Lecture 10 of the Molecular Genomics series. Predicting breeding values from markers, and why it changed animal breeding.
The previous nine lectures build to this one. We have dense markers covering the genome, we know they are in linkage disequilibrium with whatever lies near them, and Lecture 6 ended with the observation that for most traits the significant associations explain only a small part of the genetic variance. Genomic selection is the response to that observation.
Association studies can identify regions influencing a trait and, with effort, narrow them to genes. That works when effects are large. For the traits that matter in animal breeding it usually does not.
Almost every production, reproduction and functional trait is quantitative: influenced by a large number of loci, each contributing a very small fraction of the variation. Milk yield is the standard example. Detecting a locus that explains a tenth of a percent of the variance requires sample sizes that are rarely available, and even when detected the loci found account for little of the total.
The breeding question is therefore awkward. If a trait is influenced by thousands of genes, each contributing a negligible amount, how do you select the best animals? Classical quantitative genetics answered this without knowing anything about the genes at all, and that answer is worth restating before improving on it.
The established method is best linear unbiased prediction applied to the animal model.
where \(\mathbf{y}\) is the vector of records, \(\mathbf{b}\) the fixed effects such as herd, year and season, \(\mathbf{a}\) the random animal effects, and \(\mathbf{e}\) the residuals. The animal effects are assumed to follow
where \(\mathbf{A}\) is the numerator relationship matrix, whose elements are the expected fractions of genes shared between pairs of animals, computed from the pedigree.
The output is an estimated breeding value for every animal, including those with no records of their own, obtained by borrowing information from relatives. Superior sires and dams are selected and mated, and the next generation is on average better.
For a trait expressed only in females, such as milk yield, a young bull has no record of his own. His breeding value can only be estimated accurately once his daughters have calved and completed a lactation. In dairy cattle that is roughly five years from his birth, and progeny testing is built around that wait.
The relationship matrix \(\mathbf{A}\) is also, by construction, a matrix of expectations. Two full sibs have \(a_{ij}=0.5\) whatever their genotypes; the model cannot distinguish them before they have records.
The first attempt to use molecular information was marker assisted selection. It is no longer used, and is worth covering only because the terminology persists and because its failure explains the design of what replaced it.
The idea was to identify a small number of markers linked to QTL of appreciable effect and select on them alongside conventional information. Two variants existed: markers close to but distinct from the QTL, acting as proxies through linkage, and markers assumed to be the QTL itself or effectively identical to it.
The last point made the method unreliable in a way that is hard to detect in operation. It was nonetheless easy to explain and easy to market, and commercial products based on it were sold. The lesson carried into genomic selection is that any method relying on a small number of marker to QTL associations inherits the phase problem, and the way to avoid it is to use so many markers that the associations are re-estimated from data in each population rather than assumed.
Use all the markers on the chip, with no significance testing and no selection of which to include. Estimate an effect for every marker simultaneously, then predict an animal's breeding value as the sum of the effects of the alleles it carries.
The justification rests on two arguments.
Dense markers guarantee coverage. If the panel is dense enough that every part of the genome is in linkage disequilibrium with at least one marker, then every QTL is captured by some combination of nearby markers, whether or not we know where the QTL is or that it exists. Small effects and unlocated effects are picked up along with everything else.
Uninformative markers cancel. Many markers are not linked to any QTL, and their estimated effects are noise. But that noise is centred on zero: some estimates are positive by chance and some negative, and when summed across tens of thousands of markers the errors largely cancel while the real effects accumulate. Adding a marker that carries no information costs almost nothing, whereas omitting one that does costs the variance it would have explained. That asymmetry is the whole argument for not testing significance.
Note what this abandons. Genomic selection makes no claim to identify which genes affect the trait. It is a prediction method, and it is judged by how well it predicts, not by whether its marker effects are biologically interpretable.
The genomic estimated breeding value of animal \(i\) is
where \(n\) is the number of markers, perhaps 50,000, \(S_{ij}\) is the number of copies of the reference allele carried by animal \(i\) at marker \(j\), taking the values 0, 1 or 2 from Lecture 2, and \(\hat g_j\) is the estimated effect of that marker.
Written for the whole population,
with \(\mathbf{S}\) the matrix of allele counts and \(\mathbf{g}\) the vector of marker effects.
There are far more markers than animals. With 50,000 markers and 2,000 animals the system is massively underdetermined and cannot be solved by ordinary least squares. Every genomic selection method is, at bottom, a different way of handling that: by shrinking the estimates towards zero, by imposing a prior on how effects are distributed, or by reformulating the problem in terms of relationships between animals rather than effects of markers.
Implementation has three stages, involving three groups of animals.
The third stage is where the value lies. An animal with no record, no progeny and no phenotype of its own receives a breeding value based on the DNA it actually carries.
The rate of genetic gain per year is given by the standard expression
where \(i\) is selection intensity, \(r\) the accuracy of selection, \(\sigma_g\) the genetic standard deviation and \(L\) the generation interval in years. Genomic selection acts on two of these.
Accuracy \(r\). GEBVs are more accurate than pedigree-based EBVs for animals without records, because they use realised rather than expected relationships and capture Mendelian sampling, the deviation of an animal from its parent average. For animals that already have many progeny records, however, conventional accuracy is already high, so there is less to gain.
Generation interval \(L\). This is the larger effect. Genotyping can be done at birth, so selection decisions that previously waited for progeny records can be made years earlier. In dairy cattle the generation interval can be roughly halved.
Because \(L\) appears in the denominator, halving it doubles \(\Delta G\), other things equal. An increase in accuracy from, say, 0.75 to 0.80 raises \(\Delta G\) by under 7%. The effect on generation interval is therefore usually more important than the effect on accuracy, and this is the point most often missed in discussions of genomic selection.
There is a further advantage that does not appear in the formula. Progeny testing requires maintaining and testing many young bulls whose eventual breeding values turn out to be unremarkable. Genomic pre-selection removes most of them from the programme before that expense is incurred, so the same genetic progress is achieved at lower cost.
One caution belongs here. Shortening the generation interval also accelerates the accumulation of inbreeding per year, because fewer years pass per generation and because selection on GEBVs concentrates on the families predicted best. Managing that rate is not optional, and the tools for doing so are in Lecture 9.
A large number of methods exist, differing in what they assume about how marker effects are distributed. Two families cover most practice.
Genomic BLUP (GBLUP). Assumes all markers have small effects drawn from the same normal distribution. Rather than estimating marker effects directly, it replaces the pedigree relationship matrix \(\mathbf{A}\) in the animal model with a genomic relationship matrix \(\mathbf{G}\) built from the markers. The model is then the familiar animal model, which is a considerable practical advantage since existing software and infrastructure apply directly.
Bayesian methods, collectively the "Bayesian alphabet". These place a prior on marker effects that allows most to be essentially zero and a few to be large, which is closer to what is believed about genetic architecture. They can outperform GBLUP when a trait genuinely has a few loci of large effect.
In simulation studies, methods matched to the simulated architecture outperform the alternatives, sometimes substantially. On real data the differences are usually small. The reason is instructive: most real traits are close to the infinitesimal model that GBLUP assumes, and where they are not, the advantage of a better-matched prior is largely absorbed by limitations in reference population size and phenotype quality.
The practical conclusion is that effort spent enlarging and improving the reference population returns more than effort spent choosing among methods.
A practical problem arises in any real population. Relationships can be obtained from the pedigree as \(\mathbf{A}\), or from markers as \(\mathbf{G}\). Markers are preferable because they measure realised rather than expected sharing. But genotyping is not free, so in practice:
Multi-step approaches handle this by running a conventional evaluation, then a genomic evaluation on the genotyped subset, then blending the results. This works but involves approximations at each handover and can produce inconsistencies.
Combine the two relationship matrices into a single matrix \(\mathbf{H}\) that spans all animals, using genomic relationships where genotypes exist and pedigree relationships elsewhere, with the two made consistent with each other. Then run one animal model using \(\mathbf{H}\).
Genotyped and ungenotyped animals are evaluated together in a single coherent analysis, and information flows between them: a genotyped bull's evaluation informs his ungenotyped relatives, and their records inform his.
Single-step genomic BLUP is now the standard in national genetic evaluation systems for the reasons above: it uses all available information, requires no blending step, and handles the continuous arrival of new genotypes without restructuring the evaluation.
Genomic selection is the most visible application of genomic data in animal breeding and it is usually the first thing a breeding organisation asks about. It is worth being clear about when it is the right thing to attempt.
Genomic selection is not a stand-alone technology. It requires a reference population of animals that are both genotyped and reliably phenotyped, and the size and phenotype quality of that population, not the sophistication of the statistical method, determine the accuracy achieved. It requires the recording infrastructure to keep producing phenotypes, because marker effects must be re-estimated as the population changes. And it requires the organisational capacity to act on the predictions.
For a population with no existing genomic experience, attempting genomic selection first is usually the wrong order. Admixture analysis, diversity assessment, parentage verification and inbreeding monitoring, all covered in earlier lectures, deliver value from smaller datasets, build the technical capacity that genomic selection presupposes, and do not depend on a large phenotyped reference population. Genomic selection is better approached as a later step in a sequence than as a starting point.
The wider point, made in Lecture 2 and worth repeating at the end of the course, is that genotyping is now the cheap part. The phenotype is the constraint.
Where the gain comes from. A dairy programme has selection intensity \(i=2.0\), accuracy \(r=0.75\), \(\sigma_g=500\) kg and generation interval \(L=6\) years. Genomic selection raises accuracy to 0.80 and cuts the generation interval to 3 years. Compute \(\Delta G\) before and after, and decompose the improvement.
Before. \(\Delta G = (2.0 \times 0.75 \times 500)/6 = 750/6 = 125\) kg per year.
After. \(\Delta G = (2.0 \times 0.80 \times 500)/3 = 800/3 = 266.7\) kg per year.
The gain is a factor of 2.13.
Decomposition. Accuracy alone, holding \(L=6\): \(800/6 = 133.3\), a factor of 1.067. Generation interval alone, holding \(r=0.75\): \(750/3 = 250\), a factor of 2.0.
So the accuracy improvement contributes about 7% and the halved generation interval contributes a doubling. Nearly all the benefit comes from evaluating animals earlier, not from evaluating them better. This has a design implication: a genomic programme that improves accuracy but leaves the breeding structure unchanged, still waiting for progeny records before making decisions, forgoes almost all of the available gain.
Why not just use the significant markers. A GWAS on the same reference population finds 40 genome-wide significant markers. A colleague proposes building the prediction from those 40 rather than all 50,000, arguing it is simpler and avoids fitting noise. Respond.
The proposal reduces accuracy, for a reason that is the central insight of genomic selection.
The 40 significant markers are the extreme tail of the effect size distribution, and as Exercise 6.4 established, they collectively explain a small fraction of the genetic variance. The remaining variance sits in thousands of loci whose individual effects are too small to reach significance but which in aggregate dominate. Discarding all markers except the 40 discards most of the signal.
The "avoids fitting noise" argument mistakes the situation. It is true that most of the 50,000 markers contribute noise. But that noise is centred on zero and largely cancels in the sum, whereas the small true effects accumulate coherently. The cost of including an uninformative marker is therefore close to zero, while the cost of excluding an informative one is the variance it would have explained.
There is also a selection bias problem. Effects estimated at markers chosen because they were significant in the same data are overestimated, so a predictor built from them is optimistic when applied to new animals. This is the winner's curse, and it is why any such proposal must be tested in an independent validation population rather than argued from first principles.
Designing a reference population. A national programme in Ghana can genotype and phenotype 1,500 animals for genomic prediction of growth rate in an indigenous cattle population. What determines whether this will work, and what would you check before committing?
Reference population size relative to effective population size. The number of animals needed for a given accuracy scales with \(N_e\), because \(N_e\) determines how many independent chromosome segments must have their effects estimated. Indigenous populations typically have large \(N_e\), so 1,500 animals may deliver considerably lower accuracy than the same number would in a commercial breed. This is the single most important consideration and it works against the project.
Extent of linkage disequilibrium. Estimate the LD decay curve first, from a modest pilot sample. If \(r^{2}\) falls below 0.2 within a few kilobases, a 50K chip will not cover the genome and higher density or sequence data is needed, changing the budget entirely.
Phenotype quality. Growth rate must be measured consistently, with recorded birth dates and defined weighing ages, and contemporary groups must be identifiable so that management effects can be fitted. Poorly defined phenotypes cap accuracy regardless of genomics.
Population homogeneity. If the 1,500 animals come from several genetically distinct populations, marker effects estimated across them will not transfer well to any of them. An admixture analysis, as in Lecture 7, should precede the design.
What happens next. Marker effects must be re-estimated as the population changes, so a one-off reference population is a depreciating asset. If there is no plan to keep recording, the accuracy achieved will decay over subsequent generations.
A defensible alternative is to begin with parentage verification, inbreeding monitoring and diversity assessment, which deliver value from this sample size immediately, and build toward genomic prediction as the reference population accumulates.
Single-step reasoning. A national evaluation has 800,000 animals in the pedigree, of which 40,000 are genotyped. Explain why single-step is preferable to evaluating the genotyped animals separately, and identify the technical difficulty in combining the two relationship matrices.
Why single-step is preferable. Information flows in both directions. The 760,000 ungenotyped animals carry the great majority of the phenotypic records, and those records inform the estimates for genotyped animals through the pedigree. Conversely, a genotyped bull's more accurate evaluation propagates to his ungenotyped relatives. Evaluating the genotyped subset separately discards the records of the ungenotyped animals from that analysis and requires an approximate blending step afterwards, which introduces inconsistencies and can produce evaluations that do not agree with each other.
There is also a selection bias argument. Genotyped animals are not a random sample; they were chosen because they were promising. Evaluating them in isolation from the population they were selected out of gives biased estimates, whereas a joint analysis accounts for the selection through the relationship structure.
The technical difficulty is that \(\mathbf{A}\) and \(\mathbf{G}\) are on different scales and refer to different base populations. \(\mathbf{A}\) measures relationships relative to the pedigree founders, assumed unrelated; \(\mathbf{G}\) measures them relative to the allele frequencies used to construct it, typically those of the genotyped animals. Combining them naively produces a matrix that is internally inconsistent, and evaluations in which genotyped and ungenotyped animals are systematically offset from each other.
The standard remedy is to scale \(\mathbf{G}\) so that its mean diagonal and off-diagonal elements match those of the pedigree relationship submatrix for the same animals, and to blend a small proportion of \(\mathbf{A}\) into \(\mathbf{G}\) to keep it invertible. Both are approximations, and getting them wrong is a common source of bias in practice.
