One-dimensional-based tests, while computationally efficient and effective in many scenarios, have inherent limitations in capturing higher-order correlations and complex dependencies within high-dimensional data. These tests typically rely on marginal distributions, which means they assess differences in distributions along individual dimensions without considering interactions between them. As a result, they may fail to detect subtle yet significant relationships that arise from the joint behavior of multiple variables.
For instance, in cases where the data exhibits intricate structures, such as non-linear relationships or multi-modal distributions, one-dimensional tests may yield misleading results. They can overlook the effects of correlations that exist between dimensions, leading to an underestimation of the differences between the reference and alternative distributions. This limitation is particularly pronounced in high-dimensional settings, where the curse of dimensionality can exacerbate the challenges of accurately modeling and interpreting data.
Moreover, one-dimensional tests may not adequately account for the presence of outliers or noise, which can disproportionately influence the results. In contrast, multivariate tests that consider the joint distribution of data points are better suited to capture these complexities, providing a more comprehensive assessment of the underlying relationships in the data.