  In paired design studies, it is common to have multiple measurements taken
for the same set of subjects under different conditions. In observational
studies, it is many times of interest to conduct pair matching on multiple
covariates between a treatment group and a control group, and to test the
treatment effect represented by multiple response variables on well
pair-matched data. However, there is a lack of an effective test on
multivariate paired data. The multivariate paired Hotelling's $T^2$ test can
sometimes be used, but its power decreases fast as the dimension increases.
Existing methods for assessing the balance of multiple covariates in matched
observational studies usually ignore the paired structure and thus they do not
perform well under some settings. In this work, we propose a new non-parametric
test for paired data, which exhibits a substantial power improvement over
existing methods under a wide range of situations. We also derive the
asymptotic distribution of the new test and the approximate $p$-value is
reasonably accurate under finite samples through simulation studies even when
the dimension is larger than the sample size, making the new test an
easy-off-the-shelf tool for real applications. The proposed test is illustrated
through an analysis of a real data set on the Alzheimer's disease research.
