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.


Introduction
This paper implements a test for a specific form of heteroscedasticity which was defined for a general form of experimental design proposed in Clarke & Godolphin (1992).The specific case when there are covariates in the two-way layout is addressed.The computational aspects of implementing the test in this case in order to maximise power and at the same time having an exact test are discussed in detail expanding on both Clarke & Godolphin (1992) and Clarke & Monaco (2004).
Experimental designs that incorporate two-way layouts are typically used for analysing varieties of wheat and barley, respectively, for example, in the study of varieties in frost trials which are very important in the case of the Australian National Frost Program (ANFP) funded by the Grains Research and Development Corporation (GRDC).The economic impact of frost on the Australian grains industry is significant.This has lead to the establishment of research programs 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.
From a statistical point of view, it is important to ascertain estimates of variety effects.Also typically of natural importance are the confidence intervals of the 'true' variety effects, which are usually arrived at with an assumption of independent normal mean zero and variance σ 2 errors in the linear model associated with the two-way layout.Failure to satisfy an assumption of homogeneity of variance would necessarily affect any confidence intervals, and there is a need to know whether error variances can be assumed to be homogeneous.
As careful laying out of experiments and data collection is overseen by those doing the experimentation, and the forces of nature are at work, the assumption of normal errors may be reasonable, simply due to the central limit theorem.Nevertheless, there are potentially other forces at play, one which is often quoted, is that increased response values can often lead to increased error variance.See Snee (1982) and Kendall (1975) for interesting variations on this.Without correcting for this extra variation, should there be some, quoted confidence intervals for variety effects may not be valid, and variety comparisons may then be misleading.Another way that error variances can be different is when several varieties come from the same pedigree, which may then lead to increased responses on average and again extra variation.These are instances of what is generally termed heteroscedasticity.
In the context of our analysis of frost trials, we are often faced with data from two-way layouts, where the response is yield, and there are t replications (treatments) and s varieties (blocks).
However, often covariates are necessary in the model, such as when including the number of days to flowering, as measured through Zadoks scores (Zadoks, Changi & Konzak 1974).The latter would account for varieties' varied response to the environment 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.For example, estimated residuals from different varieties are correlated, which precludes straightforward use of tests that are essentially defined for one-way layouts.Testing heteroscedasticity for the two-way layout for a more general form of departure from constant variance is examined in Shukla (1972Shukla ( , 1982)), however, these tests are not easily applicable when there are covariates.
There are many approaches to testing for heteroscedasticity in the two-way layout, mostly based on asymptotic assumptions where both the number of treatments, t, and the number of blocks, s, are assumed to be large, which is not necessarily the focus of our work.Examples of works that rely on asymptotic results include research on panel data as in Bai (2009), Feng et al. (2020) and work using principal components in Bai & Ng (2002), Fan, Liao &Mincheva (2013), andCressie, Bertolacci &Zammit-Mangion (2022) which can include covariates.The advantage of our approach is that either s or t or both s and t, can be small and yet useful information can be gleaned from an exact test of heteroscedasticity even when there are covariates.Also the design does not have to be balanced.
Current Australian grains industry standard methods for agricultural field trials involve using 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.Yet the testing is done on the basis of an asymptotic approximation to the distribution of the test statistic.The size of the test in practice may not reach the nominal levels.
In this paper, we consider the statistical size and power of a test arrived at through the study of recursive residuals in Brown, Durbin & Evans (1975) which is discussed in Clarke & Godolphin (1992), where a covariate is added in the specific application to the two-way layout.The test is compared with the asymptotic likelihood ratio test based on REML.We consider a specific example of a frost trial on barley variety data as an application.
In addition in the Appendix we illustrate the formulation of the test, for a design with t = 3, s = 4 and two covariates, and with l = 1 block being considered to have a different variance from the other blocks.

Two-way layout with no covariates
The fixed effects model for the two-way layout can be where μ is some overall mean, α i is the i th treatment effect, β j is the j th variety effect, and the ij s are the independent unobserved errors which are assumed normal with mean zero and variance σ 2 under the null hypothesis.For a balanced two-way layout with one observation per cell, we shall, without loss of generality, assume that the sum of the treatment effects and the sum of the variety effects is zero.
Tests for heteroscedasticity in the two-way layout considered here are discussed in Clarke & Monaco (2004) where it is shown, at least empirically, that one test, proposed by Clarke & Godolphin (1992), has superior power to many proposed alternatives.It is noted a classical approach is to model any possible differing error structure that occurs, using ASReml-R as in Butler (2022), and use the asymptotic likelihood ratio test to test for heteroscedasticity.See Patterson & Thompson (1971) and Harville (1977) for implementation of the asymptotic likelihood ratio test generally based on error contrasts.
To be clear, one person's treatments are another person's blocks.Without loss of generality, we follow the discussion of Clarke & Godolphin (1992) and use t to be the number of treatments (rows which could be replications) and s to be the number of blocks (in this case varieties), with s − l varieties having variance σ 2 and the last l varieties having variance λ 2 .Here it is easiest to refer to those observations having error variance σ 2 as observations in section a and likewise observations having error variance λ 2 as observations in section b.The scenario is illustrated in Figure 1.The n × k design matrix for the two-way layout, ordered treatments within blocks; with the rows associated with λ 2 observations being the last p equal to l × t rows is Here 1 n is the n × 1 vector of ones, I s represents the s × s identity matrix for example, and ⊗ is the usual Kronecker product.The partitions X a and X b are formed from the rows of X † associated with the observations in section a and b respectively.For the uninitiated the formulation of a design matrix is spelled out in Clarke ( 2008, examples 1.7 and 1.8), for example.See also Clarke (2008, formula (5.6)).
Briefly the model contemplated in vector matrix notation under the null hypothesis is There are a total of n = st observations; ordered as treatments within blocks, effectively creating an st × 1 vector Y † .The vector β = [μ, α 1 , . . ., α t , β 1 , . . ., β s ] and † is a vector of independent normal errors each with error variance σ 2 .Since in this paper we are examining for a different error variance structure under the alternative hypothesis, as in Figure 1, the error variance structure is dichotomous.The observations of the last l levels of the blocking factor have an error variance of λ 2 , whereas the observations of the first s − l levels of the blocking factor have an error variance of σ 2 .That is, the last p = lt observations have an error variance of λ 2 , whereas the first n − p observations have an error variance of σ 2 .Hence the variance structure of the errors in † can be described in terms of two sections.The sections are based on the levels of the blocking factor, with the first section being associated with the first s − l levels of the blocking factor and the second section being associated with the last l levels of the blocking factor.In this example, the errors are independent and identically distributed within each section.The subsequent variance structure for the errors in this example is of the form where φ = σ 2 , λ 2 .The design matrix X † , is of rank r ≤ k .It can, as is contemplated here, be less than full rank.In reordering the rows of the above vectors, a new design matrix can be partitioned as where the rows of X † may need to be reordered to achieve the ordering of rows as implied by the partitions of X.The partitions of X, given in (3), are an r × k index set partition, X 0 , of full row rank r; and the (n − r) × k star partition, X * .The sub-partitions of each partition, denoted by the subscripts a and b, indicate that sub-partition only contains rows of the design matrix associated with the variance σ 2 observations and variance λ 2 observations respectively.The sub-partition X 0a is chosen to be of full row rank i , where i = rank (X a ).
The response vector denoted as Y † , is ordered treatments within blocks, as per the ordering of rows of the design matrix X † ; and the response vector denoted as Y is according to the ordering of the rows of X.The observations corresponding to the matrix X 0 are often called an index set.See Theil (1965); Brown, Durbin & Evans (1975).
In illustrating this, we consider as above a balanced two-way layout with one observation per cell, and later discuss situations of unbalanced designs.
An appropriate choice of the index set for the two-way layout design, equivalent to the choice of r rows of X † to form the X 0 partition, is as follows: the t rows of the design matrix X † associated with observations from block one (the first variety) with variance σ 2 , and the remaining s − 1 rows of the design matrix partition associated with observations from the first row of the two-way layout (which could be replication or treatment).Refer to Figure 2. The remaining n − r = t × s − t − s + 1 rows of X † are partitioned into X * .Finally, the rows within each of the X 0 and X * partitions are placed in the respective a or b sub-partition, if the row was associated with an observation having σ 2 or λ 2 error variance respectively to complete the partitions of X as in (3).
The exact F -test of Clarke & Godolphin (1992) requires the construction of a matrix C to satisfy the relations CX = 0 and CC = I n−r .
Figure 2. Assignment of observations to the index set for the two-way layout with one observation per cell.Cells marked with '×' or '0' have the associated rows of X † assigned to the index set sub-partition X 0a or X 0b , respectively.
To be clear several matrices which are necessary intermediary steps in calculating of a matrix C are given.These are as defined in Clarke & Godolphin (1992).The (n − r) × r matrix Z, is a matrix such that X * = ZX 0 and is defined as Next define the (n − r) × n matrix and where C * is the lower triangular solution of The choice of a lower triangular solution of the matrix C * gives a natural delineation of the information on the variance of the vector w = CY, where in fact Here in the case of the two-way layout ν 1 = (t − 1)(s − l − 1) and ν 2 = (t − 1)l .In fact var(w ν 1 ) = σ 2 I ν 1 and the cov(w ν 1 , w ν 2 ) = 0, while var(w ν 2 ) is diagonalisable to have elements with, assuming without loss of generality λ 2 > σ 2 , values strictly greater than σ 2 and less than or equal to λ 2 on the diagonal.See Theorem 6.1 in Clarke & Godolphin (1992) or, though with different notation, Problem 10-1 in Clarke (2008).Also note that under the null hypothesis var(w) = σ 2 I n−r .The key to this delineation is that the column space of the matrix X * a is contained in the column space of X 0a written as Here the fact that the i rows of X 0a are of full row rank i imply that the rows of X 0a span the row space of X a ; which necessarily implies that (9) holds.
The choice of a lower triangular solution C * is important.It corresponds to using the recursive residuals of Brown, Durbin & Evans (1975).See Clarke (2008, Chapter 10) for example.
The subsequent F -statistic for testing equality of variances is then which can be used to test the null hypothesis H 0 : λ 2 = σ 2 against H 1 : λ 2 > σ 2 in a one-tailed test, or a hypothesis of H 1 : λ 2 = σ 2 in a two-tailed test.Remarkably in the case of the former, it is shown that the test is the exact likelihood ratio test in Lemma 7.1 of Clarke & Godolphin (1992) when l = 1.When l = 1 the test is Test 2 of Russell & Bradley (1958).It is not known whether this test corresponds to the exact likelihood ratio test more generally, however, Clarke & Monaco (2004) show that the test performs more powerfully than many other tests of heteroscedasticity, including those of Shukla, when applied to this specific difference in error variances and does better than the asymptotic likelihood ratio test afforded by ASReml-R as in Butler (2022).It can be remarked that the size of the F -test is exactly the value of the significance level, whereas the asymptotic likelihood ratio tests are usually not exactly the nominal significance level in small samples under H 0 .

Two-way layout with covariates
There are many instances where there can be an extra covariate, or possibly several covariates.Briefly, we now consider the model as previously but with the addition of a covariate score which, if real valued, could be such that Again, we assume parameter restraints α • = t i =1 α i = 0 and β • = s j =1 β j = 0.The assumptions regarding the ij s errors are unchanged from the case with no covariates Here the variable z ij is the Zadoks score for the i th treatment and the j th variety.This results in an additional t + s + 2th column in the subsequent design matrix X † leading it to have rank t + s as opposed to rank t + s − 1 in the previous section.The vector β = [μ, α 1 , . . ., α t , β 1 , . . ., β s , γ ] and the linear model is as in (1) where now the design matrix X † is We can include the case of q covariates where then z changes from being a ts × 1 vector to being a ts × q matrix.Naturally, it is assumed q < (t − 1) × (s − l − 1).If there are q > 1 covariates, and if q extra observations from the second 2 down to the s − l th columns of the two-way layout can be chosen to make the 'index set' of full row rank r, so that ( 9) is satisfied, and the conditions of theorem 6.1 of Clarke & Godolphin (1992) are satisfied (See Figure 3 when q = 1).
Figure 3. Assignment of observations to the index set for the two-way layout incorporating a covariate, with one observation per cell.Cells marked with '×' or '0' have the associated rows of X † assigned to the index set sub-partition X 0a or X 0b respectively.
The test of H 0 : λ 2 = σ 2 against either the one-or two-tailed alternative, is based on the F -statistic with ν 2 = (t − 1)l and ν 1 = (t − 1)(s − l − 1) − q degrees of freedom.As a final remark, given that i = rank (X a ), the values ν 1 and ν 2 can be expressed generally as ν 1 = n − p − i and ν 2 = p − r + i respectively.An illustrative example of how to formulate the vector of response variables and associated design matrix when there are t = 3 treatments, s = 4 blocks, a section with l = 1 block having potential differing variance λ 2 , and two covariates is given in the appendix.See also Figure A1.

Linear mixed models
The linear mixed model is a generalisation of the usual fixed effects linear model to incorporate random effects, which are parameters treated as random variables.The usual fixed effects linear model includes the random component of the errors, ; and random models estimate the 'fixed' mean term μ; as such, both fixed and random models can be regarded as specific cases of the more general mixed model.

The linear mixed model equation
The linear mixed model equation is of the form: where: • Y is an n × 1 vector of observations on the response variable, • β is a k × 1 vector of fixed effects, • X is an n × k design matrix for the fixed effects, • u is a q * × 1 vector of random effects, • W is a n × q * incidence matrix for the random effects, and • is an n × 1 vector of random errors.
It is assumed that the distribution of the random vectors is of the form, where the covariance matrices G and R are functions of parameters in τ and φ respectively.Under the usual assumptions of normality, Suppose the model is a two-way linear mixed model with one fixed and one random factor, such that u ∼ N 0, σ 2 u I q * and ∼ N 0, σ 2 e I n .Then the variance of Y, following the form of ( 12) is Relating this back to (11), it can be seen that for this example G = I q * , τ = σ 2 u , R = I n , and φ = σ 2 e respectively.

R structures
The following explanation of R structures is necessarily based on the ASReml-R reference manual (Butler et al. 2017, p. 10).Variance structures for the random errors in are referred to as R-structures.Imposing such a structure is required when the errors are indexed by, and can be partitioned into, sections on based on factor(s) whereby each section has its associated variance matrix.The variance structures of any given section need not have the same form as other sections.
If there is more than one section, then the data ordered by sections have an error variance structure which can be described using the direct sum of the error variance structures of the individual sections.Hence, the errors are partitioned into sections as = 1 , 2 , . . ., v and the R structure is of the form where ⊕ is the usual direct sum.Similar to the R structure for imposing complicated variance structures on the random error term, variance structures for the random effects in u are imposed through G structures.For further information regarding G structures, and other facets of ASReml-R, the reader is referred to the ASReml-R Manual (Butler et al. 2017).

Relevance of R structures
The R structures are of particular relevance to performing tests for assessing the assumption of homogeneity of error variance through the likelihood ratio test.In the scenario in this paper R (φ) = Σ (φ) is given in (2).Complicated variance structures imposed through fitting R and G structures involve the use of software capable of fitting linear mixed models, such as ASReml-R; the estimation procedures of which are based upon (restricted) likelihood methods.Tests for heteroscedasticity which are applicable to this example, therefore include the likelihood ratio test, which would compare the likelihood of the model fitted with the R structure (2) and a model fitted with all observations having identically distributed errors (hence σ 2 = λ 2 ) to test if σ 2 and λ 2 were significantly different.

The likelihood ratio test (LRT)
Suppose that a random sample of n random variates Y = (Y 1 , . . ., Y n ) is selected from an assumed probability distribution, and that the likelihood function L (θ |Y) is a function of nuisance parameters and θ .A hypothesis test regarding the parameters in θ can be performed using the likelihood ratio test.The null hypothesis specifies that θ lies in a particular set of possible values, say, 0 ; the alternative hypothesis specifies that θ lies in another particular set of possible values 1 .The model associated with the null hypothesis is nested in the model under the alternative hypothesis; that is to say, that the parameter space of 0 is a subset of the parameter space of 1 .For the likelihood ratio test of the hypotheses where L ˆ 0 denotes the likelihood function evaluated at the maximum likelihood estimates of θ subject to the restriction that θ ∈ 0 , and L ˆ 1 denotes the likelihood function evaluated at the maximum likelihood estimates of θ subject to the restriction that θ ∈ 1 .
The rejection region of the test is determined by ≤ k , where k is chosen such that the significance level α remains at the prespecified significance level.It can be demonstrated that takes values of 0 ≤ ≤ 1 (Hocking 1996, p. 84).A value of close to 0 indicates that the likelihood of the data under the null hypothesis is small compared with the likelihood under the alternative hypothesis; that is to say that the data support the alternative hypothesis.A value of close to 1 indicates that the likelihood of the data under the null hypothesis is similar to the likelihood of the data under the alternative hypothesis.So there is insufficient evidence to reject H 0 .
Under general regularity conditions for f (Y|θ), for sufficiently large n, the distribution of −2 log is approximately χ 2 with degrees of freedom equal to the number of parameters, or functions thereof, which have been assigned specific numerical values under H 0 (West, Welch & Galecki 2014, p. 35).The value of n considered large depends on the specific problem.This result is valuable as it allows the use of χ 2 tables in determining the rejection region for a prespecified, fixed α level when n is large.

The LRT applied to REML models (REMLRT)
The likelihood ratio test is not appropriate to use on models fitted with the REML method to estimate the variance parameters, if under the null and alternative hypotheses the fixed effect specification differs (Verbeke & Molenberghs 2000).This occurs since the REML estimation procedure involved distributions free from the fixed effects, hence comparing the likelihoods from such models is not appropriate.For tests involving only variance parameters; which includes tests for heteroscedasticity performed through comparison of fitted REML models of increasingly complicated variance structures, the REMLRT is appropriate.
The appropriate degrees of freedom for the χ 2 distribution is the number of covariance parameters of the alternative hypothesis model subtract the number of covariance parameters of the null hypothesis model.So long as such a test does not involve testing if covariance parameters lie on the boundary of the parameter space, the test statistic calculated from the −2×REML log-likelihood of the model under the alternative hypothesis subtracted from the −2×REML log-likelihood of the model fit under the null hypothesis is asymptotically χ 2 distributed with the aforementioned degrees of freedom (West, Welch & Galecki 2014, p. 35).It should be noted, however, that in the case of the one-tailed hypothesis test, if we write λ 2 = σ 2 b + σ 2 , where σ 2 b > 0 under the alternative hypothesis, then σ 2 b is on the boundary under the null hypothesis.In this case, the asymptotic distribution under the null hypothesis is a mixture of (1/2)χ 2 0 + (1/2)χ 2 1 .See for example Moran (1971), Stern & Welsh (2000), Crainiceanu & Ruppert (2004).

Exact test versus the REML likelihood ratio test
To illustrate our comparison of the tests, we compare the exact F -test of the balanced two-way layout with the REML likelihood ratio test for the same when there is a covariate, in this case, the Zadoks score as mentioned in the introduction.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 2022) for comparison of models through the restricted maximum likelihood ratio test (REMLRT).
The exact F -test was most powerful for all the select values of l , the number of blocks having λ 2 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.See Figure 4.

Type I error rate (size) of the tests
The asreml procedure struggled to estimate the error variances in the simulation study under the null hypothesis, with simulated values for λ 2 = σ 2 = 1.This was especially the case for l = 1; a single block having error variance λ 2 ; resulting in the unusually low value of 0.009 for the size of the REMLRT lrt.asreml (Table 1).
The exact F -test, for both the one-and two-sided alternative, consistently demonstrated a type I error rate closer to the expected rate of α = 0.05 than the REMLRT; the latter show a type I error rate consistently above 0.05 for the selected values of l greater than 1 (Table 1).

Barley dataset example
The barley example dataset is from the Australian National Frost Program (ANFP) trial grown in 2014 at Wickepin, Western Australia.For the barley data, from the fifth time-of-sowing, there were 35 varieties, which we will consider to be the blocks, as this framework allows us to apply the Clarke & Godolphin (1992) exact F -test to compare the error variance of varieties.There were two field replicate blocks, which can be considered as the treatments.Hence for this dataset, the number of blocks is s = 35, and the number of treatments is t = 2.The number of days to flowering (days to Z49) were also recorded at the individual plot level, and is included as a covariate; q = 1.In addition the response variable is yield calculated as t/ha.
Suppose it was reasonable to suspect that the grain yield of barley varieties with larger variety effects (higher variety index) have larger variances associated with their yields.That is, in the case of heteroscedasticity considered in this paper, the variance increases dichotomously with increased mean yield.This suspicion might be warranted upon inspection of a plot of the observed yield versus the variety index (Figure 5), with the first s − l = 9 varieties appearing to have lower variability in yield.Therefore the example will proceed with l = 26 blocks (varieties) hypothesised as having an error variance of λ 2 under the alternative.
A one-tailed test, having suspected that the variability of yields for the first s − l = 9 varieties (implying l = 26) was lower based on the yield versus variety index plot (Figure 5).However, the one-tailed F -test using the statistic (10), with the relevant alternative hypothesis H 1 : σ 2 < λ 2 is not significant (P = 0.1032).The recorded F = 2.5335 and the P -value was calculated for an F -statistic with degrees of freedom ν 2 = 26 and ν 1 = 7.The REMLRT  is not significant at the 5% significance level (P -value = 0.26085) based on the mixture distribution (1/2)χ 2 0 + (1/2)χ 2 1 ; and so, we do not reject the null hypothesis based on the REMLRT.We do not have sufficient evidence that σ 2 and λ 2 are significantly different.In this example, the exact test supports the conclusion based on the approximate REMLRT test.Both tests come to the same conclusion of homogeneity of variance.The proposed F -test is more reliable because it has an exact size and demonstrates increased power.
As a final note for fitting a model with a covariate being the Zadoks score, the z -ratio for the covariate coefficient reported under the null hypothesis was −2.013 and under the alternative hypothesis of different variances the z -ratio is −1.986.Both are significant at the 5% level of significance, thus justifying the fitting of a covariate in this instance.
An example R-script for calculating the REMLRT statistic is available in the Supplementary Material.

Discussion
There are some natural extensions to this test.First, both tests can be generalised to when there are unbalanced designs, for example, if there are loss of observations in the two-way layout.Barring the unforeseen circumstance that there may be a loss of connectivity in the two-way layout, as contemplated in Godolphin & Godolphin (2001), see also Godolphin (2018), there is no impediment to calculation of the F -exact test of heteroscedasticity.Extensions to Balanced Incomplete Block designs are also possible as ) Figure A1.Assignment of observations to the index set for the unreplicated two-way layout with s = 4, t = 3, l = 1 and incorporating two covariates.Cells marked with '×' or '0' have the associated rows of X † assigned to the index set sub-partition X 0a or X 0b respectively.
Similarly, the vector of the response Y, with elements ordered according to the order of the rows of X(A2) would therefore be such that Y = 4.72 4.74 6.21 6.95 4.82 8.11 5.48 2.15 5.07 6.42 2.21 6.64 .
A test of H 0 : λ 2 = σ 2 vs. H 1 : λ 2 > σ 2 yields a P -value of 0.242 which implies F is not significant at the 5% significance level.Hence the assumption of homogeneous variances is not rejected.

Supporting information
Additional supporting information may be found in the online version of this article at http://wileyonlinelibrary.com/journal/anzs.
The first doument "File 1" is the Readme file which contains the important information concerning all the other files.Hence please read and study this file first.

Figure 1 .
Figure 1.Two-way layout with t treatments, s blocks and one observation per cell.

Figure 5 .
Figure 5. Plot of grain yield (t/ha) versus variety index for the ANFP barley TOS 5 2014 dataset.

Table 1 .
Simulated sizes of tests for design with s = 35 blocks, t = 2 treatments and q = 1 covariate.