Paired or partially paired two-sample tests with unordered samples

In paired two-sample tests for mean equality, it is common to encounter unordered samples in which subject identities are not observed or unobservable, and it is impossible to link the measurements before and after treatment. The absence of subject identities masks the correspondence between the two samples, rendering existing methods inapplicable. In this paper, we propose two novel testing approaches. The first splits one of the two unordered samples into blocks and approximates the population mean using the average of the other sample. The second method is a variant of the first, in which subsampling is used to construct an incomplete U-statistic. Both methods are affine invariant and can readily be extended to partially paired two-sample tests with unordered samples. Asymptotic null distributions of the proposed test statistics are derived and the local powers of the tests are studied. Comprehensive simulations show that the proposed testing methods are able to maintain the correct size, and their powers are comparable to those of the oracle tests with perfect pair information. Four real examples are used to illustrate the proposed methods, in which we demonstrate that naive methods can yield misleading conclusions.