Rank tests in heteroscedastic multi-way HANOVA

Abstract

This article develops rank tests for the nonparametric main factor effects and interactions in multi-way high-dimensional analysis of variance when the cell distributions are completely unspecified. The design can be balanced or unbalanced with the cell sample sizes fixed or tending to infinity. An arbitrary number of factors and all types of ordinal data are allowed. This extends the use of rank methods to the NeymannScott and triangular array problems. The asymptotic distribution of the rank statistics is obtained by showing their asymptotic equivalence to corresponding expressions based on the asymptotic rank transform. Compared with test procedures based on the original observations, the proposed rank procedures are free of moment conditions, converge to their limiting distribution faster, and have better power when the underlying distributions are heavy tailed or skewed. These advantages are demonstrated by simulations and an application to a real data set.

A distribution free test to detect general dependence between a response variable and a covariate in the presence of heteroscedastic treatment effects

“…The structure of the augmented data resembles high‐dimensional ANOVA (HANOVA) (Wang & Akritas, 2009) in which at least one factor has a large number of levels. The difference is that the data for HANOVA are independent while the augmented observations $\{U_{ict}, c=1,\ldots, N, t=1, \ldots, k\}$ are not independent since the observations are repeatedly used during the augmentation.…”

“…If $k\to\infty$ with N , then techniques from nonparametric smoothing such as locally weighted kernel regression can be borrowed as the k here would play a similar role as the bandwidth for kernel regression. For a fixed finite k , the intuition that this particular approach would work comes from the fact that the asymptotic results for HANOVA can be achieved not only for independent data (Wang & Akritas, 2009) but also for data with weak dependence (Wang & Akritas, 2010) and long range dependence (Wang, Higgins & Blasi, 2010). The augmented data vectors for a treatment level viewed in the order from the smallest to largest covariate values (from left to right in Table 1) are close to weak dependent processes as $N\to\infty$ .…”