Barnard's test

In statistics, Barnard’s test is an exact test used in the analysis of contingency tables with one margin fixed. Barnard’s tests are really a class of hypothesis tests, also known as unconditional exact tests for two independent binomials. These tests examine the association of two categorical variables and are often a more powerful alternative than Fisher's exact test for contingency tables. While first published in 1945 by G.A. Barnard, the test did not gain popularity due to the computational difficulty of calculating the p value and Fisher’s specious disapproval. Nowadays, even for sample sizes n ~ 1 million, computers can often implement Barnard’s test in a few seconds or less. Barnard’s test is used to test the independence of rows and columns in a contingency table. The test assumes each response is independent. Under independence, there are three types of study designs that yield a table, and Barnard's test applies to the second type. To distinguish the different types of designs, suppose a researcher is interested in testing whether a treatment quickly heals an infection. One possible study design would be to sample 100 infected subjects, and for each subject see if they got the novel treatment or the old, standard, medicine, and see if the infection is still present after a set time. This type of design is common in cross-sectional studies, or ‘field observations’ such as epidemiology. Another possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and see if the infection is still present after a set time. This type of design is common in clinical trials. The final possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and stop the experiment once a pre-determined number of subjects has healed from the infection. This type of design is rare, but has the same structure as the lady tasting tea study that led R.A. Fisher to create Fisher's exact test.