Generalized Independence Test

Abstract

Testing independence is a widely encountered problem in modern data analysis, especially in high-dimensional settings where complex dependency structures are common. Traditional methods often lack power in such scenarios. To address this, we propose a novel test statistic that leverages both similarity and dissimilarity information to capture intricate relationships within the data. The proposed test demonstrates strong power across a wide range of alternatives in high-dimensional settings, as shown through extensive simulation studies. Under mild conditions, we prove that the permutation null distribution of the statistic converges to the ${\chi }_4^2$ distribution, enabling straightforward control of the type I error. Our theoretical analysis further advances the moment method to establish joint asymptotic normality for a class of double-indexed permutation statistics. Moreover, we prove the power consistency of the proposed test without imposing explicit restrictions on the data dimension, under mild conditions that allow for detecting complex dependencies. We illustrate the proposed test using gene-expression data from the Genotype-Tissue Expression project, where GIT rejects independence between the GTEx category "Breast - Mammary Tissue" and the cell-line category "Cells - EBV-transformed lymphocytes."