In a manuscript draft examining postoperative wound infection, a surgical research team ran a multivariable logistic regression and obtained an adjusted odds ratio (OR) of 3.50 for patients with poorly controlled diabetes. In the Discussion section, they wrote: “Patients with poorly controlled diabetes were 3.5 times as likely to develop a postoperative infection.”
During peer review, the statistical reviewer flagged the sentence: because the infection occurred in 30% of the cohort, an odds ratio of 3.50 did not mean a 3.5-fold increase in risk—the corresponding relative risk (RR) was likely well below 3.5, possibly closer to 2–2.5 depending on the risk in the unexposed group.
Odds ratios and relative risks answer the same directional question, but they are calculated on different scales. Treating an odds ratio as a direct multiplier of probability when the outcome is common is one of the most frequent misinterpretations in clinical literature.
Probability vs. Odds: The Arithmetic Foundation
Before comparing two groups, we need to distinguish between probability (risk) and odds within a single group.
- Probability (Risk): The number of individuals who experience the event divided by the total number of individuals in that group.
Risk = Events/Total
- Odds: The number of individuals who experience the event divided by the number of individuals who do not experience the event.
Odds = Events/(Non-events) = p/(1 - p)
For example, if 20 out of 100 patients experience a complication:
- The risk (probability) is 20 / 100 = 0.20 (or 20%).
- The odds are 20 / (100 - 20) = 20 / 80 = 0.25 (or 1 to 4).
Notice that because the denominator of odds (n - events) is smaller than the denominator of risk (n), the odds of an event will exceed its probability whenever the event occurs at least once.
Defining RR and OR from a 2 × 2 Table
Consider the standard 2 × 2 epidemiological contingency table:
| Exposure Group | Outcome Present (Event) | Outcome Absent (No Event) | Total |
|---|---|---|---|
| Exposed | a | b | a + b |
| Unexposed | c | d | c + d |
From these four cells:
- Relative Risk (Risk Ratio — RR): The ratio of the risk in the exposed group to the risk in the unexposed group.
RR = (a / (a + b))/(c / (c + d))
- Odds Ratio (OR): The ratio of the odds in the exposed group to the odds in the unexposed group, which simplifies to the cross-product ratio.
OR = (a / b)/(c / d) = (a · d)/(b · c)
Worked Example: Comparing a Common Outcome vs. a Rare Outcome
Let us trace the exact arithmetic across two scenarios to see how baseline incidence changes the gap between RR and OR.
Scenario A: Common Outcome (Incidence > 10%)
Suppose we follow 200 patients in a prospective cohort study (100 exposed to a clinical risk factor, 100 unexposed) to measure postoperative nausea:
| Group | Nausea Present | Nausea Absent | Total |
|---|---|---|---|
| Exposed | a = 40 | b = 60 | 100 |
| Unexposed | c = 20 | d = 80 | 100 |
Step 1: Calculate the Relative Risk (RR)
- Risk in the exposed group: R₁ = 40 / 100 = 0.40 (40%).
- Risk in the unexposed group: R₀ = 20 / 100 = 0.20 (20%).
- Relative Risk:
RR = 0.40/0.20 = 2.00
(Interpretation: Exposed patients had twice the risk of postoperative nausea compared with unexposed patients).
Step 2: Calculate the Odds Ratio (OR)
- Odds in the exposed group: Odds₁ = 40 / 60 = 0.6667 (i.e., 2/3).
- Odds in the unexposed group: Odds₀ = 20 / 80 = 0.2500 (i.e., 1/4).
- Odds Ratio:
OR = (40 × 80)/(60 × 20) = 3200/1200 = 2.67
Step 3: Compute the 95% Confidence Interval for the OR Confidence intervals for ratios are computed on the natural log scale ((2.6667) ≈ 0.9808) and exponentiated back:
SE() = √(1/40 + 1/60 + 1/20 + 1/80) = √(0.1042) ≈ 0.3228
On the log scale, 0.9808 ± (1.96 × 0.3228) = [0.3481, 1.6135]. Exponentiating both bounds (e^0.3481, e^1.6135) gives 95% CI = [1.42, 5.02].
Scenario B: Rare Outcome (Incidence < 5%)
Now suppose the outcome is rare—occurring in 4 out of 100 exposed patients (a = 4, b = 96) and 2 out of 100 unexposed patients (c = 2, d = 98):
- Relative Risk: RR = (4 / 100) ÷ (2 / 100) = 0.04 / 0.02 = 2.00.
- Odds Ratio: OR = (4 × 98) ÷ (96 × 2) = 392 / 192 ≈ 2.04.
Why the Odds Ratio Exaggerates Relative Risk When Outcomes Exceed 10%
Compare the two scenarios above: in both cases, the true relative risk is exactly 2.00. Yet when the outcome is rare (2% vs. 4%), OR = 2.04 closely approximates RR = 2.00. When the outcome is common (20% vs. 40%), the OR rises to 2.67.
Mathematically, the relationship between OR and RR is:
OR = RR × (1 - R₀)/(1 - R₁)
When the outcome is rare in both groups (R₀ and R₁ are near 0), the fraction (1 - R₀)/(1 - R₁) is very close to 1/1 = 1, which is known as the rare disease assumption. As a practical convention in epidemiology, when the outcome incidence exceeds roughly 10%, the odds ratio pulls further away from 1.0 than the relative risk (becoming larger than RR for harmful exposures where RR > 1, and smaller than RR for protective exposures where RR < 1).
Which Study Design Gives Which Measure?
Prospective or Retrospective Cohort Studies & Randomized Controlled Trials (RCTs): Because participants are sampled by exposure (or randomized) and followed forward to observe incident outcomes, the true incidence risks R₁ and R₀ are known. You can directly calculate both Relative Risk (RR) and Odds Ratio (OR).
Case–Control Studies: Here, the investigator fixes the number of cases (a + c) and controls (b + d) by design (for instance, recruiting 100 cancer cases and 100 healthy controls). Because the sample ratio of cases to controls is chosen by the researcher rather than reflecting population prevalence, the row totals a + b and c + d do not represent population denominators. Therefore, you cannot calculate population risk or RR directly from a standard case–control table; you report the Odds Ratio (OR), which remains mathematically valid regardless of whether you calculate the ratio of exposure odds or disease odds.
Cross-Sectional Studies: These yield Prevalence Ratios (PR) or Prevalence Odds Ratios (POR) rather than incidence risk ratios.
Why Logistic Regression Outputs Odds Ratios (and Alternatives for Common Outcomes)
When you need to adjust for confounders (such as age, sex, and comorbidities) with a binary outcome, binary logistic regression is the standard workhorse model. Because logistic regression models the natural logarithm of the odds (logit(p) = (p/(1 - p))), exponentiating a regression coefficient (e^) always yields an adjusted Odds Ratio (aOR)—even if your underlying study is an RCT or a cohort study.
When reporting an aOR from logistic regression in a study with a common outcome (> 10%), be careful to state “had 2.67 times the odds” rather than “was 2.67 times as likely.” If a direct adjusted risk ratio (aRR) is required for clinical interpretation in a cohort study, epidemiologists often use modified Poisson regression with robust error variance (Zou’s method) or log-binomial regression.
How to Report It
Always report the point estimate alongside its 95% confidence interval, specify the reference group, and use accurate language distinguishing odds from risk:
Reporting an Odds Ratio (Logistic Regression or Case–Control):
“Exposed patients had significantly higher odds of postoperative nausea than unexposed patients (OR = 2.67, 95% CI [1.42, 5.02], p = .002). In a multivariable model, report the adjusted OR (aOR) with its own 95% CI and list the covariates adjusted for.”
Reporting a Relative Risk (Cohort Study or RCT):
“The incidence of postoperative nausea was 40.0% (40/100) in the exposed group compared with 20.0% (20/100) in the unexposed group, corresponding to a twofold increase in risk (RR = 2.00, 95% CI [1.26, 3.17], p = .002).”
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