Selection of plants and animals in a breeding program deals with experimental data for which the underlying genetic model is best formulated as a mixed linear model. The genetic model is improved by including pedigree information through an additive relationship matrix, A. This matrix can be quite large and complex for large populations involving many generations, and its inverse is required when solving the mixed model equations. Efficient methods have been developed to permit routine application of this methodology. However, its application to multiple traits or environments in crop populations, where both additive and non-additive genetic variation can be measured, raises some issues to be resolved.

While pedigree information has been used extensively in animal breeding, adoption on a routine basis in the plant breeding sphere has been much slower. In cereal breeding programs, genotype performance is typically measured in a series of replicated field trials grown across multiple locations and years, and is collectively referred to as a multi-environment trial (MET), where current MET analyses assume independence between genotypes 1. Benefits from the use of pedigree information can be two-fold. Firstly, the estimates of individual genotype performance are more accurate through the use of correlated information from relatives. In addition, breeding values can be estimated for each genotype, quantifying the potential of the individual as a parent in the breeding program.

One important aspect in the use of pedigree information in plant populations is the underlying genetic model, as additive and non-additive effects can be estimated separately 2. This partitioning is possible since field crop data are generally from plots of genetically identical material, replicated both within and across environments. The additive component provides a simple covariance structure between related lines and the non-additive component is the lack of fit to the additive one. Crossa et al. 3 have fitted a genetic model including only an additive component, ignoring the non-additive variation. In our work, we investigate the performance of these different models and comment on the magnitude of the non-additive variation. The lack of fit can also be attributed to various forms of non-additivity including dominance 4 and additive by additive interaction 5 but we have not considered these more complex models.

The most general form for the genetic variance matrix from MET data is a fully unstructured matrix with p (p + 1)/2 parameters where p is the number of environments, and this matrix is, by definition, nonnegative definite. For particular data, genotype effects are often highly correlated across some environments, leading to an estimated genetic covariance matrix that violates this condition; imposing constraints to force nonnegative definiteness leads to singular matrices, but standard REML methods require non-singular variance matrices. The magnitude of the estimation problem increases with the number of environments included, and the usual response is to replace the fully unstructured matrix with a more parsimonious approximation, the simplest of which has a common correlation across all environments. The factor analytic (FA) form introduced by Smith et al. 6 is intermediate in parsimony and is widely used in the analysis of MET data from most Australian plant breeding programs. Kelly et al. 7 have shown through simulation that this FA model is a robust model with high predictive accuracy. This model can accommodate increased correlation structure through incorporation of more factors, and can accommodate the singularity issue in the sparse matrix formulation presented by Thompson et al. 8.

When fitting a pedigree model across multiple environments, both Crossa et al. 3 and Oakey et al. 4 have adopted an FA model for the genotype by environment effects. Applications of the FA methodology have also recently arisen in the animal breeding literature, for example Meyer and Kirkpatrick 9 have fitted a constrained form of the factor model to animal pedigree data across multiple traits. However, problems arise in estimation methods for pedigree models combined with a complex variance structure across multiple environments or traits. Henderson 10 has presented a simple recursive method for computing the inverse of a relationship matrix, A^-1, without the need to form the relationship matrix A itself. More recent improvements to the methodology have come from the work of Quaas 11 and Meuwissen and Luo 12, and this efficient algorithm is currently implemented in the software package ASReml 13. For more complex variance models involving both the factor analytic and pedigree structure, the average information (AI) residual maximum likelihood (REML) methodology requires the formation of both elements of A and A^-1 for the score equations and working variables. In this paper, we present the estimation approach used in ASReml for these more complex models and show how computational efficiency is maintained by only forming some elements of A.

In summary, this paper adopts the genetic model of Oakey et al. 2 with an extension to multi-environment trial data as the prototype 4. We have investigated efficient model formulation and REML estimation of variance parameters for multiple environment/trait data using the standard approach in ASReml, with an extension for the factor analytic structure. An example of a multi-environment trial with pedigree structure is presented, and the goodness of fit of differing genetic models is considered.

The model with maximum REML log-likelihood and significant improvement in REMLRT over subsequent nested models is Model 8 in Table 3, which includes an FA structure of order 3 for additive effects, and an FA structure or order 1 for non-additive effects. It comes from a class of models proposed by Oakey et al. 4, in which Models 6 and 7 are lower order FA models, and involves fitting 71 genetic variance parameters through two FA structures. Model 8 is considered the best model based on the criterion of minimum AIC. The performance of other models is now considered in greater detail.

The simplest models for genetic variance, and those currently used in plant breeding programs in Australia, are Models 1–3, assuming independence between genotypes, (Table 3). Of these three models, the model of best fit is Model 3 with an FA structure of three dimensions. However it is inferior to all models incorporating pedigree information (Models 4–14).

The second class of model fitted, Models 4 and 5, involves only additive genetic variance, and does not capitalise on replication of genotypes and partitioning of non-additive effects. While these models are superior to those assuming independent genotypes, they are still inferior to the genetic model that partitions additive and non-additive effects, (Models 6–8).

The remaining models (Models 6–14) differ purely in the model for additive effects. The first subset (Models 6–8) involves an FA structure for of increasing dimension, and the model with maximum likelihood is taken from this subset. The reduced rank (FARR) models impose a constraint on the more general FA model, and it can be noted that this constraint results in models of poorer fit for the same number of FA dimensions. In fact, six dimensions must be fitted in the reduced rank form (FARR6) to produce equivalent likelihoods to the best FA model with three dimensions. The AIC comparison also indicates that the general FA model produces a more parsimonious form than the FARR models.

In terms of the actual estimated parameters, we observed heterogeneity of both additive and non-additive genetic variance and heterogeneity of error variances across environments. Summaries of estimates of these variance parameters from the best model for and are given in Table 4. Genetic covariance in this data set was also heterogeneous and in our experience this is also typical of most multi-environment trial data. Genetic correlations were predominantly positive but there were instances where some pairs of trials had low/zero genetic correlation.

Of greatest importance to a breeding program is the impact of new analysis models on selection decisions. By examining changes in the empirical BLUPs between competing models, we can assess any changes in the ranking of the genotypes and subsequent changes in the selected subset of genotypes. Figure 1 displays the empirical BLUPs from the 'best' pedigree model, consisting of an FA3 model for additive effects combined with an FA1 model for non-additive effects, against the empirical BLUPs from the standard FA3 model assuming independence between genotypes. These plots are demonstrated using a subset of four sites. There is close agreement in rankings of empirical BLUPs for sites (c) and (d), where only six and two different genotypes are included in the top 46 genotypes (which forms the top 10%), respectively. These sites represent those with moderate and low levels of error variance, relative to additive genetic variance, (see Table 4). For sites (a) and (b) the empirical BLUPs deviate more from the one-to-one relationship, with 17 genotypes differing in the ranks of the top 46 genotypes.

The four sites in Figure 1 were chosen to demonstrate the different types of patterns evident in genotype predictions between the competing models. The relativity of additive genetic variance to error variance varies markedly between all sites, and while there is some consistency in genotype prediction for the sites with low error variance, there are many and varied patterns for sites with low to moderate levels of additive variance relative to error. Also, no consistent pattern in genotype predictions based on the relative magnitude of additive and non-additive variance is observed. For example, the site in Figure 1(a) has a very low proportion of non-additive variance estimated in the model, while plots (b) and (c) have the same proportion of non-additive variance (relative to total variance), with vastly different patterns between predictions.

It is also obvious from the banding patterns in Figure 1(a) and 1(b) that genotypes are regressing to a different underlying response in the pedigree model. The additive component provides a simple covariance structure between related lines and the non-additive component is the lack of fit to the additive one. Incorporation of the additive covariance in the model means each line is regressed toward the level predicted by its relatives, rather than to a common level for all genotypes, and reflects the theory of breeding by selection of parents for the next generation. The banding patterns in plots (a) and (b) result from the same cross, where the performance of individuals within this cross is elevated in plot (a) and depressed in plot (b). These differential predictions demonstrate the interaction between additive genetic variance and environment.

The inclusion of pedigree information in the analysis of MET data adds to the complexity of the mixed model and associated variance structure. Most plant breeding trials consist of replicated plot data across multiple environments, with an underlying variance structure for spatial effects and heterogeneity of variance at the residual level. Current analysis methods for MET data adopt a factor analytic variance structure for genetic correlation between environments. When the pedigree structure is added to model the relationship between genotypes, the resulting mixed model is quite complex, requiring the estimation of numerous variance parameters, and subsequent prediction of random genotype effects. The capacity of existing software to fit these complex models to 'real' data sets, (see ASReml) is largely due to the use of sparse matrix methodology and the AI algorithm 13.

In the analysis of the example data set, we investigate different genetic models for multi-environment data with a factor analytic variance structure. The genetic model of Oakey et al. 2, with an extension for MET data 4, adequately captures both the additive and non-additive genetic variation across environments, and is the model of best fit to the example data used in this study. Although only a small proportion of the total variation in the example data set is due to non-additive effects, a low order factor analytic model assuming independent genotypes still improved the goodness of fit. The genetic model with only additive effects 3, may be adequate when the level of non-additive genetic variance is low. Reduced rank models were less parsimonious than those with a standard FA form, requiring estimation of many more parameters from a greater number of dimensions to achieve an equivalent goodness of fit.

In theory, non-additive effects are comprised of the higher order interaction terms between additive and dominance effects 17. In practice, the partitioning of the interaction variance is seldom more than trivial when compared with the errors of estimation 17. While it is shown to be potentially beneficial to fit a simple model for non-additive variance, we surmise that partitioning into a complex model for non-additive effects 5 is unnecessary, as these often represent a relatively small proportion of the total genetic variance.

The improvement in model fit over the current model for MET data 6 is achieved through the inclusion of the numerator relationship matrix, A. In this paper, the relationship matrix is derived from pedigree information in the breeding program, but with the proliferation of molecular marker and quantitative trait loci data, elements of the genetic relationship matrix may now be derived in different ways 18. For differing applications, the inter-individual relationships may be estimated, rather than assumed to be known, and methodology is available for estimating the elements of this correlation matrix, A. In these instances, it will not have the properties that allow A^-1 to be an easily formed sparse matrix and this will limit the population size to which this empirical A matrix can be applied.

Of greatest importance to genetic gain in a breeding program is the impact of new analysis models on selection decisions. In this paper, we consider goodness of fit of each genetic model, and the impact of changes in rankings of empirical BLUPs of genotype effects between the pedigree and standard models. A large proportion of changes occur in the rankings of the genotypes at some environments, and we assume that the pedigree model would be predicting the most accurate effects. An additional benefit to selection of individuals and parents in the program is that the pedigree model estimates and adjusts for the interaction between additive genetic effects and environment.

An alternative way of assessing the impact on selection is through an improvement in prediction error variance (pev) of the empirical BLUPs from competing models. While for the pedigree model in our study the pev was reduced on average, we commonly overlook the fact that in this type of experiments, known biases are present in the pev and the empirical BLUPs themselves. The assumption of known G is violated as variance parameters must be estimated, and resulting empirical BLUPs and pev's are formed from , not G. Studies have shown that, while the properties of BLUPs do not hold under estimation of , the factor analytic models still perform well for empirical BLUPs 7. A simulation study is required to examine the performance of empirical BLUPs for these more complex genetic models.

The authors declare that they have no competing interests.

AK completed the algebra to form mixed model estimates and validate software, and carried out the data analysis. RT and BC conceived the study, and BC and JE participated in its design and coordination. AG provided support for software and estimation methods. AK, AG, BC, and JE helped to draft the manuscript.