Prediction using full sibs (1A)

For most trait × population combinations, full-sib prediction did not reach phenotypic accuracy measured as the square root of heritability (Figure 3). However, correlated response to selection also depends on selection cycle length and the intensity of selection (Falconer and Mackay 1996). The latter can be increased markedly by shifting resources from phenotyping toward production and genotyping of many more DH lines than under traditional phenotypic selection.

For full sibs, prediction accuracy was strongly dependent on TS size and moderately dependent on trait heritability. This was expected from theoretical simulations (Meuwissen 2009), as well as from empirical work in wheat (Heffner et al. 2011a,b), barley (Lorenz et al. 2012), and oats (Asoro et al. 2011). Within an individual biparental population, genomic prediction information can come from (i) variation in relatedness (i.e., Mendelian sampling) captured by the SNPs and (ii) cosegregation of loci linked to QTL (linkage information). Artifacts originating from population structure can be assumed to be absent, which is empirically supported by (i) the homogeneous data clouds of the populations in the principal components analysis (Figure S1) and (ii) the unstructured distribution of IBS values (Figure 2 and Figure S2). Hence, within populations, it should be possible to model the expected prediction accuracy () by population and trait parameters.

In dairy cattle, deterministic predictions could be successfully used as a guide for the required TS size (Hayes et al. 2009b). Our results support this also for within-population prediction using full sibs (Figure 4). One possible reason for the systematic over- and underestimations of traits lies in deviations from the polygenic genetic architecture, which is assumed in the applied deterministic formula. Indeed, genome partitioning of genetic variance indicates a less polygenic architecture for certain trait × population combinations (Figure S9). On the other side, the robustness of RR-BLUP toward the underlying number of QTL in a high LD scenario (Riedelsheimer et al. 2012b) suggests that differences in the number of underlying QTL play a less important role here. A more likely explanation lies in the imprecise heritability estimates reflected by their large confidence intervals (Table 4). This was especially true for the yield component traits that were evaluated in only 1 year and showed the strongest deviation from the expected values. This result emphasizes the need for high-quality phenotypic data for the TS for genomic prediction.

Nevertheless, for highly heritable traits, the high correlations between actual and observed values suggest that within biparental populations, prediction accuracies can be predicted well using information of (i) training set size, (ii) heritability, and (iii) effective number of loci. Given that good heritability estimates are available, it seems possible to use deterministic formulas to optimize resource allocation between TSs and VPs for prediction using full sibs.

Notably, the minimum training set sizes required to achieve reasonable prediction accuracies within biparental maize DH populations are by at least two orders of magnitude smaller than those required in animal breeding (Goddard and Hayes 2009). This results mostly because of the smaller genome size (in morgans) and the much smaller N_e of biparental populations compared to, e.g., cattle for which N_e ≈ 100 (Sørensen et al. 2005). Thus, results from simulation or empirical studies in animal breeding should therefore be examined carefully before drawing inferences for situations in plant breeding.

Influence of TS composition

Prediction within populations using full sibs is appealing because of (i) the high accuracies achievable with a moderate TS size and (ii) the good agreement of theoretical expectations with empirical results. However, in an ongoing breeding program, phenotypic and genotypic data are available for multiple related and/or unrelated populations often before the actual cross is being made. With the current body of theoretical knowledge in the field of genomic prediction, it is not possible to answer the question of how the composition of the TS affects the genomic prediction of other crosses, which is needed to help breeders in their decision-making process. We therefore followed an empirical model-based approach with statistical testing for significant differences between TS compositions, while keeping the TS size, and hence total resource capacity, constant. TS size had to be fixed to a rather small number because 84 was the maximum possible size for which balanced comparisons among all TS compositions were possible. However, relative differences between TS compositions for which larger TS sizes were possible remained relatively constant beyond this value because prediction accuracies began to level off at this TS size (Figure 5).

The strong decline in prediction accuracy from FS (1A) to HS (1B) and to UR (1C) was expected from theory (Habier et al. 2007) as well as from recent empirical studies emphasizing that the level of relatedness between genotypes in the TS and the VP has a strong impact on prediction accuracies (Habier et al. 2010; Clark et al. 2012). Having multiple related HS families, one important question is whether resources should be spent to increase the number of crosses or the number of DH lines per cross. Here, we found no statistical difference between different numbers of involved crosses once both parents of the VP are parents of at least one cross involved in the TS. In addition, increasing the number of HS families with one parent in common with the VP did not compensate for the lack of representation of the second parent of the VP. In a genomic prediction scenario with HS families, our results suggest that an optimal TS should (i) be of sufficient total size and (ii) have both parents of the VP represented. The number of families used to generate such a TS, however, does not seem to have a strong impact (Table 4). Nevertheless, real or simulated data sets with many more HS families should be analyzed to investigate this question in greater detail.

One interesting finding of our study was the negative accuracies obtained with UR families for certain traits, especially in case of π_2,4 as the VP. We note here that the effect of an UR family could not be evaluated for population π_1,2 because its UR family π_3,4 is not present in our study. It is also important that the term “unrelated” was used to distinguish this type of cross from half sibs and full sibs. UR families still showed a substantial IBS similarity with the VP (Table S2) and were therefore not unrelated in a population genetic sense. However, since prediction accuracies became increasingly negative with increasing TS size, the UR family must have provided a negative prediction signal coming from opposite linkage phases with important QTL in the VP. Genome partitioning of genetic variance following Riedelsheimer et al. (2012b) suggests that important QTL for the measured traits are indeed especially present in population π_2,4 (Figure S9). This also indicates that causal variants might not segregate in all DH populations. In addition, linkage phase similarity measured as s_LP between population π_2,4 and its UR family was among the lowest ones of all combinations and genomic correlation measured as s_GC was negative (−0.233, Table S2). If such an UR family is combined with one HS family to a mixed TS, the negative predictive signal coming from the UR family could even counteract the positive signal coming from the HS family (decreasing curve of TS composition 2C compared to 1B in Figure S4E). Therefore it is recommended to exclude such families from the TS. This finding is also in agreement with a simulation study showing that increasing accuracies by combining different populations into a single TS requires persistence of marker–QTL LD across populations (de Roos et al. 2009).

The strong correlation (0.89 < r < 0.93) among s_IBS, s_GC, and s_LP suggests that any of these measures can be used to estimate genetic similarity between populations to identify populations with negative predictive signal. However, we found the s_GC measure appealing because of its simplicity, applicability to individual genotypes, and defined range of values that fall between −1 and +1. This measure might also be suitable in identifying outlier genotypes in the TS or the VP. In the case of monogenic or oligogenic genetic architecture with known positions of major QTL, a similarity measure that takes linkage phases in the vicinity of these QTL into account might be even more accurate than a simple genome-wide measure. Future simulation studies with large sets of related populations and known QTL positions and effects could shed more light on this issue.

Replacing a HS family with an UR family led to significant reductions in prediction accuracy only if one parent of the VP was no longer represented in the TS (comparison 2A – 2C); otherwise the difference was not significant. With a constant size of the TS, a drop in accuracy might be expected if the TS is diluted with unrelated genotypes, especially if they show reverse linkage phases. However, since all line crosses in our study originated from only four parents, the DH lines from every UR family were necessarily HS to both HS families included in the TS. For a single locus with additive effect a, letThen, it follows µ_AA – µ_BB = 2a₂ + 2a₄ – 2a₃ = 2a₁. Thus, with the assumption of no epistasis, additive effects in one population can be derived from a linear combination of the additive effects in the HS families of each parent and the UR family. In a general setting with more than four parents, the inclusion of an UR family might, in contrast to our results, reduce prediction accuracy even if both parents are represented. Careful investigation of the linkage phase similarities (or genomic correlations) between UR families and the VP should therefore be carried out before such families are included in the TS.

Influence of population and trait on TS composition

Systematic population and trait effects accompany the described contrasts among the TS compositions. Variability among traits was substantially higher than variability among VPs (Table 3) and both VP and trait effects accounted for 34.9%, which was of similar magnitude compared to the variation introduced due to different TS compositions (41.7%). This was not unexpected as previous studies also found large variability in prediction accuracy attributable to traits (corrected for their heritability) and populations (Ornella et al. 2012).

Although differences between TS compositions could be quantified and tested, the population and trait variability hampered a direct quantification of interchangeabilities, e.g., how many HS from one or more families are needed to obtain the same prediction accuracy as with a certain number of FS. For example, whereas 48 HS from two families were sufficient to replace 24 FS in the TS for predicting GER severity in population π_1,4 (Figure S4C), 72 HS were needed for population π_1,3 (Figure S4B), and 168 HS were barely enough in population π_1,2 (Figure S4A). In addition, these numbers were in most cases not transferable to other traits. This variability among traits and populations has important implications for resource optimizations. If such calculations are based on only one cross or trait, results can be misleading. Therefore, any optimization calculation should be based on a large set of crosses and results for one trait should be extrapolated to another trait only with utmost caution. Further research should focus on elucidating in more detail the reasons underlying the variability of the prediction accuracy among traits and populations. Possible factors to be investigated with very large sets of simulated or real populations and traits include differences in genetic architecture with different linkage phases between TSs and VPs in regions harboring important QTL for the trait of interest.

Conclusions

From the analysis of our set of five interconnected biparental DH populations, we draw the following conclusions:

1. Within-population prediction is feasible for highly heritable traits with moderate training set sizes and agrees on average well with theoretical expectations.

2. Prediction accuracies decline strongly if FS are replaced by HS, but statistically significantly better results can be achieved if HS families are available from both instead of only one parent of the VP.

3. Once both parents are represented, it is not favorable to increase the number of crosses.

4. Unrelated families with opposite linkage phase similarities with the VP can have negative prediction accuracies or reduced prediction accuracies if combined with related material. Such crosses should be identified using linkage phase or genomic correlation measures and excluded from the TS.

5. The observed variability among populations and traits requires resource optimization that incorporates knowledge about the variability of parameters.