Finite sample t-tests for high-dimensional means

When sample sizes are small, it becomes challenging for an asymptotic test requiring diverging sample sizes to maintain an accurate Type I error rate. In this paper, we consider one-sample, two-sample and ANOVA tests for mean vectors when data are high-dimensional but sample sizes are very small. We establish asymptotic t -distributions of the proposed U -statistics, which only require data dimensionality to diverge but sample sizes to be fixed and no less than 3. The proposed tests maintain accurate Type I error rates for a wide range of sample sizes and data dimensionality. Moreover, the tests are nonparametric and can be applied to data which are normally distributed or heavy-tailed. Simulation studies confirm the theoretical results for the tests. We also apply the proposed tests to an fMRI dataset to demonstrate the practical implementation of the methods.

Nonparametric two-sample tests of high dimensional mean vectors via random integration

Testing the equality of the means in two samples is a fundamental statistical inferential problem. Most of the existing methods are based on the sum-of-squares or supremum statistics. They are possibly powerful in some situations, but not in others, and they do not work in a unified way. Using random integration of the difference, we develop a framework that includes and extends many existing methods, especially in high-dimensional settings, without restricting the same covariance matrices or sparsity. Under a general multivariate model, we can derive the asymptotic properties of the proposed test statistic without specifying a relationship between the data dimension and sample size explicitly. Specifically, the new framework allows us to better understand the test's properties and select a powerful procedure accordingly. For example, we prove that our proposed test can achieve the power of 1 when nonzero signals in the true mean differences are weakly dense with nearly the same sign. In addition, we delineate the conditions under which the asymptotic relative Pitman efficiency of our proposed test to its competitor is greater than or equal to 1. Extensive numerical studies and a real data example demonstrate the potential of our proposed test.

T2-DAG: a powerful test for differentially expressed gene pathways via graph-informed structural equation modeling

MOTIVATION

A major task in genetic studies is to identify genes related to human diseases and traits to understand functional characteristics of genetic mutations and enhance patient diagnosis. Compared with marginal analyses of individual genes, identification of gene pathways, i.e. a set of genes with known interactions that collectively contribute to specific biological functions, can provide more biologically meaningful results. Such gene pathway analysis can be formulated into a high-dimensional two-sample testing problem. Given the typically limited sample size of gene expression datasets, most existing two-sample tests tend to have compromised powers because they ignore or only inefficiently incorporate the auxiliary pathway information on gene interactions.

RESULTS

We propose T2-DAG, a Hotelling's T2-type test for detecting differentially expressed gene pathways, which efficiently leverages the auxiliary pathway information on gene interactions from existing pathway databases through a linear structural equation model. We further establish its asymptotic distribution under pertinent assumptions. Simulation studies under various scenarios show that T2-DAG outperforms several representative existing methods with well-controlled type-I error rates and substantially improved powers, even with incomplete or inaccurate pathway information or unadjusted confounding effects. We also illustrate the performance of T2-DAG in an application to detect differentially expressed KEGG pathways between different stages of lung cancer.

AVAILABILITY AND IMPLEMENTATION

The R (R Development Core Team, 2021) package T2DAG which implements the proposed T2-DAG test is available on Github at https://github.com/Jin93/T2DAG.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.