Confidence Interval for an Odds Ratio


Note that while we have discussed using the odds ratio as a measure of association in the context of a case-control study, odds ratios can also be computed in other types of study designs as well. Recall our example of a prospective cohort study to examine the association of hypertension and cardiovascular disease. In this study the risk ratio was RR=2.18, but we can also compute an odds ratio and then use these data to illustrate how to compute a confidence interval for an odds ratio.

Table - Association Between Hypertension (HTN) and Cardiovascular Disease (CVD)

CVD

No CVD

Total

HTN

992

2260

3252

No HTN

165

1017

1182

Total

1157

3277

4434

Just as we noted for risk ratios, odds ratios are also not normally distributed. They too are skewed toward the upper end of possible values. As a result, we must once again take the natural log of the odds ratio and first compute the confidence limits on a logarithmic scale, and then convert them back to the normal odds ratio scale.

The formula for the 95% Confidence Interval for the odds ratio is as follows:

The standard error for log(OR) is computed using the following equation:

We will illustrate computation of a 95% confidence interval for the data in the contingency table shown above. 

> log(2.71)

[1] 0.9969486

> exp(0.816)

[1] 2.261436

> exp(1.178)

[1] 3.247872

Interpretation: In this study, subjects with hypertension 2.71 times the odds of developing cardiovascular disease compared to non-hypertensive subjects. With 95% conficence the true odds ratio lies in the range of 2.26-3.24.