# Test for homogeneity with unordered paired observations

## TL;DR

This paper develops likelihood ratio tests for homogeneity using unordered paired observations, relaxing previous assumptions and improving accuracy with Bartlett corrections, supported by simulations and real data.

## Contribution

It introduces new likelihood ratio test procedures for unordered paired data that do not rely on variance or independence assumptions, with improved finite-sample accuracy.

## Key findings

- Proposed likelihood ratio tests perform well under various scenarios.
- Bartlett corrections improve test accuracy for small samples.
- Methods are validated through simulations and real data examples.

## Abstract

In some applications, an experimental unit is composed of two distinct but related subunits. The response from such a unit is $(X_{1}, X_{2})$ but we observe only $Y_1 = \min\{X_{1},X_{2}\}$ and $Y_2 = \max\{X_{1},X_{2}\}$, i.e., the subunit identities are not observed. We call $(Y_1, Y_2)$ unordered paired observations. Based on unordered paired observations $\{(Y_{1i}, Y_{2i})\}_{i=1}^n$, we are interested in whether the marginal distributions for $X_1$ and $X_2$ are identical. Testing methods are available in the literature under the assumptions that $Var(X_1) = Var(X_2)$ and $Cov(X_1, X_2) = 0$. 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.