Deriving a Confidence Interval for an Odds Ratio

Why Build the Interval on the Log Scale?

An odds ratio compares the odds of an outcome in one group with the odds in another. Its estimate is positive and can range from nearly zero to arbitrarily large values, so an interval built directly by adding and subtracting a margin of error can be awkward: it might even have a negative lower endpoint. Taking the logarithm changes the ratio into a difference, gives a standard large-sample standard error, and makes a symmetric interval practical. The final step is to exponentiate the interval endpoints to return to the odds-ratio scale.

This tutorial considers two independent groups and a binary outcome, such as whether a hypothetical participant experiences an event. The groups could be assigned treatments or arise from a comparison of existing groups; the odds ratio describes their association with the outcome, not by itself a causal effect.

Four Counts and the Odds Ratio

Let \(a\) and \(b\) be the numbers with and without the outcome in group 1, and let \(c\) and \(d\) be the corresponding counts in group 2. Each group total is fixed by its two counts: \(n_1=a+b\) and \(n_2=c+d\). The odds of the outcome in group 1 are \(a/b\), and those in group 2 are \(c/d\).

The comparison order matters: \(\widehat{OR}\) here is group 1 odds divided by group 2 odds. Reversing the groups gives the reciprocal, \(1/\widehat{OR}\). An odds ratio of 1 means equal odds; a value above 1 means greater estimated odds in group 1, and a value below 1 means lower estimated odds in group 1.

Deriving the Standard Error

The standard error describes the sampling variability of the log odds ratio estimate. To derive it, assume each group is an independent sample and the outcome for each individual is binary. Within group \(i\), the number of outcomes has a binomial model with sample size \(n_i\) and population probability \(p_i\). Thus the sample proportion \(\hat p_i\) has approximate variance \(p_i(1-p_i)/n_i\).

The estimated log odds in a group can be written as the logit of its sample proportion, \(g(\hat p_i)=\log\{\hat p_i/(1-\hat p_i)\}\). The derivative of \(g(p)=\log\{p/(1-p)\}\) is \(g'(p)=1/\{p(1-p)\}\). The delta method says that, for a smooth function of an estimate, its approximate variance is the variance of the estimate multiplied by the square of the function’s derivative. Applying it here gives the following approximation:

Replace the unknown probability \(p_i\) by the observed proportion \(x_i/n_i\), where \(x_i\) is the number with the outcome. Since \(n_i-x_i\) is the number without it, the estimated variance becomes

The log odds ratio is the group 1 log odds minus the group 2 log odds. Because the groups are independent, the estimated variances of these two log odds add. Substituting the four cell counts gives the standard error formula.

Constructing and Interpreting the Confidence Interval

Under the large-sample approximation, the estimate \(\log(\widehat{OR})\) is approximately normally distributed around the population log odds ratio \(\log(\theta)\), with estimated standard error given above. For a confidence level \(C\), let \(z^*\) be the standard normal critical value with central area \(C\). The confidence interval on the log scale is the estimate plus or minus \(z^*\) standard errors.

The exponential function is increasing, so exponentiating the lower and upper log-scale endpoints preserves their order and gives a confidence interval for \(\theta\). A 95% interval uses \(z^*\approx1.96\). The resulting interval is generally asymmetric around \(\widehat{OR}\), even though the interval on the log scale is symmetric around \(\log(\widehat{OR})\).

Assumptions and Common Pitfalls

Before relying on this interval, check that the data fit the comparison being described. Each individual contributes to one group and has one binary outcome; observations within a group should be independent under the sampling or study design, and the two groups should be independent of one another. The large-sample approximation also needs enough information in each of the four cells for the log odds ratio to behave approximately normally. There is no single count cutoff that guarantees good approximation in every setting. A zero cell makes the displayed estimate or standard error undefined, and a very small cell can make the interval unstable.

• Using the wrong group order. Mixing the group 1 and group 2 cells changes the odds ratio to its reciprocal. Label the groups before calculating.
• Using an unpooled proportion-difference standard error. The formula here is for the log odds ratio and uses the reciprocals of all four cell counts. It is not the standard error for a difference in proportions.
• Exponentiating only the estimate. The confidence limits come from exponentiating both log-scale endpoints, not from adding and subtracting a margin directly on the odds-ratio scale.
• Rounding too early. Keep the odds ratio unrounded when taking its logarithm, and round the reported endpoints only after exponentiation. For example, \(\log(16/41)\approx-0.9410\); computing from a rounded odds ratio can shift the last digits.
• Treating a confidence level as a probability for this interval. The 95% statement concerns the long-run performance of the interval procedure under its assumptions, not a 95% probability that the fixed population odds ratio is inside the observed limits.