Exact testing for heteroscedasticity in a two-way layout in variety frost trials when incorporating a covariate

Two-way layouts are common in grain industry research where it is often the case that there are one or more covariates. It is widely recognised that when estimating fixed effect parameters, one should also examine for possible extra error variance structure. An exact test for heteroscedasticity, when there is a covariate, is illustrated for a data set from frost trials in Western Australia. While the general algebra for the test is known, albeit in past literature, there are computational aspects of implementing the test for the two way when there are covariates. In this scenario the test is shown to have greater power than the industry standard, and because of its exact size, is preferable to use of the restricted maximum likelihood ratio test (REMLRT) based on the approximate asymptotic distribution in this instance. Formulation of the exact test considered here involves creation of appropriate contrasts in the experimental design. This is illustrated using specific choices of observations corresponding to an index set in the linear model for the two-way layout. Also an algorithm supplied complements the test. Comparisons of size and power then ensue. The test has natural extensions when there are unbalanced data, and more than one covariate may be present. Results can be extended to Balanced Incomplete Block Designs.