Medical researchers often need one number that describes the association between an exposure and a binary outcome. The two most common choices are the odds ratio and the relative risk. However, these two measures are not interchangeable, and the choice of odds ratio vs relative risk depends on the study design and on how common the outcome is. A common mistake is to report an odds ratio and then describe it as a risk. This guide explains the difference in simple steps, using one hypothetical clinical example from the beginning to the end.
Table of Contents
Odds Ratio vs Relative Risk: What Is the Difference?
Both measures compare two groups, and a value of 1 means no difference between the groups. A value above 1 indicates a higher risk or odds in the exposed group, while a value below 1 indicates a lower risk or odds (Ranganathan et al., 2015). The difference lies in what is being compared.
Relative Risk
Risk is the number of patients with the outcome divided by all patients in the group. The relative risk (RR), also called the risk ratio, is the risk in the exposed group divided by the risk in the unexposed group. Therefore, an RR of 2 means that the risk is twice as high.
Odds Ratio
Odds are the number of patients with the outcome divided by the number of patients without the outcome. The odds ratio (OR) is the odds in the exposed group divided by the odds in the unexposed group. Consequently, an OR of 2 means that the odds are twice as high, which is not the same as twice the risk.
How the Two Measures Are Related
When the outcome is rare, typically below 10%, the OR and the RR are close to each other. As the outcome becomes more common, the OR moves further from 1 than the RR (Ranganathan et al., 2015).
Importance of Odds Ratio vs Relative Risk in Medical Research
Correct Communication of the Effect Size
Clinicians and patients think in terms of risk. Therefore, an OR that is read as an RR can exaggerate the size of the association. Our guide on effect size explains why the magnitude of an association matters as much as its statistical significance.
Agreement Between the Design and the Analysis
The study design determines which measure can be calculated. Consequently, the decision on odds ratio vs relative risk should be made during planning, not after the results are available.
Meeting the Expectations of Peer Reviewers
Reviewers often ask why an OR was presented in a trial or in a cohort with a common outcome. A clear justification of the selected measure prevents this comment and improves the credibility of the manuscript.
Applications of Odds Ratio vs Relative Risk in Clinical Studies
Randomized Controlled Trials
Trials follow patients from randomization to the outcome, so the risk in each arm is known. The RR is the preferred measure for a binary outcome in this design (Knol et al., 2012).
Cohort Studies
In cohort studies, the number of patients at risk is known. Therefore, both the RR and the OR can be calculated, although the RR is easier to interpret.
Case-Control Studies
In case-control studies, the researcher selects the number of cases and controls. The number of exposed people in the source population is not available, so the RR cannot be calculated and the OR is used (Ranganathan et al., 2015).
Assumptions Behind Odds Ratio vs Relative Risk
Before calculating either measure, the following conditions should be checked:
- Binary outcome: each patient either has the outcome or does not have it during the same follow-up period.
- Independent observations: each patient contributes once. Paired or repeated data require different methods.
- Known denominators for the RR: the number of patients at risk in each group must come from the study design.
- Adequate cell counts: very small counts make the confidence interval unreliable. In this situation, Fisher’s exact test can be used to test the association.
- Control of confounding: a crude OR or RR describes an association only, and important confounders should be considered in the analysis.
How to Calculate Odds Ratio vs Relative Risk: A Clinical Example
Consider a hypothetical prospective cohort of 300 patients undergoing elective colorectal surgery. A total of 150 patients were current smokers, and 150 were non-smokers. The outcome was a surgical wound complication within 30 days. The data were created for teaching purposes, and every value below was calculated from these data.
Step 1: Build the 2×2 Table
Place the exposure in the rows and the outcome in the columns. Wound complications occurred in 90 of the 300 patients (30.0%).
| Group | Complication | No complication | Total | Risk | Odds |
|---|---|---|---|---|---|
| Smokers | 60 | 90 | 150 | 40.0% | 0.667 |
| Non-smokers | 30 | 120 | 150 | 20.0% | 0.250 |
Step 2: Calculate the Relative Risk and Its 95% CI
The relative risk equals 0.40 ÷ 0.20 = 2.00. Its standard error on the natural log scale is √(1/60 − 1/150 + 1/30 − 1/150) = 0.1915. Therefore, the 95% CI is exp(0.693 ± 1.96 × 0.1915), which gives 1.37 to 2.91. Smokers had twice the risk of a wound complication, and the risk was 100% higher. Our guide on confidence intervals explains how to read this range.
Step 3: Calculate the Odds Ratio and Its 95% CI
The odds ratio equals (60 × 120) ÷ (90 × 30) = 2.67. Its standard error on the natural log scale is √(1/60 + 1/90 + 1/30 + 1/120) = 0.2635. Therefore, the 95% CI is exp(0.981 ± 1.96 × 0.2635), which gives 1.59 to 4.47. Smokers had 2.67 times the odds of a complication. This result does not mean 2.67 times the risk.
Step 4: Test the Association and Calculate the Absolute Difference
The chi-square test uses expected counts of 45 and 105 in each row. It gives χ² = 14.29, df = 1, p < 0.001. Moreover, the risk difference equals 20.0 percentage points (95% CI, 9.9 to 30.1). The number needed to harm is 1 ÷ 0.20 = 5, so one additional complication occurred for every five smokers. A correct reading of the p-value should be presented with these estimates.
Step 5: Examine Why the Two Measures Differ
The OR in this example is 33.3% larger than the RR because the outcome is common. The next table keeps the RR fixed at 2.00 and changes only the risk in the unexposed group:
| Risk in unexposed | Risk in exposed | RR | OR |
|---|---|---|---|
| 1% | 2% | 2.00 | 2.02 |
| 5% | 10% | 2.00 | 2.11 |
| 10% | 20% | 2.00 | 2.25 |
| 20% | 40% | 2.00 | 2.67 |
| 30% | 60% | 2.00 | 3.50 |
| 40% | 80% | 2.00 | 6.00 |
Step 6: Compare with a Case-Control Sample
Suppose that a researcher had instead selected the 90 patients with complications and 105 of the 210 patients without complications. Half of the patients without complications are included, so their smoking pattern is unchanged:
| Group | Cases | Controls |
|---|---|---|
| Smokers | 60 | 45 |
| Non-smokers | 30 | 60 |
The OR remains (60 × 60) ÷ (30 × 45) = 2.67 (95% CI, 1.49 to 4.78). However, a “risk” calculated from this table gives 57.1% ÷ 33.3% = 1.71, which is wrong because the researcher fixed the number of controls. This step shows why the RR is not valid in a case-control design.
Step 7: Adjust for Confounders and Interpret the Model
Diabetes was more frequent among smokers (40.0%) than non-smokers (20.0%), and diabetes also increased the risk of complications:
| Stratum | Smokers, complications / n | Non-smokers, complications / n |
|---|---|---|
| Diabetes | 36 / 60 (60.0%) | 10 / 30 (33.3%) |
| No diabetes | 24 / 90 (26.7%) | 20 / 120 (16.7%) |
Logistic regression provides adjusted ORs. Adjusted RRs can be obtained with log-binomial regression or with Poisson regression using robust standard errors, and both approaches gave correct RRs and confidence intervals in simulations (Knol et al., 2012). Here, the RR model used Poisson regression with robust standard errors.
| Measure for smoking | Estimate | 95% CI | p-value |
|---|---|---|---|
| Crude OR | 2.67 | 1.59 to 4.47 | < 0.001 |
| Adjusted OR | 2.17 | 1.27 to 3.72 | 0.005 |
| Crude RR | 2.00 | 1.37 to 2.91 | < 0.001 |
| Adjusted RR | 1.68 | 1.15 to 2.45 | 0.007 |
For logistic regression interpretation, the smoking coefficient equals β = 0.776, and e0.776 = 2.17. Therefore, after adjustment for diabetes, smokers had 117% higher odds of a complication. In contrast, the adjusted RR of 1.68 indicates a 68% higher risk. The contrast between these two statements is the practical meaning of odds ratio vs relative risk.
How to Report Odds Ratio vs Relative Risk Results in a Manuscript
The Methods section should name the measure and the model. The Results section should present the measure that matches the design, as shown below:
Association Between Smoking and Surgical Wound Complications
Wound complications occurred in 90 of 300 patients (30.0%). Complications were significantly more frequent among smokers (40.0%) compared to non-smokers (20.0%) (χ² = 14.29, df = 1, p < 0.001). Moreover, smoking was associated with a higher risk of complications (RR = 2.00, 95% CI, 1.37 to 2.91, p < 0.001). After adjustment for diabetes, this association remained significant (adjusted RR = 1.68, 95% CI, 1.15 to 2.45, p = 0.007). (Table 2)
Notably, possible explanations of the association and comparisons with previous studies belong to the Discussion section, not to the Results.
Common Mistakes When Choosing Odds Ratio vs Relative Risk
Describing an Odds Ratio as a Risk
An OR of 2.67 does not mean a 167% higher risk. In our example, the risk was 100% higher. When an OR is presented, the word “odds” should be used in the text.
Calculating a Relative Risk in a Case-Control Study
Step 6 showed that the value depends on the number of controls selected. Therefore, only the OR should be reported from a case-control design.
Assuming That the Outcome Is Rare
The similarity between the OR and the RR holds only for rare outcomes. Researchers should check the outcome frequency in their own sample before treating the two measures as equivalent.
Reporting Relative Measures Only
A relative measure does not show the size of the absolute effect. Therefore, the risk in each group should be presented, and a risk difference should be added when possible.
Relying on a Simple Conversion Formula
The formula proposed by Zhang and Yu converts an OR to an RR. However, this method also overestimated the RR in simulations, and its use is discouraged (Knol et al., 2012). An adjusted RR should be estimated directly.
Running Odds Ratio vs Relative Risk Analysis in SPSS and R
In SPSS, choose Analyze → Descriptive Statistics → Crosstabs, click Statistics, and select Risk. The output shows the OR and the RR for each cohort. For an adjusted OR, use Analyze → Regression → Binary Logistic. For an adjusted RR, use Analyze → Generalized Linear Models, select Poisson loglinear, and choose the Robust estimator on the Estimation tab. In R, with the sandwich and lmtest packages loaded, one line gives the adjusted RR on the log scale: coeftest(glm(complication ~ smoker + diabetes, family = poisson, data = d), vcov. = vcovHC, type = "HC0").
Conclusion
The decision on odds ratio vs relative risk depends on the study design and on the frequency of the outcome. The RR is preferred in trials and cohort studies, while the OR is the suitable measure in case-control studies and is the direct output of logistic regression. If you need support in choosing the correct measure, analyzing your data, or writing your Results section, the StatisMed team of biostatisticians provides statistical analysis services for medical studies. You can request a free quote today.
Frequently Asked Questions
1. Can I report an odds ratio in a cohort study?
Yes, an OR is valid in a cohort study. However, it should be described as odds, and an RR is preferred when the outcome is common because the OR will be further from 1.
2. Why does logistic regression give odds ratios instead of relative risks?
Logistic regression models the natural logarithm of the odds. Therefore, the exponentiated coefficient is an OR. Log-binomial or robust Poisson models use a log link, so their exponentiated coefficients are RRs.
3. Should I use a hazard ratio instead?
When the time to the event is recorded, Cox regression can be used, and it provides a hazard ratio (Knol et al., 2012). The RR and the OR are suited to outcomes measured over one fixed follow-up period.
References
Knol, M. J., Le Cessie, S., Algra, A., Vandenbroucke, J. P., & Groenwold, R. H. H. (2012). Overestimation of risk ratios by odds ratios in trials and cohort studies: Alternatives to logistic regression. CMAJ: Canadian Medical Association Journal, 184(8), 895–899. https://doi.org/10.1503/cmaj.101715
Ranganathan, P., Aggarwal, R., & Pramesh, C. S. (2015). Common pitfalls in statistical analysis: Odds versus risk. Perspectives in Clinical Research, 6(4), 222–224. https://doi.org/10.4103/2229-3485.167092
