Uncertainty-Aware Rank Test with Missing Data and Application in Causal Discovery

Covariance-rank constraints are widely used in causal discovery, including settings with latent variables, but missing observations complicate their statistical testing. Pairwise covariance estimates use different, overlapping sets of rows; yet, plugging these estimates into a classical rank test with an effective sample size correction need not properly control Type-I error. In this paper, we propose the Missing-value uncertainty-Aware Rank Test (MART), which uses partially observed rows beyond those retained by test-wise deletion. MART consistently estimates the joint uncertainty of the pairwise covariance estimates and measures their weighted distance to the rank-constrained set. A Kronecker approximation makes this distance solvable by a whitened singular value decomposition, while the full estimated uncertainty calibrates the statistic through a weighted chi-square mixture as the asymptotic null distribution. For causal discovery with many rank test queries, the covariance and uncertainty estimates can be computed once and reused to reduce the computational cost. Experiments on both synthetic and real-world data validate our theoretical claims and the effectiveness of our method.