR above) with an exact CI — a convenient alternative that also gives the significance test.
| Study design | Can you estimate risk? | Report |
|---|---|---|
| Cohort / RCT (follow exposed vs unexposed forward) | yes | relative risk (and OR if you wish) |
| Case-control (start from cases vs controls, look back) | no | odds ratio (RR is not estimable) |
| Logistic regression | — | odds ratio (exp(coef)) |
The relative risk is more intuitive (“half the risk”), but it can only be computed when you can estimate the actual probability of the outcome — i.e. in a cohort or experimental study. In a case-control study you fix the number of cases and controls, so risk is not estimable and the odds ratio is the only valid measure. For a rare outcome they nearly coincide.
A forest-style plot makes the two estimates and their intervals easy to compare against the no-effect line at 1:
library(ggpubr)
a <- 104; b <- 10933; c <- 189; d <- 10845
RR <- (a / (a + b)) / (c / (c + d)); se_logRR <- sqrt(1/a - 1/(a + b) + 1/c - 1/(c + d))
OR <- (a * d) / (b * c); se_logOR <- sqrt(1/a + 1/b + 1/c + 1/d)
est <- data.frame(
measure = c("Relative risk", "Odds ratio"),
value = c(RR, OR),
lower = c(exp(log(RR) - 1.96 * se_logRR), exp(log(OR) - 1.96 * se_logOR)),
upper = c(exp(log(RR) + 1.96 * se_logRR), exp(log(OR) + 1.96 * se_logOR))
)
ggdotchart(est, x = "measure", y = "value", color = "measure", palette = "jco",
sorting = "none", rotate = TRUE, dot.size = 4, ylab = "Estimate (vs no effect = 1)", xlab = NULL) +
geom_errorbar(aes(ymin = lower, ymax = upper, color = measure), width = 0.15) +
geom_hline(yintercept = 1, linetype = "dashed")
Aspirin was associated with a reduced risk of myocardial infarction: relative risk 0.55 (95% CI 0.43–0.70) and odds ratio 0.55 (95% CI 0.43–0.69). The near-equality reflects the rarity of the outcome. Both intervals exclude 1, indicating a statistically significant protective effect.
For the 2×2 table above, risk in each row is \(a/(a+b)\) and \(c/(c+d)\), so \(\text{RR} = \dfrac{a/(a+b)}{c/(c+d)}\). Odds are \(a/b\) and \(c/d\), so \(\text{OR} = \dfrac{a/b}{c/d} = \dfrac{ad}{bc}\). Confidence intervals are built on the log scale (where the estimates are roughly normal) with standard errors \(\text{SE}(\log \text{OR}) = \sqrt{1/a + 1/b + 1/c + 1/d}\) and \(\text{SE}(\log \text{RR}) = \sqrt{1/a - 1/(a+b) + 1/c - 1/(c+d)}\), then exponentiated back. When the outcome is rare, \(a \ll b\) and \(c \ll d\), so \(a/(a+b) \approx a/b\) and the OR converges to the RR.
Compute the odds ratio and relative risk for your own 2×2 table. Edit the four counts and re-run; the sandbox boots on first Run.
Ask Prova “I have a 2×2 table — compute the odds ratio and relative risk with confidence intervals, and tell me which to report for my study design” — it answers with code you can run on your own counts. The runtime is the judge. Ask Prova →
You reported a relative risk from a case-control study. Risk isn’t estimable in a case-control design (the case/control ratio is fixed by the sampling), so the odds ratio is the only valid measure there.
You mislabelled the cells. The cross-product (a×d)/(b×c) and the risks depend on which cell is which. Keep the layout exposure in rows, outcome in columns, with the event as the first column, so OR and RR both describe the event in the exposed group.
You computed the CI on the raw scale. The OR and RR are skewed, so the confidence interval is built on the log scale and exponentiated back — computing estimate ± 1.96·SE directly is wrong.
Relative risk is the ratio of the probabilities of the outcome (exposed vs unexposed); odds ratio is the ratio of the odds. RR is more intuitive but only estimable in cohort/experimental studies; OR is valid in any design (including case-control) and is what logistic regression returns. Both equal 1 under no association.
Use the relative risk for cohort or randomized studies, where you can estimate the actual risk of the outcome. Use the odds ratio for case-control studies (risk isn’t estimable) and as the output of logistic regression. For a rare outcome the two are nearly equal.
When the outcome is rare, the number of events (a, c) is small relative to the non-events (b, d), so the probability \(a/(a+b)\) is close to the odds \(a/b\). The odds ratio (ratio of odds) then converges to the relative risk (ratio of probabilities) — which is why ORs from rare-disease studies are interpreted like risk ratios.
Fill the blank with OR. Compare the two values; here the “outcome” (event) is common, not rare. For question 2, recall what is fixed by the case-control sampling.
OR <- (9 * 7) / (1 * 3) #> 21
RR <- (9/10) / (3/10) #> 3
c(OR = OR, RR = RR)They are far apart (OR 21 vs RR 3) because the event is common here (90% vs 30%) — the OR approximates the RR only when the outcome is rare. For question 2: a case-control study fixes the number of cases and controls by design, so you cannot estimate the probability (risk) of the outcome — only the odds ratio, which is invariant to that sampling, is valid.
exp(coef)) — see logistic regression.You can now compute the odds ratio and relative risk in R for a 2×2 table — the cross-product (a×d)/(b×c) for the OR, the ratio of risks for the RR, each with a log-scale confidence interval. Choose the relative risk for cohort/experimental designs and the odds ratio for case-control studies and logistic regression, and remember they nearly coincide when the outcome is rare. They are the effect sizes that turn a significant chi-square into a meaningful clinical statement.
@online{2026,
author = {},
title = {Odds {Ratio} \& {Relative} {Risk} in {R:} {Effect} {Sizes}
for 2×2 {Tables}},
date = {2026-06-23},
url = {https://www.datanovia.com/learn/biostatistics/categorical/odds-ratio-and-relative-risk-in-r},
langid = {en}
}