Breeding and Genetics

Breeding is the directed improvement of populations across generations. A programme starts from a breeding goal (the traits to improve and their relative economic or societal weights), then repeats a cycle: measure phenotypes, estimate each candidate’s genetic merit, select the best as parents, and mate them to produce the next generation. Plant and animal breeding share this logic, differing mainly in reproductive biology, generation interval and how populations are structured.

Genetic gain per cycle depends on how intensely we select, how accurately we rank candidates, how much genetic variation exists, and how long a generation takes, the same levers explored in the companion genomic-selection course and simulator.

Those levers are summarised in the equation that runs through the whole course, ΔG = (i · r · σ_A) / L: genetic gain per year rises with selection intensity (i), the accuracy of evaluation (r) and the additive variation available (σ_A), and falls with the generation interval (L). Plant and animal breeders pull the same levers but in different ways: plants often allow many offspring, self-pollination and rapid cycling, whereas animals have long generation intervals and few offspring per dam, which is why reproductive technologies and, later, genomic prediction matter so much for raising the rate of gain in livestock.

Quantitative traits (growth, milk yield, disease resistance) are shaped by many genes of small effect plus the environment, the infinitesimal model. The core decomposition is P = G + E: an individual’s phenotype is its genotypic value plus an environmental deviation. The genotypic value splits further into additive (transmissible), dominance and epistatic parts. The additive value is the breeding value, what a parent passes on, and is the currency of selection.

At the population level, variance partitions the same way: total phenotypic variance V_P = V_A + V_D + V_E (and more). Heritability in the narrow sense, h² = V_A/V_P, is the fraction of phenotypic variation that is additive and therefore responds to selection. Relatives resemble one another because they share alleles; the degree of resemblance (covariance between relatives) is what lets us estimate genetic parameters and breeding values.

The key quantity is the average effect of an allele: the mean deviation an allele confers when passed to offspring, averaged over the genotypes it meets in the population. An individual’s breeding value is simply the sum of these average effects over all its loci, which is why it is population-specific and why it, not the genotypic value, predicts the performance of progeny. Relatives resemble one another because they share alleles identical by descent, so the covariance between relatives is a known fraction of the additive variance (one-half between parent and offspring, one-quarter between half sibs). Turning that around, the observed resemblance among relatives is exactly what lets us estimate the additive variance and heritability from data.

Before we can predict breeding values we need the genetic parameters: heritabilities, and genetic and environmental (co)variances among traits. These are estimated from the resemblance between relatives, using mixed models that separate fixed effects (herd, year, sex) from random genetic effects. The standard estimation method is REML (restricted maximum likelihood), which estimates variance components (V_A, V_E, and others) from the data and pedigree or genomic relationships.

Genetic correlations matter because traits are not independent: selecting hard on one trait drags correlated traits along, sometimes unfavourably. Reliable parameters are the foundation of both selection-index design and BLUP evaluation.

In practice the variance components are estimated by REML rather than older analysis-of-variance formulae, because REML handles the unbalanced, selected and overlapping-generation data that real herds produce and gives estimates that do not depend on the fixed effects in the model. Reliable estimates need large, well-connected datasets: animals must be linked across herds and years through shared sires and relatives, or the genetic and environmental effects cannot be separated. Estimates come with standard errors that shrink as data accumulate, and the same machinery yields not just heritabilities but the genetic, phenotypic and environmental correlations among traits that a multi-trait index needs. A classic, economically important example is the unfavourable genetic correlation between milk yield and fertility in dairy cattle, which decades of single-trait selection for yield made worse before balanced indexes corrected course.

To rank selection candidates we estimate their breeding values. The standard tool is BLUP (best linear unbiased prediction) in the animal model, y = Xb + Zu + e, which simultaneously corrects for fixed effects and predicts each animal’s additive genetic merit u. BLUP combines every available record, the animal’s own, its relatives’, across generations, weighting them by the pedigree relationship matrix A and the genetic parameters from Module 3.

The prediction’s accuracy (correlation between estimated and true breeding value) rises with heritability and with the amount of information on the animal and its relatives. BLUP’s great strength is that it accounts for selection, non-random mating and unbalanced data, which is why it underpins national genetic evaluations.

BLUP is solved through the mixed-model equations, which combine the records, the design matrices for fixed and random effects, and the inverse of the relationship matrix into one linear system whose solution gives the fixed-effect estimates and the predicted breeding values together. Its named properties matter in practice: predictions are unbiased, use all available information, and remain valid under selection and assortative mating provided the data that drove those decisions are included. Animals are compared within contemporary groups (herd-year-season), so management differences are absorbed rather than mistaken for genetics. Each EBV is reported with a reliability or accuracy that tells the breeder how far to trust it, low on a young animal with few records, high on a proven one, the figure a buyer should read alongside the EBV itself.

The relationship matrix A is itself built from the pedigree by the tabular method, processing animals in birth order so that each animal’s relationship to all others is the average of its parents’ relationships, and its inbreeding is half the relationship between its sire and dam. A has useful structure, its inverse can be written down directly from the pedigree without inverting a large matrix, which is what makes national BLUP evaluations of millions of animals computationally feasible. Breeding values can also be updated directly as new records arrive, rather than re-solving from scratch, so evaluations are refreshed routinely. These properties are not mere technicalities: they are why pedigree-based prediction scales, and they carry straight over to the genomic relationship matrix in the next module.

Dense SNP markers let us measure relationships and gene effects directly. In GBLUP, the pedigree matrix A is replaced (or blended, single-step) with a genomic relationship matrix G built from markers, capturing the realised sharing of DNA between animals rather than its pedigree expectation. Equivalently, one can fit all marker effects at once with ridge regression / SNP-BLUP or Bayesian models that allow markers to differ in effect.

Genomic prediction gives accurate breeding values on young animals before they have records or progeny, shortening the generation interval and accelerating genetic gain. Accuracy depends on the size and relatedness of the reference population and the trait’s heritability. This module connects directly to the companion course Breeding programs with Genomic selection, which covers reference-population design and the accuracy equation in depth.

Genomic prediction works because dense markers are in linkage disequilibrium with the unknown causal genes, so a marker effect estimated in a reference population of genotyped-and-phenotyped animals carries over to relatives that share the same haplotypes. The accuracy of the prediction rises with the size and relatedness of that reference population and with the trait’s heritability, which is why pooling data across herds and countries is so valuable for numerically small breeds. In modern national evaluations the pedigree and genomic relationships are usually combined in a single-step approach (ssGBLUP) that blends the A and G matrices, so genotyped and non-genotyped animals are evaluated together on one scale. The headline benefit is unchanged: accurate breeding values on young animals before they have records, which shortens the generation interval and accelerates gain, the central theme of the companion genomic-selection course.

The course is built around hands-on computing in R. The guide and practicals walk through importing data, fitting linear and mixed models, estimating heritabilities and breeding values, and carrying out genomic prediction, the methods of Modules 2 to 5 applied to real data. Work through the practicals in order; full R solutions are provided so you can check yourself.

R is used because it is free, runs on any computer, and brings statistical modelling, matrix algebra and graphics into one reproducible script, so an analysis can be re-run and audited rather than clicked through once. The practicals lean on R’s linear- and mixed-model tools and on the matrix operations behind BLUP and GBLUP, building the habit of inspecting data, fitting a model and checking assumptions before trusting a result. Learning to compute these estimates yourself, rather than treating the software as a black box, is what lets a breeder diagnose a strange result and adapt the method to a new dataset.

Individual practicals: Practical 1 · Practical 2 · Practical 3 · Practical 4 · R solutions (1–2).

Glossary