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8

Family-Based Selection

Draft Version 9 February 2000, c© 2000, B. Walsh and M. Lynch

Please e-mail any comments/corrections to [email protected]

Up to now, we have focused on individual selection, wherein selection decisionsare based solely on the phenotypes of single individuals (this is also referred to asmass or phenotypic selection, and we use these terms interchangeably). Selectiondecisions can also incorporate the phenotypic values of an individual’s measuredrelatives, and in fact most plant and animal breeding schemes do just this. Thefocus of this chapter is family-based selection — using family information to se-lect individuals. We restrict discussion in this chapter to using sib information.Chapters 21-25 develops the general theory of index selection, which can accom-modate information from arbitrary types of relatives. The culmination of thisapproach is BLUP selection using an index based on the entire known pedigree ofan individual (Chapter 24), which is the major route for artificial selection in mostdomesticated animals. While our focus here is on short-term response (formally,the single-generation response), certain family-base schemes may give a greaterlong-term response than individual selection, even when their initial response isless. Chapter 14 examines this in some detail. Likewise, a number of family-basedschemes in plant breeding use extensive inbreeding or crossbreeding and theseare examined in Chapter 9.

There are a variety of reasons for using family-based schemes. Mass selectionmay be impractical in many settings due to difficulties in measuring trait values inindividuals. Family-based designs can also provide greater accuracy in predictingan individual’s breeding value and hence give a larger (short-term) response. Inparticular, an appropriately weighted index of an individual’s family mean andphneotypic value always has an expected response at least as large as mass se-lection. When significant environmental heterogeneity exists (e.g., crops plantedacross a broad climatic range), replication of families over environments providesa more efficient method than mass selection for choosing higher-performing geno-types. This is a major factor why crop breeders usually favor family-based schemesover individual selection.

The structure of this chapter is as follows. We start with a few overviewremarks on the nature and types of family-based selection before consideringa number of particular designs in detail. Our treatment of the expected single-

239

240 CHAPTER 8

generation response under these schemes follows, starting with extensions of thegeneralized breeders equation to accommodate family-based schemes. We nextdevelop the variances and covariances required to apply these equations, andthen consider the major within- and between-family selection designs in detail.The relative efficiencies of within- and between-family selection compared tomass selection is examined next, and we then consider designs where familiesare replicated over environments, as is usually in the case in plant breeding. Weconclude by examining the properties of family index selection.

INTRODUCTION TO FAMILY-BASED SELECTION SCHEMES

Family-based designs are based on two components: between-family schemesthat chose entire families on the basis of their means relative to the populationmean, and within-family schemes that chose individuals based on their rela-tive performance within families. While many designs are based on just one ofthese components, a more general approach is family-index selection where in-dividuals are chosen based on a weighted index of between- and within-familycomponents. Mass selection is a special case of a family index, where the within-and between-family components are weighted equally.

Overview of the Different Types of Family-Based Selection

The key to making sense out of the often bewildering number of family-baseddesigns that have appeared in the literature is to consider the individual compo-nents that together define any particular design. The first component is the typeof sib family providing information for selection decisions. A family may consistof half-sibs, full-sibs, or full-sibs nested within half-sibs (the NC Design I, see LWChapter 18). Sibs can also be generated by one (or more) generations of selfing,and we examine such families in Chapter 9. While the family-based schemes de-veloped in this chapter are generally used with allogamous species (outcrossers),they can also be applied to facultatively autogamous species (facultative selfers)through the use of either controlled pollination and/or the introduction of male-sterile genes under open pollination (e.g., Gilmore 1964, Doggett and Eberhart1968, Brim and Stuber 1973, Burton and Brim 1981, Sorrells and Fritz 1982).

While we assume in this chapter that the parents for any particular fam-ily are from the same population, in some plant-breeding settings parents arefrom different populations. An example of this are interpopulation improve-ment schemes where the goal is to improve the performance of hybrids betweenpopulations (Chapter 9). Our focus in this chapter are family-based schemes forintrapopulation improvement (increasing the performance of the population un-der selection).

Once a particular family type has been chosen, the second component ofany family-based design is how the sib data is used for selection decisions. One

FAMILY-BASED SELECTION 241

could use between-family selection, choosing the best families. Alternatively, onecould use within-family selection, choosing either the best individuals within eachfamily (strict within-family (WF) selection), or the individuals with the largestoverall family deviations (family-deviations (FD) selection). While WF and FDselection are very similar, there are subtle differences between the two schemes,as they do not necessarily select the same individuals. One could also consideran index weighting both family mean and family deviations.

The final major component is the relationship between the measured sibs andthe individuals serving as parents for the next generation. Under either within-family or family-index selection, the selected individuals themselves are used toform the next generation. However, with between-family selection, we can useany number of relatives of the chosen families to form the next generation. Themost straight-forward is to use some (or all) of the measured sibs from the chosenfamilies (family selection). Some characters cannot be scored on living organisms,such as carcass traits in meat animals. In such cases, one can use unmeasured sibsfrom the best families to form the next generation (sib selection). Sib selectioncan also be used to improve selection on sex-specific traits. For example, milkproduction can be selected in males by choosing males from families whose sistersshow high levels of milk production. An important variant of sib selection is touse remnant seeds from the best families, which are planted and subsequentlycrossed to form the next generation. For perennial species and for annual speciesthat can be asexually propagated (or cloned), one can select the best parentsby the performance of their offspring (parental selection or progeny testing).Finally, an option available for facultatively autogamous species, is to both selfan individual to generate S1 progeny (S1 seeds) and likewise outcross it to one ormore individuals to generate a family for testing. For such species, one can grownup and intercross S1 seed from the chosen families to form the next generation(the S1 seed design).

Plant vs. Animal Breeding

While animal breeders typically use only a few standard sib-based designs (Turnerand Young 1969), an array of options are available to plant breeders (Allard 1960,Empig et al. 1971, Hallauer and Miranda 1981, Nguyen and Sleper 1983, Hal-lauer 1985, Weber and Wricke 1986, Mayo 1987, Gallais 1990, Nyquist 1991, Bosand Caligari 1995). It is perhaps not surprising then that the literatures on family-based selection in the two fields are rather divergent. Much of the animal breedingliterature is expressed in terms of the phenotypic (t) and additive-genetic (r) corre-lations among sibs. In contrast, much of the plant breeding literature is expressedin terms of variance components. As our discussion with attempt to interweaveboth approaches, we will often present response equations in both forms.

Reproductive differences between plants and animals underlie many of thedifferences in options available to breeders. Historically, plant breeders have hadmore options than animal breeders because of the reproductive flexibility of many

242 CHAPTER 8

plants (i.e., selfing, stored seed, vegetation propagation). With the recent successesin cloning several domesticated animals, animal breeders have the option of ex-ploiting some of these classic plant breeding schemes.

One obvious difference between plants and many animals is the ability to eas-ily store progeny for many generations in the form of seed. Generally speaking,plants also produce more offspring than domesticated animals, providing moreoffspring per family which allows for extensive replication of families across en-vironments. Another reproductive advantage of plants (from an experimentaldesign standpoint) is that asexual propagation (cloning) is trivial in many plantspecies, allowing individual genotypes to be preserved over many generations.

Another key difference is in control of crosses. While simple isolation will pre-vent most undesirable crosses in animals, either complete isolation or extensivemanual control may be required to prevent pollination vectors from generatingundesirable crosses in plants. For facultatively autogamous species, the investi-gator may be faced with either trying to prevent selfing or prevent outcrossing,or to allow for both while identifying which seed came from which type of cross.Options for controlled crosses range from complete manual control over pollina-tion at one extreme to open pollination at the other. Given that most plants havemultiple flowers (which are often both very numerous and very small), large-scale controlled crosses can be much more labor intensive than similar crossesamong animals, as hand pollination and control of external and/or self pollina-tors are often required. Even under open pollination (allowing seed plants to bepollinated at random), the investigator still has different levels of control over thepollen spectrum that a seed plant experiences. In a test cross or topcross design(Chapter 9), pollination is at random, but the population of plants supplying thepollen is controlled. For example, individual maize plants can be detasseled byhand (removing the pollen-producing flowers) to prevent the plants from eitherselfing or pollinating other plants. Such plants serve only as seed plants and areintergrown with rows of the tester strain which provides the pollen. Under a poly-cross design, seed parents are randomly pollinated from the population with nocontrol of the pollen parent. A consequence of open pollination is that while mosthalf-sib families in animal breeding are paternal, most half-sib families in plantbreeding are maternal (sharing the same seed parent).

There are also more subtle biological differences between plants and animalsthat flavor differences in designs. While one can usually score many traits in in-dividual animals, this is often not the case in plants. For example, many traits offorage grasses, grains, and legumes are scored as plot totals, measuring an entirefamily instead of each separate individual. When individuals cannot be directlyscored, between-family selection is possible, but within-family and family-indexselection is not. Similarly, many selective trails in plants can be scored only af-ter reproduction (yield being a prime example), and this influences the types ofrelatives that can be used to form the next generation.


Between- vs. Within-Family Selection

When the heritability of a character is high, an individual’s phenotype is an ex-cellent predictor of its breeding value, and mass selection is more efficient thaneither strict within- or between-family selection. When heritability is low, indi-vidual phenotypic value is poor predictor of breeding value, in which case anindividual’s family mean or its relative performance within its family may be abetter predictor of its breeding value.

The relative efficiencies of between- vs. within-family selection depends onthe relative magnitudes the common-family (Ec) and individual-specific (Es) en-vironmental values. A large common family effect severely comprise the pheno-type as a predictor of breeding value. However, within each family, all membersshare the same environmental effectEc and differences between individuals moreaccurately reflect differences in breeding value. In this case, selection within fami-lies (for example, by choosing the largest individuals from each family) can give alarger response than individual selection. Conversely, suppose that environmen-tal effects unique to each individual account for a large fraction of the phenotypicvariance. In this case, selecting whole families as units can give a larger responsethan individual selection, as the family mean averages out differences based onenvironmental values, exposing those families with the most extreme breedingvalues.

An important example of this family-averaging of environmental effects isthe use of between-family selection to improve performance across multiple envi-ronment. Under mass selection, a genotype is represented by a single individualin a single environment, while family-based approaches allow the performanceof different families to be compared over multiple environments. Such studies areby no means restricted to plant breeding, as animal selection experiment exam-ining phenotypic plasticity (norms of reactions), in which genotypes must alsobe assessed over multiple environments, almost exclusively use between-familyselection (e.g., Waddington 1960, Waddington and Robertson 1966, Kindred 1965,Druger 1967, Scharloo et al. 1972, Minawa and Birely 1978, Scheiner and Lyman1991, Brumpton et al. 1997).

Even when the heritability of a character is high, there may be logistical rea-sons for using family-based selection schemes. From a husbandry standpoint,some type of family selection may be a more efficient use of breeding spacethan mass selection. For example, many mouse selection experiments use within-family selection, especially for traits with suspected material effects such as bodyweight (Falconer and Latyszewski 1952; Falconer 1953, 1960a, 1973; Eisen andHanrahan 1972; von Butler et a. 1984; Nielsen and Anderson 1987; Siewerdt et al.1999), litter size (Falconer 1960b), and nesting behavior (Lynch 1980).

DETAILS OF FAMILY-BASED SELECTION SCHEMES

P1

x1 r1 r2

y

P2

x2

244 CHAPTER 8

Selection and Recombination Units

Under mass selection, individuals are scored and those with the best phenotypicvalues are used as the parents to form the next generation. The group of indi-viduals (or unit) upon which selection decisions are based and unit used forrecombination (gamete production to form the next generation) are one in thesame, and a single cycle of selection takes a single generation. In family-basedselection schemes the individuals used for selection decisions may be entirelyseparate from those used to form the next generation. Further, a single cycle ofselection may take three (or more) generations, as one must generate, score, andrecombine families.

Following the convention of plant breeders, we distinguish between the se-lection unit x (individuals upon which selection decisions are made, which weassume throughout this chapter are sibs) and the recombination unitR (individ-uals serving as parents for the next generation) whose resulting offspring are y.Even though we may not directly select on the parents (R1, R2) of y, we expectsome response in y as there is a genetic correlation between an individual xi fromthe selection unit and an offspring y due to xi andRi sharing the common relativePi (Figure 8.1).

Figure 8.1. Under family-based schemes, selection decisions are based on somefunction of the sib values xi (the selection unit) and we wish to predict theresponse in the offspring y. The parents of y, the recombination unitR1 andR2,are chosen on the basis of the selection unit. Members of the selection (xi) andrecombination (Ri) units are related as they both share the common relative Pi,the parent of sib xi.

The variety of family-based schemes arises by combining four specific com-ponents.

1. Type of sib family in the selection unit. The sibs in xi can be entirelyhalf-or full-sibs, or be full sibs nested within half-sibs (NC Design I).

2. Nature of the selection decisions based on the sib information. Selec-tion can be based on sib means, the deviations of individuals withinfamilies, or an index of both.


3. Selection on one vs. both parents. Often selection decisions involveonly one sex, with the parents of the opposite sex chosen at random(and hence unselected). For example the best plants may not be ableto be scored until after pollination, resulting on selection on females(seed parents) but not males (pollen parents). In such cases, we are onlyconcerned with one side of the pedigree, for example involvingR1 butnot R2. More generally, the parents (R1 vs. R2) may be chosen usingdifferent schemes, further increasing the variety of potential family-based schemes.

4. Nature of the relationship between x andR. These can be related assame individual (Ri = xi), as sibs, as parent-offspring (Ri = Pi), etc.

While the variety family-based selection schemes may seem a bit overwhelmingat first (especially in plant breeding), considering each design in terms of thesefour basic components greatly simplifies matters.

Variations of the Selection Unit

Once the type of family (half sib, full-sib, nested) has been specified, there is stillthe issue of how to incorporate sib information when making selection decisions.Suppose that we select the uppermost fraction p of the relevant population andlet zij denote the i-th individual from the j-th family, with m families each withn sibs, for a total of N = mn scored individuals. Four different approaches forweighting sib information are commonly used:

1. Between-family selection: Individuals are selected solely on the basisof their family means, zi, so that individuals from the same family havethe same selective rank. Here, the best pm families are chosen.

2. Strict within-family (WF) selection. The best pn individuals from eachfamily are chosen, so that individuals are ranked within each family.

3. Selection on within-family deviations (FD): Individuals are rankedsolely on the basis of their within-family deviation zij − zi. The pNindividuals with the best deviations (regardless of family) are chosen.

4. Family-index selection: Individuals are ranked using an index weight-ing within- and between-family components,

I = b1 (zij − zi ) + b2 zi

= b1 zij + (b2 − b1) zi

The pN individuals with the best index score are chosen. The index withweights (cb1, cb2) chooses the same individuals as an index with weights(b1, b2). As a result, one of the index weights is often set to one, as theindices with weights (b1, b2), (1, b2/b1), and (b1/b2, 1) are equivalent.Individual selection, between-family selection, and selection on family

r

y

x1 xn...

P

b: r is a sib of P

r

y

x1 xn...

P

d: r is a selfed-sib (S1)

y

x1 xn...

P

c:(Progeny testing)

r is the parent P

= r

y

x1

P

a: r = xi

xi xn.. ..

246 CHAPTER 8

deviations (FD) are special cases of combined selection, being indiceswith weights (b1, b2) = (1,1), (0,1), and (1,0), respectively. Note, however,than strict within-family (WF) selection cannot be expressed in terms ofan index. Family-index selection is also often referred to as combinedselection, but this term is also used by plant breeders for schemes thatusing different types of family-based selection schemes over differentgenerations during one cycle of selection.

The choice of a particular scheme has implications on the selection intensity andlong-term effective population size. When p is fixed, between-family and strictwithin-family selection have lower selection intensities than family-deviations,combined, or mass selection. The former selects the best pm of m and pn of n,while the later three select the best pN of N . Since N > n, m the finite-samplingcorrection for ı is larger when sampling from N than from n or m (Chapter 4 andbelow). Selection decisions also influence the long-term effective population size(and hence the long-term response, see Chapter 14), with schemes placing moreweight on between-family components having smaller effective population sizethan those placing more weight on within-family components.

Variations on the Recombination Unit

Under either within-family or combined selection, measured individuals (sibs)are the parents for the next generation and hence form the recombination unit. Bycontrast, with between-family selection there are a variety of options for whichrelatives comprise the recombination unit (Table 8.1, Figure 8.2)

Figure 8.2. Different types of relatives used as the recombination unit R forbetween-family selection. P denotes the shared parent(s) of x and R, while y


denotes a random offspring from the recombination unit R. The measured sibsupon which selection decisions are based are denoted byx1, · · · , xn. All pedigreesfocus on just one parent of y, with corresponding pedigrees for the other parent.A: Family selection.R is one of the measured sibs (xi =R). B: Sib selection.Ris an unmeasured sib of x. C: Parental selection.R is the parent of the sibs (R =P ). D: S1 seed selection.R is the selfed progeny of the parent of the sibs.

Table 8.1. Family-based selection schemes using outbred sibs. zHS and zHS denotes themeans of half- and full-sib families, while P is the parent of the measured sibs.

Between-Family Selection Recombination Unit Selection UnitR x

Family Selection Measured sibHalf-sib Family Selection zHSFull-sib Family Selection zFS

Sib Selection / Remnant Seed Unmeasured sibHalf-sib Sib Selection zHSFull-sib Sib Selection zFS

Parental Selection Parent P zHS

S1 Seed Selection S1 Seed of PHalf-sib S1 Seed Selection zHSFull-sib S1 Seed Selection zFS

Within-Family Selection

Family Deviation (FD) Selection Measured SibHalf-sib Family Deviation Selection zij − zHSFull-sib Family Deviation Selection zij − zFS

Strict Within-Family (WF) Selection Measured SibHalf-sib Strict Within-Family Selection zij − µHSFull-sib Strict Within-Family Selection zij − µFS

The most straightforward situation is family selection, using measured sibsfrom each chosen family as the parents for the next generation (Figure 8.2A). Un-der sib selection, unmeasured sibs are used to form the next generation (Figure8.2B). Sib selection is often used for characters that are sex-limited or that cannotbe scored without sacrificing the individual. Plants breeders routinely use sib se-lection in the form of remnant seeds. Here, seeds from a parental cross are splitinto two batches, one is planted and used to assess families, the other held inreserve. Seeds from the elite families are grown and crossed to form the next gen-eration. Under this design, a single cycle of selection takes at least two generations— one to assess the families and a second to grow up and cross remnant seeds.Given this extra generation, what is the advantage to using remnant seeds? For

248 CHAPTER 8

annual plants, any traits that are expressed during or after anthesis (flowering)can only be directly selected in (open-pollinated) females, with seeds from the bestperforming plants forming the next generation. Since these plants are pollinatedat random, selection has occurred for the seed, but not pollen, parents. By usingremnant seeds, one can chose the best families, grow up remnant seeds and allowthe resulting plants to randomly intercross. Since both seed and pollen parentshave now been selected (through their families), a single cycle of selection usingremnant seed has double the response of family selection on open pollinated seedplants. This doubling of response per cycle is exactly counters the extra genera-tion in each cycle, so that open pollination family selection and sib selection usingremnant seed both have the same expected response per generation. One poten-tial advantage is that the extra generation to grow up the remnant seed can beused for selection on other characters, for example culling those otherwise elitefamilies that show poor disease or insect resistance.

Another common between-family design is parental selection (or progenytesting), where R= P , the parent of the measured sibs (Figure 8.2C). This designtypically involves evaluation of half-sib families with selection on just one sex. Inanimal breeding this is typically sires, choosing elite males by the performance oftheir half-sib families. This is greatly facilitated by the use of artificial inseminationand frozen sperm. The recent ability to clone several domesticated animals (e.g.,sheep, Campbell et al. 1996; cattle, Wells et al. 1999a,b; goats ; chickens —REFS)is likely to further increase the importance of progeny testing in animal breed-ing settings. (The most elaborate, and widely used, extension of progeny testingis BLUP selection wherein the entire pedigree is used for information on selec-tion decisions, see Chapter 24.) Plant breeders typically perform progeny testingusing maternal half-sib families (common seed parents). Vegetative propagation(cloning) allows even annual plants to be used as parents in future generations.If the species being selected is monoecious, one can obtain elite plants for bothseed and pollen on the basis of female (seed) performance, and hence select onboth sexes.

Finally, with monoecious self-compatible species (single individuals produceboth seed and pollen), an alternative to vegetative propagation is the S1 seeddesign (Figure 8.2D). For each parent, a subset of flowers are selfed to produceS1 seed and the remainder outcrossed. The outcrossed seed is then grown toproduce the sibs in which the trait of interest is assessed. Following selection ofthe best families, the S1 seed are grown and the adults crossed to form the nextgeneration. As with remnant seed, a single cycle takes (at least) two generations.One advantage of the S1 design is that one can store seeds, rather than adults. Inmaize, the S1 seed designs requires the use of prolific plant (those with more thanone ear), as (at least) one ear is selfed and (at least) one other outcrossed. Hallauerand Mirana (1982) note that the use of such plants also results in selection forprolificacy, which by itself can increase yield.


THEORY OF EXPECTED SINGLE-CYCLE RESPONSE

Response is typically given on a per cycle, rather than per generation, basis. Acycle begins with choosing the parentsP to form the sib families and ends with theoffspring y formed by crossing members of the recombination unit. The expectedresponse is the difference in the means of these two populations (P vs. y). Whencomparing the efficiencies of different schemes, response should be convertedto a per generation (for discrete populations) or per unit time (for overlappinggenerations) basis.

Our treatment of the theory of response starts by developing several equiva-lent modifications of the Breeders’ Equation to accommodate family-based selec-tion. To apply these expressions, we require the selection unit-offspring covari-ance σ(x, y) and the variance of the selection unit σ2

x for various family-baseddesigns. The full development of these variances and covariances is somewhatdetailed, with the final results summarized in Tables 8.3 and 8.4.

Modifications of the Breeders’ Equation for Predicting Family-Based Response

Response is a function of how selection on the sib units (x1 and x2) translates intoselection on the corresponding parents (R1 andR2) of the offspring y. Making thestandard assumptions that all appropriate regressions are linear (which followsunder the infinitesimal model assumptions, see Chapters 5 and 12), the expectedresponse is given by the general form of the Breeders’ Equation (4.4a,b),

Ry =σ(xm, y )σ2xm

Sxm +σ(xf , y )σ2xf

Sxf (8.1a)

Here xm and xf correspond to the selection units associated with the male(sire/pollen) and female (dam/seed) parents (Rm and Rf ) of the offspring y.Equation 8.1a allows the male and female parents to be chosen by completely dif-ferent schemes. For example, sib selection could be used on males and individualselection on females when selecting for a female-limited character. The selectionunit-offspring covarianceσ(x, y ) can be directly computed from the pedigree con-necting P , x, andR through the use of path analysis (LW Appendix 1). The path(or correlation) between selection on the unit xf through the female parent Rfand its offspring y is

xf ← P → Rf → y

Because the path connecting xf and y is through Rf , we often write σ(x, y |Rf )in place of σ(xf , y ) to remind the reader of this fact (especially when the valueof x is a function of several sibs). Path(s) connecting xm and y through Rm aresimilarly defined. If P consists of multiple relatives, each path connecting xi andRi (and hence y) needs to counted. For example, if xi and Ri are full sibs, wemust compute the paths through each of the common parents (e.g., Figure 8.3D).

250 CHAPTER 8

If the covariances are the same for both parents, Equation 8.1a simplifies to

Ry =σ(x, y )σ2x

Sx (8.1b)

where Sx = (Sxm +Sxf )/2 is the average selection intensity on the unit(s) leadingto the parents and

σ(x, y) = σ(x, y |Rf ) + σ(x, y |Rm ) = 2σ(x, y |R1 ) (8.1c)

is the covariance between the selection unit x and the offspring y, counting thepaths through both parents (Rm and Rf ). When covariances are equal, this istwice the single parent covariance, σ(x, y |R1 ). By analogy with the Breeders’Equation, Equation 8.1b is often written as

RY = h2x,ySx (8.2a)

Here the generalized heritability of y given x,

h2x,y =

σ(x, y)σ2x

(8.2a)

is the slope of the regression of y on x (LW Chapter 3).

Example 1. Consider family selection, wherein the selection unit is the familymean (x = zi ) and the recombination unit are sibs from this family. Assumingthe covariance between the sib mean and an individual sib is independent of sex,Equation 8.1b gives the response as

Rb =2σ( zi, y |R1 )

σ2( zi )Sb

where R1 is one of the sibs (say sib j, so that R1 = zij ). This can be even morecompactly written as Rb = h2

bSb, where the between-family heritability h2b is

h2b =

2σ( zi, y |R1 )σ2( zi )

Similarly, for selection on within-family deviations, the selection unit is x =zij − zi, giving

RFD =2σ( zij − zi, y |R1 )

σ2( zij − zi )SFD


whereR1 = zij . Response can also be expressed in terms of the family deviationsheritability, with RFD = h2

FD SFD where

h2FD =

2σ( zij − zi, y |R1 )σ2( zij − zi)

Other (equivalent) versions of Equations 8.1a and 8.2a appear in the literature.The selection intensity version allows standardized comparisons of differentselection schemes. Defining the selection intensity on x by ıx = Sx/σx, Equation8.1a becomes

Ry =σ(xm, y )σxm

ıxm +σ(xf , y )σxf

ıxf (8.3a)

If the regressions are the same for both parents,

Ry =σ(x, y)σx

ıx (8.3b)

where ıx = ( ıxm + ıxf )/2 is the average selection intensity. The selection intensityversion is frequently written in terms of the selection unit-offspring correlation ρ,

Ry = σz ıx ρ(x, y) (8.4a)

where ρ(x, y) = 2ρ(x, y |R1 ). Equation 8.4a follows immediately from Equation8.3b by recalling that ρ(x, y) = σ(x, y)/(σx σy ) and that the trait variance in theoffspring y is just the phenotypic variance of the character (σ2

y = σ2z ). A variant of

Equation 8.4a commonly seen in the literature is

Ry = σA ıx ρ(x,AR ) (8.4b)

where ρ(x,AR), the correlation between the selection unit x and the breedingvalue of a parent R, is referred to as the accuracy of selection. The accuracy ofindividual selection is ρ(zR, AR) = h (the correlation between an individual’sphenotypic and breeding values). A particular family-based approach is favoredover individual selection if x is a more accurate predictor of the breeding value ofR than isR’s phenotypic value, i.e., ρ(x,AR) > h. Strictly speaking, Equation 8.4bholds only in the absence of epistasis, while Equations 8.1-8.3 hold for arbitraryepistasis. To see this, recall that the mean value of an offspring is the average ofits parental breeding values, y = µ+ARf /2 +ARm/2 + ey . Hence,

σ(x, y) =12σ(x,ARf ) +

12σ(x,ARm) + σ(x, ey)

252 CHAPTER 8

In the absence of epistasis, inbreeding, and shared environmental effects,σ(x, e) =0. If the regression is the same for both sexes, then σ(x, y) = σ(x1, AR1). Recallingthat σy = σz ,

ρ(x, y) =σ(x, y)σx σz

=(σAσz

)σ(x1, AR1)σx σA

= h ρ(x,AR1) (8.5)

Substitution into Equation 8.4a gives Equation 8.4b (as σz h = σz (σA/σz) = σA).Equations 8.1-8.4 provide equivalent expressions for computing the expected re-sponse. To apply these for any particular scheme, we need to compute the selec-tion unit-offspring covariance σ(x, y) and the variance of the selection unit σ2

x.We develop these for a number of designs over the next several pages. The casualreader may wish to skip the details and proceed directly to the section on expectedresponse.

The Selection Unit-Offspring Covariance, σ(x, y)

Recall that the genetic covariance between two relatives is a function of theircoefficients of coancestry Θ and fraternity ∆ (LW Chapter 7). Ignoring epistasis(for now), the genetic covariance between a particular sib xi and y is σG(xi, y) =2Θxiy σ

2A + ∆xiy σ

2D (LW Equation 7.12). In the absence of inbreeding (the parents

R1 and R2 are from different, unrelated families), ∆xy is zero. For dominanceeffects to be shared by relatives, there must be paths wherein both alleles from anindividual in the selection unit are passed onto the offspring y. This cannot occurif the parents of y (R1 andR2) are unrelated.

Table 8.2. Coefficients of coancestry Θ between an offspring y (of parent R1) and amember of the selection unit x1. Genetic covariances σG(x1, y ) are computed assumingno epistasis.

Relationship between x1 andR1 Θx1y σG(x1, y )

x1 =R1 (the measured sib is also a parent of y) 1/4 σ2A/2

x1 andR1 are half-sibs (Figure 8.3a) 1/16 σ2A/8

x1 andR1 are full-sibs (Figure 8.3d) 1/8 σ2A/4

R1 is the parent of both x1 and y (Figure 8.3b) 1/8 σ2A/4

R1 is an S1 offspring of the parent of x1 (Figure 8.3c) 1/8 σ2A/4

The coefficient of coancestry between x1 and y depends upon the relation-ship between R1 and x1. The designs covered in Table 8.1 involve four differentrelationships (Figure 8.2): (i) x1 = R1 (measured sib is a parent of y), (ii) x1 andR1 are sibs, (iii) R1 = P1 (the parent of x1), (iv) R1 is the selfed-progeny of theparent of x1. The path diagrams for computing Θx1y for these four relationships

(a) x and R 1 are half-sibs

P 1

x r1

y

1/2

1/21/2

1/2P1

x r1

y

P2 P3

r2

P 4 P5

(b) x and y have a common parent, P 1

P 1

x

y

P2

r2

P4 P 5

P1

x

r1

y

1/21/2

1/2

=

(c) half-sib-S 1

P1

x r1

y

P2

r2

P 4 P 5

P1

x r1

y

1/2

1/2

1/2

1/2

1/2

P1

x r1

y

P 2

r2

P 4 P5

P1

x r1

y

1/2

1/21/2

1/2

P2

1/2

x r1

y

1/2

1/21/2

(d) x and r 1 are full-sibs


are given in Figure 8.3, and Table 8.2 summaries the resulting genetic covariances.Consider family selection first. Ignoring epistasis,

σ( zi, y |R1 = zij ) =1nσ(zij , y) +

(1− 1

n

)σ(zik, y)

= σ2A

(1/2n

+(

1− 1n

)2Θziky

)(8.6)

This follows since the first covariance is for parent and offspring (σ2A/2), while the

second covariance follows using the appropriate value of 2Θ from Table 8.2 (1/8for half-sibs, 1/4 for full-sibs). Using the results from Table 8.2, expressions forthe sib selection, parental selection (progeny-testing), and S1 seed designs followusing similar logic. These are summarized in Table 8.3.

Figure 8.3. Pedigrees (left) and associated path diagrams (right) for computingthe coefficient of coancestry Θ between a measured sib (x1) and an offspringy from the parent R1. P1 to P5 are assumed to be unrelated and non-inbred.

254 CHAPTER 8

(a): x andR1 are half-sibs. Taking the product of the path coefficients gives thecoefficient of coancestry as Θx1y = (1/2)4 = 1/16. (b): x1 and y are half-sibs,with Θx1y = (1/2)3 = 1/8. (c) : R1 is a selfed progeny from the common parentP1. Here there are two separate paths between x and y (as there are two differentroutes through P1), giving Θx1y = 2·(1/4)4 = 1/8. (d): x1 andR1 are full-sibs.Again there are two paths between x1 and y (one through each parent), againgiving Θx1y = 2 · (1/4)4 = 1/8.

In much of the animal breeding literature, Wright’s coefficient of relation-ship, r, is usually used in place of 2Θ. Assuming no inbreeding, r = 1/4 forhalf-sibs and 1/2 for full sibs. Using Wright’s coefficient, Equation 8.6 simplifiesto

σ( zi, y |R1 = zij ) = rnσ2A

2where rn = r +

1− rn

(8.7a)

Considering the paths through both parents (R1 andR2) of y,

σ( zi, y) = 2σ( zi, y |R1 ) = rn σ2A (8.7b)

Likewise, the covariance between an individual’s family deviation and itsoffspring’s phenotypic value is

σ( zij − zi, y |R1 = zij ) = σ( zij , y |R1 )− σ( zi, y |R1 ) = (1− rn)σ2A

2(8.8a)

as σ( zij , y |R1 ) is the parent-offspring covariance, σ2A/2. Doubling the single-

parent contribution gives the total contribution (considering both parents of y)as

σ( zij − zi, y ) = (1− rn)σ2A (8.8b)

The covariance for strict within-family (WF) selection is slightly different, as theappropriate covariance here is σ( zij−µi, y ), withµi in place of zi. The rankings ofindividuals under WF selection is simply their ranking within each family, whilethe ranking under FD selection further depends on how much an individualactually deviates from its family mean. Thus the top ranked individuals in twofamilies are always chosen under WF selection, but may not be under FD selectionunless the actual value of their deviations are sufficiently large. As a consequence,FD selection is influenced by the observed family mean zi, while WF selection isa function of the true mean µi (Dempfle 1975, 1990; Hill et al. 1996).

Table 8.3. Summary of the covariances between the selection unit and one parent (R1)from the recombination unit.

Between-family Selection:


Family selection (R1 is a measured sib from family i)

σ( zi, y |R1 ) = rn (σ2A/2) =

{(1 + 3/n) (σ2

A/8) half-sibs

(1 + 1/n) (σ2A/4) full-sibs

Sib Selection / Remnant Seed (R1 is a unmeasured sib from family i)

σ( zi, y |R1 ) = r (σ2A/2) =

{σ2A/8 half-sibs

σ2A/4 full-sibs

Parental Selection / Progeny testing (R1 is a the parent of the sibs)S1 seed design (R1 is a selfed progeny of the parent of the sibs)

σ( zi, y |R1 ) = σ2A/4

Within-family Selection:

Selection on Family Deviations (FD)

σ( zij − zi, y |R1 ) = (1− rn) (σ2A/2) =

{(1− 1/n) (3/8)σ2

A half-sibs

(1− 1/n) (σ2A/4) full-sibs

Strict Within-family Selection (FW)

σ( zij − µi, y |R1 ) = (1− r) (σ2A/2) =

{(3/8)σ2

A half-sibs

σ2A/4 full-sibs

A few simple rules emerge from Table 8.3. The number n of measured sibsonly influences the covariance for family selection and family deviations selection.Even in these cases its effect is small unless the number of sibs is small. Under sibselection (and family selection ignoring terms of order 1/n), the selection unit-offspring covariance contributed through one parent is σ2

A/8 for half-sibs andσ2A/4 for full sibs. For parental selection and S1 seed designs, this covariance isσ2A/4 (independent of whether full or half sibs are used). The covariance under WF

within-family selection (and FD selection ignoring terms of order 1/n) is 3σ2A/8

for half-sibs and σ2A/4 for full-sibs.

Variance of the Selection Unit, σ2x

The variance σ2x of the selection unit is a function of the within- and between-

family variances. To obtain these, we assume that the total environmental valueis partitioned as E = Ec + Es, a common family effect (Ec) plus an individual-specific effect (Es). This decomposes the total environmental variances intobetween- and within-family components, σ2

E = σ2Ec

+ σ2Es

. When families arereplicated over plots/environments, the environmental variance contains addi-tional structure and is partitioned into further components (Equations 8.38-8.40).

256 CHAPTER 8

For families with a very large number of sibs, the between-family varianceσ2b (the variance in the expected family means µi ) is

σ2b = σ2(µi) = σ2

GF + σ2Ec

where σ2GF is the between-family genetic variance (the variance in the expected

mean genotypic value of each family), and is developed below (Equations 8.11a,8.26a). Likewise, the within-family variance when the number of sibs is large is

σ2w = σ2(zij − µi) = σ2

Gw + σ2Es

where σ2Gw is the within-family genetic variance (Equations 8.11b, 8.26b). Note

that σ2b and σ2

w are functions of µi, while the variance of the selection unit usuallyrequires the variances about the observed mean zi of each family. Replacing µiby zi results in a slight inflation of the between-family variance and a slightreduction in the within-family variance (this is formally shown below in Example2). With n sibs in each family, the between-family variance based on the observedmeans becomes σ2( zi ) = σ2

b + σ2w/n (the between-family variance σ2

b plus theerror in estimating µi from zi ), while the within-family variance is correspondingreduced, σ2(zij − zi) = (1− 1/n)σ2

w. Thus

σ2( zi ) = σ2GF + σ2

Ec +σ2Gw + σ2

Es

n(8.9a)

In the animal breeding literature, this equation is often more compactly writtenin terms of t, the correlation between sibs (the intraclass correlation coefficient,see LW Chapter 7). The covariance between sibs equals tσ2

z = σ2b = σ2

GF + σ2Ec

(Example 2), implying thatσ2( zi ) = tn σ

2z (8.9b)

wheretn = t+

1− tn

(8.9c)

Likewise, the within-family variance is

σ2( zij − zi ) =(

1− 1n

) (σ2Gw + σ2

Es

)(8.10a)

which is usually written as

σ2( zij − zi ) = (1− tn)σ2z (8.10b)

Table 8.4 gives these family variances in terms of genetic and environmentalvariance components.

Table 8.4. Within- and between-family variances as functions of the genetic and envi-ronmental variance components. Epistasis is assumed absent and the environmental valueis assumed to equal E = Ec + Es, a common family plus an individual-specific value.


Half-sib between-family variance

σ2( zHS ) =σ2A

4+

(3/4)σ2A + σ2

D + σ2Es

n+ σ2

Ec(HS)

Full-sib between-family variance

σ2( zFS ) =σ2A

2+σ2D

4+

(1/2)σ2A + (3/4)σ2

D + σ2Es

n+ σ2

Ec(FS)

Half-sib with nested full-sibs between-family variance(f females per male, n offspring/female)

σ2( zHS(FS) ) =σ2A

4

(1 +

1f

+2fn

)+σ2D

4f

(1 +

3n

)+σ2Es

fn+ σ2

Ec(HS)

Half-sib within-family variance

σ2( zij − zi |HS ) =(

1− 1n

) (34σ2A + σ2

D + σ2Es

)

Full-sib within-family variance

σ2( zij − zi |FS ) =(

1− 1n

) (12σ2A +

34σ2D + σ2

Es

)

The expressions in Table 8.4 follow by expressing the within- and between-family genetic variances in terms of additive and dominance variance compo-nents. Recalling from Analysis of Variance (ANOVA) theory that the between-group variance equals the within-group covariance (e.g., LW Chapter 18), thebetween-family component σ2

GF equals the genetic covariances between sibs. Ig-noring epistasis,

σ2GF =

14σ

2A half-sibs

12σ

2A + 1

4σ2D full-sibs

(8.11a)

Since the total genetic variance σ2G equals the between-family genetic variance

plus the within-family variance,

σ2Gw = σ2

G − σ2GF =

34σ

2A + σ2

D half-sibs

12σ

2A + 3

4σ2D full-sibs

(8.11b)

258 CHAPTER 8

where we have again ignored epistasis. When epistasis is present, Equations8.26a,b replace 8.11a,b.

Example 2. To obtain the within- and between-family variances for familieswith n sibs, decompose the phenotypic value of the jth individual from family ias

zij = Gij + Eij = µ+GFi +Gwij + Eci + Esij

where the genotypic value Gij = µ+GFi +Gwij has both a family genotypiceffectGFi (the expected genotypic value for a random sib from that family) and adeviation Gwij of the j-th individual’s genotypic value from its family average.The environmental value is similarly decomposed, with Eij = Eci + Esij , anenvironmental effect Eci common to family i and an environmental effect Esijunique to the j-th individual from this family. GFi + Eci = bi are the effectscommon to a family, giving the between-family variance as

σ2b = t σ2

z = σ2GF + σ2

Ec

The equalityσ2b = t σ2

z follows from the ANOVA identity that the between-groupvariance equals the covariance between group members (e.g., LW Chapter 18).

Likewise, Gwij + Esij = wij are the within-family effects, giving the within-family variance (in an infinitely-large family) as

σ2w = (1− t)σ2

z = σ2Gw + σ2

Es

The equality σ2w = (1 − t)σ2

z again follows from ANOVA theory, as the totalvariance equals the between- plus within-group variances, σ2

z = σ2b +σ2

w . Usingthese results, for a family of size n,

zi =1n

n∑j=1

zij = µ+GFi + Eci +n∑j=1

(Gwij + Esij)n

Recalling that the Esij and Gwij (being deviations from the mean) are uncorre-lated with each other gives

σ2( zi ) =(σ2GF + σ2

Ec

)+

1n2

n∑j=1

(σ2Gw + σ2

Es

)= σ2

b +nσ2

w

n2=(t+

1− tn

)σ2z = tn σ

2z

Now consider the variance of within-family deviations. Recalling the expressionfor the variance of a sum (LW Equation 3.11a),

σ2( zij − zi ) = σ2z + σ2( zi )− 2σ(zij , zi )


Since σ2( zi ) = tn σ2z , and the covariance term simplifies to

σ(zij , zi ) =1n

σ(zij , zij) +n∑k 6=j

σ(zij , zik)

=σ2z

n+n− 1n

t σ2z = tn σ

2z

the variance of within-family deviations reduces to

σ2( zij − zi ) = σ2z + tn σ

2z − 2 tn σ2

z = (1− tn)σ2z

Finally, under a nested sib design (the North Carolina Design I of Comstockand Robinson 1948) one sex (typically a sire/pollen plant) is mated to each of fdams/seed plants, each of which produces ns sibs for a total of n = fns half-sibs.The expression in Table 8.4 for the between-family for this type of half-sib familyfollows using similar logic as in Example 2, with

σ2( zHS(FS) ) = σ2GF (HS) +

σ2G(f/m)

f+σ2Gw(FS)

nsf+ σ2

Ec(HS) (8.12a)

where σ2G(f/m), the genetic variances of females nested within males is

σ2G(f/m) = σ2

GF (FS) − σ2GF (HS) =

σ2A + σ2

D

4(8.12b)

When epistasis is present, Equation 8.26a provides the appropriate additionalgenetic variance terms in σ2

G(f/m). The between-family variance under a nesteddesign is bounded below by the half-sib variance (f = n and ns = 1) and aboveby the full-sib variance (f = 1 and ns = n).

RESPONSE FOR PARTICULAR DESIGNS

The formal development of the response equations for any particular design fol-lows from the generalized breeders’ equation (8.1-8.4) and the appropriate vari-ances (Table 8.4) and covariances (Table 8.3). Results for a number of standardbetween- and within-family designs are developed below, with family index se-lection examined at the end of the chapter. Before considering the details, again afew general comments are in order.

Overview of Between- and Within-Family Response

To hopefully provide a guide so that the reader is not lost in the details, wecan motiviate the structure of the response equations for particular designs byconsidering how the additive-genetic and total variances are partitioned within-and between-families. When the number of sibs per family (n) is large, thesevariances are partitioned as

260 CHAPTER 8

Within-family Between-family

Breeding values (1− r)σ2A r σ2

A

Phenotypic values (1− t)σ2z t σ2

z

where t and r are the phenotypic and additive-genetic correlations between sibs(r = 1/4 for half-sibs, 1/2 for full-sibs). When the number of measured sibs withineach family is small, tn = t + (1 − t)/n replaces t and rn (similarly defined)replaces r. Since the response to selection depends on the ratio of the availableadditive genetic variance to the phenotypic variance, the response to between-family selection is of the form

Rb =rn σ

2A

tn σ2z

S = σA

(σAσz

)(rn√tn

) (S√tn σz

)= σA h

rn√tn

ı (8.13a)

Equation 8.13a is the exact expression for family selection and is due to Lush (1947).The response under other designs (e.g., sib and parental selection, S1 seed) arevery similar (see below).

Likewise, the response to within-family selection is a function of the within-family additive-genetic and phenotypic variances, leading us to expect that re-sponse is in the form of

RFD =(1− rn)σ2

A

(1− tn)σ2z

S = σA h1− rn√1− tn

ı (8.13b)

Indeed, this is the exact expression for selection on family deviations (FD), whilethe response under strict within-family (WF) selection is given by replacing rnand tn by r and t.

Equations 8.13a and 8.13b are the standard response equations that appearin much of the elementary animal breeding literature. The use of r and t allowsthese results to be presented in a very compact fashion. When the design is morecomplicated, such as replication of families or the use of nested sibs families, theresponse can not longer be cleanly expressed in terms of t, and expressions aregiven in terms of variance components.

Between-Family Selection

Here the selection unit is z, the mean of a half-, full-, or nested-sib family. Thetype of sib family, together with the relatives used to produce the next generation,specifies each particular between-family design (Table 8.1). Tables 8.3 and 8.4 andEquation 8.1a gives the response to a single cycle of selection as

Rb =θ√tn

σA2h(ıxf + ıxm

)=

θ

σ( z )σ2A

2(ıxf + ıxm

)(8.14a)


The left equality holds when sib families are not nested and families are notreplicated, while the rightmost expression is completely general (using σ2( z ) inplace of tnσ2

z ). The selection unit-offspring covariance is θσ2A/2, where

θ =

rn = r + (1− r)/n family selection

r sib selection

1/2 parental, S1 seed selection

(8.14b)

The variance of the selection unit σ2( z ) = tnσ2z depends only on the types of

sibs measured and is independent of the types of relatives used to form the nextgeneration. The theory of expected response to between-family selection tracesback to Lush’s classic 1947 paper, and Equation 8.14 is a generalization of hisresults. Table 8.5 expresses the response in terms of variance components.

Table 8.5. Variance components versions of the expected response for the major types ofbetween-family selection schemes using outbred sibs. The number n of measured sibs isassumed sufficiently large that terms of order 1/n can be ignored (i.e., rn ' r and tn ' t).We also assume no epistasis and the simple structure E = Ec + Es for environmental values.

Half-sibs Full-sibs

Family, Sib(σ2A/8) ( ıxm + ıxf )√σ2A/4 + σ2

Ec(HS)

(σ2A/4) ( ıxm + ıxf )√

σ2A/2 + σ2

D/4 + σ2Ec(FS)

Parental, S1 seed(σ2A/4) ( ıxm + ıxf )√σ2A/4 + σ2

Ec(HS)

(σ2A/4) ( ıxm + ıxf )√

σ2A/2 + σ2

D/4 + σ2Ec(FS)

Several variants of Equation 8.14 are common in the literature. Noting thatσA h = σzh

2, the response is often expressed as

Rb =θ√tnσz h

2 ı (8.15a)

where ı = ( ıxf + ıxm)/2. Similarly, response can be expressed in terms of thebetween-family heritability,

Rb = h2b,θS, where h2

b,θ =θ

tnh2 (8.15b)

and S = (Sf + Sm)/2 is the average selection differential on the parents.

262 CHAPTER 8

Turning now to particular between-family designs, we start with family se-lection. Here measured sibs (all or a random subset) from the chosen families formthe parents for the next generation. To reduce the effects of inbreeding, crossesbetween sibs from the same family are typically avoided. With family selection,Equation 8.14 becomes

Rb =

(1 + 3/n)√tn(HS)

σA8h(ıxm + ıxf

)half-sibs

(1 + 1/n)√tn(FS)

σA4h(ıxm + ıxf

)full-sibs

(8.15c)

as first obtained by Lush (1947). While full-sibs have twice as much usablebetween-family additive variance (σ2

A/2 vs. σ2A/4), this advantage is reduced

because half-sibs have a smaller between-family variance than full sibs, withtHS/tFS < 1. This inequality follows by recalling that σ2( z ) = tσ2

z and notingthat (tFS − tHS)σ2

z = σ2( zFS)− σ2( zHS), where

σ2( zFS)− σ2( zHS) =σ2A + σ2

D

4+(σ2Ec(FS) − σ2

Ec(HS)

)> 0 (8.16)

Given that full-sibs share a common mother (and hence potentially share materialeffects), we expect σ2

Ec(FS) ≥ σ2Ec(HS) and hence σ2( zFS) > σ2( zHS). The ratio

of response for full- vs. half-sib family selection is thus 2√tn(HS)/tn(FS) < 2.

If the character can only be measured after reproduction, females (seed par-ents) from the chosen families have already been pollinated, and hence selectionhas occurred on only one sex. Planting these seeds and evaluating the resultingfamilies allows for half-sib selection. Full-sib selection can also be accomplished,but each cycle takes an additional generation. Here seeds from open-pollinatedselected females are grown up and controlled crosses made between the offspringfrom different seed parents to create full-sib families for the next cycle of selection.

Example 3. Clayton et al. (1957) examined family selection on abdominal bristlenumber in Drosophila melanogaster (LW Figure 14.1). Their estimated intraclasscorrelations for half- and full-sibs were 0.121 and 0.265, the estimated additivevariance was 5.59, and the heritability was estimated to be 0.52. Hence

tHS = 0.121, tFS = 0.265, σA h =√

5.59 · 0.52 = 1.70

Clayton et al. performed selection in two different settings: (i) the top two often half-sib families were saved, and (ii) the top four of 20 full-sib families wereselected. The expected selection intensities under these two schemes are ıHS =1.27and ıFS =1.33 (Example 6). The family sizes used were 20 half-sibs and 12 full-sibs. Because of the design, there was a one in ten chance that the half-sibs areactually full-sibs, resulting in a slight inflation of r from 0.25 to 0.275 (= 0.25 +0.5/10). Summarizing:


Half-sibs Full-sibsr 0.275 0.5n 20 12tn 0.165 0.326rn 0.311 0.542

Equation 8.13b gives the expected response to half-sib family selection as

Rb(HS) = ıHS · (σA h)rn√tn

= 1.27 · 1.700.311√0.165

= 1.67

while the expected response to full-sib family selection is

Rb(FS) = 1.33 · 1.700.542√0.326

= 2.15

Clayton et al. obtained slightly different estimated responses (1.33 and 2.02 forhalf- and full-sibs, respectively). This occurred because they usedR = h2

b Sb withSb = σbı computed by taking the observed between-family variance σ2

b (in placeof the estimates σ2

A, t, and h2). The observed responses (averaged over the firstfive generations) were 1.38 and 0.94 for up- and down-selected half-sibs, and 1.62and 1.36 for up- and down-selected full sibs. The authors noticed a fairly sizablereduction in the estimated additive variance during generations two throughfour, and this (in addition to sampling error) likely accounts for the discrepancybetween observed and predicted response.

Under sib selection, unmeasured sibs from each chosen family are used toform the next generation. The most common response equation for sib selectionin the literature, due to Robertson (1955b), is

Rsib = ı σA hn r√

n(1 + [n− 1] t)(8.17)

where ı denotes the average selection intensity on both parents. As mentioned,the use of remnant seed is a variant of sib selection, which allows for selection onboth sexes of parents, at the expense of an extra generation per cycle.

Under parental selection, parents are chosen based on the performance of atrail set of their offspring. Typically half-sib families are used and selection is ona single sex. In this case, the expected response is

Rpt =σA/4√tn(HS)

h ı (8.18a)

In monoecious species, the expected response is double that given by Equation8.18a if one uses the selected parents for both seed and pollen. The use of maternal

264 CHAPTER 8

half-sib families (as commonly occurs in plant breeding) is expected to inflatet(HS) relative to paternal half-sibs (and hence reduces response) as the commonfamily environmental effects can be rather significant due to maternal effects.

If males (sires/pollen plants) are progeny-tested using a nested sib design,wherein each male is crossed to f females (dams/seed plants) each of which hasn sibs, the appropriate between-family variance is given in Table 8.4, and theresponse becomes

Rpt = h ıσ2A/4√

σ2GF (HS) + σ2

G(f/m)/f + σ2GW (FS)/nf + σ2

Ec(HS)

= h ıσ2A/4√

σ2A

4

(1 +

1f

+2fn

)+σ2D

4f

(1 +

3n

)+σ2Es

fn+ σ2

Ec(HS)

(8.18b)

For progeny testing of females using a nested design, the roles of males andfemales are exchanged in the above expression. Since σ2( zHS ) ≤ σ2( zHS(FS) ) ≤σ2( zFS ), the response using a tested progeny test in intermediate to that fordesigns using half- or full-sibs. All above comments for parental selection equallyapply to the S1 seed design, as the expected response is the same.

Given the number of options for between-family selection, which schemeshould be used? Biological and/or economic restriction may preclude certaindesigns and make others more feasible. These logistical concerns aside, there arethree issues that must be weighted: (i) cycle time versus selection on one or bothsexes, (ii) performance evaluation using half- vs. full-sib families (the value of tn,and more generally σ2( z )), and (iii) choice of relatives for the recombination unit(the value of θ in Equation 8.14a). As mentioned above, often the reason using atwo-generation cycle (e.g., remnant seed) is the inability to select on both sexes.In such cases, doubling the cycle time is countered by selection on both sexesdoubling the response per cycle, giving both approaches same rate of progress ona per-generation basis. In many cases, a multigeneration method is used becauseselection on other characters beside the primary one of interest is also performedin one (or both) generations of the cycle.

The second choice is the type of family. While the type of sibs changes therelatives under family- and sib-selection, it does not influence relatives underparental or S1 selection. Indeed, for these last two designs it is more efficient touse half-sib families, as the ratio of response to parental half-sib versus parentalfull-sib is t(FS)/t(HS) > 1.

Provided the same type of families (half-, full-, nested-sibs) are measured,choosing relatives that increase the recombination unit-offspring covariance (in-crease θ) increase the expected response. For half-sib families, parental and S1

selection gives twice the response per cycle as sib or family selection (assumingthe same number of sexes are selected in comparisons). With full-sibs, the re-sponse per cycle under all four methods is the same (again assuming the same


number of sexes are selected). While the response to selection using full-sib fami-lies is greater than that of family or sib selection using half-sibs, the use of full-sibsdoes not result in a doubling the response as 2

√tn(HS)/tn(FS) < 2. This less

than two-full increase in response per cycle using full-sibs is thus not sufficientto cover the cost of the extra generation often required to create full-sib families.

Once one has chosen a particular design, there is also the issue of allocationof the number of sibs per family n given constraints on the total number of sibsN measured. One increases the accuracy of the method by increasing the numberof sibs per family, but does so by decreasing the selection intensity. Robertson(1957, 1960) and Rendel (1959) have examined this problem of optimal familysize. To maximize the response, the breeder usually has two fixed constraints:the total number of sibs N examined and the number S of parents (or families)used to form the next generation. A low S increases inbreeding and thus not onlyinvites inbreeding depression, but also reduces the eventual long term response(Chapter 14). For fixed S and N , the goal is the find the number of sibs n perfamily that maximizes response. Noting that σz h2 is fixed, while S = mp and thatm = N/n, Equation 8.15a shows that the single-generation response is maximizedby maximizing

θı(N/n,S)√

tn

with respect to n. With the exception of family selection (where θ = rn) θ is afixed constant. Maximization given the particular values for any experiment caneasily be done numerically in any standard spread-sheet program (for example,by using Equation 4.17a to approximate ı(N/n,S)).

Within-Family Selection

Within-family selection chooses individuals based on their relative performancewithin families. Under family-deviations (FD) selection, individuals with thelargest deviations are chosen, independent of which family they come form. Incontrast, strict within-family (WF) selection chooses the largest individuals fromeach family, independent of how much they actually deviate from their familymeans. Suppose in family one the deviations are 4, 3, and -7, while the deviationsin family two are 1, 0, -1. If we select the upper 1/3, then under WF selection, thetop individual from each family are chosen, while under FD selection, two indi-viduals from family one, and none from family two, are chosen. The result of thisrather subtle difference is that FD selection is influenced by the observed meanzi, while WF selection is not. Family deviations and strict within-family selectionhave been confused in the literature, with the correct expression for WF selectiondue to Dempfle (1975, 1990) and Hill et al. (1996). Since WF selection ensuresequal representation of families, while FD selection does not, WF selection has alarger effective population size and hence an expected larger long-term response(Chapter 14).

Under family-deviations (FD) selection, the selection unit is the value of an

266 CHAPTER 8

individual’s within-family deviation, zij−zi. Using the results from Table 8.3 and8.4, Equation 8.1 gives the expected response as

RFD =σ( zij − zi, y |Rf )

σ( zij − zi )ıxf +

σ( zij − zi, y |Rm)σ( zij − zi )

ıxm

=1− rn√1− tn

σA h

(ıxf + ıxm

2

)(8.19)

The last equality follows from σ2A/σz = σA h.

Under strict within-family (WF) selection, individuals are chosen enitrely ontheir rank within each family, resulting in the observed mean zi being replacedby the true (and unobserved) mean µi. The response becomes

RWF =σ( zij − µi, y |Rf )

σ( zij − µi )ıxf +

σ( zij − µi, y |Rm)σ( zij − µi )

ıxm

=1− r√1− t

σA h

(ıxf + ıxm

2

)(8.20)

Noting that

1− rn√1− tn

=(1− 1/n)(1− r)√

(1− 1/n)(1− t)=

1− r√1− t

√1− 1

n

it follows that

RFD = RWFıFDıWF

√1− 1

n(8.21)

Thus, when the number of measured sibs in each family is modest to large (so thatthe selection intensities are essentially equal, ıFD ' ıWF ), the difference betweenthe expected responses under WF vs. FD selection is very small. Equation 8.20gives resulting response for strict within-family selection using half- and full-sibfamilies as

RWF =

(3/8)σ2

A√1− t(HS)

h(ıxm + ıxf

)half-sibs

(1/4)σ2A√

1− t(FS)h(ıxm + ıxf

)full-sibs

(8.22a)

Expressed in terms of variance components,

RWF =

3σ2A/8√

(3/4)σ2A + σ2

D + σ2Es

(ıxm + ıxf

)half-sibs

σ2A/4√

σ2A/2 + (3/4)σ2

D + σ2Es

(ıxm + ıxf

)full-sibs

(8.22b)


For half-sibs, the within-family additive variance is (3/4)σ2A, only half of which

is passed from parent to offspring, giving the (3/8)σ2A term. For full-sibs, the

within-family additive variance is (1/2)σ2A, again only half of which is passed

onto offspring, giving σ2A/4.

The within-family heritability, h2w, is the same under both FD and WF

within-family selection as

1− rn1− tn

=(1− 1/n)(1− r)(1− 1/n)(1− t) =

(1− r)(1− t)

Hence,

h2w =

2σ( zij − zi, y |R1 )σ2( zij − zi)

=(1− rn)σ2

A

(1− tn)σ2z

=(1− r)(1− t) h

2 (8.23)

Example 4. Using the data of Clayton et al. (1957) from Example 3, what isthe expected response under the two within-family selection schemes? Supposestrict within-family (WF) selection was performed on the full-sib families, withthe upper 20 percent chosen from each family (the top four of the 20 measuredsibs). The expected selection intensity is ı(20,4) = 1.33 and from Equation 8.20the predicted response is

RWF = ı · (σA h)1− r√1− t

= 1.33 · 1.701− 0.275√1− 0.121

= 1.75

Using within-family deviations (FD), selecting the uppermost 20 percent givesa corrected selection intensity of 1.39 (Example 6), and Equation 8.19 gives thepredicted response as

RFD = ı · (σA h)1− rn√1− tn

= 1.39 · 1.701− 0.311√1− 0.165

= 1.78

Realized Heritabilities

By analogy with individual selection, one can estimate the realized heritabilityassociated of a particular family-based scheme from the ratio of observed responseto selection differential,

h2x(r) =

RxSx

(8.24a)

Falconer and Latyszewski (1952) used this approach to estimate a realized within-family heritability for response to selection on body size in mice. These authorscomputed the standard error of this estimate by noting that

σ2(h 2wf (r)

)= σ2

(RwfSwf

)=σ2 (Rwf )S2wf

(8.24b)

268 CHAPTER 8

The last equality follows by assuming that the variance in measuring S can beignored.

One is faced with several options for estimated the realized heritability whenpresented with several generations worth of data (Chapters 6, 7). The simplest isto use the ratio of total response to total selection differential. Alternatively, cu-mulative response can be regressed on cumulative selection differential, with theslope being the estimate of the realized heritability. As with mass selection, an ap-propriately weighted GLS regression must be used to obtain an unbiased estimateof the standard error. Unweighted (OLS) regressions severely underestimate thestandard error due to correlations among the residuals (Chapter 6, 7). Anotherconcern is that the underlying genetic parameters can change each generation.The general assumption is that these change are small enough to be ignored, butthis may not be a valid assumption in many cases (Chapters 5, 7, 10). Provided theinfinitesimal model applies, mixed-model (MM) analysis (Chapter 7) deals withthese concerns. MM analysis is completely general and can easily accommodatefamily-based selection, as the relationship matrix A accounts for all appropriatecovariances between relatives.

Since between- and within-family heritabilities can be expressed as a func-tion of the individual heritability h2 (Equations 8.15 and 8.23), we can similarlytranslate a realized heritability estimate for a particular family-based designs intoa realized individual heritability. With between-family selection,

h 2(r) =(tnθ

)h 2b (r) (8.25a)

while for within-family selection,

h 2(r) =(

1− t1− r

)h 2wf (r) (8.25b)

These expressions apply for a single generation of selection. Additional uncer-tainly is introduced into the estimate if the sib phenotypic correlation t is unknownand must itself be estimated. Equations 8.25a, b should be used only with extremecaution when multiple cycles of selection have taken place, as the sib additive-genetic correlation r increases each successive generation due to inbreeding. Thesechanges in genetic variances also change the phenotypic correlation t.

Accounting for Epistasis

The response to within- and between-family selection in the presence of epistasishas been briefly developed by Nyquist (1991), and we expand upon his resultshere. As with individual selection, additive epistasis contributes to the initialresponse, but its contribution to the ultimate response rapidly decays with timeas recombination breaks up favorable combinations of alleles at different loci(Chapter 4). We first consider the single-generation response and then brieflyexamine the transient dynamics.


The between-family genetic variance σ2GF with arbitrary epistasis immedi-

ately follows from the genetic covariance between sibs (LW Table 7.2),

σ2GF =

14σ

2A + 1

16σ2AA + 1

64σ2AAA + · · · half-sibs

12σ

2A + 1

4σ2D + 1

4σ2AA + 1

8σ2AD + 1

16σ2DD + 1

8σ2AAA + · · · full-sibs

(8.26a)Likewise, the within-family genetic variance, σ2

Gw = σ2G − σ2

GF , becomes

σ2Gw =

34σ

2A + σ2

D + 1516σ

2AA + σ2

AD + σ2DD + 63

64σ2AAA + · · · half-sibs

12σ

2A + 3

4σ2D + 3

4σ2AA + 7

8σ2AD + 15

16σ2DD + 7

8σ2AAA + · · · full-sibs

(8.26b)The between- and within-family variances, σ2( zi ) and σ2( zij−zi ), immediatelyfollow by substituting Equation 8.26 into Equations 8.9a and 8.10a.

The selection unit-offspring covariances under epistasis follow using LWEquation 7.12,

σG(x, y) = (2Θxy)σ2A + (2Θxy)2

σ2AA + · · · =

∑u=1

(2Θxy)u σ2Au

We assume ∆xy = 0, so that terms involving dominance are not included. Usingthe values of Θxy from Table 8.2, the parent-offspring covariance is

σ(R1, y) =σ2A

2+σ2AA

4+σ2AAA

8+ · · · =

∑u=1

(12u

)σ2Au (8.27a)

Table 8.2 shows that Θxy = 1/16 and 1/8 when x is a half- or full-sib (respectively)ofR. Expressed in terms of Wright’s coefficient of relationship,

σ(x1, y |R1 ) = (r/2)σ2A+(r/2)2σ2

AA+(r/2)3σ2AAA+ · · · =

∑u=1

(r2

)uσ2Au (8.27b)

Substituting Equation 8.27a and 8.27b into Equation 8.6 gives the covariance forfamily selection as

σ( z, y |R1 ) =1n

∑u=1

(12u

)σ2Au +

(1− 1

n

)∑u=1

(r2

)uσ2Au

=∑u=1

(12

)u(ru)n σ

2Au (8.28)

where (ru)n = ru + (1 − ru)/n. For large family size, the coefficient for u-foldadditive epistasis approaches ru/2u, which is the value under sib selection and

270 CHAPTER 8

(taking r = 1/2) parental and S1 seed selection. Equation 8.28 gives the single-parent covariance for half-sib family selection as

σ( zHS , y |R1 ) =(

1 +3n

)σ2A

8+(

1 +15n

)σ2AA

64+(

1 +63n

)σ2AAA

512+ · · · (8.29a)

Likewise, the single-parent covariance for full-sib family selection is

σ( zFS , y |R1 ) =(

1 +1n

)σ2A

4+(

1 +3n

)σ2AA

16+(

1 +7n

)σ2AAA

64+ · · · (8.29b)

For sib-selection, σ( z1, y |R1 ) is given directly from Equation 8.27b, and Equa-tions 8.29a,b apply if terms of order 1/n are ignored. For between-family selectionusing parental selection or S1 seed, the covariance is the same as that for full sibsunder sib selection (as all three have the same Θxy value).

The covariance for between-family deviations (again considering the contri-bution through a single parent of y) becomes

σ( zij − zi, y |R1 ) = σ(R1, y )− σ( zi, y |R1 )

=∑u=1

(12

)u(1− (ru)n

)σ2Au

=(

1− 1n

)∑u=1

(12

)u(1− ru)σ2

Au (8.30)

where we have used the identity (1−rn) = (1−1/n)(1−r). Ignoring the common(1− 1/n) factor found in all terms, for half-sibs we have

σ( zij − zHS , y |R1 ) =(

38

)σ2A +

(1564

)σ2AA +

(63512

)σ2AAA + · · · (8.31a)

while for full-sibs,

σ( zij − zFS , y |R1 ) =(

14

)σ2A +

(316

)σ2AA +

(764

)σ2AAA + · · · (8.31b)

Equations 8.29 and 8.31 show that additive epistasis contributes to the short-term response. However, as with individual selection, this contribution is tran-sient and decays over time as recombination breaks up linkage groups of favorablealleles (Chapter 4). For u-locus additive epistasis (σ2

Au ), the per generation decayrate for unlinked loci is (1/2)u−1, the probability that a parental gamete containingspecific alleles at u unlinked loci is passed onto an offspring. The probability thatsuch a gamete remains unchanged after τ generations is 2−τ(u−1), which rapidly


converges to zero. Thus, if RAu is the contribution due to u-locus additive epis-tasis, after τ generations the contribution from a single generation of selectionbecomes 2−τ(u−1)RAu .

EFFICIENCY OF FAMILY-BASED vs. INDIVIDUAL SELECTION

Our intuition suggests that individual selection is better than either within- orbetween-family selection when h2 is modest to large. When h2 is small, we expectwithin-family selection to be more efficient if there is a large common familyenvironmental effect (σEc ' σ2

z ) and between-family selection to be more efficientif the individual-specific environmental effects are large (σEs ' σ2

z ). To moreformally develop these points, recall that the expected response under individualselection is Rm = ım σA h . Equation 8.14a implies that the ratio of response ofbetween-family vs. individual selection is thus:

RbRm

=(ıbım

) (θ√tn

)(8.32a)

Likewise, for within-family (family deviations) selection,

RFDRm

=(ıFDım

) (1− rn√1− tn

)(8.32b)

Finally, Equation 8.21 gives the response ratio for strict within-family selection as

RWF

Rm=(ıWF

ıFD

)(RFDRm

)√n

n− 1(8.32c)

Equations 8.32a-c shows that the relative efficiency of any particular family-based scheme is the product of the ratio of selection intensities (the first term) andthe accuracy of selection relative to individual selection (the second term). Theaccuracy ratio measures how well (relative to individual selection) the selectioncriteria predicts the breeding values of the parents. We focus first on the accuracyratio as the selection intensity ratio is generally close to one unless sample sizesare very small (Table 8.6).

The Relative Accuracies of Family-based vs. Individual Selection

Relative accuracies are typically expressed in terms of the phenotypic correlationt between sibs and the coefficient of relatedness r. Since under the simple envi-ronmental model (E = Ec+Es) σ2( z ) = tσ2

z = σ2GF +σ2

Ec, Equation 8.11a showsboth how t and r are related and how they can be expressed in terms of variancecomponents. In the absence of epistasis,

t = rh2 + c2, where σ2zc

2 =

σ2Ec(HS) half-sibs

14σ

2D + σ2

Ec(FS) full-sibs(8.33)

20181614121086420.0

0.2

0.4

0.6

0.8

1.0

Family size, n

t, P

heno

typi

c co

rrel

atio

n be

twee

n si

bs

Half Sibs

Within Family

Individual

Between Family

20181614121086420.0

0.2

0.4

0.6

0.8

1.0

Family size, n

t, P

heno

typi

c co

rrel

atio

n be

twee

n si

bs Within-Family

Between Family

Individual

Full Sibs

272 CHAPTER 8

where c2 is the residual between-family variance (upon removal of any additivevariance). Figures 8.4 and 8.5 plot the relative accuraries and relative responsesunder between-family selection (family selection) and within-family selection(family deviations).

Figure 8.4. Regions of the family size (n) - sib correlation (t) space where indi-vidual, within-family (family selection) and between-family (selection of familydeviations, FD) are the most accurate. If t is sufficiently large, within-family se-lection gives the largest response (for large n, t > 7/16 =0.4375 for half-sibs andt >3/4 for full-sibs), between-family selection is best when t is sufficiently small(for large n, t < 1/16 = 0.0625 for half-sibs, t< 1/4 for full-sibs), and individ-ual selection gives the largest response for intermediate values of t. For large n,family selection equals sibs selection and (using the curve for full-sibs) parentalselection, while family-deviations selection approaches strict within-family (WF)selection.

What are the exact conditions for a particular method to be more accuratethan individual selection? Equation 8.32a shows that between-family selection ismore accurate when θ/

√tn > 1, or

tn = t+1− tn

=(rh2 + c2

)(1− 1

n

)+

1n< θ2 (8.34a)

For n moderate to large, between-family selection is more accurate than massselection when the fraction c2 of total variance due to residual between-familyeffects is sufficiently small. Substituting the appropriate values of θ into Equation8.34a (for moderate to large n) yields

c2 <

116(1− 4h2

)half-sibs (for family, sib selection)

14(1− h2

)half-sibs (for parental and S1 seed selection)

14(1− 2h2

)full-sibs

(8.34b)

1.00.90.80.70.60.50.40.30.20.10.00.0

0.5

1.0

1.5

2.0

2.5

3.0

n = 2

n = 10

large n

Phenotypic correlation t between sibs

Res

pons

e R

elat

ive

to M

ass

Sele

ctio

n

Half SibsBetween Family Selection

1.00.90.80.70.60.50.40.30.20.10.00.0

0.5

1.0

1.5

2.0

2.5

3.0

n = 2n = 10

large n


Res

pons

e R

elat

ive

to M

ass

Sele

ctio

n

Full Sibs

Between Family Selection

1.00.90.80.70.60.50.40.30.20.10.00.0

0.5

1.0

1.5

2.0

2.5

3.0

n = 2n = 5

large n


Res

pons

e R

elat

ive

to M

ass

Sele

ctio

n

Within Family, Half Sibs

1.00.90.80.70.60.50.40.30.20.10.00.0

0.5

1.0

1.5

2.0

2.5

3.0

n = 2n = 5n large


Res

pons

e R

elat

ive

to M

ass

Sele

ctio

n

Within Family, Full Sibs


If h2 > 1/2, between-family selection using full-sibs is always less efficient thanindividual selection. With half-sibs, family- and sib-selection is always less effi-cient than individual selection when h2 > 1/4. If heritability is small, a necessarycondition for between-family selection to be more efficient is that the commonfamily environment amounts to only a small fraction of the total variance (Equa-tion 8.33).

Figure 8.5. Accuracies of family selection (top row) and selection on familydeviations (bottom row) relative to mass selection. Values exceeding one indicatean increased response relative to individual selection.

Turning to within-family selection, family deviations (FD) gives a larger re-sponse than individual selection when (1 − rn)/

√1− tn > 1. When families are

large (n >> 1), this condition reduces to

t = rh2 + c2 > 1− (1− r)2 (8.35a)

or

c2 > 1− (1− r)2 − rh2 =

716− h2

4half-sibs

34− h2

2full-sibs

(8.35b)

274 CHAPTER 8

Since h2 + c2 ≤ 1 (both being fractions of the total variance due to differencesources), there is an additional constraint that 1−h2 ≤ c2. Whenh2 > 0.085 within-(half-sib) family selection is less efficient than individual selection, as c2 ≤ 1−h2 =0.915, while the critical c2 value that must be exceeded is 0.916. For full-sibs,individual selection is more efficient than within-family selection whenever h2 >0.5. Equations 8.35a and 8.35b are also the conditions for strict within-family (WF)selection. Within-family selection is thus more efficient than individual selectiononly when the heritability is low and the residual between-family variance (c2σ2

z )accounts for a very significant fraction of the total variance, i.e., the individual-specific environmental effects account for much of the phenotypic variance.

Example 5. Wilson (1974) examined family selection (using full-sibs) on larvaland pupal weight in Tribolium castaneum. Correlations among full-sibs were esti-mated to be t = 0.20 for pupal weight and t = 0.16 for larval weight. Familysize was n = 12. Since these are full-sibs, r = 1/2, giving the relative accuracyof family selection (θ = rn) on larval weight as

θ√tn

=r + (1− r)/n√t+ (1− t)/n

=0.5 + 0.5/12√0.16 + 0.84/12

= 1.13

Likewise, the relative accuracy for pupal weight is

1/2 + 1/(2 · 12)√0.2 + 0.8/12

= 1.05

Both characters are thus expected to show a larger response under family selectionthan under mass selection. However, Wilson observed that individual selectiongave a larger (though not significant) response for both characters. Note fromEquation 8.14b that the expected response for sib selection using full sibs is thesame as for parental selection (both have θ = 1/2). The relative accuracy of thesemethods on larval weight is

θ√tn

=0.5√

0.16 + 0.84/12= 1.04

while their relative accuracy for pupal weight is

θ√tn

=0.5√

0.2 + 0.8/12= 0.97

Thus for pupal weight, family selection is slightly more accurate than mass selec-tion, while sib and parental selection are slightly less accurate.


Campo and Tagarro (1977) also compared full-sib family and individual selectionon Tribolium pupal weight, using experiments with family sizes of four and ten.For n = 4, the predicted relative accuracy is 0.87. In both experiments, familyselection gave the larger single-generation response, while mass selection had thelarger response after six generations. None of these differences were significant.

Two other studies have compared individual and between-family selection,both using half-sib family selection in chickens. Garwood et al. (1980) examinedlaying rate (h2 = 0.22) and egg weight (h2 = 0.55), finding that individual se-lection gave a greater single-generation response for both characters, but the dif-ference for egg weight was not significant. Kinney et al. (1970) examined severalcharacters, finding individual selection exceeded family selection, although againnone of the differences were significant.

Finally, it is also informative to compare methods using the heritability ver-sion of response, Rx = h2

x Sx. From Equation 8.23, the within-family heritabilityexceeds the individual heritability when 1− r > 1− t. Hence,

h2w > h2 when t > r (8.36a)

Likewise, from Equation 8.15, the between-family heritability satisfies

h2b > h2 when θ > tn (8.36b)

These conditions are rather different from those given by Equations 8.34a and8.35a. For example, Equation 8.34a implies that between-family selection gives alarger response than individual selection when θ >

√tn, Equation 8.36b implies

that the between-family heritability is greater than h2 when θ > tn. What is thediscrepancy between these two approaches (accuracies vs. heritabilities)?

The key is that the variances of the groups being selected differ. Since σ2z =

σ2b + σ2

w, the between-family and within-family variances are each less than thephenotypic variance of a random individual. Since Sx = ıxσx, larger selectionintensities are required to give a family-based approach the same selection differ-ential as individual selection. Since the within- and between-family variances are(1− tn)σ2

z and tnσ2z , respectively, it follows that

SbSm

=ıb σbım σz

=ıbım

√tn and

SwSm

=ıw σwım σz

=ıwım

√1− tn

Under identical selection intensities, the differentials for between- and within-family selection are

√tn and

√1− tn of the differential under mass selection.

Thus even when h2w or h2

b exceeds h2, this advantage is partially countered bysmaller selection differentials due to smaller variances. The initial comparison

276 CHAPTER 8

assuming the same selection differential thus has a hidden assumption of moreselection (a larger selection intensity) under family-based selection.

Comparing Selection Intensities: Finite Size Corrections

While not nearly as dramatic as differences in the selection differentials, the se-lection intensities can differ across methods even if the same fraction p is saved.These differences arise from the finite sample size correction of ı (Chapter 4). Sup-pose a total of nine individuals are measured, three from each of three families. Ifwe select for the upper 1/3, we keep the best one of three families under between-family selection, and the best of the three individuals within each family underWF selection, giving an expected selection intensity of ı(3,1) = 0.846 (the expectedvalue of the largest of the three order statistics). Under family deviations (FD)and mass selection, we chose the largest three of nine values, giving an expectedselection intensity of ı(9,3) = 0.996. Table 8.6 summarizes the selection intensitiesfor the different methods, showing that ıb, ıWF ≤ ım ≤ ıFD.

Table 8.6. Selection intensities for various forms of family-based selection corrected forfinite sample size. The upper p of the population is saved and the population consists of mfamilies each with nmembers, for a total of N = mn individuals. Tables of exact values forı(N,K) (the average value of the top K of the N standardized order statistics, see Chapter4) are given by Becker (1992). Approximations for ı(N,K) are given in Equations 4.17a-d.

Selection Type Corrected Selection Intensity

Individual ım = ı(N,pN)

Between-family ıb = ı(m,pm)

Family-deviations ıFD = ı(N,pN)

√1 +

1N − 1

Within-family ıWF = ı(n,pn)

An additional subtly in adjusting the selection intensity was pointed out byHill (1976, 1977c). The expected selection intensity is computed by taking the ex-pected value of the largest standardized order statistics (Chapter 4), under theassumption that the order statistics are uncorrelated. With family deviations (FD),family index, and even mass selection, there is the potential for correlations be-tween order statistics. Here different families can contribute different numbersof individuals, resulting in correlations between those measures from the samefamily and hence correlations between some of the order statistics. The correctionfor mass selection is generally very small and will be ignored here (see Equa-tion 8.54b). Within-family deviations are negatively correlated within a family,(ρ = −1/(n − 1) for a family of size n ), as they are deviations from a common


family mean. Dempfle (1990) and Hill et al. (1996) show that the resulting selectionintensity for within-family deviations is thus slightly larger than the intensity formass selection ım,

ıFD = ım

√1 +

1N − 1

(8.37)

where N is the total number of measured sibs. Again, this correction is only im-portant when family size is very small. On the other hand, with selection on afamily index, the correlations between index scores are positive and can be con-siderable even for large n (Equation 8.55). We consider the appropriate correctionfor ı in our treatment of family index selection at the end of this chapter.

Example 6. Consider the selection intensities for Clayton et al.’s (1957) study onabdominal bristle number in Drosophila melanogaster (Examples 3, 4). Here the toptwo of ten half-sib families and the top four of 20 full-sib families were selected.The expected selection intensities for between-family selection are ı(10,2) = 1.27 forhalf sibs and ı(20,4) = 1.33 for full sibs. The family sizes used were 20 half sibs and12 full sibs, for total sample sizes of 200 and 240, respectively. If the same fraction(20 percent) is saved under individual selection, the half- and full-sib experimentsboth have an expected selection intensity of 1.39 (as ı(200,40) = ı(240,48) = 1.39).Under selection on within-family deviations (FD), ı is also 1.39 as the correctionfactor (Equation 8.37) is essentially 1 (1.002) under either sample size.

RESPONSE WHEN FAMILIES ARE REPLICATED OVER ENVIRONMENTS

Family members are often raised in multiple plots and/or environments. If theinvestigator is not aware of this underlying environmental structure, it inflatesthe between-family means relative to designs that take this structure into account.Carefully designed family replication offers two potential advantages. First, theyallow selection of genotypes that perform best over a range of environments, evenwhen extensive genotype-environment interactions are present. Second, replica-tion within an environment reduces the effects of microenvironmental differences,increasing the predictability of a family’s breeding value (and hence the response).

Since family replication is a hallmark of plant breeding, we examine severalschemes used by crop breeders in detail in this section (Chapter 9 examines re-lated designs involving inbreeding and line crossing). Detailed reviews of plantbreeding methodology are given by Allard (1960), Namkoong (1979), Hallauerand Miranda (1981), Nguyen and Sleper (1983), Hallauer (1985), Wricke and We-ber (1986), Gallais (1990), and Bos and Caligari (1995). Additional experimentalresults are reviewed by Hallauer (1981) and Hallauer et al. (1988).

278 CHAPTER 8

Between-family Variance Under Replication

The expected response to between family selection under replication follows fromEquation 8.14a using the appropriate between-family variance σ2( z ) given thereplication design.

In the simplest case, only a single macroenvironment (such as a growingregion) is considered, and the family is replicated by raising n sibs in each of pseparate plots (for a total ofN = np sibs per family). Here the total environmentalvalue can be partitioned as E = Ec + Ep + Ew(p), a common-family effect (Ec),a plot-specific effect (Ep), and individual within-plot effects (Ew(p)). Followingsimilar logic to that in Example 2, the resulting variance is

σ2( z ) = σ2F +

σ2Ep

p+σ2w

pn(8.38a)

where σ2Ep

is the plot-to-plot variance (the environmental variance between plotsin the same macroenvironment), σ2

F = σ2GF + σ2

Ecis the between-family variance

and σ2w = σ2

Gw+σ2Ew(p)

is the within-plot variance of individuals from their familyaverages. Noting that σ2

G = σ2GF + σ2

Gw, we can express Equation 8.38a as

σ2( z ) = σ2GF

(1− 1

N

)+σ2Gw

N+ σ2

Ec +σ2Ep

p+σ2Ew(p)

N(8.38b)

showing that while the weighting on the genetic components (σ2GF , σ

2Gw ) depends

only on the total number of sibs N , weighting of the environmental componentsis a function of the particular experimental design (the values of p and n = N/pused).

More generally, if the family is replicated (in p plots of size n) over e distinctmacroenvironments for a total of N = pne individuals,

σ2( z ) = σ2GF

(1− 1

N

)+σ2Gw

N+ σ2

Ec +σ2F×Ee

+σ2p

e p+σ2Ew(p)

N(8.39a)

where σ2GF is the genetic variance among family means over this set of environ-

ments, and σ2F×E is the variance from the family-environment interaction (LW

Chapter 22). Note that if we measured the family in only a single environment,σ2GF also includes a σ2

GF×E term. Plant breeders often use an alternative partitionof the environment into location L and year Y effects. If a family is replicated in` locations over y years, the resulting family variance is

σ2( z ) = σ2F +

σ2F×L`

+σ2F×Yy

+σ2F×L×Y`y

+σ2Ep

`yp+

σ2w

`ypn(8.39b)

where σ2F×L, σ2

F×Y , and σ2F×L×Y are the family by environment (year, location,

and year-location) interactions (Lonnquist 1964, Nyquist 1991).


Equation 8.39 shows the importance of replication and provides some guid-ance as to how one should allocate resources — for a fixed number of sibs perfamily (N ), how should one chose e, p, and n to minimize σ2( z )? With N fixed,the relative weighting on the within-family genetic variance, total genetic, andwithin-plot individual environmental variance are fixed. When the genotype-environment interaction variance (σ2

GF×E) is large, its effect on the selection re-sponse can be reduced by replication of families across more environments (in-creasingly e). When the between-plot variance (σ2

Ep) is large, its effect is reduced

by increasing p. With preliminary estimates of the variance components in hand,one can numerically search for the optimal values of e, p, and n that give thesmallest σ2( z ) for a fixed value of N . Using replication can result in a consider-able improvement over mass selection. For example, using variance componentsestimated for maize strains grown in several locations in Indian, Sanghi (1983)estimated that full-sib selection with replication would be three to six times moreefficient than mass selection.

One consequence of replication is that the between-family heritability, h2b =

θ σ2A/σ

2( z ) is now a complex function of the design (i.e., eandp in additional to thetotal number of sibs). Thus with replication, a between-family heritability does notsimply translate into an individual heritability (Hanson 1963, Nyquist 1991). Evenwith the same variance components, the heritability changes as a function of thereplication design. Hanson suggests that the between-family heritability whenreplication is present needs to be defined with respect to a particular standarddesign, such as two locations each with two replications (e = r = 2) for soybeans.

Finally, consider the between-family variance under a nested sib design withreplication. Suppose (as before) that there are f females per male, but now thateach full-sib family is replicated as n sibs over e environments. The resultingvariance becomes

σ2( z ) = σ2GF (HS)+

σ2G(f/m)

f+σ2GF (HS)×E

e+σ2G(f/m)×Efe

+σ2Gw(FS) + σ2

e

N(8.40a)

where N = fer is the total number of half-sibs per male (Robertson et al. 1955,Webel and Lonnquist 1967, da Silva and Lonnquist 1968). Assuming no epistasis,we can express this between-family variance as

σ2A

4+σ2A + σ2

D

4 f+σ2A×E4 e

+σ2A×E + σ2

D×E4 fe

+(1/2)σ2

A + (3/4)σ2D + σ2

e

N(8.40b)

Example 7. Eberhart et al. (1966) estimated genetic variance components forseven characters in two different open-pollinated maize varieties. Using individ-uals grown in two locations in North Carolina (which serve as the two different

280 CHAPTER 8

environments), they obtained the following estimates for yield in the variety“Jarvis”:

σ2A = 120, σ2

A×L = 114, σ2D = 270, σ2

D×L = 98, σ2e = 508

Estimates of epistatic variances were not significantly different from zero. Con-sider the expected response under a design with 25 half-sib families, each with atotal of 50 offspring scored over five environments (e = 5). The top five familiesare selected, using S1 seed to from the next generation (allowing for selection onboth sexes). The expected response is

R =2 ı25,5 (σ2

A/4)σ( z )

=2 · 1.345 · 30

σ( z )=

80.7σ( z )

Using the above variance estimates, Equation 8.26 gives σ2GF = σ2

A/4 = 30,σ2Gw = (3/4)σ2

A + σ2D = 360, and σ2

GF×E = σ2A×L/4 = 28.5. If the families

being scored are strict half-sibs (all offspring from a pollen parent each have adifferent seed parent, f = N = 50), Equation 8.39a gives

σ2( zHS ) = σ2GF

(1− 1

N

)+σ2Gw + σ2

e

N+σ2GF×Le

= 30(

1− 150

)+

360 + 50850

+28.5

5= 52.46

The expected response is thus 80.7/√

52.46 = 11.14.

Now suppose that the sibs are from a nested design with each male pollinatingfive seedplants, with each cross producing 10 offspring (f = 5, N = 50). Usingthe above variance components, Equation 8.40b gives σ2( z ) as

σ2A

4+σ2A + σ2

D

4 f+σ2A×L4 e

+σ2A×L + σ2

D×L4 fe

+(1/2)σ2

A + (3/4)σ2D + σ2

e

N

=1204

+120 + 270

20+

11420

+114 + 98

100+

(1/2)120 + (3/4)270 + 50850

= 72.73

giving the expected response as 80.7/√

72.73 = 9.47. The strict half-sib designhas a smaller between-family variance, and hence a larger expected response,being 118% of that expected under a nested design.

Another example of family selection with replication is provided by selectionfor increased yield in maize by the International Maize and Wheat ImprovementCenter (CIMMYT), summarized by Pandey et al. (1986, 1987) and Crossa and


Gardner (1989). The goal of the CIMMYT selection schemes was to develop va-rieties of maize that perform well for yield over a wide range of environments.Starting in 1974, 250 full sib families along with six local checks were evaluated atsix lowland tropical locations (with two replications per location) in the northernand southern hemispheres. A total of 28 countries were used during the course offive cycles of selection. Selection (initially) was strictly between-families with theinternational field trails conducted on full-sib families, while the recombinationunit consisted of S1 seed from the superior families. Selection was later modifiedto allow for within-family selection as well. Roughly 50% of the families wereselected based on the international trails, about 20% of which were subsequentlyrejected given their poor performance in disease and insect resistance trials inseparate nurseries. The average gain in yield per cycle was around 2%.

Ear-to-Row Selection

One of the earliest examples of family-based selection in plants is ear-to-rowselection in maize, first used by Hopkins (1899) to start his classic long-termselection experiments (Chapter 11). Here the seeds from each maize ear are plantedin a single row, with individuals from the best rows chosen as seed parents for thenext generation. Plants within rows being scored are detassled, removing theirability to produce pollen. As a result, these plants can neither self nor pollinateother members of their (or any other) row. Pollen is provided by rows plantedwith bulk of all seeds (a polycross mating design). Assuming open pollination,the seeds on a single ear are half-sibs (with a common mother), so that the ear-to-row method is an example of half-sib family selection, with selection on only onesex (the seed parent). Suppose a total of N = e p n sibs per family are scored, bygrowing p rows of n sibs replicated over e distinct environments. The expectedresponse when choosing the top K of M families is

RER = ı(M,K)(1 + 3/N) (σ2

A/8)σ( zHS)

' ı(M,K)σ2A/8

σ( zHS)(8.41)

where σ2( zHS ) is given by Equation 8.39. For largeN (in the absence of epistasis)we have

RER = ı(M,K)σ2A/8√

σ2A

2+σ2F×Ee

+σ2Ep

e p+σ2Ew(p)

e p n

(8.42)

Modified Ear-to-Row Selection

The ear-to-row method has the advantages of being fairly easily to implementand of testing a family (and hence controlling the environmental variance), cou-pled with the same cycle time as mass selection (one generation). As a result,

E1 E2 E3 E4E1 E2 E3 E4 E1 E2 E3 E4

E1 E2 E3 E4

Field trials to select best familiesEach family replicated over several environments

Best individuals within the families selectedfrom field trails are used for recombination

282 CHAPTER 8

this method was commonly used by early maize breeders, e.g., Hopkins (1899),Smith (1908, 1909), Montgomery (1909), Williams and Walton (1915), Kiesselback(1916), and Hume (1919). While it proved effective at modifying highly heritabletraits (such as kernel protein and oil content), ear-to-row selection was generallynot successful in improving yield (Kiesselbach 1922, Richey 1922, Smith and Bru-son 1925), and it was not regarded as a practical scheme for yield improvement.Spragne (1955) suggested that the failure for yield improvement was largely theresult of insufficient control over environmental variance, resulting in σ2

E largelyobscuring the additive variance. (For this same reason, mass selection was alsoregarded as being impractical for improving maize yield.) An alternative hypoth-esis was suggested by Hull (1945, 1952), who thought that the lack of responsein yield was a result of most the genetic variance being nonadditive. The findingof considerable additive variance in yield by a number of maize geneticists moti-vated the development by Lonnquist (1964) of the modified ear-to-row selectionscheme, which combines both between-family (ear-to-row) and within-family(within row) selection (Figure 8.6).

Figure 8.6. Lonnquist’s modified ear-to-row selection scheme. Half-sib families(represented here by the maize ears in the middle of the figure) are planted bothas rows in multiple environments (the yield trails here over environments E1 toE4 at the bottom of the figure) and as a single row in yet another location (thecrossing block). For the best families in the yield trails (here families one andfour), one chooses the best individuals from the crossing block (indicated by thecircled plants) to form the next generation.


Under Lonnquist’s design, seed from each family is planted as rows in sev-eral environments (usually with replications within each environment). Theseform the yield or performance trails for selecting the best-performing familiesaveraged over these environments. On a separate plot (the crossing block), seedfor each family is also planted as a single row. Again, each row is detassled andpollen is provided from rows consisting of a bulk of all families. Within the cross-ing block, the best individuals from the rows corresponding to the families withthe best performance in the yield trails are used as the seed parents for the nextgeneration. One advantage of this scheme is that one can use bulk measures overrows in the yield trails and more detailed (and labor-intensive) individual plantmeasures in the smaller crossing block.

Under Lonnquist’s original design, the replicated field trials and the cross-ing block are grown contemporaneously (planting of the crossing block may bedelayed slightly to ensure that all field information from the yield trials can begathered). Thus, one cycle of modified ear-to-row can be carried out in a singlegeneration. Plants in the crossing block are open pollinated from a random bulkof all the initially planted families. The expected total response is the sum of theexpected gains at each step in the cycle, RER(m) = RER + RER(w). The responseunder the first step (choosing the best families) is the same as for ear-to-row (Equa-tions 8.41-8.42). Since plants in the crossing block are open pollinated using a bulkof all families, selection is only on females within each row. If one chooses the bestk ofm plants within each selected row, Equation 8.20 gives the expected responseto within-row selection as

RER(w) = ı(m,k)(3/8)σ2

A

σw(HS)(8.43)

Since families are not replicated within the crossing block, we have

σ2w(HS) = σ2

GW (HS) + σ2Ew(p)

so that (in the absence of epistasis), the component of response from within-rowselection becomes

RER(w) = ı(m,k)(3/8)σ2

A√(3/4)σ2

A + σ2D + σ2

Ew(p)

(8.44)

Ignoring any potential changes in σ2A due to the first step of selection, the expected

response becomes

RER(m) = RER +RER(w)

= ı(M,K)σ2A/8

σ( zHS)+ ı(m,k)

(3/8)σ2A

σw(HS)(8.45a)

284 CHAPTER 8

where we have chosen the best K of M families in the yield trails and the bestk of m within each selected family in the crossing block. With a large number ofsibs/row (n large) and a roughly equal selection within- and between rows, theexpected response to modified ear-to-row selection is

RER(m) =ı σ2

A/8√σ2GF +

σ2F×Ee

+σ2Ep

e p+σ2Ew(p)

e p n

+ı (3/8)σ2

A√σ2Gw(HS) + σ2

Ew(p)

(8.45b)

Inspection of Equation 8.45b shows that it is not obvious which component(within- vs. between-family) contributes more to the total response. The three-hold increase in usable additive variance in the within-family component can bepartly or fully offset by the fact that σ2

Gw > σ2GF (the within-family genetic vari-

ance is greater than the between-family variance, see Equations 8.26a,b). Like-wise, it is not clear whether the between- or the within-family environmentalvariance is expected to be larger. Some fine-tuning is possible on the between-family component, as if estimates of the appropriate environmental variances areavailable, changing the experimental design (the values of p, n, and e) can reducethe between-family variance.

Example 8. Webel and Lonnquist (1967) used modified ear-to-row selection foryield in the “Hays Golden” variety of US maize. Performance of each family wasevaluated using single rows grown in three different locations. Based on theseyield trails, the best 44 of roughly 220 families were identified. In the crossingblock the best five of the 25 (or so) plants are chosen in each of the 44 rows corre-sponding to the selected families. The resulting expected selection intensities forthe between- and within-family components are ı220,44 = 1.40 and ı25,5 = 1.35,respectively. Over the first four cycles of selection, Webel and Lonnquist observeda 9.4 % increase in yield per cycle, compared with the 3% per cycle observed undermass selection (Gardner 1973). The predicted response was 8.4%, with expectedcontributions of 4.6% from among-families (54% of predicted response) and 3.8%from within-families. The results for ten cycles of selection are summarized byCompton and Bahadur (1977).

Paterniani (1967) also used modified ear-to-row for three cycles of selection foryield in Brazilian maize populations. The average yield increased by 42% overthe course of the experiment.

Compton and Comstock (1976) suggested a variant of Lonnquist’s design.Families are again planted ear-to-row in replicated performance trails, but rem-nant seed from each family is stored. The best families are chosen and the remnant


seed for these families is planted to form the crossing block. The pollen plants inthe crossing block are a bulk of the selected families. Hence, both parents in thecrossing block were subjected to half-sib selection, doubling the response fromthe between-family component, giving

RER(m) = ı(M,K)σ2A/4

σ( zHS)+ ı(m,k)

(3/8)σ2A

σW (HS)(8.46)

The Compton-Comstock modified ear-to-row scheme requires two generationsper cycle, but offers increased response (per cycle) as the pollen is also fromselected parents. Using the predicted values of Webel and Lonnquist (Example8), the expected response per cycle under the Compton-Comstock design wouldbe 2· 4.6 + 3.8 = 13, for an expected 155% increase per cycle over the Lonnquistdesign (which had a predicted response 8.4). However, the Compton-Comstockdesign also requires two generations per cycle, so that the response per generationis 6.5, 77% of that expected under the Lonnquist design.

SELECTION ON A FAMILY INDEX

While our focus to this point has been on schemes that use either within- orbetween-family selection, the modified ear-to-row approach points out the ad-vantage of using selection schemes containing both within- and between-familycomponents. Approaches incorporating both components are referred to as com-bined selection. Modified ear-to-row is an example of combined selection wherethe components are sequentially selected in different generations (and/or plots),and several such schemes are used by plant breeders. Alternatively, one can useboth within- and between-family information to select individuals within a singlegeneration. The most general way to do this is to select on a family index,

I = b1 (zij − zi) + b2 zi (8.47a)

where the index value I is for a individual j from family i. Individuals with thelargest index scores are mated (avoiding within-family crosses) to form the nextgeneration. Recall that individual (I = zij), family (I = zi) and family-deviation(I = zij − zi) selections are all special cases of this general family index. Thefamily index is often written as

I = zij +B zi (8.47b)

where B is the relative weight on family mean compared to an individual’s phe-notype. As the reader can easily verify with a little algebra, this is equivalent tothe index given by Equation 8.47a, with

B =b2b1− 1 (8.47c)

286 CHAPTER 8

Response to Selection on a Family Index

One again, Equations 8.1a and 8.4a can be used to predict the short-term (single-generation) response to selection. Taking x = I gives

RI =σ(I, y |R1)

σ2I

(SIm + SIf

)= ıI σz ρ(I, y) (8.48)

The variances and covariances in Equation 8.48 are obtained as follows. Using thecovariances summarized in Table 8.3,

σ(I, y |R1) = b1 σ(zij − zi, y |R1 = zij) + b2 σ( zi, y |R1 = zij)= b1(1− rn)(σ2

A/2) + b2rn(σ2A/2)

= [ b1 + rn(b2 − b1) ] (σ2A/2) (8.49a)

Likewise, recalling that σ2(x + y) = σ2x + σ2

y + 2σx,y (LW Equation 3.11a), thevariances summarized in Table 8.4 give

σ2(I) = b21 σ2(zij − zi) + b22 σ

2(zi) + 2b1 b2 σ(zij − zi, zi)= b21 (1− tn)σ2

z + b22 tn σ2z + 2b1 b2 σ(zij , zi)− 2b1 b2 σ2(zi)

=(b21 (1− tn) + b22 tn + 2b1 b2 tn − 2b1 b2 tn

)σ2z

=[b21 + tn(b22 − b21)

]σ2z (8.49b)

The heritability of the index thus becomes

h2I =

2σ(I, y |R1)σ2(I)

= h2

[b1 + rn(b2 − b1)b21 + tn(b22 − b21)

](8.50a)

Finally, the correlation between an individual’s index score (I) and the value ofits offspring (y) is

ρ(I, y) =2σ(I, y |R1)σ(I)σ(y)

= h2

[b1 + rn(b2 − b1)√b21 + tn(b22 − b21)

](8.50b)

The term in the brackets represents the accuracy of the index relative to mass se-lection. Substituting Equation 8.50b into Equation 8.4a (and recalling that σzh2 =hσA) gives the expected response as

RI = ıI hσAb1 + rn(b2 − b1)√b21 + tn(b22 − b21)

(8.50c)


Example 9. Again consider the selection experiment of Clayton et al. (1957) onabdominal bristle number in Drosophila (Examples 3, 4). Here rn = 0.542, tn =0.326, andσA h = 1.70. What is the expected response if we place three times theweight on within-family deviations as we do on family means (b1 = 3, b2 = 1)?Suppose individuals with index scores in the upper twenty percent are chosen.Since 20 families each with 12 sibs are scored, the expected selection intensity isı240,48 = 1.39 (Example 5), and Equation 8.50c gives the expected response as

RI = 1.39 · 1.70

(3 + 0.542(1− 3)√32 + 0.326(12 − 32)

)= 1.79

This is not as efficient as between-family selection (whereRb = 2.15, see Example3). Likewise, the response under individual (mass) selection isRm = ım σA h =2.36. Since individual selection is a special case of the general index, we canalways chose the index weights to give at least as large an expected response asindividual selection. For example, placing twice the weight on family means as isplaced on within-family deviation (b1 = 1, b2 = 2), gives an expected responseof R = 2.59, 110% of the expected response under individual selection.

Lush’s Optimal Index

As the previous example shows, by the appropriate choice of index weighs wecan always obtain a response at least as larger as under mass selection. Notefrom Equation 8.48 that σA and ıI remain constant under different index weights,so that the maximal response occurs by choosing the weights that maximize tocorrelation ρ(I, y) between the index and offspring value (Equation 8.5 shows thatthis is equivalent to maximizing the correlation ρ(I, A) between the index andbreeding values of an individual). Lush (1947) showed that the correspondingindex weights are

b1 =1− r1− t , b2 =

1 + (n− 1) r1 + (n− 1) t

(8.51)

The formal derivation is given in Chapter 22. We refer to the family index usingthese weights as the Lush Index. Note that the weight b1 on family deviations isindependent the family size n, while the weight on the family mean b2 dependson n, approaching r/t for large families. Figure 8.7 plots the ratio of between- towithin-family weights (b2/b1) for the Lush index as a function of t and n. For asmall between-sib correlation, more weight is placed on family mean, while moreweight is placed on within-family deviation when the sib correlation is large. Wecan rearrange the Lush index as IL = zij +BLzi, where Equation 8.47c gives

BL =(r − t)n

(1− r)[ 1 + (n− 1)t ](8.52)

1.00.90.80.70.60.50.40.30.20.10.00

1

2

3

4

5

6

7


Rat

io o

f In

dex

wei

ghts

, b2/

b1

n = 2

n = 5

n = 10

n = 20

Half Sibs

1.00.90.80.70.60.50.40.30.20.10.00

2

4

6

8

10

12

14

16

18

20


Rat

io o

r in

dex

wei

ghts

, b2/

b1

n = 2

n = 5

n = 10

n = 20

Full sibs

288 CHAPTER 8

Figure 8.7. Weights for the Lush index (the family index weights for the optimalsingle-generation response) as a function of the phenotypic correlation t betweensibs and number n of sibs.

Using the optimal weights, Equation 8.48 simplifies to give the responseunder Lush’s index as

RLI = ı σA h

√1 +

(r − t)2(n− 1)(1− t)(1 + [n− 1]t)

(8.53a)

The increase is response over that expected under individual selection is thus

RLIRm

=

√1 +

(r − t)2(n− 1)(1− t)(1 + [n− 1]t)

(8.53b)

Figure 8.8 plots this ratio as a function of t and n for half- and full-sibs. Since thequantity in the square root exceeds one, the expected response under Lush’s indexexceeds the response under individual selection, except at r = t (i.e., t = 0.25 forhalf-sibs, t = 0.5 for full-sibs) in which case the expected responses are equal.

1.00.90.80.70.60.50.40.30.20.10.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

n = 2n = 5n = 10n = 20


Res

pons

e re

lati

ve t

o M

ass

Sele

ctio

n Half-Sibs

1.00.90.80.70.60.50.40.30.20.10.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

n = 2n = 5n = 10n = 20


Res

pons

e re

lati

ve t

o M

ass

Sele

ctio

n Full Sibs


Figure 8.8. Response of Lush’s index relative to individual selection. Except atr = t (where the expected response equals that of individual selection), Lush’sindex gives a larger expected response than individual selection.

Example 10. Recalling (Example 3) that t = 0.265 and r = 0.5 for full-sibs inClayton et al.’s bristle experiments, the Lush weight on family deviations is

b1 =1− r1− t =

1− 0.51− 0.265

= 0.680

Further recalling that the family size was n = 12, the Lush weight on familymeans becomes

b2 =1 + (n− 1) r1 + (n− 1) t

=1 + (12− 1) 0.5

1 + (12− 1) 0.265= 1.66

We can rescale the weights as b1 = 1 and b2 = 1.66/0.680 = 2.44, giving aresulting response of

RLI = ıLI σA h ρ(I, y) = 1.39 · 1.70 ·(

1 + 0.542(2.44− 1)√12 + 0.326(2.442 − 12)

)= 2.60

290 CHAPTER 8

The expected response under individual selection is Rm = 2.36 (Example 9), sothat the expected response under the Lush index is 110% that of mass selection.

The Lush index weights hold for a single generation, and potentially needto be adjusted each generation as t and r change. Both drift and gametic-phasedisequilibrium can be important when several generations of selection are consid-ered. As selection proceeds both these forces increase the importance of within-versus between-family selection (Chapters 5, 12, 14), so that individual valuebecomes weighted more and family mean less. Wray and Hill (1989) note thatwhile the relative efficiency of combined selection over individual selection maybe greatly diminished by gametic-phase disequilibrium, the relative rankings ofthe methods still hold.

A second concern is that, as with any index, population parameters have to becorrectly estimated else the index constructed from these estimates has incorrectweights and is less than optimal. For the Lush index only the intraclass correlationt must be estimated, and Sales and Hill (1976a) have shown that the efficiency ofcombined selection is quite robust to estimation errors in t (as initially suggestedby Lush 1947).

Based on these concerns, it is not surprising that experimental verificationof the advantage of the Lush index over individual or family selection is mixed.McBride and Robertson (1963) and Avalos and Hill (1981) found that index se-lection gave a larger response than individual selection for abdominal bristles inDrosophila melanogaster. More conclusive results for selection on the same charac-ter were those of James (cited in Frankham 1982), who found that the observedincrease in response under index selection (relative to mass selection) was 133 ±9.7% and 111 ± 7% in two replicates, very consistent with the expected increaseof 121%. Results for selection for egg production in poultry were less conclusive,with Kinney et al. (1970) finding that individual selection gave a larger (but notsignificant) response than index selection, while Garwood and Lowe (1981) foundthat index selection gave a larger response (again not significant) that family se-lection. Larval and pupal weight in Tribolium showed similar mixed results, withWilson (1971) finding that individual selection gave the largest response, whileCampo and Tagarro (1977) did not find any significant differences (index selec-tion gave a larger response in a replicate with large family size, while individualselection showed the larger response in a replicate with small family size).

We note in passing that more a general family index was considered byOsborne (1957b, c) for the nested sib design which separately weights informationfrom full- and half-sib families. If zijk denotes the kth full sib from dam j and sirei, an index weighting both half- and full-sib information is

I = b1 (zijk − zij) + b2 ( zij − zi·) + b3 zi·

where b1 is the weight on within full-sib family deivations, b2 the weight on


deviations among dams within a sire, and b3 the sire weights (half-sib means).Chapter 22 examines this (and other indices) in some detail.

Correcting the Selection Intensity for Correlated Variables

As mentioned previously, expressions for the selection intensity in finite popu-lations make the assumption that the order statistics are uncorrelated. Selectionof multiple individuals from the same family results in correlations among theorder statistics due to the correlation between sibs. Our treatment follows that ofHill (1976, 1977c).

Suppose the population from which individuals are drawn consists of mfamilies each with n sibs, for a total of N = mn individuals. If phenotypic valuesare uncorrelated between all members of the sample (the sib correlation t is zero),Burrow’s correction (Equation 4.17c) gives the finite-population size adjustedselection intensity as

ıN,K = ıp −1− p

2 ıp p(N + 1)

where a fraction p = K/N of the population is saved and ıp is the infinite-population selection intensity associated with the fraction p saved. When somemembers are correlated, the net result is to reduce the effective number of inde-pendent variables, so that the correct value is somewhat below N . This rangesfrom mn = N with no correlation between sibs (t = 0) to m with a perfect cor-relation between sibs (t = 1). Using this observation, Hill (1976a) suggested thelinear approximation for the effective number Ne of independent variables as

Ne = N(1− t) +mt (8.54a)

Substituting into Burrow’s correction gives the expected selection intensity ad-justed for correlations as approximately

ıN,K(t) = ıp −1− p

2 ıp p[N(1− t) +mt+ 1 ](8.54b)

Simulation studies by Hill shows that this is a reasonable approximation, andHill (1976a) provides tables of exact values (over a limited set of n and t values).Note that ı decreases as t increases, being most extreme when only a few familiesare chosen (here Ne approaches m). An alternative approximation is offered byRawlings (1976), while Tong (1982) considers unequal family size.

The effect of sib-correlations on the selection intensity for individual selectionis generally small, as t is typically less that 0.5, which has only a modest effecton reducing ı. In contrast, the presence of the family mean zi in the index scoresgreatly inflates the correlation between the sib index values over the correlationamong phenotypic values. Hill (1976a) showed that if selection occurs on theindex I = zij + B zi (Equation 8.47b), the intra-class correlation T among the

292 CHAPTER 8

index values of sibs is given by

T = 1− n (1− t)n+B(2 +B)[ 1 + (n− 1)t ]

(8.55a)

where t is the intra-class correlation of individual phenotypic values among sibs.Note that for large B, T approaches one. Hence, for schemes that place consider-able weight on family means, sibs within are almost perfectly corrected (in termsof being chosen by selection), and the effective number of independent orderstatistics approaches the number of families m chosen. This is very reasonable,as I approaches zi for large B, which is simply between-family selection. In thiscase the number of independent order statistics is simply the number of families.

Using the value of B (from Equation 8.52 ) under Lush index weights, Hill(1976a) showed (for large n) that

TLush '

1− t full-sibs

1− t1 + 8t

' 11 + 2h2

half-sibs(8.55b)

Example 11. Once again, consider Clayton et al’s experiment on Drospholiabristle number. From Example 10, the Lush index weights are b2/b1 = 2.44,giving B = 1.44. Recalling that t = 0.265 and n = 12, Equation 8.55 gives thecorrelation T among index values of sibs as

T = 1− 12 (1− 0.265)12 + 1.44(2 + 1.44)[ 1 + (12− 1)0.265 ]

= 0.72

The correlation among index values of sibs is thus 2.7 times the correlation amongsib phenotypic values. Note that under strict family selection (T = 1), the corre-lation among the index value increases to 3.8 times the sib phenotypic correlation.Suppose we select on a family index using four families (m = 4). The resultingtotal number of individuals becomesN = 12·4 = 48, while the effective numberof independent variables is

Ne = 48(1− 0.72) + 4 · 0.72 = 16.3

Thus the effective number is only 34% of the actual number of total individuals.Here p = 0.2, with ıp = 1.40, giving the corrected selection intensity as

ı = 1.40− 1− 0.22 · 1.40 · 0.2 · (16.3 + 1)

= 1.32

family-based selectionnitro.biosci.arizona.edu/zdownload/volume2/chapter08.pdf · chapters 21-25...

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