Two-way layouts incorporating covariates are widely used in agricultural experimentation, where the objective is the precise estimation of variety effects while accounting for environmental variation. Such experiments are routinely analysed using mixed linear models fitted with residual (restricted) maximum likelihood (REML) methods following Patterson and Thompson [Reference Patterson and Thompson3]. Although these models readily accommodate random row effects, for example, replications and complex covariance structures, they generally assume homogeneous residual variances, despite environmental stresses or even delineation in variety types either latent or not, frequently producing heterogeneous variability across sections of an experiment. Detecting such heteroscedasticity is therefore an essential component of model building, as misspecification of the residual variance structure may influence estimates of variety effects, their standard errors and associated significance tests [Reference Pilkington, Clarke and Diepeveen6].

The motivating application is the Australian National Frost Program (ANFP), established to improve the assessment of frost tolerance in Australian wheat and barley varieties. Since plant susceptibility depends strongly upon developmental stage, flowering time provides an important explanatory variable in frost trial analyses. The ANFP estimated flowering dates from repeated observations of crop development using the decimal growth-stage code of Zadoks et al. [Reference Zadoks, Changi and Konzak7], allowing predicted days to flowering (Z45 for barley, Z65 for wheat) to be incorporated as covariates in mixed-model analyses.

The thesis [Reference Pilkington4] addresses testing for two-section residual heteroscedasticity in balanced or unbalanced mixed two-way layouts incorporating covariates, although examples in the thesis are illustrated only for balanced designs. Clarke and Godolphin [Reference Clarke and Godolphin2] derived an exact finite-sample F test for comparing two error variances in balanced experimental designs by constructing orthogonal contrasts that partition residual information into independent quadratic forms whose ratio follows an exact F distribution. Although elegant, the original development applies to fixed-effect models without covariates and cannot be directly implemented within the mixed-model framework used in agricultural research. Extending this exact procedure to mixed models while retaining its finite-sample properties constitutes the principal methodological contribution of the thesis.

The thesis reviews linear mixed models and residual (R) variance structures appropriate for sectional heterogeneity, emphasising that while REML provides considerable flexibility for fitting complex variance models, inference for variance parameters generally relies upon likelihood ratio tests with only asymptotic validity. Boundary constraints on variance parameters further complicate hypothesis testing, motivating the development of an exact finite-sample alternative. A major theoretical contribution is the extension of the Clarke and Godolphin exact F test [Reference Clarke and Godolphin2] to mixed two-way layouts incorporating covariates. The principal challenge is constructing the uncorrelated residual vector required by the original theorem after accounting simultaneously for random effects and fixed covariates. The resulting procedure preserves the exact F distribution of the test statistic under the null hypothesis of homogeneous residual variances, while extending the method to the mixed-model framework used in practice.

The derivation further demonstrates how design-specific orthogonal contrasts are obtained through an index-set formulation determined entirely by the experimental design. This preserves the independence of the quadratic forms while accommodating covariates and extends the framework developed for fixed effects by Pilkington et al. [Reference Pilkington, Clarke and Diepeveen5]. The thesis also shows that the exact-test formulation uses more residual information than analyses based on two separate two-way layouts, improving estimation of the two variance components and clarifying the associated degrees of freedom. The finite-sample properties of the proposed procedure are examined through Monte Carlo simulation studies based on mixed-model representations of ANFP wheat trials. Simulated data were generated under both homogeneous and heterogeneous residual variance structures, with and without covariates, and the proposed exact F test was compared with the Restricted Maximum Likelihood Ratio Test (REMLRT) implemented in ASReml-R [Reference Butler1]. Whereas the REMLRT depends upon asymptotic theory, the proposed test maintains exact finite-sample inference, and is proved optimal when a section involves only one variety and is potentially optimal otherwise.

Simulation results demonstrate that the exact F test maintains its nominal Type I error rate across the scenarios considered, whereas the REMLRT may depart from its nominal size, particularly when relatively few varieties contribute to the heterogeneous variance section. The exact F test also exhibits consistently greater empirical power while preserving exact size. The apparent power of the REMLRT in some settings is shown to result largely from inflated false-positive rates arising from its approximate reference distribution. These findings indicate that the exact F test is preferable whenever experimental designs permit its application.

The methodology is illustrated using Western Australian ANFP wheat variety trials, with mixed models incorporating flowering time as a covariate together with random block effects and competing residual variance structures. The analyses demonstrate that identifying residual heterogeneity is an important preliminary step before interpreting varietal differences and that improved modelling of the residual variance structure leads to more reliable standard errors, confidence intervals and tests of fixed factor effects.

The thesis concludes that extending the Clarke and Godolphin exact F test [Reference Clarke and Godolphin2] to mixed two-way layouts incorporating covariates provides an important addition to the statistical methodology available for agricultural experimentation. The procedure combines exact finite-sample inference with the flexibility of modern mixed-model analysis while overcoming the limitations of asymptotic likelihood ratio testing. Although motivated by frost variety trials, the methodology is applicable to a broad class of experimental designs as shown through the software provided, giving a foundation for future extensions to unbalanced designs, multiple covariates, balanced incomplete block designs and robust procedures for variance-component testing.