$U$-tests for variance components in one-way random effects models

TL;DR

This paper introduces a $U$-test for assessing the variance component in one-way random effects models, offering advantages over the traditional $F$-test especially in nonnormal conditions.

Contribution

The paper develops a new $U$-test for variance components, deriving its asymptotic distribution under mild conditions and comparing its performance to the $F$-test through simulations.

Findings

$U$-test performs better than $F$-test under nonnormal distributions.

$U$-test is preferable with small sample sizes and nonnormal data.

$F$-test works well under normality and balanced designs.

Abstract

We consider a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null. This test is based on a decomposition of a $U$-statistic. Its asymptotic null distribution is derived under the mild regularity condition that the second moment of the random effects and the fourth moment of the within-treatment errors are finite. Under the additional assumption that the fourth moment of the random effect is finite, we also derive the distribution of the proposed $U$-test statistic under a sequence of local alternative hypotheses. We report the results of a simulation study conducted to compare the performance of the $U$-test with that of the usual $F$-test. The main conclusions of the simulation study are that (i) under normality or under moderate degrees of imbalance in the design, the $F$-test behaves well when compared to the $U$-test, and (ii) when the distribution of the random effects and within-treatment errors are nonnormal, the $U$-test is preferable even when the number of treatments is small.