In observational studies, achieving covariate balance in pair matching between treatment and control groups or exposed and unexposed groups is essential. This balance enables testing treatment effects or examining {associations between exposures and} multivariate response variables in pair-matched data. Paired design studies involve taking multiple measurements for the same subjects under different conditions. All these call for an effective test for multivariate paired data. However, current methods for assessing covariate balance in matched observational studies often ignore the paired structure, leading to reduced performance in some cases. The multivariate paired Hotelling's $T^2$ test can be used for paired data, but its power decreases rapidly as dimensions increase. To address these issues, we propose a new non-parametric test for paired data, significantly improving power across various scenarios. We also derive the test's asymptotic distribution, making it user-friendly for practical applications. Our proposed test's effectiveness is demonstrated through an analysis of real data on Alzheimer's disease research.

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