Odds Ratio Calculator

What the odds ratio calculator does

The odds ratio compares the odds of an outcome between two groups. This calculator works it out from a two-by-two table, the events and non-events in an exposed group and a control group, and also reports a confidence interval, a p-value, and a z-score. It is a mainstay of medical and social research.

Odds are a slightly different way of expressing chance than risk, and the odds ratio compares them between groups. Below is how it works and how it relates to the more intuitive relative risk.

How to use it

1. Enter the exposed group counts: events and non-events.
2. Enter the control group counts in the same way, and set a confidence level.
3. Press Calculate for the odds ratio and its statistics, or Reset to clear it.

How the odds ratio is worked out

Within each group, the odds of the outcome are the number of events divided by the number of non-events. The odds ratio is one group's odds divided by the other's, which simplifies to a tidy cross-multiplication of the four cells:

Odds ratio = (events exposed × non-events control) ÷ (non-events exposed × events control)

This cross-product form, the top-left times the bottom-right over the top-right times the bottom-left, is why the odds ratio is quick to compute from a table. The result compares the odds in the exposed group against the odds in the control group in a single number.

Reading the odds ratio

Like relative risk, the odds ratio is read against 1. An odds ratio of exactly 1 means the odds are the same in both groups, so there is no association between the exposure and the outcome. Above 1 means the exposed group has higher odds of the outcome, and below 1 means lower odds.

So an odds ratio of 2 means the odds of the outcome are twice as high in the exposed group, and an odds ratio of 0.5 means they are half as high. The direction and rough strength read much like relative risk, which is part of why the two are sometimes confused, even though they are not the same quantity.

Odds ratio versus relative risk

These two measures answer similar questions but are not interchangeable. Relative risk compares probabilities, while the odds ratio compares odds, and the two only agree closely when the outcome is rare. When the outcome is common, the odds ratio is always further from 1 than the relative risk, overstating the apparent effect.

The reason the odds ratio is used at all, given that relative risk is easier to interpret, is study design. In a case-control study, where researchers start with people who already have or do not have the outcome and look back at their exposure, the relative risk cannot be calculated directly, but the odds ratio can. So the odds ratio is the natural measure there, and for rare outcomes it doubles as a good approximation to the relative risk anyway.

The confidence interval

A single odds ratio from a sample is an estimate, so the calculator also gives a confidence interval, the range of values consistent with the data at your chosen confidence level. The width of that interval reflects how much the estimate might vary, and small samples give wide intervals.

The most useful thing to check is whether the interval includes 1. If it does, the data is consistent with no association at all, so the result is not statistically significant. If the whole interval sits above 1, or entirely below it, the association is significant at that confidence level, which is the same conclusion the p-value points to.

A worked example

Take the same table as a relative risk example: among the exposed, 30 events and 70 non-events; among the controls, 10 events and 90 non-events. The odds ratio is (30 times 90) divided by (70 times 10) = 2,700 divided by 700 = about 3.86.

Now compare: on this same data the relative risk is 3.0, but the odds ratio is 3.86, noticeably larger. That gap is the odds ratio overstating the effect because the outcome here is common, at 30 and 10 percent. Had the outcome been rare, the two would have come out almost equal. This is exactly why it pays to know which measure you are looking at.

Entering your values

Enter the event and non-event counts for both groups as whole numbers, and a confidence level. The calculator returns the odds ratio along with a confidence interval, p-value, and z-score. If any cell is zero the cross-product breaks down, so all four counts should be positive for a stable result.

Questions people ask

What is an odds ratio?

The ratio of the odds of an outcome in an exposed group to the odds in a control group, found by cross-multiplying a two-by-two table. It measures the association between exposure and outcome.

What does an odds ratio of 1 mean?

That the odds are the same in both groups, so there is no association. Above 1 means higher odds in the exposed group, below 1 means lower.

How is it different from relative risk?

Relative risk compares probabilities, the odds ratio compares odds. They agree closely only for rare outcomes; for common ones the odds ratio is further from 1, overstating the effect.

Why use it instead of relative risk?

Because in case-control studies the relative risk cannot be calculated but the odds ratio can, and for rare outcomes it approximates the relative risk closely.

References

A quick note on where the methods here come from. The odds ratio, its confidence interval, and its relationship to relative risk are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering odds, proportions, and association.

1. NIST/SEMATECH e-Handbook of Statistical Methods (odds ratios and contingency tables). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (odds, proportions, and association). https://openstax.org/details/books/introductory-statistics-2e