Mann–Whitney test for two‐phase stratified sampling

Abstract

We consider the Mann–Whitney test for two‐phase stratified sampling. In this design, the independent and identically distributed (i.i.d.) sample is obtained at the first phase and then stratified based on auxiliary variables. At the second phase, stratified subsamples are obtained without replacement to collect variables of interest. The resultant data are biased and dependent sample due to stratification and sampling without replacement. This setting is different from the one considered by the existing method called the van Elteren test that treats multiple i.i.d. samples from nonoverlapping subsets of the infinite population. We propose the inverse probability weighted Mann–Whitney statistic and study its asymptotic properties. The proposed test is shown to have a correct size and power 1 in the limit. Simulation study is presented to illustrate the finite sample performance of our test.