A Note on Exact Tests for Variance Components in Unbalanced Random and Mixed Linear Models

Abstract

Ofversten (1993) proposed a method for constructing exact tests for variance components in unbalanced mixed linear models. The method is a generalization of a technique employed by Khuri (1987), Khuri and Littell (1987), and Khuri (1990) for testing variance components in random models. This approach employs a partitioning of the residual sums of squares to derive a set of quadratic forms that are independently distributed as chi-squared random variables. These quadratic forms are used to construct an exact test statistic for testing the null hypothesis that a variance component is equal to zero. Although the actual value of the test statistic depends on the particular partitioning, the distribution of the test statistic is invariant to this choice. Ofversten refers to this technique as resampling. Resampling tests are recommended with slightly unbalanced data because the properties of these tests are similar to the corresponding uniformly most powerful tests with balanced data. An inequality that can be used to place a bound on the power of a resampling test is derived in this paper.