Genomic prediction involves the use of genotypes or haplotypes to predict genetic merit. Conceptually, it involves two phases, a training phase where the genotypic or haplotypic effects are estimated, typically as random effects, in a mixed model scenario, followed by an application phase where the genomic merit of selection candidates is predicted from the knowledge on their genotypes and previously estimated effects from the training phase. The ideal data for training would be true genetic merit data observed on unrelated animals in the absence of selection. In that case, the model equation would be:

where g is a vector of true genetic merit (i.e. breeding value BV) with var(g) = T, the scalar is the genetic variance and T can be constructed using the theory from combined linkage disequilibrium and linkage analyses 10, μ is an intercept, M is an incidence matrix whose columns are covariates for substitution, genotypic or haplotypic effects, a are effects to be estimated, var(Ma) = , G is a genomic relationship matrix 111213, ε is the lack of fit, var(ε) = , hopefully small and will be 0 if BV could be perfectly estimated as a linear function of observed marker genotypes. In different settings, a might be defined as a vector of fixed effects 14 or a vector of random effects 1. Even when a is fixed, Ma is random because M, which contains genotypes, is random. However, in genomic analyses M is treated as fixed because the analysis is conditional on the observed genotypes. The philosophical issues related to the randomness of M and a are discussed in detail by Gianola 15 but for our context it is sufficient to define var(Ma) = without explicitly specifying distributional properties of M or a.

Genotypes used as covariates in Ma are unlikely to capture all the variation in true genetic merit, either because they are not comprehensively covering the entire genome, or because linkage disequilibrium between markers and causal genes is not perfect. Knowledge of E is required in the analysis whether a is treated as a fixed (e.g. GLS) or random effect (e.g. BLUP). In practice with experiments that involve related animals, it is unreasonable to assume E has a simple form such as a diagonal matrix since that implies a zero covariance between lack of fit effects for different animals, however, it can be approximated using knowledge on the pedigree using the additive relationship matrix, A 16. These lack of fit covariances can be accommodated by fitting a polygenic effect for each animal, in addition to the marker genotypes 17, or accounted for by explicitly modeling correlated residuals. For a non-inbred animal, , therefore and the proportion of the genetic variance not accounted for by the markers can be defined to be . The scalar c, will be close to 0 if markers account for most of the genetic variation and close to 1 if markers perform poorly.

In practice we do not have the luxury of using true BV as data in genomic prediction. A more common circumstance might involve training based on phenotypic observations that include fixed effects on phenotype denoted Xb where X is an incidence matrix for fixed non-genetic effects in b. An appropriate model equation for phenotypes is

where e is a vector of random non-genetic or residual effects. In comparison to (1), the use of y for training involves the addition of the vectors Xb and e to the left- and right-hand side, inflating the variance and giving

with since cov(ε, e') = 0. This model can be fitted by explicitly including a random polygenic effect for ε, or by accounting for the non-diagonal variance-covariance structure of the residuals defined as var (ε + e). Including a polygenic term is not typically done in genomic prediction analyses 1218, and when undertaken does not seem to markedly alter the accuracy of genomic predictions [Habier D. Personal communication]. Assuming var (ε + e) is a scaled identity matrix facilitates the computing involved in fitting this model, as the relevant mixed model equations can be modified by multiplying the left- and right-hand sides by the unknown scale parameter as is typically done in single trait analyses. However, this is not an option if residuals are heterogeneous, for example, because they involve varying numbers of repeated observations.

Consider the circumstance where the training observations are a vector representing observations that are the mean of n observations on the individual with n potentially varying. In that case, equation (3) becomes

With , a diagonal matrix with elements with being the phenotypic variance, heritability h², and repeatability t. Ignoring off-diagonal elements of E, the elements of the inverse of R with R = var(ε) + D would for non-inbred animals be . In fixed effects models, this matrix can be arbitrarily scaled for convenience. In univariate random effects models, a common practice is to formulate mixed model equations using the ratios of residual variance to variances of the random effects. Here, it makes sense to factor out the residual variance of one phenotypic observation, i.e. , from the expression for the residual variance of the mean of n observations. In this circumstance, a scaled inverse of the residual variance being or equivalently

which can be used for weighted regression analyses treating marker effects as fixed or random. When c = 0, the genetic effects can be perfectly explained by the model, and for n = 1, a single observation on the individual, the weight is 1 for any heritability. Scaling the weights is convenient because records with high information exceed 1 and the weights are trait independent which is useful when analysing multiple traits with identical heritability and information content.

In some cases the training data may represent the mean of p individual measurements on several offspring, rather than the mean phenotype of the genotyped animal. In that circumstance, the residual variance includes a genetic component for the mate and Mendelian sampling. For half-sib progeny means with unrelated mates and no common environmental variance, . However, the half-sib progeny mean contains only half the genetic merit of the parent, therefore the genotypic covariates need to be halved, or the mean doubled, in order to analyse data that includes records on genotyped individuals and records on offspring of genotyped individuals. The variance for twice the progeny mean is , and adding , factoring out and inverting gives

For full-sib progeny means the intraclass correlation of residuals will include a genetic component and perhaps a common environmental component (e.g. litter, with variance and giving for unrelated parents. Adding variation due to factoring out and inverting gives

This expression can be used as weights in the fixed or random regression of full-sib progeny means on parent average marker genotypes.

An estimated breeding value, typically derived using BLUP, can be recognised as the true BV plus a prediction error. That is, . Accordingly, training on EBV might be viewed as extending the model equation in (1) by the addition of the prediction error, in the same way that (3) was derived by the addition of a residual nongenetic component. The model equation would therefore be

There are at least two issues with this formulation of the problem, which may not be immediately apparent, and which both result from properties of BLUP. The first issue is that the addition of the prediction error term to the left- and right-hand side of (8) actually reduces rather than increases the variance, despite the fact that diagonal elements of must exceed 0, in contrast to the addition of non-genetic random residual effects in (3). That is , whereas var(g_i) < var(y_i), due to shrinkage properties of BLUP estimators 19. Generally, but for BLUP so that implying . The reduction in variance of the training data comes about because prediction errors are negatively correlated with BV as can be readily shown since . This means that superior animals tend to be underevaluated (i.e. have negative prediction errors) whereas inferior animals tend to be overevaluated. This is a consequence of shrinkage estimation and prediction errors being uncorrelated with EBV, i.e. . In order to account for the covariance between the prediction errors and the BV, a model that accounted for such covariance would need to be fitted. Such models are computationally more demanding compared to models whereby the fitted effects and residuals are uncorrelated. The second issue resulting from the properties of BLUP, is that it is a shrinkage estimator, that shrinks observations towards the mean, the extent of shrinkage depending upon the amount of information. This is apparent if one considers the regression of phenotype on true genotype (i.e. BV) which is 1, whereas the regression of EBV on BV is equal to ≤ 1, where is the reliability of the EBV (for animal i) or squared correlation between BV and EBV. In the context of any marker locus, the contrast in EBV between genotypes at a particular locus is shrunk relative to the contrast that would be obtained if BV or phenotypes were used as data, with the shrinkage varying according to . We are, however, interested in estimating the effect of a marker on phenotype, but we get a lower value for the contrast if EBV with ≤ 1 are used as data, rather than using phenotypes. A further complication is that training data based on EBV typically comprise individuals with varying . This problem can be avoided by deregressing or unshrinking the EBV.

The solution to the model fitting problems associated with the reduced variance of EBV and the inconsistent regression of EBV on genotype according to reliability can both be addressed by inflating the EBV. Rather than fitting (8), we will fit the linearly inflated data represented as K for some diagonal matrix K. That is, we will fit:

for some matrix K chosen so that and is a constant. Since then this expression will be 0 when . For this value k_i, , a constant for all animals regardless of their reliability. Accordingly, the deregression matrix is K = diagonal and the deregressed observations are . Note in passing that the nature of the deregression will depend upon the EBV base. Genetic evaluations are typically adjusted to a common base before publication, by addition or subtraction of some constant. The EBV should be deregressed after removing the post-analysis base adjustment or by explicitly accounting for the base in the deregression procedure 20. To show the dependence of the deregression to the post-analysis base, supposes that EBV are adjusted to a base, b. Then a linear contrast in deregressed EBV without removing the base effect is unless . Marker effects are typically estimated as linear combinations of data, and will therefore be sensitive to the base adjustment.

A deregressed observation represents a single value that encapsulates all the information available on the individual and its relatives, as if it was a single observation with h² = r². This can be shown by recognising that h² is the regression of genotype on phenotype. Taking the deregressed observation to be the phenotype, . Training on deregressed EBV is therefore like training on phenotypes with varying h². Provided > h², training on deregressed EBV is equivalent to having a trait with higher heritability. However, as explained later, we recommend removing ancestral information from the deregressed EBV.

Deregressed observations have heterogeneous variance when r² varies among individuals. The residual variance of a particular deregressed observation is but and so the residual variance expression simplifies to . Ignoring the off-diagonal elements of var(ε) as before, the diagonals of the inverse of the residual variance after factoring out are which simplifies to give

an expression analogous to (5) with n = 1 and h² = . Note that the weight in (10) approaches as →1 in which case the weight tends to infinity as c→0. This is the same as would occur when the number of offspring p→∞, and p is used as a weight.

Animal model evaluations by BLUP using the inverse relationship matrix shrink individual and progeny information towards parent average (PA) EBV 21. It makes sense to remove the PA effect as part of the deregression process for two reasons. First, some animals may have EBV with no individual or progeny information. These animals cannot usefully contribute to genomic prediction. This is apparent if one imagines a number of halfsibs with individual marker genotypes and deregressed PA EBV. These animals cannot add any information beyond what would be available from the common parent's genotype and EBV. Second, if any parents are segregating a major effect, about half the offspring will inherit the favourable allele and the others will inherit the unfavourable allele. However, the EBV of both kinds of offspring will be shrunk towards the parent average. Parent average effects can be eliminated by directly storing the individual and offspring deregressed information and corresponding r² during the iterative solution of equations carried out for the purposes of genetic evaluation 2. In some cases researchers do not have access to the evaluation system used to create the EBV on their training populations. In those circumstances, it is necessary to approximate the evaluation equations and backsolve for deregressed information free of the effects of parent average. This can be done for one training animal at a time, given h² and knowledge of only the EBV (unadjusted for the base) and r² on the animal, its sire and its dam. First, compute parent average (PA) EBV and reliability for animal i with sire and dam as parents: , and . Assuming sire and dam are unrelated and not inbred, the additive genetic covariance matrix for PA and offspring is with inverse . Using this result, recognise that the equations to be solved are:

where is information equivalent to a right-hand-side element pertaining to the individual, and reflects the unknown information content of the parent average and individual (plus information from any of its offspring and/or subsequent generations), λ = (1 - h²)/h² is assumed known. Define

then using the facts 19 that and leads to , and . Rearranging these equations, , and . The formula to derive the inverse of a 2 × 2 matrix applied to the coefficient matrix from (11) gives , and for .

Equating these alternative expressions for c^{PA, PA} leads to

and equating the expressions for c^{i, i} leads to

Second, solve these nonlinear equations for and . Although not obvious, there is a direct solution for and . It can be derived by dividing (12) by (13), defining , and rearranging to get

Substituting the expression for in (14) into the denominator of (13), defining , and rearranging leads to a quadratic expression in , namely , which has a positive root that can rearranged to

Application of (15) provides the solution for that can be substituted in (14) to solve for , together enabling reconstruction of the coefficient matrix of (11).

Third, the right-hand side of (11) can be formed by multiplying the now known coefficient matrix by the known vector of EBV for PA and individual. The right-hand side on the individual, free of PA effects is The equation to obtain an estimate of EBV for animal i, free of its parent average, , based only on , is and the corresponding for use in constructing the weights in (10) is given by . The deregressed information is , which simplifies to and is analogous to an average. An iterative procedure using mixed model equations to simultaneously deregress all the sires in a pedigree, while jointly estimating the base adjustment and accounting for group effects was given by Jairath et al 20. However, that method requires knowledge on the numbers of offspring of each sire.

Genetic evaluation of animal populations results in EBV that are a weighted function of the parent average EBV, any information on the individual, adjusted for fixed effects, and a weighted function of the EBV of offspring, adjusted for the merit of the mates 2. The previous section has argued for the removal of parent average effects in constructing information for genomic analyses. It could be argued that information from genotyped descendants should also be removed to avoid double counting. This can be achieved during the evaluation process, and is desirable in the absence of selection. If the genotyped descendants are a selected subset, the removal of their information will lead to biased information on the individual. Simulation suggests that the double counting of descendants performance has negligible impact on genomic predictions (results not shown).

Comparative weights for individual and average of n individual observations using (5), and for progeny means of p halfsibs using (6) and deregressed EBV of varying reliability using (10) are in Table 1.

Suppose genomic training is to be undertaken for a trait using EBV available from national evaluations that have yet to be deregressed. Widely-used bulls have been genotyped and the EBV and r² of those bulls are available, along with corresponding information on the sire and dam of each bull. Such a trio might have values of = 10, = 0.97; = 2, = 0.36; and = 15, = 0.68. Given h² = 0.25, λ = 0.75/0.25 = 3, the PA information is , and . Using (15), with α = 5.97, δ = 0.523, then = 9.16 which substituted in (14) gives = 5.08.

Substituting these information contents into the coefficient matrix or left-hand side of (11) is with inverse . These values correspond to = 0.5 - 3 × 0.0558 = 0.33 and = 1.0 - 3 × 0.1066 = 0.68 the reported and confirming the equations used to determine the information content. The right-hand side of (11) can then be reconstructed by multiplying the coefficient matrix by the vector of EBV as . The element of interest is the right-hand side element corresponding to the individual, obtained as = -6 × 6 + 11.08 × 15 = 130. The deregressed information for use in subsequent analysis is obtained as and the corresponding reliability of this information free of PA effects is = 1.0 - 3/(5.08 + 3) = 0.63. The relevant scaled weight for use with the deregressed information on this individual assuming c = 0.5 can be found using (10) as . This implies that the deregressed information is 2.76 times more valuable than a single record on the individual.