The accuracy and bias of single-step genomic prediction for populations under selection

ABSTRACT

In single-step analyses, missing genotypes are explicitly or implicitly imputed, and this requires centering the observed genotypes, ideally using the mean of the unselected founders. If genotypes are only available on selected individuals, centering on the unselected founder mean is impossible. Here, computer simulation is used to study an alternative analysis that does not require centering genotypes but fits the mean µ_g of unselected individuals as a fixed effect. To improve numerical properties of the analysis, centering the entire matrix of observed and imputed genotypes, using their sample means can be done in addition to fitting µ_g. Starting with observed diplotypes from 721 cattle, a 5 generation population was simulated with sire selection to produce 40,000 individuals with phenotypes of which the 1,000 sires had genotypes. The next generation of 8,000 genotyped individuals was used for validation. Evaluations were undertaken: with (J) or without (N) µ_g when marker covariates were not centered; and with (JC) or without (C) µ_g when all marker covariates were centered. A pedigree based evaluation was less accurate than genomic analyses. Centering did not influence accuracy of genomic prediction, but fitting µ_g did. Accuracies were improved when the panel comprised only QTL, models JC and J had accuracies of 99.2%; and models C and N had accuracies of 85.6%. When only markers were in the panel, the 4 models had accuracies of 63.9%. In panels that included causal variants, fitting µ_g in the model improved accuracy, but had little impact when the panel contained only markers.

In pedigree based analyses, the expected value of breeding values is zero. In order to achieve similar properties in whole-genome analyses, marker genotype covariates are often transformed. When all individuals are genotyped, it has been shown that inference on genotype effects does not depend on how the covariates are transformed (Strandén and Christensen 2010). However, when data includes genotyped and non-genotyped individuals, inference on marker effects from single-step analyses may depend on how the covariates are transformed. In single-step analyses using marker effects models, the breeding values of non-genotyped individuals are partitioned into components representing the prediction of non-genotyped individuals conditional on their genotyped relatives and an independent deviation (Fernando et al. 2014). The prediction of non-genotyped individuals conditional on their genotyped relatives is done based on best linear prediction, which requires the first moments to be known without error. This is straightforward if the mean of the genomic breeding value is zero in the absence of selection. Centering the observed genotype covariates using what their means would be in the absence of selection would result in genomic breeding values with null means. However, such genotype covariate means are typically unavailable. Fernando et al. (Fernando et al. 2014) proposed a solution for the marker effects model that involves fitting an additional fixed covariate that estimates the mean µ_g of the linear component of the genotypic value, which is denoted by a_i in (1) below, in a population where selection is absent. Using that approach, even when there is selection, the selection process can be ignored (Goffinet 1983; Gianola and Fernando 1986; Im et al. 1989; Sorensen et al. 2001). In Markov chain Monte Carlo analyses, centering results in better mixing (Strandén and Christensen 2010), reducing the number of iterations required to obtain converged genomic predictions. In practice, centering the entire matrix of genotype covariates, including the observed and imputed genotypes, using their sample means can be done in addition to the Fernando et al. (Fernando et al. 2014) approach of fitting µ_g. This type of centering of the entire genotype matrix does not affect inference on marker effects.

The same issue with centering of the observed genotype covariates that we described above for the marker effects model is also implicit for the single-step breeding value model (single step GBLUP), and a similar solution was proposed by Vitezica et al. (Vitezica et al. 2011). In their proposed solution, the observed genotype covariates are centered using their means, and in addition the genomic covariance matrix is corrected for the change in the mean breeding value of the genotyped individuals (Vitezica et al. 2011). It was shown in that paper that this is equivalent to fitting the change in breeding value due to selection as a random effect.

Here we use simulated data to compare the accuracy and bias in genomic prediction applied to populations under selection with and without centering the entire matrix of genotype covariates, and with and without fitting µ_g as a fixed effect. Further, we will show that when the observed genotype covariates are centered using means calculated from selected individuals rather than means from all individuals, the meaning of µ_g changes from the mean of unselected individuals to become the mean breeding value in selected individuals as claimed by Vitezica et al. (Vitezica et al. 2011).

Materials and Methods

Theory

To simplify the presentation of the genetic model, without loss of generality, we will assume that the unconditional expectation of the phenotypic value for all individuals is the same. Let denote the row vector of genotypes for individual i. Then, under additive gene action, the genotypic value, g_i, which is the expected phenotypic value of an individual with genotypes can be written as where β is the value of g_i when = 0′ and α is the vector of substitution effects. Recognize the scalar β and the vector α are constants, but g_i will be a random variable because of randomness in a_i = α, due to the randomness in the genotypes for a randomly sampled individual. Note that the expected value of the linear component a_i of the genotypic value in (1) is E(a_i) = E() α = k′ α = µ_g, where k′ = E(), which may not be equal to zero. Thus, it is customary to write the model for the genotypic value, as can be derived from (1), as follows: where (β + µ_g) is a constant, representing the E(g_i), and u_i = ( −k′) α is a random variable that has null expectation, which is the breeding value predicted in a pedigree-based BLUP evaluation. When genotypes are observed and used in a genomic analysis, they may be transformed or coded by subtracting their expectations, k′, from the observed values, . In both (1) and (2), α_j is the same substitution effect for locus j. The intercepts in these models, however, are different. In (1), the intercept is β, and it is the value of g_i when = 0′. In (2), on the other hand, the intercept is (β + µ_g), and it is the value of g_i when = k′.

More generally, k′ is not known, so genotypes are coded by subtracting a different vector v′ from the observed genotypes as −v′. Still, α_j is the substitution effect for locus j, but the intercept will change to become (β + v′α), which is the value of g_i when = v′. Thus, as more rigorously shown in (Strandén and Christensen 2010), inference about α does not depend on how the genotypes are coded. A simpler but rigorous proof is given in the appendix of this paper.

In single-step analyses, where some individuals are not geno-typed, the missing genotypes are imputed either implicitly (Legarra et al. 2009) or explicitly (Fernando et al. 2014) using best linear prediction. Let M_g denote the matrix of genotypes for individuals that were genotyped. Then, the genotypes of the individuals with missing genotypes are imputed as where A_ng is the matrix of pedigree based additive relationships between the non-genotyped and genotyped individual and A_gg is the matrix of additive relationships among genotyped individuals. Now, the model for the genotypic values, when genotypes are coded as in (2), becomes where ∊ is that part of g_n that cannot be imputed from knowledge of the breeding values of genotyped relatives. In practice, the true value of k′ is not known, and data for its estimation may not be available. Rearranging these equations in terms of the uncentered M_g rather than the centered matrix of geotype covariates (M_g − 1k′), results in and substituting µ_g = k′α, as previously defined, results in which suggests that µ_g = k′ α could be treated as an unknown constant and estimated as a fixed effect from the data (Fernando et al. 2014). The covariate vector for µ_g is denoted by J_n = for non-genotyped individuals and by J_g = − 1 for genotyped individuals. So, (3) becomes which can be combined as where .

When the vector α represents the substitution effects of a large number of loci containing positive and negative effects, µ_g = k′ α will tend to have a value close to zero. Accordingly, we have simulated some scenarios with positive µ_α = E(α_i) so that the entire α vector is positive to exacerbate the impact of µ_g = k′ α. Nevertheless, when marker rather than causal alleles are fitted in the model, the sign of the substitution effects depends on the phase relationship between marker and causal allele, which may be equally likely to be positive or negative.

Even if µ_α = E(α_i) = 0, in a population undergoing selection, it is expected that E(a_i) = E() α ≠ 0 in non-founders. Suppose v′ is the mean of the observed genotype covariates in such a population undergoing selection and these means are used to center the matrix M_g of observed genotypes. Then, the model for the genotypic values can be written in terms of the matrix of centered covariates as and using J for the covariate corresponding to µ_g, (5) can be written as which can be combined as

Note that the regression coefficients for J, µ_g in (4) and + v′ α in (6) must be equal. This implies that = µ_g − v′α. Similarly, the intercepts of these two models must be equal too, and this implies β* = β + v′α.

Simulations

Phenotypic and genotypic data were simulated based on haplo-types from ten regions of 721 US Hereford beef cattle that were genotyped with the Illumina 770K BovineHD BeadChip and reported in terms of the number of copies of the A allele at each locus. The selected regions came from choosing the 5,001st to 5,100th single nucleotide polymorphisms (SNPs) from chromosomes 1 to 10 (BTA1-BTA10), after eliminating SNP with MAF < 0.01. These remaining 1,000 SNP represent ten 0.1M chromosomes. Average LD between adjacent SNPs was 0.511. Half of these SNP were randomly chosen to represent QTL. The QTL effects were sampled from a Normal distribution with mean µ_± = 0.2 and multiplied by the number of copies of the A allele to produce the true breeding value (TBV). The TBV were added to a Normally distributed residual term scaled by the sample variance of the TBV to simulate a trait with a heritability of 0.5. The first 10 SNP (i.e. the 5,001st to 5,010th) from each of the ten chromosomes were also used to simulate a smaller panel, with 5 QTL and 5 markers per chromosome. Average LD between adjacent SNPs was 0.459. TBV were simulated in the same way as for the 1,000 SNP scenario, then scaled to simulate traits with heritabilities (h²) of 0.1, 0.3 or 0.5. An additional scenario with µ_α = 0 was used to simulate TBV for a trait with heritability 0.5.

Half the observed diplotypes from US Hereford cattle were assigned to represent males and the remainder to represent females. Those 360 males and 361 females were sampled in pairs, with replacement, to produce 4,000 male and 4,000 female off-spring representing generation G-4. There were no mutations. Four more non-overlapping generations of random mating were carried out with one male and one female offspring per dam mated to randomly chosen sires to produce the founder population (G0).

The G1 generation was produced by mass phenotypic selection of the top 200 G0 males, and this was repeated for 5 more generations. Each female was randomly mated twice to selected males to produce 1 offspring of each sex each generation. Across non-overlapping generation G0 to G5, a total of 48,000 individuals with phenotypes, genotypes and TBV were simulated for each scenario.

The training data included phenotypes from all individuals in G0 to G4 (n = 40,000), and genotypes from all 1,000 sires and all 8,000 G5 animals. Fixed loci, if any, were filtered from the panel before genomic prediction analyses. The genetic and residual variances used in genomic prediction were the sample variance of the TBV in G0 and the corresponding residual variance used to define the desired heritability in the founder population.

Models

Five statistical models were compared for differences in accuracy and bias of prediction. These include models with or without µ_g and with or without centering of marker covariates, and a model that used pedigree relationships but not marker covariates.

1. Mixed Linear Model

Accuracy of pedigree-based best linear unbiased prediction (PBLUP) was quantified using the correlation of TBV and estimated breeding values (EBV), where TBV was as simulated and EBV were obtained by fitting the mixed linear model (Henderson 1973, 1984): where y is a vector of phenotypic observations, 1 is a vector of 1s, µ = β + µ_g is a general mean, u is a vector of random direct additive genetic effects, e is a vector of random residual effects, and Z is a known incidence matrix relating observations to u. In this model, E(u) = 0, E(e) = 0, so that E(y) = 1µ. Further, , for A being the numerator relationship matrix, and , so that .

2. Single-Step Bayesian Regression Model

Genomic EBVs (GEBV) were obtained by Single-Step Bayesian regression (Fernando et al. 2014) with BayesC priors for marker effects with π = 0. The model was implemented in Julia (http://julialang.org) based on the SSBR package (http://QTL.rocks) to construct an MCMC chain of 50,000 samples. Individuals were separated into 2 groups designated with subscripts g or n according to whether or not simulated genotypes were assumed to be observed or missing. The single-step bayesian regression model including a covariate J for µ_g (Model J) was: where y_n and y_g are vectors of phenotypes for nongenotyped and genotyped individuals, 1 is a vector of 1s, µ is a general mean, µ_g is the expected value of the linear component a_i of the genotypic value if selection was absent, α is a vector of random substitution effects of markers, ∊ a vector of imputation residuals, Z_n and Z_g are incidence matrices relating the breeding values of non-genotyped and genotyped individuals to their phenotypes, J_g, which is defined for geno-typed individuals, is a vector of −1s, J_n, which is defined for non-genotyped individuals, is a vector computed as is the matrix of imputed marker covariates, M_g is the matrix of observed marker covariates, e_n and e_g are vectors of random residual effects for non-genotyped and genotyped individuals. This model can be represented as where , the 0 matrix in U is required because ϵ does not appear in the model for genotyped individuals, and e is a vector of random residual effects.

There were four variants of the single-step Bayesian analysis depending on whether or not the covariate J corresponding to the mean µ_g was in the model, and whether or not the columns in the marker covariate matrix M were centered using their observed means. The analyses with J or without J are are denoted as J or N when covariates were not centered, and as JC or C, when the entire matrix of imputed and observed genotype covariates were centered, respectively (Table 1).

Table 1

Four combinations of the single-step bayesian regression analyses

Accuracy of genomic prediction was quantified using the correlation of TBV and GEBV (r_g,), where GEBV were obtained from each of the 4 analyses described above. Bias of genomic prediction was quantified using the deviation from unity of the coefficient of regression of TBV on GEBV (b_g,). In models JC and J, GEBV are obtained using equation (24) in (Fernando et al. 2014): where is the GEBV, is the best linear unbiased estimate of the mean of breeding values, is the best linear unbiased predictor (BLUP) of the vector of random substitution effects of all markers, and is the BLUP of the imputation residual.

In model J the matrix M_g contains the uncentered number of copies of the A allele at each locus, and the uncentered version is used to impute . In model JC the entire matrix of imputed, , and observed, M_g, genotype covariates is centered. In model C and N the GEBV are computed in a corresponding manner except that the covariate J and its coefficient µ_g are not included in the model.

The four analyses JC, J, C and N were all applied to 3 different genotype panels, comprising the causal QTL plus markers, just the causal QTL, or just the markers. All 12 combinations of 4 analyses and 3 genotype panels were applied to data simulated with 100 loci comprising 50 QTL whose effects were sampled from a Normal distribution with µ_α = 0.2 to construct phenotypes with h² = 0.5. The four analyses JC, J, C and N were repeated using only genotype panels comprising QTL plus markers for four other scenarios: 100 loci, h² = 0.1, µ_α = 0.2; 100 loci, h² = 0.3, µ_α = 0.2; 100 loci, h² = 0.5, µ_α = 0; and 1,000 loci, h² = 0.5, µ_α = 0.2. Every scenario was repeated for 10 replicates with each replicate having been constructed starting from the sampling of G-5 which represented simulated offspring from the haplotypes of real animals. Every phenotypic dataset was also fitted using the PBLUP model. All reported correlations and regression coefficients are the means of 10 replicates. These are presented along with the standard errors of those means.

In single-step GBLUP (Legarra et al. 2009; Aguilar et al. 2010; Christensen and Lund 2010), the missing genotypes are not explicitly imputed and only the observed genotype covariates are centered using their means. So, in addition to the above, analyses with and without J (models JC* and C*), were applied to a marker panel with 100 loci, h² = 0.5, and µ_α = 0, when the matrix = M_g − 1v′ of observed genotype covariates were centered using their means, v′ = 1′M_g. Recall that when the matrix of observed genotype covariates is centered, the model for the genotypic values can be written in terms of the matrix M_g of uncentered covariates as shown by model (6). We will compare the estimates of in this model with those of µ_g from (4) where the means of observed genotype covariates are not used for centering.

The genotypes representing G0 from each replicate and scenario are available at: https://figshare.com/s/d7798b811a9a6a4172fc. These genotypes and the methodology described previously are sufficient to reproduce the simulations used in this study.

Results and Discussion

Effect of fitting a genotyped mean and centering marker covariates

1. Accuracy

The accuracies of genomic prediction as assessed by validation in G5 after training using G0-G4 for a trait with 50 QTL whose effects were sampled from a Normal distribution with µ_α = 0.2 and h² = 0.5 are in Table 2. The accuracy of PBLUP in predicting breeding values for individuals without phenotypes, was 41.5%, accounting for less than 20% of genetic variance.

Table 2

Correlations (%, ±SE_s) between TBV and (G)EBV^a for alternative analyses^b

All analyses using genotypes resulted in more accurate predictions than using pedigree alone. Centering had no affect on the accuracy of the genomic analyses regardless of the nature of the marker panel. In this study, selection resulted in successive advance in mean TBV from G0 to G5 being 10.35, 10.77, 11.31, 11.82, 12.30 and 12.80. The mean genotypic value was not zero in G0 because QTL genotypes were not centered and the mean QTL effect was µ_α = 0.2. Since the QTL effects do not change by selection the advance in TBV reflects changes in the frequencies of the favorable alleles of the 50 QTL. So centering using the allele frequency means of the genotyped sires in G1-G4 and all individuals in G5 does not closely approximate the centering that would have occurred if the allele frequency means had been obtained from the unselected population. In contrast fitting µ_g in the model estimates the relevant mean from the data.

In panels that included causal variants (QTL), fitting µ_g in the model substantially improved the accuracy to being near perfect. This is not surprising given there were only 50 QTL, the heritability was 0.5 and there were 40,000 phenotyped ancestors, including 200 genotyped sires per generation in the training. However, in the panel that contained only markers with no causal variants, fitting µ_g in the model had little impact.

Using one replicate as an example, for the panel including both QTL and markers, the estimate of µ was about 10.51 for both analyses J and N. The estimate of µ_g was 7.66 for the analysis using J. For the genotyped individuals, the covariate values in J_g are all −1, so 1 + J_g is a vector of values equal to 10.51 − 7.66 = 2.85. For non-genotyped individuals, the covariate values in J_n can vary widely but many are close to −1 while others are close to 0. This means that 1 + J_n will contain values that range from 10.51 to 2.85, accounting for variation in accuracy of imputation. When µ_g is not included in the model, these effects are ignored which can reduce the accuracy of predicting nongenotyped individuals. Failing to account for these effects will propagate errors in and , the latter impacting the accuracy of predicting genotyped individuals. Collectively, these errors reduced accuracy from 98% to 92% for the panel including QTL and markers and from 99% to 85% for the panel including only QTL. However, when the panel comprised only markers, the estimates will include both positive and negative values because the phase of markers and QTL are equally likely to take either sign, in which case will be close to zero as confirmed in the above mentioned replicate where the estimate was 0.41.

Here, the QTL model was used with an intercept of β = 0.0 to simulate the data. When only QTL are on the panel, the true value of β is zero. Thus, in analysis J because µ = (β + µ_g), both and are estimates of µ_g, and could be pooled which for the replicate above would be (10.51 + 9.04)/2 = 9.78. In that replicate, the actual mean of a_i in G0 was 10.8, which was estimated in the analysis to be 9.78. On the other hand, the mean of the breeding value u_i in the 9,000 genotyped individuals was 2.2, which is clearly not near the pooled estimate of 9.78. These genotyped individuals included 1,000 selected sires of which 200 were genotyped in each generation from G0 to G4, and 8,000 offspring from G5. The mean values of u_i for the selected sires were 0.97, 1.41, 1.80, 2.24, and 2.74, respectively, for G0 through G4, and 2.27 for the offspring in G5. It is apparent that the µ_g parameter corresponding to the covariate J is the mean of the founder population and not the mean breeding value of selected individuals. In analysis JC with the covariates centered, the intercept β is the value of g_i when ( − ) α = 0, which is the case when = . The estimate was 17.33 in this analysis, but remained about the same value, namely 9.05. This shows that has the same interpretation whether the entire matrix of observed and imputed genotypes is centered or not. In neither case does it represent the mean breeding value of selected individuals.

2. Bias

Table 3 shows the regression coefficients of TBV on (G)EBV for h² = 0.5 and µ_α = 0.2, the same scenarios represented in Table 2. The regression coefficients of TBV on GEBV for each scenario were close to 1 with very low SE, which indicates the genomic predictions exhibited almost no bias. The differences in regression coefficients between analyses were very small, but the marker panel comprising only markers were biased upwards whereas the marker panels that included causal mutations were biased slightly downwards.

Table 3

Regression coefficients (±SE_s) of TBV on (G)EBV^a

Sensitivities to trait heritability

Accuracy of PBLUP increased with heritability, as expected (Table 4). Genomic predictions using panels that include causal mutations were near perfect when µ_g was included in the model. These high accuracies are a reflection of these phenotypes being influenced by only 50 QTL and there being a large training dataset. Accuracy was reduced when µ_g was not fitted in the model. There was no advantage in terms of accuracy to centering the covariates but MCMC mixing may have been improved although this was not investigated.

Table 4

Correlations (%, ± SE_s) between TBV and (G)EBV^a for alternative analyses^b for different heritabilities

Effect of mean QTL effect (µ_α = 0 vs µ_α = 0.2)

We had hypothesized that the impact of omitting µ_g from the model will be greatest when µ_g departs significantly from 0 which is more likely to occur when µ_α departs from 0. For that reason our base simulation used µ_α = 0.2. Results are shown in Table 5 for the panel including QTL and markers with h² = 0.5 for µ_α = 0.2 compared to µ_α = 0. These results confirmed the benefit of fitting µ_g was greatest when µ_α = 0.2 but there was still an advantage to fitting µ_g when µ_α = 0. That advantage is likely to erode as the number of QTL increases. Changing the mean QTL effect had no impact on bias, except for a slight influence on PBLUP.

Table 5

Accuracy^a and bias^b of genomic prediction (±SE_s) for alternative QTL distributions^c and analyses^d

Effect of more QTL and markers (100 SNP vs 1,000 SNP)

We had hypothesized that the improvement of accuracy from adding an extra covariate for µ_g will reduce as the number of QTL increases because µ_g is likely to be closer to zero for a trait that is more polygenic. Table 6 shows that PBLUP was largely unaffected by changes to genetic architecture but the accuracy of genomic prediction declined slightly as the number of QTL increases. This reflects the fact that precision of estimating QTL effects is greater when the effects are large and polygenic traits with more QTL must have on average smaller effects when compared at the same genetic variance. The benefit of fitting µ_g in the model was virtually eliminated when the number of substitution effects to estimate increases from 100 to 1,000. In contrast to the results of accuracy, there was no impact of QTL number on bias. Centering had no impact on accuracy or bias.

Table 6

Accuracy^a and bias^b of genomic prediction (±SE_s) for different numbers of QTL^c and alternative analyses^d

Centering using the entire matrix of genotype covariates or only the observed genotype covariates

Table 7 shows the accuracies and regression coefficients of TBV on G(EBV) for the genotype panel with 50 markers, h² = 0.5 and µ_α = 0. The analyses were performed after centering: the entire matrix of imputed and observed genotype covariates (M* = M −11′M); only observed genotype covariates ( = M_g − 11′M_g), which is the type of centering done in single-step genomic best linear unbiased prediction (GBLUP); or not centering the covariates (M* = M). The accuracy of any genomic analyses were about 8% higher than that one based on covariates centered as but without J (model C*). However, when J was included in the model with covariates centered as , the accuracy of prediction was markedly improved.

Table 7

Accuracy^a and bias^b of genomic prediction (±SE_s) when centering for all genotypes or observed genotypes^c

As explained previously, µ_g = k′α, where k′ is the expected value of the covariates in the founders, will tend to zero for the marker panel that does not include QTL even with µ_α ≠ = 0. However, even if µ_α = 0, in a population undergoing selection when selected individuals are genotyped, = µ_g − v′ α ≠ = 0, where v′ is the expected value of the observed genotype covariates. In this study, selection was used to increase the mean of the trait. Thus, is expected to be negative because most of the genotyped individuals were from G5, whereas µ_g is expected to be zero. The negative estimate of from 10 replicates of the JC* analysis, −2.84, confirms that < 0. On the other hand, the mean of from 10 replicates of the JC analysis was −0.23. This explains why fitting J in the model improved the accuracy of genomic prediction when covariates were centered as in single-step GBLUP.

Fernando et al. (Fernando et al. 2014) found that centering using improves the accuracy without J in the model when the population was not under selection and the genotyped individuals were unselected. In that study, mating was random with no selection, so the allele frequency means of the genotyped individuals were a reasonable approximation of the allele frequency means in the founder population. In contrast, our simulation here shows that centering using can reduce the accuracy when the population is under selection, unless J is fitted in the model.

In single-step GBLUP, the observed genotypes are commonly centered by subtracting their mean and used to construct a genomic relationship matrix, such as using the first method proposed by VanRaden (VanRaden 2008). Using that genomic relationship matrix in the single-step GBLUP formula in Aguilar et al. (Aguilar et al. 2010) does not account for J. This was recognised in Vitezica et al (Vitezica et al. 2011) who proposed a modification for populations under selection that involved adding a constant to all elements of the genomic relationship matrix that they derived by equating the sum of the elements of the genomic relationship matrix to the the sum of the elements of the numerator relationship matrix. In the appendix of that paper they showed this modification is equivalent to fitting a covariate Q = − J and treating − as a random effect. In addition to this modification, Christensen et al. (Christensen et al. 2012) proposed a multiplicative scaling to the genomic relationship matrix such that its diagonals have the same mean as the diagonals of the numerator relationship matrix. Vitezica et al. (Vitezica et al. 2011) claimed that − represents the mean breeding value of selected individuals, and we have confirmed here that this is true provided the observed genotype covariates are centered by their mean.

Most populations are under natural or artificial selection. In many cases, genotypes are only available on selected individuals. In single-step genomic analysis that combine genotyped and non-genotyped individuals in a joint analysis, the mean of observed genotypes are available for centering. If the observed genotypes include QTL, the accuracy of genomic prediction can be severely compromised, unless the J covariate is fitted in the model. If the observed genotypes are only markers, the accuracy of genomic prediction may not necessarily be improved by fitting J in the model, but it doesn’t do any harm. However, if centering is applied only to the observed genotypes, which is the type of centering used in single-step GBLUP, accuracy could be severely compromised, unless the J covariate is fitted in the model or an equivalent approach is adopted.

Acknowledgements

The authors are grateful to Bruce L. Golden and Hao Cheng for assistance in the implementation of SSBR models, and to Jack C. M. Dekkers for his constructive comments in design of simulation. This work was supported by the US Department of Agriculture, Agriculture and Food Research Initiative National Institute of Food and Agriculture Competitive grant no. 2015-67015-22947.

Appendix

Here we show that inference about α does not depend on how the genotypes are coded. The marker effects model can be described by the following general model: where y is a vector of observed phenotypes, 1 is a vector of 1s, µ is a general mean, M is a matrix of marker covariates, coded 0, 1, 2, which represents the number of copies of the A allele, α is a vector of random substitution effects of markers, and e is a vector of residuals. Henderson’s mixed model equations (MME) that correspond to equation (7) are: where is the best linear unbiased estimate of the mean, and is the best linear unbiased predictor of the vector of random substitution effects of all markers. Now we can eliminate from the equations for , by subtracting from those equations the equation for pre-multiplied by . Then, the MME are transformed to: and substituting 1′ 1 = n and 1′ M = n′ and its transpose, the transformed MME become: where ′ is the row vector of column means of M as in ′ = .

Now, consider the coding obtained by centering the marker genotypes as M − 1′. Then the model can be written as: where µ* = µ + ′α. The MME that correspond to equation (9) are: but 1′(M − 1′) = n′ − n′ = 0′, and, similarly, its transpose is (M − 1′)′ 1 = 0. Then the MME become:

Expanding (M − 1 ′)′(M − 1 ′) gives M′ M −1′ M − M′ 1 ′ + 1′ 1 ′, but because 1′ M = n′, M′ 1 = n, and 1′ 1 = n, as previously shown, the second term, 1′ M = n′, which is equal to the last term in the expansion. Thus, the MME become:

The equations for in (8) and (10) are identical, and this proves that centering with m′ doesn’t affect inference about α.

Now suppose an arbitrary vector v′ is used to transform the genotypes as (M − 1v′). The the model becomes:

Adding and subtracting 1 ′α, the above equation can be written as: which is identical to equation (9), proving inference about α does not depend on how the genotypes are coded.