In studies with clustered or hierarchical data structures, quantifying between-cluster heterogeneity, referred to as contextual effects, is crucial for valid cluster-level inference. The median odds ratio (MOR), derived from random effects (RE) logistic regression models for clustered binary data, provides an intuitive assessment of contextual effects. Most existing research focuses on point estimation of the MOR for two-level models, with limited exploration of its statistical properties under complex multilevel structures. However, the development of corresponding interval estimators is essential for statistical inference. Moreover, many real-world datasets, particularly those from multistage surveys, involve hierarchical structures beyond two levels, where contextual effects at each level are of interest. This paper discusses the estimation of MOR for both the two-and three-level binary data, with particular emphasis on interval estimation. Since the MOR is a post-estimation measure based on variance components of the RE logit model, its confidence interval is derived using the Delta method, treating the log-transformed MOR as asymptotically normal. The approach is demonstrated across different model specifications in two-and three-level settings. An extensive simulation study evaluated the performance of the MOR estimators across diverse scenarios in hierarchical data settings. The results showed that the estimators exhibited negligible bias and satisfactory coverage probability of a 95% confidence interval for moderate to large samples, with small-sample bias mainly due to variance component estimation. An application of the methods for estimating the contextual effect on C-section delivery demonstrated that the proposed framework enhances interpretability and supports more informed statistical and policy-oriented analyses.

翻译：在具有聚类或分层数据结构的研究中，量化群组间异质性（即背景效应）对于有效的群组级推断至关重要。中位比值比（MOR）源自针对聚类二元数据的随机效应（RE）逻辑回归模型，可直观地评估背景效应。现有研究主要关注两水平模型MOR的点估计，对其在复杂多水平结构下的统计特性探讨有限。然而，发展相应的区间估计量对于统计推断至关重要。此外，许多真实世界数据集（特别是来自多阶段调查的数据集）涉及超过两水平的层次结构，其中各水平的背景效应均值得关注。本文讨论了二水平和三水平二元数据MOR的估计方法，重点聚焦于区间估计。由于MOR是基于RE Logit模型方差分量的后估计量，其置信区间通过Delta方法推导，将对数变换后的MOR视为渐近正态分布。该方法在二水平与三水平的不同模型设定中均得到验证。通过广泛的模拟研究，评估了MOR估计量在分层数据多元场景下的表现。结果表明，对于中至大样本，估计量具有可忽略的偏倚和满意的95%置信区间覆盖概率，小样本偏倚主要源于方差分量估计。通过将上述方法应用于剖宫产背景效应的估计实例，证明所提框架增强了解释性，并支持更明智的统计分析与政策导向研究。