Clustered binary data occur frequently in many application areas. When analyzing data of this form, ignoring key features, such as the intracluster correlation, may lead to inaccurate inference; e.g., inflated Type I error rates. For clustered binary data, Gerard and Schucany (2007) proposed an exact test for examining whether the marginal probability of a response differs from 0.5, which is the null hypothesis considered in the classic sign test. This new test maintains the specified Type I error rate and has more power, when compared to both the classic sign and permutation tests. The test statistic proposed by these authors equally weights the observed data from each cluster, regardless of whether the clusters are of equal size. To further improve the performance of the Gerard and Schucany test, a weighted test statistic is proposed and two weighting schemes are investigated. Seeking to further improve the performance of the proposed test, empirical Bayes estimates of the cluster level success probabilities are utilized. These adaptations lead to 5 new tests, each of which are shown through simulation studies to be superior to the Gerard and Schucany (2007) test. The proposed tests are further illustrated using data from a chemical repellency trial.
Keywords: Binomial; Clustered binary data; Exact test; Permutation test; Power; Sign test.