Exact Methods of Homogeneity Test of Proportions for Bilateral and Unilateral Correlated Data

Subjects in clinical studies that investigate paired body parts can carry a disease on either both sides (bilateral) or a single side (unilateral) of the organs. Data in such studies may consist of both bilateral and unilateral records. However, the correlation between the paired organs is often ignored, which may lead to biased interpretations. Recent literatures have taken the correlation into account. For example, Ma and Wang (J Biopharm Stat 31:686–704, 2021) proposed three asymptotic procedures for testing the homogeneity of proportions of multiple groups using combined bilateral and unilateral data and recommended the score test. It is of importance to notice that the asymptotic behavior is not guaranteed if the sample size is small, resulting in uncontrolled type I error rates. In this paper, we extend their work by considering exact approaches and compare these methods with the score test proposed by Ma and Wang (J Biopharm Stat 31:686–704, 2021) in terms of type I errors and statistical powers. The E approach, which identifies a rejection region and calculates p values based on maximum likelihood estimations, is recommended based on simulation studies. Additionally, two real-world examples are used to illustrate the application of the proposed approaches.