REVIEW

Applications of Population Genetics to AnimalBreeding, from Wright, Fisher and Lush to

Genomic PredictionWilliam G. Hill1

Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JT, United Kingdom

ABSTRACT Although animal breeding was practiced long before the science of genetics and the relevant disciplines of population andquantitative genetics were known, breeding programs have mainly relied on simply selecting and mating the best individuals on theirown or relatives’ performance. This is based on sound quantitative genetic principles, developed and expounded by Lush, whoattributed much of his understanding to Wright, and formalized in Fisher’s infinitesimal model. Analysis at the level of individual lociand gene frequency distributions has had relatively little impact. Now with access to genomic data, a revolution in which molecularinformation is being used to enhance response with “genomic selection” is occurring. The predictions of breeding value still utilizemultiple loci throughout the genome and, indeed, are largely compatible with additive and specifically infinitesimal model assump-tions. I discuss some of the history and genetic issues as applied to the science of livestock improvement, which has had and continuesto have major spin-offs into ideas and applications in other areas.

THE success of breeders in effecting immense changes indomesticated animals and plants greatly influenced Dar-

win’s insight into the power of selection and implications toevolution by natural selection. Following the Mendelian re-discovery, attempts were soon made to accommodate withinthe particulate Mendelian framework the continuous natureof many traits and the observation by Galton (1889) ofa linear regression of an individual’s height on that of a rel-ative, with the slope dependent on degree of relationship. Apolygenic Mendelian model was first proposed by Yule(1902) (see Provine 1971; Hill 1984). After input from Pear-son, Yule again, and Weinberg (who developed the theorya long way but whose work was ignored), its first full expo-sition in modern terms was by Ronald A. Fisher (1918) (bi-ography by Box 1978). His analysis of variance partitionedthe genotypic variance into additive, dominance and epi-static components. Sewall Wright (biography by Provine1986) had by then developed the path coefficient methodand subsequently (Wright 1921) showed how to computeinbreeding and relationship coefficients and their conse-

quent effects on genetic variation of additive traits. His ap-proach to relationship in terms of the correlation of unitinggametes may be less intuitive at the individual locus levelthan Malécot’s (1948) subsequent treatment in terms ofidentity by descent, but it transfers directly to the correla-tion of relatives for quantitative traits with additive effects.

From these basic findings, the science of animal breedingwas largely developed and expounded by Jay L. Lush (1896–1982) (see also commentaries by Chapman 1987 and Ollivier2008). He was from a farming family and became interestedin genetics as an undergraduate at Kansas State. Although hismaster’s degree was in genetics, his subsequent Ph.D. at theUniversity of Wisconsin was in animal reproductive physiol-ogy. Following 8 years working in animal breeding at theUniversity of Texas he went to Iowa State College (nowUniversity) in Ames in 1930. Wright was Lush’s hero: ‘I wishto acknowledge especially my indebtedness to SewallWright for many published and unpublished ideas uponwhich I have drawn, and for his friendly counsel” (Lush1945, in the preface to his book Animal Breeding Plans).Lush commuted in 1931 to the University of Chicago toaudit Sewall Wright’s course in statistical genetics and con-sult him. Speaking at the Poultry Breeders Roundtable in1969: he said, “Those were by far the most fruitful 10 weeksI ever had.” (Chapman 1987, quoting A. E. Freeman). Lushwas also exposed to and assimilated the work and ideas of

Copyright © 2014 by the Genetics Society of Americadoi: 10.1534/genetics.112.147850Manuscript received August 7, 2013; accepted for publication October 18, 20131Address for correspondence: University of Edinburgh, Institute of Evolutionary Biology,University of Edinburgh, W. Mains Road, Edinburgh EH9 3JT, United Kingdom.E-mail: w.g.hill@ed.ac.uk

Genetics, Vol. 196, 1–16 January 2014 1

R. A. Fisher, who lectured at Iowa State through the sum-mers of 1931 and 1936 at the behest of G. W. Snedecor.

Here I review Lush’s contributions and then discuss howanimal breeding theory and methods have subsequentlyevolved. They have been based mainly on statistical meth-odology, supported to some extent by experiment and pop-ulation genetic theory. Recently, the development ofgenomic methods and their integration into classical breed-ing theory has opened up ways to greatly enhance rates ofgenetic improvement. Lush focused on livestock improve-ment and spin-off into other areas was coincidental; buthe had contact with corn breeders in Ames and beyondand made contributions to evolutionary biology and humangenetics mainly through his developments in theory (e.g.,Falconer 1965; Robertson 1966; Lande 1976, 1979; see alsoHill and Kirkpatrick 2010). I make no attempt to be com-prehensive, not least in choice of citations.

J. L. Lush and the Science of Livestock Improvement

Lush was interested in practical application and in how tomake the most rapid improvement. “Heritability” is an oldword and evolution of its use is discussed by Bell (1977),who includes a long letter from Lush. He was first to adoptheritability in the narrow quantitative genetics sense as theratio of additive genetic to phenotypic variation, and there-fore also the square of Wright’s correlation of (additive)genotypic and phenotypic value, and to use “accuracy” ofa predictor of breeding value to compare alternative selec-tion schemes. Crucially Lush developed what has becomeknown as the “breeder’s equation” for predicting responsein terms of selection differential, R = h2S. The expression isimplicit in his book Animal Breeding Plans (first edition1937, third and last edition 1945, of which my 1962 copyis the eighth printing). He writes, “for each unit which theselected parents average above the mean [. . .], their off-spring will most probably average about sH

2=ðsH2 þ sE

2Þ asfar above. This would be literally true if all genes act addi-tively” (Lush 1937, p. 84; Lush 1945, p. 100). Here sH

2 is thetotal genetic (genotypic) variance, but he clarifies in a foot-note, “This formula would be more nearly correct if thenumerator were only the additive genetic portion of thevariance but that is a slight understatement of the case sincea proportion of the epistatic variance belongs in the numer-ator.” This comment has a subtlety I discuss later.

Lush addressed what proportion of the variation inproduction traits in livestock was genetic, the resource forgenetic progress. His first major article on quantitativegenetic applications in animal breeding was on “Factors af-fecting birth weights of swine” (Lush et al. 1934). He sawthe need to obtain estimates of parameters such as the her-itability free of confounding by environmental covariancesand proposed using daughter dam within sire regression toavoid bias by herd effects (Lush 1940). He considered prac-tical questions such as the relative accuracy of selection ona cow’s own performance vs. that of her progeny mean (Lush

1935). Together with his colleague L. N. Hazel, Lush devel-oped selection index principles to make best use of the data.In the plant breeding context, Smith (1936) had deriveda discriminant function, i.e., a selection index, and thanks“Fisher for guidance and inspiration. [. . .] section I [. . .] islittle more than a transcription of his suggestions.” Hazel(1943) introduced the idea of genetic correlations andshowed how to use these to compute multitrait selectionindices, and Lush (1947) derived how best to weight anindex of records on an individual and its sibs. He and col-leagues recognized also that rates of progress should bemaximized per year rather than per generation and consid-ered the tradeoff between the high accuracy of a progenytest and the shorter generation time by selecting on ownperformance (Dickerson and Hazel 1944).

Lush’s research was closely focused on practical problemsof short-term improvement, mainly on how to select the bestanimals to breed the next generation. Over such a time scaleof a few generations, issues of finite population size, size ofgene effects, and epistasis are not important, so Lush couldjust as well be following Fisher as Wright. Further, he dis-cusses in Animal Breeding Plans (Lush 1945, Chap. 11) howselection can change variability in a population: he arguesthat, although the selected individuals are phenotypicallyand somewhat genetically less variable, most will recovervariation in subsequent generations, and so an assumptionof multiple loci and of near constancy of response is a reason-able approximation in the medium term. (This was formal-ized later by Bulmer, discussed below.)

Nevertheless Lush shows the influence of his mentorWright, who had participated in multigeneration selectionexperiments for his Ph.D. with Castle and subsequentlyundertook breeding experiments at U.S. Department ofAgriculture (USDA) and analyzed pedigree records ofShorthorn cattle, all of which took him to the shifting-balance theory (Wright 1931, 1932). Summarizing later: “Itwas apparent, however, from the breeding history of Short-horn cattle [. . .] that their improvement had actually oc-curred essentially by the shifting balance process ratherthan by mere mass selection. There were always many herdsat any given time, but only a few were generally perceivedas distinctly superior [. . .]. These herds successively madeover the whole breed by being principal sources of sires”(Wright 1978, pp. 1198–9). Still in chapter 11, Lush (1945)discusses selection for epistatic effects and presents a two-dimensional “peaks of desirability” and a contour diagram ofWright’s adaptive landscape (from Wright 1932). Lush con-cludes the chapter with practical advice: “Only rarely is massselection completely ineffective, as when selection is for a het-erozygote, when selection has already carried the popula-tion into stable epistatic peak, or when selection is within anentirely homozygous line. Often, however, the rate of prog-ress by mass selection is slow and could be made more rapidby a judicious use of relatives and progeny or by more care-ful control of the environment.” Later in the book, after de-scribing how to increase selection responses, he considers

2 W. G. Hill

inbreeding, assortative mating, and the like and returns tosome of the epistatic themes.

As clearly shown in Animal Breeding Plans, Lush (1945)had both a practical insight and a deep understanding of thequantitative genetic principles and statistical ideas behindboth Wright’s and Fisher’s work. More theory and detailedargument is given in Lush’s lecture notes “The genetics ofpopulations,” issued as mimeo and dated 1948 (Lush 1948).They were somewhat modified subsequently but, I judge,rather little. Although not then formally published, the noteswere influential because so many students took Lush’s courseand because they cover so much basic population and quan-titative genetics and the relevant statistics. After his death theywere retyped and published (Lush 1994), edited by A. B.Chapman and R. R. Shrode with comments by J. F. Crow.

Lush had many very able colleagues in Ames, both inanimal science and statistics. Among the latter was O.Kempthorne, who had worked at Rothamsted, was a greatadmirer of Fisher, and took an interest in genetics. Inter aliahe extended Fisher’s partition of variance for multiple-alleleand -locus epistatic effects and published a book on geneticstatistics (Kempthorne 1957). The many graduate studentsat Iowa State were therefore exposed both to Lush’s use ofthe path diagram approach of Wright and to Kempthorne’semphasis on the variance component approach of Fisher.Lush commented in a letter to Bell (1977), however: “Leav-ing it in variance components, rather than as fractions ofthose, seems to have certain technical advantages whenone is concerned only with statistical significance. Fisher andSnedecor were stressing unduly the testing for significance inthe early 1930s [. . .]. I merely mention that as part of theexplanation for the widespread preference for expressing ourfindings in variance components, rather than to express themas fractions of variance.”

Lush was an outstanding communicator to students,scientists, and breeders. He had 126 Ph.D. and 26 M.S.students (Chapman 1987), including C. R. Henderson. In1963, between master’s and Ph.D., I was briefly a student ofKempthorne at Ames where there was a great aura of practicalinquiry and activity. I took part in Lush’s course, albeit thenbecoming a little dated. I have never found path diagramsuseful, but others have done so, not least economists. Indeedmost subsequent developments have been undertaken interms of variances and covariances rather than their ratios.

Developments in Quantitative Genetic and BreedingPrediction Theory

Quantitative genetic principles as applied to breeding werebeing adopted and developed elsewhere in the 1940s forplants and for animals. K. Mather and colleagues atBirmingham (United Kingdom) focused mainly on design andinterpretation of crosses of plants and on inferences about theindividual parental lines. They wrote in Biometrical Genetics(Mather 1949, subsequent editions with Jinks) and wererather little influenced by Lush’s work. Comstock and Robinson

at Raleigh were concerned also with plants but gave moreemphasis to variation within lines. The work on design ofbreeding programs was concerned particularly with line-cross improvement, which had more limited impact in live-stock breeding, with the notable exception of reciprocalrecurrent selection (RRS) proposed by Comstock et al. (1949).

I. M. Lerner and E. R. Dempster at the University ofCalifornia, Berkeley, undertook both theoretical and exper-imental studies, using poultry as both a model and com-mercial species. They proposed and tested experimentallythe use of part-year laying records on sibs to replace progenytesting of males so as to reduce generation interval andincrease selection intensity. In Population Genetics and AnimalImprovement, subtitled “As illustrated by the inheritance ofegg production in poultry,” Lerner (1950) outlines the popu-lation and quantitative genetic principles and practice, and hespecifically acknowledges his indebtedness to Wright and toLush. His interests extended well beyond poultry breeding,however, including genetic homeostasis (Lerner 1954).

In part to apply the new operational research techniquesto animal breeding, a group was brought together inEdinburgh after World War II. Early theoretical work wasimmediately relevant to animal improvement in the Lushtradition. Notably, Rendel and Robertson (1950) providedgeneral formulae for rates of progress for overlapping gen-erations with different strengths of selection in male andfemale parents. They used these to contrast possible ratesof progress in dairy cattle for traditional within-herd selec-tion and for the new opportunity provided by progeny test-ing bulls by artificial insemination (AI) with daughters inmany herds. Robertson was a chemist by training, but spentsome months with Wright and with Lush prior to coming toEdinburgh. He developed the “contemporary comparison”method for genetic evaluation of sires used in AI, whichwas important until superseded by Henderson’s best linearunbiased prediction (BLUP)methodology. (I was a Ph.D. stu-dent and subsequent colleague of Alan Robertson, whosepersonal input and coffee sessions were a great source ofquestions, insight, and debate. I apologize for any Edinburgh-centric bias in coverage.)

Lerner also spent a sabbatical leave at Edinburgh before his1950 book was published, but interactions were wider. TheAmes and Berkeley groups both advised a Californian poultrybreeder in the late 1940s, and a unique collection of talentworked on the genetic improvement of threshold traits, such assurvival of poultry, using the model originally formulated byWright (1934) for polydactyly in guinea pigs. Articles wereauthored by combinations of Lush, Hazel, Lerner, Dempster,and Robertson (e.g., Dempster and Lerner 1950, which includesan appendix by Robertson deriving the well-known relationshipbetween heritability on the binary and continuous scales).

These ideas were extended later by D. S. Falconer atEdinburgh who developed a simple method for estimatingheritability of threshold traits by adapting the breeder’sequation rather than using variance components (Falconer1965), work that attracted much attention in the human

Review 3

genetic community as a model for genetic study of disease.Falconer also did much to expand the use and understand-ing of quantitative genetics and its applications through hisbook Introduction to Quantitative Genetics (Falconer 1960aand subsequent editions).

Theory developed for livestock populations by Lush andothers has also had an important influence on evolutionaryideas and theory. Breeders appreciated that Lush’s breeder’sequation R= h2S applied only to the trait on which selectionwas practiced, even if evolutionary biologists did not. Theequation was generalized, however, by Lande (1979) in termsof selection gradients consequent on fitness differences. In thebreeders’ world, the equivalent would be described as a ret-rospective index, asking after the event what the selectionwas actually on, but both apply only to traits that are actuallyrecorded. In an analysis of culling theory in dairy cattle,Robertson (1966) showed that changes in traits equaledtheir genetic covariance with fitness (i.e., the hidden index),in what has become known as the secondary theorem ofnatural selection. It was subsequently generalized by Price(1970) and is also known as the Price or Robertson–Priceequation. Further discussion of impacts of animal breedingon evolutionary theory is given elsewhere (Hill and Kirkpatrick2010). Indeed, for a discussion on this and any other issues onselection and beyond, see completed sections of the magnumopus by Walsh and Lynch (2009).

Lush and colleagues at Ames mainly utilized field data onlivestock, with estimates of genetic parameters needed formaximizing short-term response in terms of rate and di-rection. He had sufficient understanding of the theory andmodels that he had expounded and on the results of earlyexperiments to be confident that responses would continueat similar rates. Neither he nor most of his colleagues undertookselection experiments to check whether predictions fromtheory actually worked in practice, however. These wereundertaken elsewhere and I turn to them later.

Simple Polygenic Models: The Infinitesimal Model

Fisher (1918) had introduced a genetic model with “a greatnumber of different factors, so that s [standard deviation] islarge compared to every separate a [gene effect] (Fisher1918, p. 402),” the so-called “infinitesimal model.” It wasformalized by Bulmer (1971, 1980), so properties of thenormal distribution could be used: “normal theory presup-poses an effectively infinite number of unlinked loci withinfinitesimal effects; in consequence a finite change in themean can be brought about by an infinitesimal change in genefrequencies” (Bulmer 1980, p. 150). The variance among se-lected individuals is less than in the population as a whole,but in the infinitesimal model the variance contributed ateach locus, and thus the genic variance, can be assumed toremain constant despite the selection. Changes in varianceunder selection result not from gene frequency change butfrom gametic disequilibrium among loci, which is transientas a consequence of recombination.

An important feature of the infinitesimal model is thattheory for multivariate normality can proceed, exactly or togood approximation, in the quantitative genetic context.Variances and covariances among relatives for single andmultiple traits in the next generation are predictable solelyfrom the standard methods of multiple regression and linearmodels dating back to Pearson (1903). Regression theoryenables computation of the reduction in the phenotypic var-iance in a mass selected group, the variance in their breed-ing value, and thus the variance between families in the nextgeneration. This reduction, the “Bulmer effect,” is due togametic phase disequilibrium between loci that affect thetrait induced by the selection (Bulmer 1971, 1980). TheMendelian sampling variance, which comprises the geneticvariance within families, remains unchanged because, withfree recombination, gametic equilibrium for unlinked lociwithin families is recovered. The departure from normalitycaused by truncation selection has little impact on thechange predicted from the normal distribution in subse-quent generations (Turelli and Barton 1994). The Mende-lian sampling variance component declines predictably inproportion to the mean inbreeding coefficient of the parents.

Effects are necessarily additive within loci and, if all n lociare contributing, their individual effects must decline ap-proximately in proportion to 1=

ffiffiffi

np

as n / N. With direc-tional dominance, however, it is not possible to specifya model whereby both additive variance and inbreeding de-pression remain finite as n increases. Epistasis is formallyexcluded, but additive 3 additive terms can be accommo-dated provided the epistatic effects among the pairs of locidecline approximately in proportion to 1/n. Griffing (1960),who had worked with Lush, showed how epistatic variancecould contribute to response. Genetic gain arises from ga-metic disequilibrium among the epistatic loci but, for un-linked loci, asymptotes under continued selection becausehalf is lost each generation by recombination, like theBulmer effect for variance, and so does not contribute tosubsequent response. Lush understood much of this, albeitnot all at a formal level, as sections in animal breeding plansshow; for example, see the footnote quoted earlier. The lack ofimpact of epistasis on recurrent response is a general feature,however, not just of the infinitesimal model (Crow 2010).

When we look at both previous and subsequent develop-ments in the application of population genetics to animalbreeding we find that much of it has explicitly or implicitlybeen based on the infinitesimal model: it has very powerfulsimplifying statistical properties and avoids the need tospecify individual gene effects, information on which hasuntil recently been impossible to obtain.

Breeding Value Prediction Using the InfinitesimalModel: Henderson, BLUP, and the Animal Model

To utilize additive effects, genetic improvement requiresidentifying the animals with the highest breeding value andselecting them, then replacing them as better ones come

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along. A major intellectual insight into prediction ofbreeding value came from C. R. Henderson, a student ofLush and Hazel (although there is little sign to the outsiderthat they interacted much). He recognized that there were“fixed” effects such as herd or season, which had to be in-cluded but not estimated, and others, “random” effects,which were samples from a distribution. In the standardindex theory of Hazel and Lush it was assumed that envi-ronmental effects were properly corrected for, straightfor-ward to achieve when all animals are reared together, butnot for bulls with daughters got by AI and distributed un-equally around many herds. The breeder’s aim is not toestimate the current mean, but to predict the performanceof future daughters. The methodology was formalized as BLUPby Henderson. It first appears in his Ph.D. thesis (Henderson1948), but he developed it greatly subsequently when atCornell, completing a book after retirement (Henderson1984). S. R. Searle, Henderson’s colleague there, was animportant contributor to his work and also did much fordevelopments in variance component estimation (Searle1971). Alternative early methods used contemporary compar-isons or regressed least squares, which were more computa-tionally tractable. In these the process of estimation of actualeffects and regression to predict the breeding values weredone sequentially and assumptions made that the sires ofcontemporaries were randomly sampled from the population.

Henderson proposed the mixed model equations, whichlook like least-squares equations but include the criticalshrinkage term for the random effects, and are more compactand tractable to solve than the equivalent maximum likeli-hood equations. They yield best linear unbiased estimates(BLUE) of fixed effects and predictors (BLUP) of randomeffects. To many statisticians the idea of predicting a re-alization of random effects, i.e., a breeding value, in a statis-tical model was unsound, but scarcely troubled quantitativegeneticists. The critical requirement is that there is a knowndistribution from which the breeding values are sampled,assumed to be the normal with variance VA, and BLUP is nodifferent in that sense from a simple predictor from pheno-type such as A ¼ h2P. Indeed breeding value prediction, theselection index, BLUP, and the like can be given a simpleBayesian interpretation (e.g., A. Robertson 1955). The mixedmodel has provided a basis for approaching many areas ofvariance partition and prediction, including generalized non-linear models such as those for threshold traits and for mod-els in which the variance is itself subject to variation.

The first widely used BLUP models were in terms of siregenetic effects and within sire deviations, which fulfilled theprimary need for dairy sire evaluation and were computa-tionally feasible. But the complete analysis requires the “an-imal model” (e.g., Henderson 1976 described it, but did notthen name it), in which the breeding value of each individ-ual is defined in terms of effects and the covariances amongbreeding values of different individuals. The covariances areexpressed in terms of Wright’s numerator relationship, i.e.,the covariance of uniting gametes or twice coancestry (kinship)

coefficient, as off-diagonals in the relationship matrix (A),with the diagonal elements depending on the individual’sinbreeding coefficient. Solution of the BLUP equationsrequires the inverse A21, however, but Henderson (1976)saw that it had a simple form and could be obtained directlyfrom pedigrees without ever computing A.

BLUP predictions allow comparisons among individualsin the population that differ in age and amounts ofphenotypic information on them and their relatives andincorporate multiple traits. Thus, for example, genetictrends over years and generations can be estimated free ofenvironmental changes (Henderson et al. 1959). Inter aliatwo important assumptions are made, however. The first isthat all selection is accounted for, and so formally it has toinclude all traits on which selection is based (but unre-corded traits, such as on animals prior to the existing ped-igree or dead before recording remain a problem), andselection is not confounded with fixed effects (a debatedtechnical issue, see e.g., Thompson 1979, but not one forthis article). The second is implicit, that the infinitesimalmodel holds, such that, for example, the variance is notchanged by selection other than through gametic disequilib-rium and the genic variance declines in proportion to theinbreeding coefficient. Mutation is usually ignored, but canbe accommodated by adding a series of relationship matri-ces back to successive generations of new variance frommutation (Wray 1990).

Notwithstanding the strong assumptions invoked, theflexibility and successful practical adoption of the BLUP andanimal model framework are such that it became allpervasive in livestock improvement. It copes with multipletraits and also enables other major additions to thequantitative genetic model developed over the decades tobe incorporated in the same structure. These include mater-nal genetic effects, such as of dam on calf’s phenotype forbirth weight introduced by Willham (1963), a colleague ofLush; competitive effects, e.g., impact on the growth of ananimal of its pen mates (Bijma et al. 2007), which developideas for analysis of plant competition (Griffing 1967); ran-dom regression models for traits such as weight at differentages in which the regression coefficient varies genetically(Schaeffer and Dekkers 1994) and that can also beexpressed in terms of a covariance function across ages(Meyer and Kirkpatrick 2005); and dominance effects(Smith and Maki-Tanila 1990). Subsequently methods toincorporate genomic information have been put into theBLUP framework as discussed later.

Henderson (1953) also developed methods for estima-tion of variance components, e.g., additive genetic variance,in designs with an unbalanced structure typical in livestockbreeding, These methods were standard for many years un-til restricted (aka residual) maximum likelihood (REML,which incorporates BLUP) was developed (Patterson andThompson 1971) and computing power became adequateto use it for large data sets. REML is now standard in quan-titative genetic analysis and beyond (Thompson 2008).

Review 5

Bayesian methods for mixed models in quantitative geneticshave been stimulated particularly by Gianola and colleagues(e.g., Gianola and Fernando 1986) and, following the de-velopment of Markov chain Monte Carlo (MCMC) techni-ques to facilitate computation, are increasingly adopted,although still computationally intense (Sorensen and Gianola2002). Whatever one’s philosophical bent, the Bayesianand MCMC methods have advantages of flexibility in ana-lyses of nonnormally distributed (e.g., categorical) traitsmore generally and in fitting Bayesian genomic predictionmodels.

The animal model also provides a structure in which tovisualize and analyze selection experiments, because theaverage covariance between individuals within a generationessentially equals the genetic drift variance accumulated tothat generation (Sorensen and Kennedy (1986). Conse-quently, the changes in mean and in genetic and other var-iances can be estimated simultaneously.

Following Shaw (1987) and stimulated by the book byLynch and Walsh (1998) and by Kruuk (2004) the mixedmodel framework with the animal model has also becomewidely adopted in analyses of data from pedigreed naturalpopulations, not just for estimating quantitative geneticparameters such as heritability and correlations, but alsothe strength of selection acting on each trait. This has ledto new approaches and understanding. Researchers in wildpopulation biology, however, face problems of obtainingdata on sufficient numbers of related animals to obtain un-biased estimates with low variance, not least in a contextwhere natural selection is likely to be acting via many traits,some or most of which may be unrecorded (Hadfield et al.2010).

Incorporation of Finite Population Size into Predictionsof Response

How long and how fast selection could continue over manygenerations have always been of concern, but Lush and hisschool focused on selecting accurately and effectively eachgeneration, de facto assuming the infinitesimal model withchanges in VA proportional to inbreeding. After Wright therewas little formal analysis of the effects of selection in finitepopulations before the work of Kimura in the 1950s at theindividual locus level (Crow and Kimura 1970).

Direct application of population genetic methods includ-ing individual locus effects, selection, and population sizefor predicting long-term response was led by Alan Robertson,who had previously considered the effects of inbreeding onvariation due to recessives genes in unselected populations(Robertson 1952). He resurrected (Robertson 1960) a for-mula of Haldane (1931) for the selective value s of a traitunder artificial selection, s = ia/sP, in terms of selectionintensity (i) and the gene effect relative to the phenotypicstandard deviation (a/sP). Gene frequency change couldthen be predicted and he extended the formula to includeselection on an index of relatives’ records. Using Kimura’s

diffusion equation methods and results, Robertson showedthat the fixation probability of a favorable additive gene ina population of constant effective size Ne was a function ofthe product Nei. Thus, while more intense selectionincreases short-term response, if fewer parents are therebyselected and Ne is reduced, there is a cost in long-term re-sponse. Indeed for selection on phenotype, the limit is max-imized with half the population selected (i.e., very weak),and if family information is included to increase accuracyand immediate gain, the limit is reduced because coselectionof relatives reduces Ne. The impact of small Ne on fixationprobability or long-term response is greatest for genes thathave very small effect or are initially at low frequency. Ex-perimental tests of the theory were undertaken, with gen-erally concordant results (Jones et al. 1968).

While the basic premises are clear, the practical problemwith this theoretical approach is that one cannot predict theactual limit or indeed the response after the first fewgenerations without knowledge of, or making assumptionsabout, s values, i.e., gene effects, and their initial frequen-cies, let alone considering complexities of dominance, epis-tasis, and any counteracting natural selection.

The ideas in Robertson’s 1960 selection limits articlewere extended to include linkage (Hill and Robertson1966), using Monte Carlo simulation as a mathematical toolrather than to represent a binary word as a chromosome. Asa personal aside, this article has had negligible influence onanimal breeding practice but was picked up, initially byFelsenstein (1974), in discussions of the evolutionaryadvantages of recombination and sex by increasing the sur-vival of advantageous mutant genes. The 1966 and subse-quent articles (Hill and Robertson 1968; Sved 1968) drewattention to production of linkage disequilibrium (LD) bydrift in finite populations, but it was then essentially anacademic exercise because many years passed before muchLD data could be collected.

Population genetic arguments were also applied to theimpact of selection on effective population size. High-performing individuals are likely to have high-performingrelatives so the variation in family size is increased and Ne

reduced by artificial selection. Robertson (1961) computedthe impact using a model in terms of variation in fitness. Theideas were extended by Nei and Murata (1966) to the evo-lutionary context and later revised to account for the im-proving performance of competitors such that fitnessbenefits decay over generations (Wray and Thompson1990). These have led to increasingly sophisticated and dy-namic methods based on the infinitesimal model to optimizebreeding structure, maximizing response relative to inbreed-ing, by balancing selection intensity, accuracy, and matingscheme (Wray and Goddard 1994; Meuwissen 1997).

Frankham (1980) showed that mutation was importantin contributing response in long-term artificially selectedpopulations. Clayton and Robertson (1957) had previouslyobtained estimates of its magnitude for quantitative traits,and the upper end of their estimate (�0.1% new heritability

6 W. G. Hill

per generation) turned out to be roughly the modal estimatefor many species and traits (Houle et al. 1996). Asymptoticresponse from mutations is in theory proportional to Ne, butthe short-term impact is negligible if mutant effects are in-finitesimally small (Hill 1982). If some effects are large, re-sponse is less predictable but expected sooner (Caballeroet al. 1991). Mutation is relevant to maintenance of long-term variation and response and to the comparisons be-tween infinitesimal and finite locus models, which we dis-cuss subsequently.

Selection Experiments to Check onand Develop Theory

Multigeneration selection experiments have been usedwidely with the aim of learning about the genetic architec-ture of traits, not least their polygenic basis, and inevaluating selection methods for breeding programs. Theydate back to those of Castle, Sturtevant, and others. Forexample, Mather and colleagues in Birmingham attemptedto draw inferences about, e.g., the relations among geneeffects along the chromosome (Mather 1941). Nevertheless,experiments that tested quantitative genetics selection the-ory based on formulations of Lush and colleagues did notfeature significantly much before 1950. Lush himself seemedconfident enough of the theory and was perhaps reassuredby the early experiments of Castle and others not to beconcerned about testing the effectiveness of selection, forexample, in practice. Indeed it is worth noting that annualcycles of selection in the Illinois lines of maize were startedbefore 1900 and responses have now continued for .100generations (Dudley and Lambert 2004).

A straightforward check was made of whether selectionresponse over multiple generations met predictions madefrom parameter estimates in the base population by Claytonet al. (1957) in Edinburgh. For abdominal bristle number inDrosophila, predictions (e.g., R = h2S) were quite good overearly generations; most of the response achieved was notlost quickly on relaxation of selection; and inbred lines andreplicate selection lines varied as expected from geneticdrift. This provided some reassurance on Wright’s and Lush’stheories of inbreeding and selection response. Further, theselection experiments showed that, as expected from thepolygenic model, if selection was continued for very manygenerations, then response continued such that the means ofboth the high and low lines were well outside the phenotypicrange in the base population (Clayton and Robertson 1957).

Bristle number is a highly heritable trait for which mostof the genetic variance is additive (Clayton et al. 1957) andis not strongly associated with fitness. Other experiments atEdinburgh, however, e.g., by Reeve and F.W. Robertson,were focused more on fitness-associated traits, for whichthe patterns and continuation of response were more poorlypredicted and plateaus were obtained (F. W. Robertson1955). Even so, Falconer (1960b) showed that continuedselection to increase ovulation rate was effective in raising

the mean, although more so in reducing it. Lerner andDempster were able to show the effectiveness of part recordselection in poultry, as predicted, but had already foundclear examples of natural opposing artificial selection, forexample, in shank length (discussed in Lerner’s 1957 bookThe Genetic Basis of Selection). The difficulties with suchwere, and remain, to make adequate predictions from basepopulation parameters.

The selection experiments also led to developments intheory. Notably Falconer (1952) introduced “realized heri-tability” to describe the selection response and extended theconcept of genetic correlation between traits as used byHazel (1943) to that between the same trait in individualsreared in different environments and so having phenotypicrecord in only one (Falconer 1952). He also showed that,contrary to some animal-breeding dogma, larger responsesmight be obtained in a good environment by selection ina poorer one than in the good one itself. The principle ofthe genetic correlation across environments is, for example,a fundamental component of methods used for evaluation ofdairy sires in each of many countries by Interbull (http://interbull2.slu.se/www/v1/).

In farm livestock a number of multigeneration experi-ments were started. The nicest early example of simple massselection working in livestock was for fatness in pigsreported by Hetzer and Harvey (1967) of USDA, followingproposals by Hazel, in which they obtained a high–low linedivergence in backfat depth of 68% of the initial mean after10 generations in one breed and 44% after 8 generations inanother. The realized heritabilities accord with estimates onthe trait from elsewhere by variance component analysis.Several selection lines of sheep were started in Australia inthe 1950s, and substantial responses were obtained for in-dividual traits (Turner and Young 1969).

Although most of these studies showed that an additivemodel did quite a good job, the obvious inbreeding de-pression and heterosis in fitness led to attempts to utilize thelatter, as was being practiced in maize, both by generatinginbred lines and test crossing and by using RRS, assessingindividuals by their crossbred sibs or progeny. An example ofusing selection experiments to compare breeding schemeswas reported by Bell et al. (1955) at Purdue, who assessedpure-line and cross-line selection alternatives for improve-ment of crossbred performance, including recurrent inbreed-ing. While the more complex schemes had potential benefits,the use of inbreeding and selection on crossbreds as in maizewas not really feasible in livestock: without selfing, inbreed-ing can be built up only slowly, and the high-inbreeding de-pression leads to much loss of lines, precluding intenseselection and rapid turnover of new improved crosses; so itfell out of use. Similarly RRS involves a more complex breed-ing program with the need for family rather than individualselection. Although commercial poultry and pig products arecrosses, selection seems to be primarily or exclusively onpurebred performance, focusing on selection intensity andgeneration turnover rather than accuracy.

Review 7

Experiments were also undertaken to assess the effec-tiveness of some of Wright’s ideas by comparing continuousselection in a single large population with selection in manysmall lines, selecting among these, and from a cross amongthem, continuing with later cycles of inbreeding and selec-tion. These were undertaken using bristle number in Dro-sophila, however, an “additive” trait, and so, in hindsight,the lower response from the inbreeding scheme is unsurpris-ing (Madalena and Robertson 1975).

Looking back at these experiments, mainly from theperspective of the early 1960s, what do they tell us? Selectionworks and keeps working over many generations to give largephenotypic change provided that correlated changes in fitnessdo not interfere. They are not very discriminatory in terms ofmodels of gene action, e.g., many or few important loci. The-ory in terms of average effects (not necessarily additive geneaction per se) gives useful predictions. Drift occurs amonglines and inbreeding within lines in accordance with simpleadditive and dominance model predictions respectively. Therewas no real upset to the models and arguments of Wright andLush, or indeed of Fisher, although he had mostly discounteddrift effects working in the evolutionary context. What we didnot get from the experiments is much detail of the geneticarchitecture, beyond the obvious conclusion that many locimust contribute and that there are often fitness-associatedeffects (i.e., unfavorable fitness–trait genetic correlations).Furthermore, drift-sampling variation precludes fine-scale dis-crimination of models from selection experiments of manage-able size (Hill 1971; Falconer 1973).

How Well Does the Infinitesimal Model Fit?

Theoretical analyses show the sensitivity of predictions overmultiple generations to, for example, genes of large effect,but we do have to ask how well results fit predictions fromthe simplistic and unrealistic infinitesimal model. It isprobably fair to say that most early selection experimentsgave results that did not depart far from such predictions;but many did not span a sufficient time scale for departuresto be detected, were too small (in terms of Ne) to avoid largedrift sampling errors, or were confronted not so much withfixation of favorable genes as with natural selection oppos-ing artificial selection. Certainly there is little published ev-idence of departure from infinitesimal model predictions inlivestock breeding programs based on BLUP models, but thetime horizon in generations is not long, and objectives inbreeding programs change over time, complicating evaluation.

More comprehensively, Weber (2004) put togetherresults of many experiments with Drosophila undertakenat different population sizes, including his own with up to1000 selected parents, and expressed results as the ratio ofselection limit to first-generation response, which would be2Ne with an infinitesimal model (Robertson 1960). The fitwith this prediction is remarkably good, especially if allow-ance is made for mutation; but the discriminatory power ispoor against other genetic models, for example, incorporat-

ing a broad distribution of gene effects and frequencies(Zhang and Hill 2005). Indeed for Drosophila, where theselection effect on variance is more important with no re-combination in males and few chromosomes, we showedthat it would be difficult to distinguish an infinitesimalmodel with one of a U-shaped allele frequency distributionof a finite number of loci. In an analysis of the long-termIllinois maize selection experiment Walsh (2004) showedthat responses over 100 generations were not inconsistentwith an infinitesimal model provided mutation was allowedfor.

Another approach for assessing the fit of the infinitesimalmodel is to estimate genetic variances using an animalmodel after several generations of selection, but takingaccount of the inbreeding to date and the selection of theparental generation and asking whether the changes invariance corresponded to infinitesimal model predictions.Results of our analyses differed somewhat according to thetrait selected: we obtained a near-perfect fit (for log-transformeddata) to infinitesimal model assumptions for selected linesof mice differing fourfold in a fat measure between high andlow after 20 generations (Martinez et al. 2000), but selec-tion for body weight from the same founder stock revealeda QTL with large effect on the X chromosome (Rance et al.1997) and poorer fit.

At a meeting in 1987 I suggested to Henderson that, to beconservative, one should exclude or down weight old datato reestimate parameters for BLUP in a multigenerationbreeding program. He disagreed, arguing that selectionwould lead to bias in such estimates, so all must be includedin the parameter estimation. I think he regarded theinfinitesimal model as the real world, whereas Lush thoughtthat it was adequate for short-term predictions becausechanges in variance associated with changes in genefrequency “. . .will be very slow, especially if heritability isnot high. It is not often, except when the amount of epistaticvariance is large, that the rate of progress will declinesharply after only a generation or two” (Lush 1945, p.152). The latter caveat appears to reflect the influence ofWright rather than any calculations Lush undertook himself.

The infinitesimal model is not true of course, but seemsto do a good job at least over the span of generations wherebreeding programs have maintained constant plans. Analy-ses using dense genomic data have subsequently providedincreasingly more information, revealing individual loci oflarge effect but also showing very many loci affect quanti-tative traits.

The Rise of Gene-Orientated Studies

Breeders and quantitative geneticists assumed that manyloci were acting on each trait, and indeed were often usingthe infinitesimal model implicitly. Nevertheless many hopedto do better by locating them with genetic markers and thenutilizing individual loci by marker-assisted selection (MAS)among young animals without phenotypic records. The first

8 W. G. Hill

markers, blood groups, became available in the 1950s, andmethods for linkage analysis were soon developed (Neimann-Sorensen and Robertson 1961). Such markers would be im-portant indicators if the markers per se either were trait geneswith major effect or were closely linked to them. But bloodgroups marked little of the genome, so it was fortuitous thatthe chicken B locus, which is in the MHC complex, was foundto be associated with resistance to Marek’s disease (Briles andAllen 1961). It remains an important marker.

Linkage analysis for QTL detection did not really take offuntil many microsatellite markers became available andmaximum likelihood (Lander and Botstein 1989) and re-gression methods of prediction were developed for inbredline crosses (Haley and Knott 1992), with the latter ex-tended to enable analyses in crosses of noninbred popula-tions (Knott et al. 1996). Extensive analysis into how tooptimize the design of such studies was undertaken. Forexample the “grand-daughter design” utilizes the structurein dairy cattle whereby the effects of QTL segregating in thegrandfather are estimated from his sons’ progeny tests basedon field records of many granddaughters (Weller et al.1990). Extensive simulations and other theoretical analyseswere undertaken to find how best to utilize findings inbreeding practice (e.g., Smith and Simpson 1986).

Numerous linkage studies have been undertaken (Weller2009), and hundreds of QTL, some of very large effect, werefound in livestock species (http://www.animalgenome.org/).The number confirmed by repeat studies is much less, and inthese the same causal locus may be mapped to somewhatdifferent locations. With few recombinants per chromosome,the within-family methods lack power and precision, andestimates of effects of those QTL found significant are biasedupward (the Beavis effect; Beavis 1998). Markers for con-ditions known to be due to major genes such as for porcinestress syndrome (PSS) were identified (Fujii et al. 1991),enabling rapid elimination, and “double muscling”in cattlewas shown to be determined by the MSTN gene recentlyidentified in mice (Grobet et al. 1997). New major geneswere found, including DGAT1 for milk fat percentage incattle (Grisart et al. 2002).

Nevertheless these QTL accounted for a small part of thegenetic variability and the ones easiest to detect are typicallythose for easily recorded traits with high heritability, such asgrowth rate or fat percentage, rather than the more lowlyheritable, sex limited, traits of mature animals such asfertility or longevity where use of QTL could have greaterbenefit. Therefore, in view of the genetic progress that couldbe made by conventional selection on phenotypic recordsusing BLUP in a well-designed program, the impact ofidentified QTL on achieved rates of progress was small(Dekkers 2004) and has remained so.

The development of SNP technology and availability ofthousands of markers enables LD association mapping ratherthan linkage mapping and so higher precision in QTLdetection and location can be achieved (e.g., Meuwissenand Goddard 2000). The understanding and training in pop-

ulation genetics stimulated by linkage and LD mapping andthe dearth of useful QTL obtained, however, stimulateda more sophisticated view of the genetic architecture andof how to utilize the knowledge in improving polygenicquantitative genetic traits. This has led to what is alreadybecoming a major revolution in livestock improvement prac-tice, and one with much more in common with the approachof Lush in combining all relevant data to inform selectiondecisions.

Genomic Prediction and Selection

There had been some analysis previously of what might beachieved by utilizing identified variation at all loci (notablyby Lande and Thompson 1990), but there were not thegenomic tools to effect it. As dense SNP markers were be-coming available and affordable, the landmark article byMeuwissen et al. (2001) showed how whole-genome markerdata could be incorporated effectively in a breeding programfor a polygenic trait. Subsequent modeling showed howlarge an impact on genetic progress such a scheme mighthave (e.g., Schaeffer 2006) and the ideas were rapidlybrought into commercial practice (Hayes et al. 2009). Thisopportunity has stimulated more intense interest and activ-ity in both development of statistical inference and method-ology and its integration with population and quantitativegenetics than ever before in the breeding context.

Principles

In conventional methodology, breeding value prediction foranimals without records has to be made from pedigree, butits accuracy is limited because the Mendelian samplingvariation (which comprises half the additive genetic vari-ance for unselected parents, more in a population underselection) cannot be utilized, so full sibs get the samepredictions. The genomic information on young animals andthe genomic and production data on their older relativesenable prediction both within and across families of breed-ing values for animals without phenotypes. Benefits arelikely to be greatest for traits that are sex limited, such asmilk yield and egg production, or not recorded till late in lifeor post mortem such as longevity or meat content, butincreases in accuracy can also be achieved for animals thatdo have records. Thus greater rates of progress, up todouble, can be made and the costs of genotyping can be atleast partly offset by reducing or eliminating progenytesting.

The critical idea of Meuwissen et al. (2001) was to pre-dict breeding value using trait effects bk estimated for (i.e.,associated with) all the markers as a linear function Sxikbkfor individual i, where xik denotes genotype, e.g., 0, 1, 2 atlocus k according to its genotype aa, Aa, or AA, utilizingtheir LD with nearby trait genes. They assumed a model inwhich the trait genes were dispersed throughout the ge-nome. SNP genotypes for all loci are then included in a BLUPor similar analysis, with their associated effects as random

Review 9

variables, necessary not least because very many moremarkers may be fitted than there are animals with data.

The markers both incorporate long-standing LD with traitgenes but also help establish the realized or actual relation-ship between close relatives for individual segments of thegenome. While, say, half sibs have a 50% chance that thealleles inherited from their common parent at any genomicsite are identical by descent (IBD) (i.e., their pedigree re-lationship is 0.25), they actually vary in realized relation-ship, sharing IBD regions of varying length scatteredthrough the genome that markers can identify. Thus thepedigree relationship matrix can be replaced by an (addi-tive) genomic relationship matrix (GRM), with elementscomputed in terms of identity in state (IBS) as a predictorof IBD, with elements obtained by averaging over all markerloci quantities such as (xik – 2pk)(xjk – 2pk)]/[2pk(1 – pk)],where pk is the gene frequency at locus k, for individuals iand j. These have expectation equal to Wright’s numeratorrelationship for neutral loci, but on a shifted and pragmaticreference point, the current population on which gene fre-quencies are estimated (Powell et al. 2010), rather thanback as far as pedigrees are available, even though all allelesultimately coalesce.

The procedure in which the pedigree relationship matrixin BLUP is replaced by the genomic relationship matrix inBLUP is commonly known as GBLUP and the analysis canproceed otherwise essentially as in traditional BLUP (VanRaden2008). Computation is much heavier since the GRM doesnot have a simple inverse, but is generally feasible. Basicallythe method, as indeed is BLUP, is a form of ridge regressionstatistical analysis with shrinkage of the least-squares equa-tions according to the degree of genomic relationship. Use ofGBLUP is equivalent to assuming that the distribution ofmarker-associated effects on the trait is the same for allmarkers fitted across the genome and each is sampled froma normal distribution with the same variance. It would alsoarise from assuming an infinitesimal model of trait effects.The models based on marker LD and genomic relationshipare then computationally equivalent (Stranden and Garrick2009), with GBLUP capturing the average effects of all ge-nomic regions simultaneously.

Before considering details further, an example shows thatthe impact of using genomic prediction can be large. Accu-racies of predicting the breeding values of young dairy bullscomputed solely from pedigree information or by incorpo-rating genomic information can be compared by correlatingeach with those realized subsequently by progeny testing. Inan early report using the large U.S. data set, published R2

(“reliability” or squared accuracy) for prediction of milkyield of daughters of young bulls were 0.28 for parent av-erage, 0.47 for GBLUP, and 0.17 and 0.26, respectively, forproductive life (VanRaden et al. 2009).

Models

Meuwissen et al. (2001) recognized that the optimalweights to give to individual markers in the analysis depend

on the distribution of effects of genes on the trait. This couldrange from very many of small effect contributing varianceevenly distributed throughout the genome, to just a few oflarge effect, perhaps concentrated in a few chromosomes,and on the distribution of LD between markers and traitgenes across the genome. These are unknowable with finitedata sets, so in the analysis it is necessary to make priorassumptions about how gene effects (formally marker-associatedeffects of genes) on the trait are distributed. Meuwissenet al. proposed two Bayesian models: Bayes A, which inpractice assumes a t distribution (longer tail of large effectsthan the normal) and Bayes B, which is as A but with a de-fined proportion of markers having no effect. (This startedan industry, the “Bayesian alphabet”; Gianola et al. 2009.) Incontrast with GBLUP, some form of MCMC simulation isthen needed to obtain predictions.

Alternative models proposed subsequently include moreextreme prior distributions of marker effects than the t thatis used in Bayes A. These include the Bayesian LASSO,others allowing for the number of non-zero-weighted markersused in Bayes B to be estimated, and models with a mixtureof distributions of effects with different variances. Gianola(2013) gives a recent comprehensive review and reminds usthat the numbers of marker effects being fitted relative tothe number of observations is so large that inferences canstill be greatly influenced by the chosen prior. There hasbeen extensive simulation of alternative models, but realdata with proper separation of training data sets used toestablish the prediction equations and test data sets to eval-uate them are needed to evaluate models and methods.Choice of priors remains an active (and contentious) area.

An alternative approach is to avoid distributional assump-tions by using a nonparametric method to obtain a predictionalgorithm numerically from the training and test sets. Asa linear model is not fitted, such predictions also incorporateany epistatic interactions. Methods have been discussed byGianola and colleagues, and in a study comparing linear andnonparametric approaches for dairy and wheat data, thenonparametric method gave more accurate predictions(Morota et al. 2013). A weakness of this approach seems,however, to be that as a linear additive model is not used inthe analysis, predicted breeding value of unselected off-spring cannot be assumed to be simply half that of theparent.

For the dairy cattle data mentioned above VanRadenet al. (2009) also fitted Bayes A. For most traits, R2 valuesusing Bayes A (e.g., 0.49 vs. 0.47 for milk yield) were closeto those for GBLUP, implying the normal model fitted, butwere higher for milk fat percentage (0.63 vs. 0.55). Forpercentage fat in the milk, a gene of large effect (DGAT1)is segregating, and in a different data set Hayes et al. (2010)found that one-quarter of its variance was explained bythree QTL and that predictions using Bayes B, in whicha small number of genomic regions are included, were moreaccurate. For overall type, however, fitting ever more SNPs(i.e., approaching GBLUP) continued to increase accuracy.

10 W. G. Hill

Factors affecting accuracy of prediction

The accuracy of prediction depends both on operationalfactors, such as the density of markers fitted and the size ofthe training data set, and on broader factors, such as thepopulation history and demography and the genetic archi-tecture of the trait. As the training set is likely to be farsmaller than the number of SNPs to be fitted, increases in itssize are always likely to lead to increases in accuracy andability to discriminate between the effectiveness of alterna-tive Bayesian models. Increasing marker density alone is notenough.

Accurate prediction requires that the LD structure is thesame in the data used for training the model as that in whichit is applied in practice. Thus retraining is needed regularlyover generations of selection in a closed population as therelationships become more distant (Wolc et al. 2011). Het-erogeneity in population structure generates heterogeneousmarker QTL associations through LD. Hence accuracies ofprediction from one breed or distinct population to anotherare lower, as is accuracy of prediction within a structuredpopulation such as a breed and breed cross mix. Daetwyleret al. (2012) discuss how to deal with such issues and howeffective they are. The problem is critical for populations forwhich large training sets are not available.

Among the biological factors, accuracy increases ofcourse with the heritability of the trait, i.e., the informationfrom individual records. The accuracy of GBLUP depends onthe extent to which realized relationship varies about pedi-gree relationship. The latter depends on the length of seg-regating chromosome segments and is an inverse function ofNeL, where L is chromosome length (Goddard et al. 2010).Together with the number of markers used, this indicateshow useful they are at identifying genomic regions. Addi-tionally, particularly in Bayesian models, the accuracydepends on the magnitude of LD between markers and traitgenes, which in turns depend on their joint frequency dis-tributions. Thus trait genes with low minor allele frequency(MAF) are likely to be poorly marked because the SNPs usedhave higher minor allele frequency. Hence there is interest inusing ever more dense markers and in sequencing of impor-tant animals in the breeding pyramid, such as sires used inAI (Meuwissen and Goddard 2010), with imputation of gen-otypes in animals down the pedigree for economy. Sequenc-ing also increases the number of markers fitted and hencethe need to have large training sets.

Information comes from close relatives and via LD frommore distant ones. That from close relatives is less de-pendent on the joint distribution of marker and trait genefrequencies. Indeed de los Campos et al. (2013) argue thatthe critical factor driving accuracy is the extent to whichmarker-based relationships properly describe the unobservedgenetic relationships at trait loci. Hence if the training andtest data sets have related individuals the markers can begood predictors even if the LD between markers and traitgenes is weak (see also Wray et al. 2013). This is exemplified

by their comparisons showing quite high accuracy of predic-tions for traits of humans in which the training and test indi-viduals are from the same local population, but low accuracyfor sets of unrelated individuals from the whole population.

Interest in genomic prediction is not restricted solely tolivestock breeding. For example, maize breeders who aredeveloping new lines founded from crosses want to obtainreliable indicators from early generations of their perfor-mance in subsequent generations of inbreeding and, criti-cally, as a parent of a commercial hybrid. Theory/simulationstudies (e.g., Jannink et al. 2010), and also experimentaltrials, have been undertaken. Results to date for maize cor-respond to those in livestock, in that predictions are betterwithin families (i.e., specific F2 of initial cross) than acrossfamilies (0.72 vs. 0.47 for grain yield; Albrecht et al. 2011),and in another data set predictions of cross performance hadessentially zero accuracy (Windhausen et al. 2012). Wimmeret al. (2013) provide analyses in three species and find thatmethods involving marker selection (in contrast to BLUPemploying ridge regression) can be unreliable unless datasets are large.

Although genome wide association studies (GWAS) studiesin humans have identified significant genetic lesions or highlydisease-associated SNP markers, the numbers detected haveincreased as sample sizes and marker density have increasedand thus power to find those of ever smaller effect risen(Visscher et al. 2012). Even so, all those detected (e.g., .150for human height; Lango Allen et al. 2010) typically accountfor a small proportion (�10%) of the genetic variance inquantitative and disease traits estimated from pedigree stud-ies. That many of small effect are missed is shown by the factthat if all SNPs are fitted together, whether significant or not,they can account for half of the genetic variance (Yang et al.2010) in height, and similarly for other traits. There is there-fore interest in human genetics in using all the markers inwhole-genome prediction based on the animal breeders’methods. There are limitations, however, not least becauseprediction of individual phenotype is necessarily less accuratethan for genotype for any trait. Further, Ne in humans is muchgreater than in cattle, indicating that it would take.145,000records with humans to achieve the same accuracy, 0.65, ofgenetic prediction as for about 2500 cattle (Kemper and God-dard 2012). Thus accuracies of prediction may be very small,�0.1 or less, if training and test sets properly comprise un-related individuals (de Los Campos et al. 2013; Wray et al.2013). Overcoming these limitations may prove difficult: sim-ply increasing marker density does not resolve it, unless thedata sets become correspondingly large and informative onthe correspondence of marker and trait loci and there aresubstantial differences among genomic regions in their effectson the trait.

Remarks

In view of the range of models available and differences inresults obtained using them, further work is required beforea consensus on the optimum approach is likely to be

Review 11

reached. Again it must be emphasized that simulation ofmodels is not enough; results may not apply to the realworld architecture. While GBLUP is a conservative option, itis perhaps also the most robust against departure fromassumptions about the architecture and inadequate data intraining sets (as further evidenced by results of dataanalysis; Wimmer et al. 2013). While the recent studiesshow (no surprise there) that not all trait loci have infini-tesimally small effect, typically few account individually formuch of the variation; it remains a good a priori model. AsSNP density rises, data sets increase in size, and the genomeanalysis becomes more fine scale, it seems likely that thissimple model will become increasingly superseded. The de-sign of breeding programs using genomic prediction, theprediction method, and the genetic architecture also influ-ence the potential long-term responses, and some issueshave already been discussed (Goddard 2009). These arelikely to be further developed as research continues andprograms become more established.

The principles and practice of genomic prediction area highly active area, and for want of space and competence Icannot do full justice to the issues, approaches to theirresolution, findings, disappointments, and triumphs. Thosewishing to become more deeply immersed will find thearticle of Meuwissen et al. (2001) useful for setting thescene and those of Goddard et al. (2010), Habier et al.(2013), de los Campos et al. (2013), and Gianola (2013)for providing more current views. While these focus on us-ing whole-genome prediction, the recent discussion by Wrayet al. (2013) of GWAS analysis for finding the actual traitgenes points to pitfalls in design, analysis, and interpretationthat can influence both objectives.

Progress and Perspectives

Very substantial and continued genetic improvement hasbeen made in livestock over the past several decades (Hilland Bunger 2004; Hill 2008). Broiler chickens increased in8-week weight over fourfold as shown by a contemporarycomparison of 1957 control and 2001 commercial stock,with a further twofold difference in breast meat yield, andresponse continues. Analysis of dairy cattle records showsmore than doubling in milk yields over 50 years, with atleast half attributable to genetic change. Worsening offitness-associated traits such as fertility in dairy cattle andleg defects in broilers have been reversed by increasing em-phasis on these traits (Kapell et al. 2012). In contrast, judg-ing by the winning times of classic flat races, Thoroughbredracehorses and Greyhounds have hardly increased in speedfor well over half a century; but it is not clear why (Hill1988; Hill and Bunger 2004).

These changes were obtained before genomic methodswere introduced, so we can hope for more rapid progress,particularly for sex-limited traits. There are two importantcaveats. The first is that good phenotypic records are stillneeded. Technological improvements will facilitate their

collection, but while the costs of genotyping continue to fallrapidly, the costs of maintaining and individually recordinganimals are less likely to do so. The second is thatsubstantial and useful genetic variation remains. We con-tinue to find similar heritabilities for commercial traits (thatfor juvenile growth weight remains about one-quarter inpoultry and indeed in other species). Much is made of con-servation of unselected breeds and populations as materialbut the real problem with using these is in identifying anyuseful genes. Variation is coming into the populations frommutation, which we continue to utilize unknowingly; per-haps we have to reconsider enhancing mutation rates, if wecan do so selectively. Unfortunately, genomic methods ofutilizing them are likely to be rather ineffective because boththe beneficial and deleterious mutants are too rare to locatewith any power in GWAS.

Lush’s standpoint of concentrating on the short term hasbeen effective, not just to get breeders started in sensibledirections but continues to be so because incorporating ge-nomic prediction and selection follows his aim of using allavailable data on individuals and their relatives optimally tomake breeding decisions. Wright’s theoretical develop-ments, such as of inbreeding and relationship, of gene fre-quency distributions under selection and mutation, and ofthreshold traits have had continuing impact. In contrast hisshifting balance theory has remained controversial amongevolutionary biologists; see, for example Coyne et al. (1997)for the attack and Wade (2013) for the defense. Lushexplained the theory in Animal Breeding Plans and its rele-vance to breeding programs, discussing “Ideal breeding sys-tem for rapid improvement of the whole breed” in terms oflocal groups subject to selection and introgression amongthem. As indeed with Wright’s work, however, there is a clearcontrast between the quantitative genetics theory that Lushuses to maximize genetic change and the discursive sectionsbased on assumptions of important interactions. Ironically,Lush’s main contributions more closely followed the ideas ofFisher (1930), albeit over a short time horizon.

The ability to fit multiple markers is also concentratingattention again on the genetic architecture, not just on thedistribution of gene effects, but also epistasis. For example,it has been argued that epistatic variance provides anexplanation of why much, say one-half, of the geneticvariation estimated from pedigrees is unaccounted for byfitting SNPs in GWAS for disease and quantitative traits (Zuket al. 2012). Nevertheless even if the loci show epistasis,population genetic theory tells us to expect low amountsof epistatic relative to additive variance from low-heterozygositygenes (Hill et al. 2008), and models proposed by Zuk et al.(2012) seem biologically unrealistic (Stringer et al. 2013).In any case, epistatic variance is hard to utilize in breedingprograms and can largely be ignored in predicting response(Crow 2010), so we can perhaps be content to work withthe polygenic additive model for within-population improve-ment, even perhaps in genomic prediction using the ge-nomic relationship matrix. It has been the foundation for

12 W. G. Hill

successful prediction of breeding value and selection re-sponse and, like its infinitesimal special case, does not haveto be true to be useful.

In practice there has been migration (upgrading or otherexpressions in the breeding context) or breed replacementto utilize desirable properties of other breeds, consequent ondrift, selection history, and particular desirable attributes.(The Shorthorn breed Wright analyzed is now a rare breed.)While there is little if any direct evidence that impacts onperformance traits have been other than linear, so muchbreed choice is undocumented that it is very hard to tell.Breed diversity has been widely used in crossbreeding toutilize heterosis and complementary traits (e.g., reproduc-tive rate in a dam line, meat yield in a sire line) of popula-tions to generate commercial crosses, and any epistaticinteractions would just be counted in with the dominance.

Thus Lush’s ideas have been influential and lasting. He isbest known for simple formulae like the breeder’s equationand ideas like accuracy of selection, but he had a great depthof understanding of population genetics that he learnedfrom the work and influence of Wright and to a lesser extentof Fisher. They have provided food for thought and, indi-rectly, our stomachs.

Acknowledgments

I am grateful to Michael Turelli for suggesting this articleand for much stimulation and advice. I am also grateful toAsko Maki-Tanila, Daniel Sorensen, Armando Caballero, twoanonymous referees, and Daniel Gianola (extensively) forcritical, helpful, and contradictory comments on earlier drafts.I apologize for factual errors, misrepresentations and omis-sions. Opinions we can debate.

Literature Cited

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Communicating editor: M. Turelli

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