Test for homogeneity with unordered paired observations

Abstract

In some applications, an experimental unit is composed of two distinct but related subunits. The response from such a unit is (X1,X2) but we observe only Y1 =min{X1,X2} and Y2 =max{X1,X2}, i.e., the subunit identities are not observed. We call (Y1,Y2) unordered paired observations. Based on unordered paired observations {(Y1i,Y2i)}ni=1,weare interested in whether the marginal distributions for X1 and X2 are identical. Testing methods are available in the literature under the assumptions that var(X1)=var(X2)andcov(X1,X2) = 0. However, by extensive simulation studies, we observe that when one or both assumptions are violated, these methods have inflated type I errors or much lower powers. In this paper, we study the likelihood ratio test statistics for various scenarios and explore their limiting distributions without these restrictive assumptions. Furthermore, we develop Bartlett correction formulae for these statistics to enhance their precision when the sample size is not large. Simulation studies and real-data examples are used to illustrate the efficacy of the proposed methods. © 2021, Institute of Mathematical Statistics. All rights reserved.