Permutation-Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
This addresses a practical issue in fields like psychometrics where discretization is common, offering a method that handles mixed data for causal discovery, though it appears incremental as it builds on existing rank tests.
The paper tackles the problem of rank tests for cross-covariance matrices in causal discovery when variables are discretized, proposing MPRT which theoretically controls Type I error and is validated empirically on synthetic and real-world data.
Recent advances have shown that statistical tests for the rank of cross-covariance matrices play an important role in causal discovery. These rank tests include partial correlation tests as special cases and provide further graphical information about latent variables. Existing rank tests typically assume that all the continuous variables can be perfectly measured, and yet, in practice many variables can only be measured after discretization. For example, in psychometric studies, the continuous level of certain personality dimensions of a person can only be measured after being discretized into order-preserving options such as disagree, neutral, and agree. Motivated by this, we propose Mixed data Permutation-based Rank Test (MPRT), which properly controls the statistical errors even when some or all variables are discretized. Theoretically, we establish the exchangeability and estimate the asymptotic null distribution by permutations; as a consequence, MPRT can effectively control the Type I error in the presence of discretization while previous methods cannot. Empirically, our method is validated by extensive experiments on synthetic data and real-world data to demonstrate its effectiveness as well as applicability in causal discovery.