Odds Ratio Calculator
Enter a 2×2 table to get the odds ratio with its confidence interval, the odds in each group and the chi-square and Fisher exact p-values. A zero cell can be handled with the Haldane-Anscombe correction.
In a case-control study the odds ratio is the measure of association; it approximates the relative risk only when the outcome is rare. For cohort data use the relative risk calculator; the difference between the two is explained in relative risk vs odds ratio.
Paste a 2×2 table of counts, one row per line, the counts separated by tabs, commas or spaces:
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What the odds ratio measures
The odds of an event are the number of subjects who have it divided by the number who do not: 50 with the outcome and 20 without give odds of 2.5, which is a probability of 50 / 70 = 71%. The odds ratio (OR) divides the odds in one group by the odds in the other. For the table [[50, 20], [30, 100]] the odds are 2.5 and 0.3, so OR = 2.5 / 0.3 = 8.3333. OR = 1 means the outcome is equally likely in both groups, OR > 1 that the odds are higher in the first group and OR < 1 that they are lower.
The odds ratio is symmetric: comparing the odds of the outcome between exposed and unexposed subjects gives the same number as comparing the odds of exposure between subjects with and without the outcome. That is why it is the measure of case-control studies, which fix the numbers of cases and controls and so cannot estimate risks, and why logistic regression coefficients are log odds ratios. For cohort studies and trials the relative risk is easier to interpret.
How to use the odds ratio calculator
- Enter the counts. Row 1 is the exposed group and row 2 the unexposed group; column 1 counts the subjects with the outcome (the cases) and column 2 those without it. You can rename the rows and columns, or paste a table.
- Choose the confidence level. If a cell can be 0, decide whether to apply the Haldane-Anscombe correction, which adds 0.5 to every cell when a cell is 0.
- Press Calculate Odds Ratio. The results follow every later change to the table.
- Load example enters the worked example below, Clear empties the table and Copy link to this calculation shares your table, labels and settings.
Odds ratio formulas
Odds₁ = a / b, Odds₂ = c / d
OR = Odds₁ / Odds₂ = (a × d) / (b × c)
SE(ln OR) = √( 1/a + 1/b + 1/c + 1/d )
CI for OR = exp( ln OR ± z × SE(ln OR) ), z = 1.96 for 95%
Haldane-Anscombe: a, b, c, d → a + 0.5, b + 0.5, c + 0.5, d + 0.5
The interval is Woolf's logit interval (Woolf 1955): it is computed for ln(OR), which is approximately normal with the standard error above, and transformed back, so it is not symmetric around the odds ratio. It is an approximation that needs reasonably large counts in all four cells.
With an empty cell the sample odds ratio is 0 or infinite and the standard error is infinite. The Haldane-Anscombe correction (Haldane 1956, Anscombe 1956) adds 0.5 to every cell so that an estimate and an interval exist; it biases the odds ratio toward 1, so treat the result as a rough guide. The calculator applies it only when you choose it and only when a cell is 0.
Worked example: 50 of 70 exposed, 30 of 130 unexposed
Press Load example to enter a = 50, b = 20, c = 30, d = 100.
Working
Odds (exposed) = 50 / 20 = 2.5 and odds (unexposed) = 30 / 100 = 0.3.
OR = (50 × 100) / (20 × 30) = 8.3333 and ln(OR) = 2.1203.
SE = √( 1/50 + 1/20 + 1/30 + 1/100 ) = √0.113333 = 0.3367.
95% CI = exp( 2.1203 ± 1.96 × 0.3367 ) = 4.3079 to 16.1204.
The chi-square p-value is 2.7852e-11 and the Fisher exact p-value 3.1353e-11.
The interval does not include 1, so the exposure is significantly associated with the outcome. SciPy's odds_ratio(kind="sample") and statsmodels' Table2x2.oddsratio_confint give the same odds ratio and interval, and SciPy's fisher_exact gives the same p-value. R's fisher.test reports a slightly different estimate, about 8.23, because it is the conditional maximum likelihood estimate rather than the sample odds ratio (SciPy's conditional estimate is 8.2262).
Worked example with an empty cell
Suppose 5 of 20 exposed subjects and none of 20 unexposed subjects have the outcome: enter 5, 15 in the first row and 0, 20 in the second. The sample odds ratio is infinite and the calculator reports it as undefined. With the Haldane-Anscombe correction the counts become 5.5, 15.5, 0.5 and 20.5, so OR = (5.5 × 20.5) / (15.5 × 0.5) = 14.5484, ln(OR) = 2.6775 and SE = 1.5150, with a 95% interval of 0.7469 to 283.3702.
The interval includes 1 and is enormously wide, yet the Fisher exact p-value of the entered counts is 0.047124 (the chi-square p-value 0.016827 is unreliable here, because two expected counts are below 5). Sparse tables are exactly where the Woolf interval is weakest: report the counts, and prefer an exact method such as the Fisher exact test, which also gives an exact interval.
Odds ratio or relative risk?
Both measure association, but they answer different questions and are not interchangeable. For the table above the odds ratio is 8.3333, while the relative risk (71.43% against 23.08%) is 3.0952: the odds ratio is further from 1 whenever the event is common, so calling it "8 times as likely" would be wrong. When the event is rare the two nearly coincide: with 120 events among 10,000 exposed and 90 among 10,000 unexposed subjects, RR = 1.3333 and OR = 1.3374.
Use the odds ratio for case-control studies and for results that come from logistic regression, and the relative risk with the risk difference and the number needed to treat for cohort studies and trials. The article relative risk vs odds ratio compares them in detail.
How to read the result
- The odds ratio and its interval. If the interval excludes 1 the association is significant at the level 100% minus the confidence level. The width of the interval shows how precisely the odds ratio is known; small counts give wide intervals.
- Direction. Swapping the two rows (or the two columns) turns the odds ratio into its reciprocal: 8.3333 becomes 0.12. Check that the row you call exposed is the first one.
- Crude, not adjusted. The odds ratio of a 2×2 table ignores every other variable. Confounding can change it, and an adjusted odds ratio needs logistic regression.
- The two p-values. The chi-square p-value is an approximation that needs every expected count to be at least 5. The Fisher exact p-value has no such condition, so prefer it for small tables. The calculator warns when an expected count is below 5.
- Empty cells. The odds ratio is undefined until you choose a correction, and even then the interval is unreliable. The calculator says so instead of printing a number that looks precise.
Odds ratio in Excel, R, Python and SPSS
| Software | How to get it |
|---|---|
| Excel | With a, b, c, d in A2:D2: =(A2*D2)/(B2*C2) for the odds ratio and =SQRT(1/A2+1/B2+1/C2+1/D2) for the standard error of ln(OR). If E2 holds the odds ratio and F2 the standard error, the limits of the 95% interval are =EXP(LN(E2)-1.96*F2) and =EXP(LN(E2)+1.96*F2) |
| R | library(epitools); oddsratio(matrix(c(50, 20, 30, 100), nrow = 2, byrow = TRUE), method = "wald") prints the odds ratio with the Woolf (Wald) interval and the mid-p, Fisher and chi-square p-values. Base R: fisher.test(tbl) reports the conditional maximum likelihood estimate (about 8.23 here), not the sample odds ratio |
| Python | from scipy.stats.contingency import odds_ratio; result = odds_ratio([[50, 20], [30, 100]], kind="sample"); result.statistic and result.confidence_interval(0.95). The default kind is "conditional", the fisher.test estimate. statsmodels: Table2x2(np.array([[50, 20], [30, 100]])).oddsratio and .oddsratio_confint() |
| SPSS | Analyze > Descriptive Statistics > Crosstabs, then Statistics > Risk. The Risk Estimate table starts with the odds ratio and its 95% confidence interval; check which row and column SPSS treats as the exposed group and the outcome |
Frequently Asked Questions
What is an odds ratio?
The odds ratio compares the odds of an outcome between two groups. The odds are the number with the outcome divided by the number without it, and OR = (a × d) / (b × c) for a table with a, b in the first row and c, d in the second. OR = 1 means no association.
How do I calculate an odds ratio from a 2×2 table?
Divide the counts of the first row to get its odds, divide those of the second row to get its odds, and divide the first odds by the second; this equals (a × d) / (b × c). With 50, 20 in the first row and 30, 100 in the second, the odds are 2.5 and 0.3 and OR = 8.3333.
How do I interpret an odds ratio?
OR above 1 means higher odds of the outcome in the first group, OR below 1 lower odds, and OR = 1 no difference. OR = 8.3333 means the odds are about 8.3 times as large. Read the confidence interval with it: an interval that excludes 1 means the association is statistically significant.
What is the difference between odds and probability?
Probability is the share of subjects with the outcome, p = events / total; odds are events divided by non-events, p / (1 − p). A probability of 0.5 is odds of 1, a probability of 0.8 is odds of 4. Odds are unbounded above, probabilities are not.
What is the difference between odds ratio and relative risk?
Relative risk is a ratio of probabilities, the odds ratio a ratio of odds. They are close only when the outcome is rare; for a common outcome the odds ratio is further from 1, so 8.3333 in the example corresponds to a relative risk of only 3.0952. Relative risk needs cohort data, the odds ratio also works for case-control data.
When is the odds ratio approximately equal to the relative risk?
When the outcome is rare in both groups, because then the odds p / (1 − p) are almost equal to the risks p. With risks of 1.2% and 0.9% the relative risk is 1.3333 and the odds ratio 1.3374. As the outcome becomes common the odds ratio overstates the relative risk.
How is the confidence interval of an odds ratio calculated?
With Woolf's method: ln(OR) is approximately normal with standard error √(1/a + 1/b + 1/c + 1/d), so the interval is exp(ln OR ± z × SE). It is not symmetric around the odds ratio, and it is unreliable when a cell count is small.
What if a cell of the table is zero?
The sample odds ratio is then 0 or infinite and has no Woolf interval. The Haldane-Anscombe correction adds 0.5 to every cell so that an estimate and an interval exist, at the price of pulling the estimate toward 1. For sparse tables an exact method such as Fisher's exact test is more trustworthy.
Should I use the chi-square or the Fisher exact p-value?
The chi-square p-value is an approximation that needs every expected count to be at least 5. The Fisher exact p-value is exact for any 2×2 table, so it is the safer choice for small tables, and the two agree for large ones. The calculator warns when an expected count is below 5.
Can an odds ratio be negative?
No. The odds ratio runs from 0 to infinity, with 1 meaning no association. Its logarithm, the coefficient of logistic regression, can be negative, which corresponds to an odds ratio between 0 and 1.
What does an odds ratio below 1 mean?
The odds of the outcome are lower in the first group than in the second. An odds ratio of 0.25 means the odds are a quarter as large, and swapping the two rows gives the reciprocal, 4. If you prefer to talk about the group with the higher odds, put it in the first row.
Why is the odds ratio used in case-control studies?
A case-control study chooses how many cases and controls to include, so the share of cases in the sample says nothing about the risk. The odds ratio of exposure among cases and controls equals the odds ratio of the outcome among exposed and unexposed subjects, so it can be estimated from such data.
Why does fisher.test in R give a different odds ratio?
R's fisher.test reports the conditional maximum likelihood estimate, which is based on the distribution of the first cell given the row and column totals. For the example it is about 8.23 (SciPy's odds_ratio with kind='conditional' gives 8.2262 with an interval of 4.1066 to 17.0585), while the sample odds ratio (a × d) / (b × c) is 8.3333. Both are valid estimates of the odds ratio; the Fisher p-value is the same.
How is this different from an adjusted odds ratio?
This calculator gives the crude odds ratio of a single 2×2 table. An adjusted odds ratio comes from logistic regression, where it is exp(coefficient), and it controls for the other variables in the model. Confounding can make the crude and the adjusted odds ratio quite different.
How do I calculate an odds ratio in Excel?
With the counts a, b, c, d in cells A2 to D2, type =(A2*D2)/(B2*C2). For the 95% interval take the exponential of LN(OR) ± 1.96 × SQRT(1/A2+1/B2+1/C2+1/D2). The calculator does this in one step and adds the two p-values.
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