Exact Tests and Exact Confidence Intervals for the Ratio of Two Binomial Proportions

Abstract

Testing for the ratio of binomial proportions (often called the relative risk) is quite common in clinical trials and epidemiology study or more generally in the pharmaceutical setting. Although this is an easy problem when we have large sample sizes, it becomes more challenging when sample sizes are small to moderate. In this type of situations asymptotic methods often lead to tests that are very liberal, i.e., have a very high Type I error. Hence one has to resort to exact methods. Although one can use Fisher's exact test if testing for unity relative risk, for the more general problem of testing for non-unity relative risk the only form of exact inference possible is by using exact unconditional tests. The standard exact unconditional test used for this problem is quite conservative, i.e., results in tests with very low power. We have proposed a test for this problem (based on the method suggested by Berger and Boos) which not only maintains the nominal size but is uniformly more powerful than the standard test (in most of the cases). A detailed comparison has been done between the two tests and various examples (from the pharmaceutical setting) have been used to compare the two methods.

Along with testing for the relative risk, researchers are also interested in obtaining confidence intervals for this parameter. Again due to small sample sizes the asymptotic methods often result in intervals that have poor coverage. We compare the confidence intervals generated from inverting the standard exact test and the test that we are proposing. Since both these tests are exact they result in intervals that are guaranteed to maintain the nominal coverage. We show that the standard intervals are quite conservative and our intervals in general have shorter lengths and coverage probabilities closer to the nominal coverage.

Although exact tests are desirable, it is often hard to implement them in practice because of computational complexities. In the last Chapter we compare the performance of the two exact tests discussed earlier with an approximate test (based on the idea of Storer and Kim) and two Bayesian tests for the hypothesis of efficacy. The hypothesis of efficacy is a special case of the relative risk testing problem and is often used in vaccine studies. For this specific problem we see that the approximate and the Bayesian tests perform quite well in terms of maintaining a low Type I error and results in tests with high power. Also these tests have a nice practical appeal because of the ease with which they can be implemented.