Weighted k-Sample Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling Tests for Assessing Covariate Balance

Abstract

Weighted distributional tests for covariate balance are currently limited to two-group comparisons. We extend the Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises tests to an arbitrary number of weighted groups k >= 2, using existing k-sample generalizations and a shared permutation-inference procedure; each statistic reduces exactly to its two-group counterpart at k = 2. An accompanying post-hoc pairwise procedure with four multiple-comparison adjustments localizes which groups differ following an omnibus rejection. In a four-scenario simulation study at k = 3, Type I error remained close to nominal, and the comparative advantages established for two groups were preserved: Kolmogorov-Smirnov was most powerful against a centrally located discrepancy, Anderson-Darling against a tail-located discrepancy, and Anderson-Darling and Cramer-von Mises performed comparably against a diffuse discrepancy. Omnibus power was nonetheless uniformly lower than in the matching two-group setting - not because the underlying discrepancy is diluted, but because adding groups enlarges the null distribution itself, so the post-hoc procedure, whose pairwise statistics carry no such penalty, is often the more sensitive tool whenever a specific group's imbalance is suspected. The methods are implemented in the Stata commands kstest, adtest, and cvmtest.