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.

The Model Parameter Randomisation Test (MPRT) is highly recognised in the eXplainable Artificial Intelligence (XAI) community due to its fundamental evaluative criterion: explanations should be sensitive to the parameters of the model they seek to explain. However, recent studies have raised several methodological concerns for the empirical interpretation of MPRT. In response, we propose two modifications to the original test: Smooth MPRT and Efficient MPRT. The former reduces the impact of noise on evaluation outcomes via sampling, while the latter avoids the need for biased similarity measurements by re-interpreting the test through the increase in explanation complexity after full model randomisation. Our experiments show that these modifications enhance the metric reliability, facilitating a more trustworthy deployment of explanation methods.

The Model Parameter Randomisation Test (MPRT) is widely acknowledged in the eXplainable Artificial Intelligence (XAI) community for its well-motivated evaluative principle: that the explanation function should be sensitive to changes in the parameters of the model function. However, recent works have identified several methodological caveats for the empirical interpretation of MPRT. To address these caveats, we introduce two adaptations to the original MPRT -- Smooth MPRT and Efficient MPRT, where the former minimises the impact that noise has on the evaluation results through sampling and the latter circumvents the need for biased similarity measurements by re-interpreting the test through the explanation's rise in complexity, after full parameter randomisation. Our experimental results demonstrate that these proposed variants lead to improved metric reliability, thus enabling a more trustworthy application of XAI methods.