Definitions: Probability, Risk, and Odds
Before comparing relative risk and odds ratio, you need to be clear on three foundational concepts that underpin both measures: probability, risk, and odds. These three quantities describe the same underlying phenomenon — how often an event occurs — but express it in different scales.
Probability and Risk
Probability (or risk, when applied to disease development) is the number of people who experience the event divided by the total number of people in the group. If 30 out of 100 smokers develop hypertension over 10 years, the risk is 30/100 = 0.30, or 30%. Risk is always between 0 and 1 (or 0% and 100%).
Odds
Odds express the same information differently: the number of people who experience the event divided by the number who do not. From the same example, if 30 smokers develop hypertension and 70 do not, the odds are 30/70 = 0.43 — or equivalently, "43 events for every 100 non-events." Odds are always positive and have no upper bound: they can be 0.01, 1.0, 5.0, or 100.
Notice how probability and odds track closely at low values (5% vs 0.053) but diverge dramatically at high values (90% vs 9.000). This divergence is the root cause of the difference between relative risk and odds ratio when outcomes are common.
What Is Relative Risk?
Relative risk (RR), also called the risk ratio, is the ratio of the probability of an outcome in the exposed group to the probability of the same outcome in the unexposed group. It directly compares two risks.
Requires knowing both absolute risks
Valid for cohort studies and RCTs
Only needs exposure-disease cross-tabulation
Valid for all designs; required for case-control
The RR interpretation is intuitive: RR = 2 means the exposed group has twice the risk compared to the unexposed group. RR = 0.5 means the exposed group has half the risk — a 50% risk reduction. Clinicians, policy-makers, and patients understand relative risk naturally.
What Is Odds Ratio?
The odds ratio (OR) is the ratio of the odds of an outcome in the exposed group to the odds of the outcome in the unexposed group. It is calculated from a 2×2 contingency table using the cross-product formula:
The OR has a crucial mathematical property: it is invariant to the sampling ratios in a case-control study. If you enrol twice as many controls, the OR calculated from the table remains the same (the RR calculated from the same table would change). This invariance makes OR the only valid measure when cases and controls are selected in researcher-determined ratios that do not reflect the natural disease frequency — exactly the situation in case-control studies.
The OR is also the natural output of binary logistic regression. Every Exp(B) value in SPSS output is an odds ratio, not a relative risk. Logistic regression is the most widely used multivariable method for binary outcomes, which explains why ORs appear in the majority of observational medical literature.
Key Mathematical Differences
The critical mathematical relationship between OR and RR depends on the baseline risk (outcome frequency in the unexposed group). The formula connecting them is:
This formula reveals everything you need to know about when OR and RR diverge. When P₀ is near zero, the denominator is approximately 1, and RR ≈ OR. When P₀ is large (common disease), the denominator grows, and RR becomes substantially smaller than OR for positive associations (OR > 1), or larger than OR for protective associations (OR < 1).
Numerical Illustration: How OR and RR Diverge by Baseline Risk
Suppose OR = 3.0 in all cases. Watch what happens to the corresponding RR as baseline risk in the unexposed group increases:
When the unexposed group baseline risk is 50%, an OR of 3.0 corresponds to a true RR of only 1.5. Reporting OR = 3.0 as if it were a relative risk doubles the apparent magnitude of the association. This is not an edge case — baseline risks of 20–40% are common for outcomes like hospital readmission, postoperative complications, medication adherence, and chronic disease recurrence.
The Rare Disease Assumption: When OR ≈ RR
The rare disease assumption states that when the outcome is sufficiently rare — typically defined as less than 10% incidence or prevalence — the odds ratio closely approximates the relative risk. This is the theoretical justification for using OR from case-control studies to infer population-level relative risks.
The mathematics: when P is small, 1 − P ≈ 1. Therefore:
Odds = P / (1−P) ≈ P / 1 = P when P is small
So odds ≈ probability ≈ risk when the disease is rare. This means the OR — a ratio of two odds — approximates the RR — a ratio of two probabilities — when both probabilities are small. The assumption holds well below 10% and reasonably well below 15%, after which the approximation begins to fail.
Study Design and the Choice of Measure
The single most important factor determining whether you should use RR or OR is your study design. This is not a stylistic choice — it is a validity issue.
| Study Design | Can You Calculate RR? | Correct Primary Measure | Reason |
|---|---|---|---|
| Case-control study | No | OR | Case:control ratio set by researcher — does not reflect natural risk |
| Cohort study — rare outcome (<10%) | Yes | RR (or OR — similar) | OR ≈ RR; both valid; RR more intuitive |
| Cohort study — common outcome (>10%) | Yes | RR preferred | OR overestimates; RR or risk difference more interpretable |
| Randomised controlled trial | Yes | RR, risk difference, or NNT | Absolute and relative risk can both be calculated; NNT adds clinical utility |
| Cross-sectional study | Yes (prevalence ratio) | Prevalence ratio (PR) or OR | PR is the RR equivalent for cross-sectional prevalence data; OR acceptable |
| Logistic regression (any design) | Not directly | OR (Exp[B]) | Logistic regression produces ORs; use log-binomial for adjusted RR |
| Meta-analysis of case-control studies | No | Pooled OR | Individual studies only have ORs — RRs cannot be pooled across case-control designs |
Logistic Regression: An Important Special Case
When researchers run binary logistic regression on cohort data — which is extremely common because of logistic regression's flexibility for confounding adjustment — the output is always an OR, even though the underlying study design could support RR calculation. This creates a problematic situation when the outcome is common: the ORs from the regression model overestimate the corresponding RRs.
Three solutions exist for this problem:
- Log-binomial regression: Uses a log link with binomial distribution. Directly produces adjusted RR. Available in SPSS (Generalized Linear Models, Log link, Binomial distribution) and R (
glm(family=binomial(link="log"))). May fail to converge for some data configurations. - Modified Poisson regression: Uses Poisson regression with robust sandwich standard errors. Almost always converges, produces valid RR estimates for binary outcomes. Preferred by many epidemiologists.
- Zhang & Yu conversion: Apply the conversion formula to each logistic regression OR to obtain an approximate RR, using the outcome prevalence in the unexposed reference group as P₀.
Clinical Examples: RR and OR Calculated Side by Side
The following four examples calculate both RR and OR from the same data, making the divergence visible and clinically concrete.
Smoking and Lung Cancer — Rare Outcome (OR ≈ RR)
Cohort study over 20 years • Outcome: lung cancer (rare: ~10% in smokers) • Illustrates OR ≈ RR when disease is rare
| Lung Cancer (+) | No Cancer (−) | Total | ||
|---|---|---|---|---|
| Smoke | Smoker | 120 (a) | 1080 (b) | 1200 |
| Non-smoker | 30 (c) | 1770 (d) | 1800 | |
| Total | 150 | 2850 | 3000 | |
Risk(non-smokers) = 30/1800 = 0.0167 (1.67%)
RR = 0.100 / 0.0167 = RR = 5.99 ≈ 6.0
Odds Ratio (OR) OR = (120 × 1770) / (1080 × 30) = 212,400 / 32,400 = OR = 6.56
Discrepancy OR overestimates RR by: (6.56 − 6.0) / 6.0 × 100 = 9.3% — modest, as expected for ~10% baseline risk
Obesity and Surgical Complication — Common Outcome (OR Substantially Overestimates RR)
Prospective cohort • Outcome: postoperative complication (common: 30% baseline) • Shows OR-RR divergence at high baseline risk
| Complication (+) | No Complication (−) | Total | ||
|---|---|---|---|---|
| BMI | Obese (BMI≥30) | 168 (a) | 232 (b) | 400 |
| Non-obese | 180 (c) | 420 (d) | 600 | |
| Total | 348 | 652 | 1000 | |
Risk(non-obese) = 180/600 = 0.300 (30.0%) ← P₀ = 0.30
RR = 0.420 / 0.300 = RR = 1.40
Odds Ratio (OR) OR = (168 × 420) / (232 × 180) = 70,560 / 41,760 = OR = 1.69
Verification via Zhang & Yu RR = 1.69 / [(1−0.30) + (0.30×1.69)] = 1.69 / [0.70 + 0.507] = 1.69 / 1.207 = 1.40 ✓
Discrepancy OR overestimates RR by: (1.69 − 1.40) / 1.40 × 100 = 20.7% — substantial divergence
Breastfeeding and Infant Otitis Media — Protective Association
Prospective cohort • Protective exposure • Shows OR underestimates protective RR (OR further from 1 in the protective direction)
| Otitis Media (+) | No Otitis Media (−) | Total | ||
|---|---|---|---|---|
| Feed | Breastfed (≥6 months) | 45 (a) | 255 (b) | 300 |
| Formula-fed | 110 (c) | 390 (d) | 500 | |
| Total | 155 | 645 | 800 | |
Risk(formula-fed) = 110/500 = 0.220 (22.0%) ← P₀ = 0.22 (reference = formula-fed)
RR = 0.150 / 0.220 = RR = 0.682
Odds Ratio (OR) OR = (45 × 390) / (255 × 110) = 17,550 / 28,050 = OR = 0.626
Discrepancy for Protective Associations RR = 0.68 (32% risk reduction) OR = 0.63 (37% odds reduction)
OR exaggerates the protective effect: 37% vs 32% — OR inflates protection for common outcomes
Diabetes and Peripheral Neuropathy — Case-Control Study (Only OR Valid)
Hospital case-control study • Only OR valid by design • OR approximates RR because outcome is rare in the population
| Neuropathy Cases | Controls | Total | ||
|---|---|---|---|---|
| DM | Diabetes | 110 (a) | 72 (b) | 182 |
| No diabetes | 70 (c) | 108 (d) | 178 | |
| Total | 180 | 180 | 360 | |
95% CI: SE(ln OR) = √(1/110+1/72+1/70+1/108) = √0.04639 = 0.2153
95% CI = e^(0.859 ± 0.422) = 1.54 to 3.62
What we CANNOT calculate Apparent risk in "cases" row = 110/182 = 60.4% — but this reflects our 1:1 sampling, not population risk
If we had enrolled 1:4 (cases:controls), this "risk" would be ~20% — not the same disease. The OR would remain 2.36.
Side-by-Side Comparison: Complete Reference Table
| Feature | Relative Risk (RR) | Odds Ratio (OR) |
|---|---|---|
| What it measures | Ratio of two probabilities (risks) | Ratio of two sets of odds |
| Formula | [a/(a+b)] / [c/(c+d)] | (a × d) / (b × c) |
| Null value | 1.0 | 1.0 |
| Range | 0 to ∞ (but bounded by P₀) | 0 to ∞ (unconstrained) |
| Valid study designs | Cohort studies, RCTs, cross-sectional | All designs; required for case-control |
| Logistic regression | Not produced directly (need log-binomial) | Direct output — Exp(B) in SPSS |
| Rare disease (<10%) | More intuitive | ≈ RR; both acceptable |
| Common disease (>10%) | Preferred; accurately reflects effect | Overestimates RR if OR > 1; use with caution |
| Protective associations (OR <1) | Accurately reflects risk reduction | Overestimates protection (appears lower than true RR) |
| Intuitive for clinicians | Yes — directly answers "how much more likely?" | Less intuitive — requires understanding of odds |
| Meta-analysis of case-control studies | Cannot be pooled from case-control data | Standard pooled measure |
| Symmetry property | Not symmetric | Symmetric — invariant to sampling design |
| Conversion to other measure | RR → OR: complex | OR → RR: Zhang & Yu formula with P₀ |
When the Odds Ratio Overestimates: Recognising the Problem in Published Papers
Several contextual clues should alert you that an OR reported in a paper likely overestimates the true relative risk:
Outcome prevalence or incidence is above 10%
Any time the outcome occurs in more than 10% of the reference group, the OR will meaningfully exceed the RR for positive associations. Check the event rates in the methods or baseline characteristics table. Outcomes like postoperative complications, medication adherence, hospital readmission, comorbidity recurrence, and quality-of-life endpoints often exceed 20–40% baseline risk.
The design is cohort or cross-sectional but the authors report ORs from logistic regression
This is the most common scenario in the literature. Logistic regression is applied to cohort or cross-sectional data — producing ORs — even though both the design and the data support direct RR calculation. The authors may not acknowledge the overestimation, especially if the outcome is common.
The OR is large (e.g., OR > 2.5) and the outcome is common
Large ORs amplify the divergence from RR. An OR of 4.0 with baseline risk 30% in the unexposed group corresponds to RR = 4.0 / [(1−0.30) + 0.30×4.0] = 4.0/1.90 = 2.10. The OR says four times higher odds; the true risk is only 2.1 times higher. A clinician reading OR = 4.0 would be very alarmed; the true RR = 2.1 would be concerning but substantially less dramatic.
The paper confusingly uses "risk" language alongside ORs
Phrases like "patients with diabetes were 2.4 times more likely to develop the complication (OR 2.4)" conflate OR language (odds ratio) with RR language (times more likely = risk ratio). The OR of 2.4 is not the same as saying the risk is 2.4 times higher — unless the outcome is rare. When you see this in a paper, check the outcome frequency and apply the conversion formula if needed.
Common Interpretation Mistakes
Mistake 1: Treating OR as RR when reporting results to patients or policy-makers
Clinicians communicating risk to patients — "this medication triples your risk" when the study found OR = 3.0 — may be substantially exaggerating for common outcomes. If the baseline outcome risk is 25%, OR = 3.0 corresponds to RR = 1.88, meaning the medication nearly doubles risk — not triples it. This discrepancy affects informed consent and shared decision-making.
Mistake 2: Computing RR from a case-control table and reporting it as relative risk
Calculating risk as a/(a+b) in a case-control study and reporting it as "relative risk" is a fundamental error. The apparent risk in the case row is determined by how many cases were enrolled, not by actual disease incidence. If you enrolled twice as many cases, the apparent "risk" in the exposed group would double, but the OR would remain the same. Any RR calculated from case-control proportions is meaningless.
Mistake 3: Ignoring the OR-RR distinction in systematic reviews and meta-analyses
Meta-analyses pooling odds ratios from logistic regression across multiple cohort studies produce pooled ORs that are potentially inflated estimates of the true pooled RR. This problem compounds when studies with different baseline risks are pooled: each study's OR overestimates its own RR by a different amount, making the pooled OR a distorted summary of the underlying risk relationships.
Mistake 4: Assuming that OR and RR always move in the same direction by the same amount
While OR and RR are always on the same side of the null value (both above 1 or both below 1 for the same association), their magnitudes diverge in predictable but non-linear ways. An OR that decreases from 2.0 to 1.5 after adjustment may correspond to a change in RR from 1.75 to 1.40 in different amounts for different baseline risks. Comparing changes in OR across studies with different baseline risks is not straightforward.
Mistake 5: Concluding that RR is always preferable and OR should be abandoned
Some researchers, upon learning about OR overestimation, conclude that OR should never be reported. This overcorrects. OR is the only valid measure for case-control studies, the natural output of logistic regression (the most powerful tool for multivariable adjustment of binary outcomes), and a technically superior measure for meta-analysis because of its symmetry properties. For rare diseases, OR and RR are interchangeable. OR should be reported with appropriate context about the outcome frequency, not abandoned.
Reporting Examples in Medical Journals
The following examples show how to correctly report RR and OR in different contexts, including the language that signals which measure is being used and how the study design justifies the choice.
Frequently Asked Questions
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