AN EXACT TEST OF TWO SECTION ERROR VARIANCE HETEROSCEDASTICITY IN MIXED TWO-WAY LAYOUT INCORPORATING COVARIATES EXPERIMENTAL DESIGNS APPLICABLE TO AGRICULTURAL VARIETY FROST TRIALS

Abstract

Two-way layouts with covariates are common in grain industry research, where assessing extra error variance structure is essential when estimating fixed effects. This thesis illustrates an exact F test for heteroscedasticity in such settings, using data from Western Australian frost trials. While the fixed-model algebra for the test exists in earlier literature, computational challenges arise when extending it to two-way mixed models with covariates. The proposed exact F test applicable to mixed models shows superior power and maintains exact size, making it preferable to the commonly used Restricted Maximum Likelihood Ratio Test (REMLRT) with its approximate distribution. Test formulation involves constructing design-specific contrasts by selecting observations via an index set. Size and power comparisons with the REMLRT are presented. A graphical method is also introduced to help select l, the number of blocks in the second section, when block ordering is available but l is unknown. The exact F test can be extended to unbalanced data, designs with multiple covariates, and Balanced Incomplete Block Designs. Future work aims to develop robust versions of the exact test.

The authors' abstract, as published at the source. Bulletin of the Australian Mathematical Society, 2026 · DOI ↗