Abstract Motivation Testing for differences in within-group dispersion is a fundamental problem in multivariate data analysis, with direct implications for interpreting group structure and validating statistical assumptions of other analysis such as ANOVA. Existing methods typically construct test statistics either based on the distance of each observation from the group center or on the mean of pairwise dissimilarities among observations within a group. Both approaches can fail when the mean within-group distance is similar across groups but the distributions of the within-group distances differ. This issue is particularly relevant in high-dimensional microbiome data, where outliers and overdispersion can distort the performance of mean-dissimilarity-based tests. Results We introduce the non-parametric Distance-based Test for Homogeneity (DTH), which measures dispersion of a group by computing within-group dissimilarity. Difference in dispersion across groups is tested by comparing the distributions of the within-group dissimilarity across different groups. A combination of Kolmogorov-Smirnov and Wasserstein distances are used to construct the difference between the distributions. For more than two groups, pairwise group tests are combined using a permutation-based p-value. Through simulations, we show that our method has higher power than existing tests for homogeneity in certain situations and comparable power in others. For continuous covariates, we offer an heuristic extension of DTH that showed good performance in simulations. Availability and implementation The DTH package, along with the code for reproducing all simulations, analyses, and an accompanying vignette, is available at https://github.com/asmita112358/DTH.

In genome-wide epigenetic studies, determining how exposures (e.g., Single Nucleotide Polymorphisms) affect outcomes (e.g., gene expression) through intermediate variables, such as DNA methylation, is a key challenge. Mediation analysis provides a framework to identify these causal pathways; however, testing for mediation effects involves a complex composite null hypothesis. Existing methods, such as Sobel's test or the Max-P test, are often underpowered in this context because they rely on null distributions determined under only a subset of the null space and are not optimized for the multiple testing burden inherent in high-dimensional data. To address these limitations, we introduce MLFDR (Mediation Analysis using Local False Discovery Rates), a novel method for high-dimensional mediation analysis. MLFDR leverages local false discovery rates, calculated from the coefficients of structural equation models, to construct an optimal rejection region. We demonstrate theoretically and through simulation that MLFDR asymptotically controls the false discovery rate and achieves superior statistical power compared to recent high-dimensional mediation methods. In real data applications, MLFDR identified 20%-50% more significant mediators than existing methods, demonstrating its ability to uncover biological signals missed by conventional approaches.

MOTIVATION: Genomic data are subject to various sources of confounding, such as demographic variables, biological heterogeneity, and batch effects. To identify genomic features associated with a variable of interest in the presence of confounders, the traditional approach involves fitting a confounder-adjusted regression model to each genomic feature, followed by multiplicity correction. RESULTS: This study shows that the traditional approach is suboptimal and proposes a new two-dimensional false discovery rate control framework (2DFDR+) that provides significant power improvement over the conventional method and applies to a wide range of settings. 2DFDR+ uses marginal independence test statistics as auxiliary information to filter out less promising features, and FDR control is performed based on conditional independence test statistics in the remaining features. 2DFDR+ provides (asymptotically) valid inference from samples in settings where the conditional distribution of the genomic variables given the covariate of interest and the confounders is arbitrary and completely unknown. Promising finite sample performance is demonstrated via extensive simulations and real data applications. AVAILABILITY AND IMPLEMENTATION: R codes and vignettes are available at https://github.com/asmita112358/tdfdr.np.