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Tests for heteroscedasticity in two-way layouts incorporating a covariate applicable to agricultural frost trials

Pilkington, Angelika Aimee (2020) Tests for heteroscedasticity in two-way layouts incorporating a covariate applicable to agricultural frost trials. Honours thesis, Murdoch University.

PDF - Whole Thesis
Embargoed until December 2021.


The economic impact of frost upon the Australian grains industry is significant. This has lead to the establishment of research programs, such as the Australian National Frost Program (ANFP) to perform a broad range of experimentation including identification of frost tolerant varieties. Such experimentation necessarily involves field trials, and given the regular rectangular array of plots in the field, many agricultural field trials can readily be considered as a variant of the two-way layout experimental design.

The two-way layout is a common experimental design consisting of two experimental factors referred to as the treatment and block factor; of t and s levels respectively. Subsequent analysis through the linear model framework or ANOVA requires assumption of homogeneity of error variance. Assessing the validity of this assumption is important to avoid inconclusive or erroneous experimental results. Hypothesis tests for heteroscedasticity that are applicable to the two-way layout are well documented in the literature.

However, covariates may need to be included in the model, such as including the number of days to flowering, as measured through Zadoks scores (Zadoks et al.; 1974), to account for varieties’ plastic response in phenology. Inclusion of covariates causes issues in assessing the assumption of homogeneity of error variance as most hypothesis tests; such as those of Bartlett (1937) and Levene (1960), cannot handle covariates. Current Australian grains industry standard methods for the analysis of agricultural field trials involves use of restricted maximum likelihood (REML) methods. The likelihood ratio test is capable of handling covariates, and can be performed on models of increasingly complex error variance structure as a means of testing for heteroscedasticity.

The Clarke and Godolphin (1992) test is an exact F test based on error contrasts termed recursive residuals (Brown et al.; 1975) which tests for the presence of two error variances. Under the alternative hypothesis, the level of the block factor determines which of the two error variances an observation hypothetically has; with the last l blocks having an error variance of λ2, and the first s − l blocks having an error variance of σ2.

Being based upon the recursive residuals, the test of Clarke and Godolphin (1992) is applicable to experimental designs beyond the two-way layout and random incomplete blocks designs demonstrated in the original paper. This thesis demonstrates the application of the exact F test of Clarke and Godolphin (1992) to experimental designs that can be considered a two-way layout incorporating covariates, including elucidation of what is termed the index set, which is important to the construction of the F statistic.

The power of the exact F test for the two-way layout of 2 treatments, 35 blocks, and incorporating 1 covariate is compared with the industry standard REML methods using ASreml-R (Butler; 2018) for comparison of models through the likelihood ratio test (REMLRT).

The exact F test was most powerful for all the select values of l, the number of blocks having λ2 error variance, that were tested. The disparity between the two-sided alternative of the exact F test and the REMLRT increased for lower values of l. The exact F test can be formulated with a one-sided alternative, which proved most powerful in all cases of l tested.

Item Type: Thesis (Honours)
Murdoch Affiliation(s): Information Technology, Mathematics and Statistics
Supervisor(s): Clarke, Brenton and Diepeveen, Dean
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