Reliability of pedigree-based and genomic evaluations in selected populations

Background Reliability is an important parameter in breeding. It measures the precision of estimated breeding values (EBV) and, thus, potential response to selection on those EBV. The precision of EBV is commonly measured by relating the prediction error variance (PEV) of EBV to the base population additive genetic variance (base PEV reliability), while the potential for response to selection is commonly measured by the squared correlation between the EBV and breeding values (BV) on selection candidates (reliability of selection). While these two measures are equivalent for unselected populations, they are not equivalent for selected populations. The aim of this study was to quantify the effect of selection on these two measures of reliability and to show how this affects comparison of breeding programs using pedigree-based or genomic evaluations. Methods Two scenarios with random and best linear unbiased prediction (BLUP) selection were simulated, where the EBV of selection candidates were estimated using only pedigree, pedigree and phenotype, genome-wide marker genotypes and phenotype, or only genome-wide marker genotypes. The base PEV reliabilities of these EBV were compared to the corresponding reliabilities of selection. Realized genetic selection intensity was evaluated to quantify the potential of selection on the different types of EBV and, thus, to validate differences in reliabilities. Finally, the contribution of different underlying processes to changes in additive genetic variance and reliabilities was quantified. Results The simulations showed that, for selected populations, the base PEV reliability substantially overestimates the reliability of selection of EBV that are mainly based on old information from the parental generation, as is the case with pedigree-based prediction. Selection on such EBV gave very low realized genetic selection intensities, confirming the overestimation and importance of genotyping both male and female selection candidates. The two measures of reliability matched when the reductions in additive genetic variance due to the Bulmer effect, selection, and inbreeding were taken into account. Conclusions For populations under selection, EBV based on genome-wide information are more valuable than suggested by the comparison of the base PEV reliabilities between the different types of EBV. This implies that genome-wide marker information is undervalued for selected populations and that genotyping un-phenotyped female selection candidates should be reconsidered. Electronic supplementary material The online version of this article (doi:10.1186/s12711-015-0145-1) contains supplementary material, which is available to authorized users.

selection. For this purpose, it is important to keep distinction between generations, selection candidates (all individuals in a generation) and selected individuals, and which information is used to obtain the EBV when using (S2). It is assumed throughout that the EBV P is based only on old information from the parental generation that has been used to select the parents, while the EBV is based on both the old information from the parental generation and the new information from the current generation. Following (S1) the variance of BV P and BV in progeny in the generation equals: and variance of EBV P and EBV correspondingly equals: where and generically stand for the sum of covariance terms between components and their estimates, respectively. Note that the variance of parental EBV changes when both the old and the new information are used to obtain EBV in the generation Recognizing that denominator in (S7) is equal to ! and that ! ! = gives the commonly used formula to compute the reliability of EBV P in progeny of unselected individuals: which assumes that the covariance between the parental EBV is zero and that parents are not inbred. Deviation from these assumptions can be seen in Table 4 of the manuscript. For example, in the random selection scenario the reliability of EBV in females and males in generation 20 was respectively 0.42 and 0.79, which would according to (S9) give the reliability of EBV P of 0.30 for progeny in generation 21, while accounting for covariance between the parental EBV and inbreeding gave the reliability of EBV P of 0.36. In the BLUP selection scenario this deviation was even larger due to more inbreeding in parents caused by selection.
Introduction of selection reduces genetic variability passed to the next generation and this has two effects on the reliability of EBV P and EBV for progeny of the selected individuals. The first effect is due to reduced variance of EBV P and EBV in progeny, caused by smaller variance of EBV in the selected individuals: where represents reduction due to selection, 0 < < 1 [15,31]. With truncation selection on normally-distributed EBV, = ( − ) , where denotes selection intensity and the standardized truncation point [31]. For common selection intensities, is around 0.8. The second effect is due to reduced additive genetic variance in progeny, ! , caused by smaller additive genetic variance that is passed to the next generation by the selected individuals: where ! !!! is the reliability of EBV of selection candidates [11,15,31]. The reliability of EBV P and EBV in progeny of the selected individuals is then: Another way to compute reliabilities in the selected population is using the property of BLUP that PEV: is not affected by selection [1] and equating the reliabilities in base generation and generation with the corresponding additive genetic variances [15], which gives: For example, consider a base generation of individuals that are used to generate the next (first) generation without selection, while the second generation is generated from the first generation with a selection of 20% of individuals (Table S1). It is  (Table S1).
The expressions for the reliability of EBV P (S12) and EBV (S13) are recursive, using variances from the previous generation. Over several generations of continuous selection an equilibrium is attained and additive genetic variance stabilizes at the so called equilibrium additive genetic variance [15,11]. The equilibrium reliability of EBV P and EBV in progeny of selected candidates is then [16]: (S17) These expressions (S16-17) give the same values as the expression (S15). The advantage of expressions (S16-17) is that they can be used without knowing the value of additive genetic variance in the generation of interest, which is commonly unknown. Comparison of (S16) and (S17) over a range of selection intensities shows how selection reduces the equilibrium reliability of EBV P and EBV ( Figure S1). As shown in the example this reduction is greater for the equilibrium reliability of EBV P than of EBV, i.e., a reduction factor for EBV P is 1 − and for EBV it is 1 , This difference arises because EBV P is based on the selected parental EBV that have reduced variance and this old information has low predictive ability of the true BV. At the extreme, when only a single pair of parents is selected, reliability of EBV P is zero as there is no variation in EBV P , while BV and EBV of progeny vary due to recombination and segregation of the parental genomes.
In the example (Table S1) the reliability of EBV P in equilibrium is reduced to ! !,! ! = 0.069, a reduction of 0.306, while the reliability of EBV is reduced to ! ! ! = 0.628, a reduction of 0.122.
When selection intensity and reliabilities are different in male and female selection candidates (S16) can be generalized to [16]: where ! and ! respectively represent reduction due to selection of sires and dams, and: