Adjusted Clustered Rank Tests for Clustered Data in Unbalanced Design

Abstract

This paper proposes an adjusted rank test-T1 to determine that two independent samples with clustered data in unbalanced design are drawn from the same population. For a large number of clusters, the paper finds that the test statistic-T1 converges to normal distribution. In addition, the clustered rank sum test-T2 is presented for three or more independent samples with clustered data. An adjusted rank test-T3 is also proposed by adjusting the clustered rank sum test. For each adjusted rank test, the same critical value is used for data sets with equivalence between the numbers of clusters and cluster sizes, but the observations might differ. The critical values of two adjusted test statistics for some numbers of clusters and cluster sizes are given at the significance levels of 0.10 and 0.05. To compare the performances of the adjusted rank tests with the alternative tests, a simulation study is necessary. Results show that the two adjusted tests can maintain the size of the tests for all situations. For three samples, the Kruskal-Wallis test based on the observation mean of cluster gives the estimated size of about 25% at the true significance level of 5%. The adjusted test-T1 has a higher power than the Wilcoxon test based on the observation mean of cluster. The power of both adjusted rank tests increases when the number of clusters, the number of observations per cluster, and the effect size all increase. However, the power of the adjusted tests decreases when the correlation coefficient between observations in a cluster increases.