*3.3. Accuracy of GP Modelling by Environment, Training Population and Statistical Methods*

G × E interactions, which affects the accuracy of trait assessment, are common for plant traits. A strong G × E interaction was observed in flax PR [4]. As a consequence, different PS QTL were

identified for individual years and for the 5-year average [4]; similarly, GP efficiencies differed when individual yearly and average PS data sets were used as training sets (Table 3). The highest accuracies were obtained when the 5-year mean phenotypic data was used as training data (Table 4), suggesting that the average phenotypic data across multiple environments should be used for GP model construction. Because phenotypic values of genotypes in each year had one replication, the average phenotypic data across multiple years is actually equivalent to the best linear unbiased prediction values (BLUPs) or the best linear unbiased estimators (BLUEs). Therefore, the means across multiple environments estimate or reflect the true breeding values of a trait.

Some studies report that prediction accuracy of GP is highly affected by the size of the TP. In general, the prediction accuracy increases with TP size [21,28,29,34–36]. In the GP of seed weight in soybean, for example, prediction accuracy was sensitive to changes in TP size, which may have led to changes of relatedness between training and validation sets [21]. Lorenzana and Bernardo observed that, in an Arabidopsis family, prediction accuracy improved by 0.10 when TP size increased from 48 to 96, by an additional 0.07 when TP size was increased to 192 and by a further 0.05 with a TP size of 332 [37]. Here GP accuracy >0.9 was observed when the TP size reached 185 which slightly increased to 0.921 with a TP size of 314 (Figure 4). Large TPs provide the statistical power needed to improve prediction accuracy [38], especially for traits with low heritability [34,39]. When TP size is sufficiently large, even low heritability traits can be accurately predicted [28,40], including the low heritability PS studied therein. Diversity of the population also affect prediction accuracy [21,29,34,41–43]. A diverse TP may contain more QTL associated with selective traits and increase the correlation of the TP with validation populations (VPs) or test/prediction populations (PPs), resulting in a subsequent increase in prediction accuracy. Although some breeding lines [11,30,44] and bi-parental derived lines [25,41,45,46] are used for TPs, many studies have opted for a more diverse TP germplasm [29,41–43]. Our core collection TP preserves the variation present in the world collection of 3378 accessions maintained by Plant Gene Resources of Canada (PGRC) and represents a broad range of geographical origins, different improvement statuses (landraces, historical and modern cultivars, breeding lines) and two morphotypes (linseed and fibre types) [1,3]. This collection also contains most parents of modern Canadian flax cultivars [25]. Therefore, diverse phenotypic and genetic variabilities within the flax core collection render it useful as a resource for breeding and as a TP for GP model construction.

A variety of statistical methods have been proposed to estimate marker effects for GP. In general, GP methods are based on additive genetic models and their accuracies may vary depending on genetic architecture of target traits. According to the assumptions for statistical distributions of the marker effects, two groups of GP models have been proposed. The first group of models, such as RR-BLUP, genomic BLUP (GBLUP) and BRR, assume that all markers have some effects on the target trait and the same variance, that is, all makers contribute to the variation of the trait. The second group of models, including BayesA, BayesB, BayesC and BL, assume a specific variance for each marker. Some of these models such as BayesB, BayesC and BL, also allow variable (marker) selection when some of markers have very small or no effects. Based on these assumptions, the first group of models are expected to be useful for complex quantitative traits that have a polygenic architecture, while the second group of models are suitable for traits that controlled by a small number of genes or QTL with large effects. Several studies have shown better performance of BayesB for traits controlled by a few of genes with large effect [47–50]. Some simulation studies have also shown that BayesB outperformed GBLUP that is equivalent to RR-BLUP, when the number of QTL underlying a trait are small [47,51]. However, BayesB, RR-BLUP and other models had a similar prediction accuracy under the infinitesimal model [51] or for some complex traits [19,49]. In this study, no difference among RR-BLUP, BRR and BL was observed (Figure S1), primarily because flax pasmo resistance is a complex and polygenic trait and most of QTL associated with it had similar and small effects (Figure 2). RR-BLUP is most commonly used because of some superior features [11,14,42,52–54]. For example, RR-BLUP successfully recognized complex patterns with additive effects and delivered good GP in wheat disease resistance [55]. RR-BLUP also has a clear-cut computational efficiency compared with any other statistical models [11,54,56,57]. Here

the RR-BLUP model with the 500 QTL markers and the 5-year mean PS produced high prediction accuracy and is therefore recommended for the prediction of PR in flax.
