Asymptotically distribution free tests in heteroscedastic unbalanced high dimensional ANOVA

Abstract

In this paper, we develop the asymptotic theory for hypotheses testing in high-dimensional analysis of variance (HANOVA) when the distributions are completely unspecifed. Most results in the literature have been restricted to observations of no more than two-way designs for continuous data. Here we formulate the local alternatives in terms of departures from the null distribution so that the responses can be either continuous or categorical. The asymptotic theory is presented for testing of main factor and interaction effects of up to order three in unbalanced designs with heteroscedastic variances and arbitrary number of factors. The test statistics are based on quadratic forms whose asymptotic theory is derived under non-classical settings where the number of variables is large while the number of replications may be limited. Simulation results show that the present test statistics perform well in both continuous and discrete HANOVA in type I error accuracy, power performance, and computing time. The proposed test is illustrated with a gene expression data analysis of Arabidopsis thaiana in response to multiple abiotic stresses.