Understanding Odds Ratios and Risk Ratios

An Interactive Explanation

An interactive article about odds ratios, risk ratios (relative risk), and risk differences, and why the interpretation of odds ratios is perhaps more odd than expected.

Nov 2, 2026 8 min read Statistics

When the outcome is binary, the default analysis in many fields is logistic regression, and the default effect size is the odds ratio. These are generally not the most intuitive numerical summaries of a result. In this article I will explain odds ratios, risk ratios, and risk differences using interactive visualizations, including why an odds ratio of 2 does not mean “twice as likely”. The visualizations also let you explore even less intuitive oddities of odds, such as why odds ratios change when adjusting for a prognostic variable, even in a randomized trial (non-collapsibility). The running example throughout will be a randomized trial of psychotherapy for depression, in which the outcome is whether a patient remits.

Risk and odds

Let’s start with 100 patients without considering treatment status: 30 remit and 70 do not. This can be expressed as either risk or odds. Drag the slider to change how many patients remit, and toggle the figure to see this visualized as risk or odds.

Summarize the group by

Patients who remit, of 100 30

Risk 30 / 100 = 0.30

Odds 30 / 70 = 0.43

The risk of remission is the proportion of all patients who remit: 0.30, or 30%.

Each dot is a patient. Risk divides the remitters by everyone; odds divide them by those who did not remit.

The risk and odds above describe a single group; next, we will focus on treatment effects, that is, group comparisons.

Risk ratios, odds ratios, and risk differences

Back in the RCT, we want to compare the control and treated arms, so we calculate the risk and odds for each arm, just as above. Drag the sliders to change the number of remissions, and toggle between risks and odds.

Compare the arms by their

Control: patients who remit, of 100 30Treated: patients who remit, of 100 50

Risk difference 0.50 − 0.30 = 0.20

Risk ratio 0.50 / 0.30 = 1.67

20 more of every 100 treated patients remit, so treating about 5 patients yields one extra remission. Treated patients remit 1.67 times as often as control patients.

Each dot is a patient, 100 per arm. The risks view shows the two risks that the risk difference subtracts and the risk ratio divides. In the odds view each arm splits into remitters facing non-remitters, and the second slider sets the odds ratio: the treated arm changes so that its odds equal the control odds times that ratio. Switching views keeps the same patients, so all three measures describe one trial.

The figure gave us three ways to summarize the same RCT result. Let $p_1$ and $p_0$ be the remission risks in treated and control patients:

\begin{aligned} \mathrm{RD} &= p_1 - p_0 \\ \mathrm{RR} &= \frac{p_1}{p_0} \\ \mathrm{OR} &= \frac{p_1/(1-p_1)}{p_0/(1-p_0)} \end{aligned}

The risk difference (RD) and risk ratio (RR) compare probabilities; the odds ratio (OR) compares odds. Interpreting an odds ratio as “times as likely” applies a ratio of odds to risks, which works only when the outcome is rare and odds and risks nearly coincide (Cummings, 2009; Davies et al., 1998). Suppose 30 of 100 control patients remit and the odds ratio is 2.33. “2.33 times as likely” would mean 70 treated patients remitting, but only 50 do. Set the figure above to 55 control remitters and an odds ratio of 4 in the odds view, and “4 times as likely” would mean 220 remitters among 100 treated patients, which would be great for power, though ontologically awkward.

Baseline risk matters

Odds ratios, risk ratios, and risk differences do not translate into one another one-to-one: for a given OR, the risk difference and risk ratio depend on the baseline risk, that is, the proportion of patients who remit in the control group (Grant, 2014).

The figure below places both arms on the logistic curve, whose X axis is the log-odds, or logit. Logistic regression is linear on this scale: the treatment coefficient is the distance between the two arms’ log-odds, which is the logarithm of the odds ratio. A fixed OR is therefore a fixed horizontal gap between the arms. The risk difference is the vertical gap between the same two arms on the risk axis, and its size depends on the arms’ position along the curve.

The figure can hold the risk difference, risk ratio, or odds ratio fixed. Start with the odds ratio, drag the control-arm risk, and watch the risk difference annotated along the Y axis change. Switch the risk axis to log, and the vertical gap shows the risk ratio instead. Hold the risk ratio fixed, and that gap stays constant while the horizontal gap, the odds ratio, changes.

The same odds ratio gives different risk differences and risk ratios at different baseline risks

Hold fixed

Risk axis

Odds ratio 2.33 Control-arm risk 0.30

Risk difference +0.20 held fixed

Risk ratio 1.67 held fixed

Odds ratio 2.33 held fixed

30% of the control arm remit and 50% of the treated arm (odds 0.43 and 1.00).

Hold one measure fixed and move the control arm, or drag the pair: the other two change with the baseline risk.

For a fixed odds ratio and control risk $p_0$ , the risk ratio is (Grant, 2014; Zhang & Yu, 1998):

\mathrm{RR} = \frac{\mathrm{OR}}{1 - p_0 + p_0\,\mathrm{OR}}

You probably noticed that at low baseline risk, the risk ratio approaches the odds ratio; as baseline risk rises, it moves toward 1 (Davies et al., 1998). The risk difference also varies with baseline risk, and is greatest near the middle of the logistic curve. The formula holds only for an unadjusted odds ratio; to convert an adjusted one, which we turn to next, average the model’s predicted risks instead (Greenland, 2004).

Adjustment and non-collapsibility

Trial analyses often add a prognostic covariate to the logistic regression, which increases statistical power (FDA, 2023; Kahan et al., 2014; Robinson & Jewell, 1991). Let’s assume half of each arm has mild and half severe depression at baseline, and that we adjust for this binary variable. The adjusted, or conditional, odds ratio compares patients within the same severity group; the unadjusted, or marginal, one compares the arms as a whole trial. What’s the relationship between the subgroup estimates and the whole-trial (unadjusted) estimate? The whole-trial risk difference is a weighted average of the risk differences in the two severity groups, so it falls between them, and the same holds for the risk ratio (Cummings, 2009; Greenland et al., 1999). This is not the case for the odds ratio, which is called non-collapsibility (Greenland, 2021). Move the slider to vary how strongly severity predicts remission.

The whole-trial odds ratio is not an average of the severity groups' odds ratios

How strongly severity predicts remission (control arm) mild 50% severe 10%

Control arm
Treated arm
Odds-ratio step

In each severity group, treatment multiplies the odds of remission by 3.00. In the whole trial, it multiplies them by 2.33.

	Control	Treated	Risk difference	Risk ratio	Odds ratio
Mild	50%	75%	0.25	1.50	3.00
Whole trial	30%	50%	0.20	1.67	2.33
Severe	10%	25%	0.15	2.50	3.00

Each arm's mild and severe groups sit on the left curve at their log-odds of remission, and the dashed lines average each arm's two risks into its whole-trial point on the right; hover a dot to follow its line. Both panels use the same log-odds scale, so the orange odds-ratio steps compare by length: the whole trial's is shorter than either group's. The slider sets how strongly severity predicts remission. The whole trial stays at 30% versus 50%, so only the groups' steps change; with the slider at its left end, all three match.

The marginal and conditional OR are two different estimands (FDA, 2023; Morris et al., 2022). The marginal odds ratio differs from the conditional one even without confounding (FDA, 2023; Greenland, 1987; Mood, 2010). Risk differences and risk ratios, by contrast, are collapsible: their whole-trial values in the table fall between their values in the two severity groups (Greenland et al., 1999). The table also shows that the risk differences and risk ratios vary between the severity groups, even though the odds ratio is the same in both groups. This is an example of effect modification that depends on the scale: the effect varies on the risk-difference and risk-ratio scales, but not on the odds-ratio scale (VanderWeele, 2009).

Case-control studies: when odds are all we have

You will likely encounter odds ratios in case-control studies, which select patients by their outcome: those with it (cases) and, for comparison, a sample of those without it (controls). In these studies, the risks in the sample depend on how many controls the investigator sampled, so the sample cannot estimate the risk difference or the risk ratio, but it can estimate the odds ratio (Cornfield, 1951; Cummings, 2009). The visualization below lets you explore why sampling a proportion of the controls (non-remitters) still allows you to estimate the odds ratio. For simplicity, we assume a population of 200 patients, 100 treated and 100 untreated, of whom 50 and 30 remit, and we want to know the odds ratio of remission. Drag the slider to change how many controls we sample.

Sampling fewer controls changes the risk ratio but leaves the odds ratio alone

Controls (non-remitters) kept in the sample 100%

Odds ratio $\dfrac{50 \,/\, (1.0 \times 50)}{30 \,/\, (1.0 \times 70)} = 2.33$ All 200 patients: 2.33, the same

Risk ratio $\dfrac{50 \,/\, (50 + 1.0 \times 50)}{30 \,/\, (30 + 1.0 \times 70)} = 1.67$ All 200 patients: 1.67, the same

Keeping 100% of the non-remitters multiplies both groups' odds by 1/1.0 = 1: untreated 0.43 → 0.43, treated 1.00 → 1.00.

Each dot is a patient. Every remitter (case) is kept, but only some of the non-remitters (controls); the faint dots are the non-remitters left out. Treated and untreated patients keep the same share of their non-remitters, in steps of 10% so that each keeps whole patients, and both ratios are computed from the patients kept.

Treated and untreated patients keep the same share of their non-remitters (controls), so both groups’ odds are multiplied by the same factor, which cancels in the odds ratio; the odds ratio is the same as if we had included all controls. The risks also count the remitters in their denominators, so the factor does not cancel, and the sample risk ratio moves toward 1 as the share of controls shrinks. When the outcome is rare, the case-control odds ratio also approximates the risk ratio (Cornfield, 1951); remission is common, so here the odds ratio of 2.33 is far from the risk ratio of 1.67.

Which effect to report?

The point of this post is not to claim there is anything wrong with odds ratios or logistic regression. The goal is to help you understand them, and to know that there are multiple options for how to communicate results even if you fit a logistic regression. Broadly speaking, the options vary in two ways. The first is the measure:

Relative effects: odds ratios and risk ratios
Absolute effects: each group’s risk, and their difference (risk difference)

The second is the target, which applies to any of these measures: a marginal (population-averaged) effect or a conditional effect within patients who share the covariates (Morris et al., 2022).

You do not have to pick one: to communicate how much a treatment helps, you can report a relative effect (the OR or the RR) together with an absolute one (each arm’s risk and the risk difference) (Hopewell et al., 2025). Likewise, you can adjust for prognostic covariates in the logistic regression and still report a marginal effect, using standardization (g-computation) (Morris et al., 2022). Getting into these details is beyond the scope of this post, but if you are interested I listed some practical resources under “Further reading” below.

How to cite

Cite this page according to your favorite style guide.

The current version is 1.0.0 (changelog).

APA 7

Magnusson, K. (2026). Understanding odds ratios and risk ratios: An interactive explanation (Version 1.0.0) [Interactive article]. R Psychologist. https://rpsychologist.com/odds-ratio

BibTeX

@online{magnussonOdds-ratio,
  author = {Kristoffer Magnusson},
  title = {Understanding odds ratios and risk ratios: An interactive explanation},
  url = {https://rpsychologist.com/odds-ratio},
  version = {1.0.0},
  date = {2026-11-02},
}

References

Cornfield, J. (1951). A Method of Estimating Comparative Rates from Clinical Data: Applications to Cancer of the Lung, Breast, and Cervix. Journal of the National Cancer Institute, 11(6), 1269–1275. https://doi.org/10.1093/jnci/11.6.1269

Cummings, P. (2009). The Relative Merits of Risk Ratios and Odds Ratios. Archives of Pediatrics & Adolescent Medicine, 163(5), 438–445. https://doi.org/10.1001/archpediatrics.2009.31

Daniel, R., Zhang, J., & Farewell, D. (2021). Making Apples from Oranges: Comparing Noncollapsible Effect Estimators and Their Standard Errors after Adjustment for Different Covariate Sets. Biometrical Journal, 63(3), 528–557. https://doi.org/10.1002/bimj.201900297

Davies, H. T. O., Crombie, I. K., & Tavakoli, M. (1998). When Can Odds Ratios Mislead? BMJ, 316(7136), 989–991. https://doi.org/10.1136/bmj.316.7136.989

FDA. (2023). Adjusting for Covariates in Randomized Clinical Trials for Drugs and Biological Products: Guidance for Industry (Final Guidance FDA-2019-D-0934). Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research. https://www.fda.gov/regulatory-information/search-fda-guidance-documents/adjusting-covariates-randomized-clinical-trials-drugs-and-biological-products

Grant, R. L. (2014). Converting an Odds Ratio to a Range of Plausible Relative Risks for Better Communication of Research Findings. BMJ, 348, f7450. https://doi.org/10.1136/bmj.f7450

Greenland, S. (1987). Interpretation and Choice of Effect Measures in Epidemiologic Analyses. American Journal of Epidemiology, 125(5), 761–768. https://doi.org/10.1093/oxfordjournals.aje.a114593

Greenland, S. (2004). Model-Based Estimation of Relative Risks and Other Epidemiologic Measures in Studies of Common Outcomes and in Case-Control Studies. American Journal of Epidemiology, 160(4), 301–305. https://doi.org/10.1093/aje/kwh221

Greenland, S. (2021). Noncollapsibility, Confounding, and Sparse-Data Bias. Part 1: The Oddities of Odds. Journal of Clinical Epidemiology, 138, 178–181. https://doi.org/10.1016/j.jclinepi.2021.06.007

Greenland, S., Pearl, J., & Robins, J. M. (1999). Confounding and Collapsibility in Causal Inference. Statistical Science, 14(1), 29–46. https://doi.org/10.1214/ss/1009211805

Harrell, F. (2021, June 28). Avoiding One-Number Summaries of Treatment Effects for RCTs with Binary Outcomes. Statistical Thinking. https://www.fharrell.com/post/rdist/

Hopewell, S., Chan, A.-W., Collins, G. S., Hróbjartsson, A., Moher, D., Schulz, K. F., Tunn, R., Aggarwal, R., Berkwits, M., Berlin, J. A., Bhandari, N., Butcher, N. J., Campbell, M. K., Chidebe, R. C. W., Elbourne, D., Farmer, A., Fergusson, D. A., Golub, R. M., Goodman, S. N., … Boutron, I. (2025). CONSORT 2025 Statement: Updated Guideline for Reporting Randomised Trials. The Lancet, 405(10489), 1633–1640. https://doi.org/10.1016/S0140-6736(25)00672-5

Kahan, B. C., Jairath, V., Doré, C. J., & Morris, T. P. (2014). The Risks and Rewards of Covariate Adjustment in Randomized Trials: An Assessment of 12 Outcomes from 8 Studies. Trials, 15, 139. https://doi.org/10.1186/1745-6215-15-139

Mood, C. (2010). Logistic Regression: Why We Cannot Do What We Think We Can Do, and What We Can Do about It. European Sociological Review, 26(1), 67–82. https://doi.org/10.1093/esr/jcp006

Morris, T. P., Walker, A. S., Williamson, E. J., & White, I. R. (2022). Planning a Method for Covariate Adjustment in Individually Randomised Trials: A Practical Guide. Trials, 23(1), 328. https://doi.org/10.1186/s13063-022-06097-z

Robinson, L. D., & Jewell, N. P. (1991). Some Surprising Results about Covariate Adjustment in Logistic Regression Models. International Statistical Review / Revue Internationale de Statistique, 59(2), 227–240. https://doi.org/10.2307/1403444

VanderWeele, T. J. (2009). On the Distinction Between Interaction and Effect Modification. Epidemiology, 20(6), 863–871. https://doi.org/10.1097/EDE.0b013e3181ba333c

Zhang, J., & Yu, K. F. (1998). What’s the Relative Risk? A Method of Correcting the Odds Ratio in Cohort Studies of Common Outcomes. JAMA, 280(19), 1690–1691. https://doi.org/10.1001/jama.280.19.1690

Source on GitHub · CC BY 4.0