Selection Response in Finite Populations

Selection Responsein Finite Populations

Ming Wei, Armando Caballero and William G. Hill

Institute of Cell, Animal and Population Biology, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JT, United Kingdom Manuscript received November 28, 1995 Accepted for publication September 5, 1996

ABSTRACT Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Accountwas taken of genetic drift, gametic (linkage) disequilibrium (Bulmereffect), inbreeding depression, common environmental variance, and both initial segregating variance within families (a;wo)and mutational (&) variance. The cumulative response to selection until generation t(C8) can be approximated as

where N, is the effective population size, = Ne& is the genetic variancewithin families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. & is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. P is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, ie., their relative rate of change. & is the major determinant of the short term selection response, but aL, Ne and p are also important for the long term. A selection method of high accuracy using family information gives a small N, and will lead to a larger- response in the short term and a smaller response in the long term, utilizing mutation less efficiently.

ENETIC response to one cycle of selection for a 1975) using all information available are formally the G quantitative character is solely a function of selec- methods to achieve the maximum response for one tion accuracy (the correlation between the selection generation of selection. Assuming an infinite popula- criterion and breeding values of individuals), selection tion, WRAYand HILL (1989) computed the asymptotic intensity and additive genetic variance in the popula- response (the constant rate of response) after account- tion (FALCONERand MACKAY 1996). However, for long ing for the Bulmer effect for various index selection term response to selection in a finite population, other schemes. DEKKERS(1992) and VILLANUEVAet aZ. (1993) factors such as genetic drift, gametic linkage disequilib extended the theory to BLUP selection. These works, rium, mutationalvariance and effective population size however, did not consider the reduction in selection are all variables depending on character, population response due to the finite size of the populations (HILL structure and selection strategy, which should be incor- 1985), which can be accounted for by means of the porated to predict thecumulative response to selection. effective population size. In populations under selec- While long term response is also a function of the distri- tion, the effective size can be predicted depending on bution of the effects and frequencies of individual loci, selection methodsand mating schemes. ROBERTSON such information is not available, so it is necessary to (1961), WRAYand THOMPSON(1990), WOOLLIAMSet aZ. consider only known, albeit restricted, parameters. (1993), and SANTIAGOand CABALLERO (1995) devel- ROBERTSON(1960) developed the theory of selection oped methods to predict the rate of inbreeding when limits under mass selection. He did not consider, how- a population is undergoing mass selection. WRAYet al. ever, that additive variance is reduced by selection due (1994) extended the theory to handle the situation of to gametic linkage disequilibrium, the “Bulmer effect” index selection. For a review of the principles of the (PEARSON 1903; BULMER1971). To predict long term above methods see CABALLERO (1994). response, the effects of gametic linkage disequilibrium Selection intensity is also a factor that depends on and genetic drift on additive variance need to be consid- population size. For the same proportion selected, the ered simultaneously. selection intensity is reduced when there are a small A selection index (LUSH1947) and animal model number of families and by the increased correlation best linear unbiased prediction (BLUP) (HENDERSON between individuals’estimated breeding values, particu- larlywith selection using family information (HILL Curresponding authur: William G. Hill, Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd., Edin- 1976; RAWLINGS 1976; MEUWISSEN1990). burgh EH9 3JT, United Kingdom. E-mail: [email protected] Different methods of selection (mass selection, index

Genetics 144: 1961-1974 (December, 1996) 1962 M. Wei, A. Caballero and W. G. Hill selection, etc.) differ in the accuracywith which individ- omitted for simplicity if a formula or parameter applies ual breeding values are estimated. A high selection ac- for any generation. curacy gives a high response in the short term but be- In this study, mass selection refers to truncation selec- cause of coselection of family members it also gives a tion on individual phenotypic values. Family selection high rate of inbreeding, so that it reduces the selection is defined as the selection of the best families based on response in thelong run (ROBERTSON1960, 1961). phenotypic family means. Within-family selection is the Thus methodsachieving maximum short term response selection of the best progeny within each family based do not necessarily optimize long term response. Some on individual phenotype. Index selection refers to the simulation studies of long term selection response have selection of individuals based on an indexof individual, beenconducted (e.&, BELONSKYand KENNEDY 1988; full-sib and/or half-sibfamily information, optimally VEWER et al. 1993) that give some insight into the weighted every generation (FALCONERand MACKAY comparison of short and longterm selection response. 1996). BLUP selection is carried out by the selection Spontaneous mutation has been found to be an im- of individuals with highest breeding values estimated portant source of new variation (CLAETONand ROBERT- by the “pseudo-animal model BLUP”, ie., an index of SON 1955; FRANKHAM1980; LYNCH1988). HILL (1982) individual, full- and half-sib information plus estimated developed the theory for genetic response from new breeding values of the dam, sire and mates of the sire mutations, but the model including both initial segre- (WRAYand HILL 1989).Pseudo-BLUP is almost identi- gating and mutational variances to predict selection re- cal to true BLUP when there are no fixed effects other sponse has not been developed, except under simple than the overall mean (WRAYand GODDARD1994), and assumptions (HILL1985; KEIGHTLEYand HILL 1992). we will simply call it BLUP henceforth. The aim of this study is to develop the analytical There are equations available to approximate effec- theory to predict cumulative response to selection un- tive population size under theselection schemes under- der various selection schemes (mass, family, within-fam- lined above (WRAYet al. 1994). In this paper, however, ily, index and BLUP selection) taking into account ge- we use values obtained by simulation, for simplicity. For netic drift, gametic linkage disequilibrium, initial and a description of the simulation procedure and examples mutational variances, inbreeding depression, common of predictions using the equation of WRAYet al. (1994), environmental variance and reduced selection intensity see CABALLERO et al. (1996b). The reduced selection due to correlation of family members. The main objec- intensity due to the correlationbetween index of family tive is not to give the most accurate predictions of re- members is calculated by using the formulae of HILL sponse to selection but to incorporate all the factors (1976) and MEUWISSEN(1990) to correct for finite pop- that affect short, medium and long-term response in ulations. relatively simple equations as functions of known pa- Prediction of genetic variance: Based on the infini- rameters, whichallow the effect of such factors and tesimal model, the additive genetic variance at genera- parameter values to be evaluated. Results are discussed tion t (&) can be predicted if gametic linkage disequi- in relation to previous theory and the results of long librium is ignored. Itis then equal to the genic variance term selection experiments. (&), Le., ait = a:, = aio[(l - 1/(2N,)lt = aioe-1’/2N~, where a:o is the initial genetic variance and Ne is the THEORY effective population size. When gametic linkage disequilibrium is accounted for, the prediction of ail is Modeland selectionschemes assumed: A single not straightforward. However, the within-family additive quantitative character is considered that is controlled variance (aiwl)is not affected by the linkage disequilib- by an infinitesimal model of gene effects (ie., the char- rium (BULMER1971) and therefore can still be pre- acter is controlled by genes at many unlinked loci, each dicted, ie., aiw = uiuoe-t/2Ne,so wewill use it as the of small additive effect). A closed and finite population reference point in all predictions. We now add to this with a nested mating structure (full-sib families nested last equation the contributionof mutation. The within- within half-sib families), discrete generations, random family additive variance due to mutations accumulated mating, constant population structure and sizeis as- until generationtis equal to NgL(1 - Lie., half sumed. Each full-sib family is assumed to have a con- the total additive variance, given by CLA~TONand ROB- stant number of individuals (n), n/2 of each sex. At ERTSON (1955) and HILL (1982)l. Thus,the total generation t, the total phenotypic variance (a&) con- within-family genetic variance at generation t is sists of additive genetic variance (&), environmental 2 oAWl2 = aiwm + (oAWO - a~wm)e-t/2~, (1) variance (a:) and common environmental variance between full-sib family members (a:). The residual vari- where ance (a:) is defined as a: = a: + 0%and is assumed to 2 CAW^ = Ne& (2) be constant over generations. The constant input of new genetic variance per generation due to mutation is the steady state within-family additive genetic vari- is assumed to be aL. Subscripts for generationsare ance, regardless of the value of ~WO. Selection Response 1963

Prediction of response: The response to selection at generation t(&) is

R,= ZPptffAt,' (3) Note that, if mutation is ignored (a" = 0),then from (2) aiwm = 0 in (6, a and b). where i is the selection intensity and pt is the accuracy The general equations (Equation 6, a and b) allow of selection at generation t. The cumulative response the cumulative response to selection tobe approxi- to selection until generation t (CR,) is mated for a range of selection methods and parameter values. We now describe how gametic linkage disequi- CR, = R, ipraArdT. = s' (4) librium is specifically accounted for. In an infinite pop r=l 0 ulation, a limiting value of genetic variance is closely R, and CR, can, therefore, beevaluated if pt and aitare approached after the Bulmer effect occurs in a few cy- expressed in terms of the within-family additive variance cles of directional selection. The response (variance) at generation t, which is predicted from (1).Equations predicted with this limiting value was called the asymp 3 and 4 can be tedious to derive in terms of aiw,except totic response (variance) by WRAYand HILL (1989). under simple assumptions, and numerical solutions can In this study, the response (variance) is continuously be obtainedusing a mathematical package (e.g., MAPLE; increased by mutation and erased by drift and, there- see HECK1993). However, a first order Taylor series fore, it does not necessarily asymptote. We shall still approximation can always be used providing t/N, is suf- call it the asymptotic response (variance) for simplicity, ficiently small (say, t/N, < 2), however. The approach that we will follow for the derivation of parameters in generation 0 (p and &) consists of assuming that the population has already experienced similar selection (which is a realistic assumption in breeding programs) so that the initial response (&) will actually be the asymptotic response. Evenif the scheme of preselection is different, the approximation where & = ipoaAo is the selection response at genera- of response would not be significantly influenced be- tion 0 and the derivative cause the asymptotic response is closely approached d log R within a few generations. We assume that the variance P= within families at generation zero (aim),which is the d log aiw result of previous selection, mutation and genetic drift, (log indicates natural logarithm) gives the relative rate is known. Then we calculate aioas the asymptotic vari- of change inresponse (R)and aiwNo subscript gener- ance assuming thatthe initial genetic variance was ation is given in the derivative, because it can be evalu- 2aio. Analogously at any generation t we predict the ated at any generation, but here we evaluate it using genetic variance within families (aiwt)accounting for the genetic parameters at the reference generation,i.e., mutation and drift using (1), and aitas the asymptotic generation zero. Substituting (1) into theabove expres- variance assuming that the initial genetic variance was sion, 2aiW. Explicit formulae to calculate the asymptotic variance are derived in APPENDICES A.1 (Equation Al) and A.2 (Equation A3) for mass and family selection, respectively. Those for index and BLUP were obtained or approximately numerically. Thus, the genetic variance at generation t (ai,) is always predicted using as a reference the within-family variance at that generation. The response to selection at generation t is then R, = ZppAt,where both p, and assuming t/N, < 1. The cumulative response to selec- ai,account forinitial asymptotic variance, genetic drift, tion until generation t (CR,) can be obtained by inte- mutation and Bulmer effect for a further t generations. grating functions (5a) or (5b) over generations, To find solutions for the derivative p = d log R/d log o~Wneeded in (6, a and b), this can be expressed as d log R d log a: X [t - 2N,(1 - e-"'y)] , (6a) P=p1P2=dlopdloga~w' I where PI indicates the relative proportional rate of or change of R and 02. This can be derived for mass selec- 1964 M. Wei, A. Caballero and W. G. Hill

tion by setting h2 = &‘(ai + a:) and R = ihoA = 12 i&ap to give p d log R d(bA) ai 1 1- - - 1 - - h2. (7) dlog da; hoA 2 Equation 7 also applies for family (subscript f) and within-family (subscript w) selection, replacing h2 by h;, the heritability of family means, and ?& the heritability of within-family deviations, respectively(see Table 1). p2 reflects the relative rate of change of ai and aiw. Derivations of p2 for mass selection and family selection are given in APPENDICES B.l and B.2 (Equations A4 and A5). Within-familyselection is not influenced by the Bulmer effect sinceit utilizes onlythe within-family addi- = tive variance and, therefore, pw pwl.For index and 0 0.5 1 1.5 2 BLUP selection, derivatives of ,8 are more complicated GenerationlNe (APPENDICES B.3 and B.4). Note that, for BLUP selection FIGURE1.-Cumulative response (in upunits)to mass selec- (subscript B), = because the response including pB Pel, tion predicted by Formulas 6a (---) and 6b (- - -) compared the Bulmer effectis proportional to that without includ- with that from recurrent equations (Equation 8) ( - ). Pa- ing it (DEKKERS1992; see APPENDIX B.4). rameters used were as follows: 20 males, 40 females, six progeny scored per couple, N, = 46, $ = 0.2 and uk = 0. RESULTS The accuracy of selection (ps and pul) was obtained Characteristics of equations and validation: In the with the usual procedure (see FALCONERand MACKAY above section we have derived explicit equations to pre- 1996, pp. 243-244). Response to selection every gener- dict cumulative response in terms of known parameters ation was calculated from (3) using the asymptotic vari- of the population. To check the validity of these equa- ance as the initial value (aio). tions, we can compare them with more exact recur- The cumulative response (C&) predicted by (6, a rence calculations. In these, gametic linkage disequilib- and b) is compared in Figure 1 with that obtained by rium is accounted for (see, e.g., VERRIERet al. 1991) and the recurrentequations (Equation 8). Comparisons are we include mutational variance (a:). Let aist and shown only for mass selection, for simplicity, but the ahntbe the variance between half-sib families and full- results are similar for otherselection methods. Formula sib familieswithin sires at generation t, respectively, 6a fits well until about 2N, generations, after which it and a:, the genic variance at generation t, i.e., the addi- starts to overestimate response; (6b) fits well up to N, tive variance which would result in the absence of selec- generations, after which response starts to be substan- tion. Thus tially underestimated. 2 The role of the different parameters determining cu- a:, = anl-l(l- 1/2Ne) + a:, (8a) mulative response can be seen in Formula 6b. The initial response (6)depends mainly on the accuracy of selection and is the key factor in determining the early response. When t G Ne, the term [p(1 - 0~~Jai~)t~/(4N,)]G t so that CR, = t&. With increasing t, however, the roles of N, and p become more important. where pst-l is the accuracy of selection of sires at gen- The value of p determines therelative rate of change eration t - 1; Ns is the number of sires selected; (1 - in response and variance, i.e., how much the response l/Ns) is used to correct for the finite population size is reduced due to the loss of variance by genetic drift (KEIGHTLEY and HILL1987; VERRIERet al. 1993), assum- and the Bulmer effect. As the response is proportional ing sampling of parents without replacement; ks = is( is to the accuracy of selection (p) (Equation 3), p will be - xS), where is is the selection intensity and xs is the small if the accuracy is fairly insensitive to changes in standardized truncationpoint; and corresponding additive variance (or heritability), and it will be large if quantities apply for dams (subscript D). the accuracy is highly dependent on theheritability. In Note that, in Expressions 8, the mutational variance general, p lies between 0.5 and 1, being smaller with a (a&) initially arises within families(Equation 8b) but it larger p, h2, and/or family size (n).This is shown in is later redistributed among and within families (Equa- Figure 2, A-F, where the values of p and p are plotted tion 8a). In the absence of selection ai, = a:,, and for against h2 for different selection schemes and popula- t -+ a,a:= -P 2NpL and oiwm”* Nea:, as expected. tion parameters. In general, 0 is small for selection Selection Response 1 !Kl

A’

0.8 0.9

h 0.6 0 g 0.8 .-c Ea 0 .-P 0 1 4 0.4 0 0.7

0.2 0.6

0 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Heritability Heritability E1 B1

0.9 0.8

.-$! 0.8 6 0.6 U E .-!2 a 1 0 d 0.7 3 0.4

0.2 0.6

0 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Heritability Heritability c1 0.9

0.8 0.9 0.7

0.6 $! 0.8 m .-U 5 0.5 .-m> 0 1 2 0.4 0 0.7 0.3 0.2 0.6

0.1

0 0.5 4 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Heritability Heritability FIGURE2.-The accuracy (p) (A-C) and derivative, p = d log R/d log azw (D-F), of different selection methods plotted against initial heritability (Ris the initial response and uiw is the within-family additive variance). Parameters used were as follows: 20 males, 20 females. A and D: n = six progeny scored per couple, u: = 0. B and E: n = 24, u:; = 0. C and F: n = 24, u:; = 0.2 u$ Both P and p are evaluated in the initial generation. 1966 M. Wei, A. Caballeroand W. G. Hill schemes using family information (family, index and BLUP selection), particularly when family size is large, because the change of accuracy is insensitive to change in heritability (cf: Figure 2, A and B). For example, in the extreme case of family selection with large family size (Figure 2, B and E), for values of h2 < 0.1, the accuracy is a very steep function of h*, while for h2 > 0.1, it is nearly independent of h2 (Figure 2B). This means that a small reduction in additive variance when h2 < 0.1 has a large impact on the response, i.e., is large; but when h2 > 0.1 a reductionin additive variance has almost no impact on the response, i.e. p is small (see Figure 2E). For large family size (Figure 2E), the value of p under BLUP and index selection is around 0.65 and changes very little with h2 between 0.2 and 0.8. 0 10 20 30 40 For a given heritability, an increase in common envi- Generation ronmental variance (a:) implies a reduction in a: and FIGURE3.-Cumulative response (in apunits) to mass selec- thus an increase in the selection accuracy for within- tion (---) and BLUP selection (-) in case of different muta- family selection, but a decrease for family, index and tional variance: ak = 0 (lines without symbols); a" = 0.001 BLUP selection (cf: Figure 2, B and C). Accordingly, a$ (lines with X); and 0%= 0.01 a$ (lines with A). Parameters the value of p increases with increasing a: for family, used were as follows: 20 males, 40 females, 12 progeny scored index and BLUP selection, but decreases for within- per couple, @ = 0.2. family selection, especially with high h2 ($ Figure 2, E and F). to give a lower asymptotic response than another with Role of mutational variance: The role of mutational larger Ne, even if the first has a higher accuracy of selection for a given value of genetic variance. This is variance (a") in determining the cumulative response because the accuracy will eventually depend on the to selection is clearly shown in Formula 6b. If aiwo> steady-state variance (ab) that is also a function of Ne. aiwm,the term 1 - aiwm/aiwois positive and selection Thus, if Ne is low the accuracy will eventually be low at response will become slower every generation, other- the steady state. Compare, for example, mass selection wise it will accelerate. Estimates of 0%from various ex- (subscript and within-family selection (subscript w). periments (LYNCH1988) are in the range from 0.0001 m) to 0.005 of, depending on the character, species and In this case, (Nm/Nm)1/22 & in general, because N, population, although most estimates come from Dro- = 2N irrespective of selection, while N, is usually sophila. Estimates for body weight in mice range from smaller than N, particularly with high h2 and intense 0.001 to 0.006 a: (CABALLERO et al. 1995). Figure 3 selection (see, e.g., SANTIAGOand CABALLERO 1995). shows cumulative responses to mass and BLUP selection The accuracies at the steady state are (using Equation 6b) for no mutation and two extreme pmw = JYLPL/~(N~PL+ (~NLPL) values of aL (= 0.001 a; and 0.01 a;). Mutation can substantially increase the long term selection response. and The cumulative response to mass selection surpasses pm, = 2N,a~/d(2Nma~+ a:) (2N,aL). that to BLUP selection earlier when mutation is present, because selection schemes with smaller accuracy Thus, again because N, 2 2N,, it follows that pmw/pm, but larger Ne (e.g., mass selection) exploit the muta- 2 l/&. Therefore, substituting these inequalities into tional variation more efficiently than selection schemes (9), ReW/km2 1, and a larger response in the long with larger accuracy but smaller Ne (e.g., BLUP). The term is predicted for within-family selection than mass reason is that the long termresponse will eventually be selection, even though the accuracy for within-family determined by the variance atthe steady state, i.e., selection is about one half that for mass selection, for ai, = 2Np" (cf: Equation 2).Different selection meth- a given genetic variance. ods (say j and k) can be compared in terms of the Short vs. long term selection response for different ratio of long term responses at the limit [R, from (3), selection methods: Inbreeding depression needs to be assuming the same selection intensity], taken into account in long term selection (FALCONER and MACKAY 1996). Assuming a linear relationship be- 1Re = PmjffAmj - pmj&q tween inbreeding coefficient and genetic mean, the to- - (9) hk PmkffAmkPmkJNek' tal reduction of genetic response at generation t is D(1 - e-'lzNe) = Dt/ (2Ne),where D is the inbreedingdepres- where pmis the accuracy of selection at the steady state. sion per unit of inbreeding (in am units). Thus from In general, a selection method with low Ne is predicted Equation 6b, Response 1967 Selection Response

'" I .

0 10 20 30 40 0 10 20 30 40 50 60 Generation Generation FIGURE5.-Cumulative response (in cpunits) to mass selec- FIGURE 4.-Cumulative response (in apunits) to mass selection (---) and BLUP selection (-) when common environ- tion (---) and BLUP selection (-) with @ = 0.2 in case of mental variance (a;) is assumed to be 0 (lines without A) different population sizes and different proportions selected, and 0.1 a; (lines with A). Parameters used were as follows: i.e., NS = 20 males, ND = 40 females, n = six progeny scored 20 males, 40 females, 12 progeny scored per couple, @ = 0.2. per couple (lines without symbols); Ns = 20, ND = 40, n = 12 (lines with X); and = 40, ND = 80, n = 12 (lines with A). are compared in Figure 4 where no inbreeding depression is assumed. The cumulative response to mass selec- t C&=& t-P 1-7 - -D-. (10) tion surpasses that to BLUP earlier with a small popula- [ ( 2;) 2N, tion size or a high selection intensity. If inbreeding depression were considered, mass selection would catch The more general Equation 10 can now be used to up BLUP even earlier, as Ne for BLUP is smaller and, compare differentselection methods. For example, the therefore, it is more affected by inbreeding depression generation when two different selection schemes (jand than is mass selection. k) areexpected to achieve the same cumulative re- The effect of common environmental variance (a:) sponse ( Tmss)is on selection response is illustrated for mass and BLUP selection in Figure 5. With an increase in a;, the effective population size is reduced for mass selection but increased for indexand BLUP selection, for less weight is given to family information (see, e.g., CABALLERO 1994). However, because a common environmental variance substantially reduces the accuracy of BLUP se- /* lection but notthat of massselection for the same value of heritability, cumulative response to mass selection surpasses that to BLUP earlier when common environmental variance is present (Figure 5). 4% 4N* 4% Comparison of equations with previousresults: If defining a = 1 - a:wm/a:wo,for simplicity. the Bulmer effect is ignored, Formulas 3 and 4 can Selection intensity affects thelong term response easily be developed for illustration and as a reference both directly and indirectly through its influence on Ne for comparison to previous results. Let us first consider and theBulmer effect, particularly for intenseselection, mass selection, for which the accuracy of selection at low h2 and index or BLUP (HILL 1985; MEUWISSEN generation tispf = ht,the square rootof the heritability. 1990). For example, with h2 = 0.2,20 sires and 40 dams Ignoring the Bulmer effect, the total genetic variance selected every generation out of 240 progeny, i = 1.296 at generation tis azf= 20;~from (l),and theresponse for aninfinite population. If computed fora finite pop from (3) is ulation, i = 1.283 for mass selection and 1.257 for BLUP selection. The difference between i valueswould be larger with more intense selection. Predicted responses from mass and BLUP selection for different population sizes and selection intensities 1968 M. Wei, A. Caballeroand W. G. Hill

TABLE 1 Parameters used to predict cumulative response to family and within-family selection, which substitutefor the corresponding parameters in Formulas 3-7 for mass selection

Mass selection Within-family selection Family selection

where a& = ai, + af is the steady-state phenotypic (1 - e-'lnNe) + of. The resulting equation differs from variance. that derived by HILL (1982), ie., CR, = 2Nei(aL/ap) The cumulative response to selection until genera- [t - 2N,(1 - e-t/2Ne)],which also does not allow for tion t can be obtained by integrating (12), change in a;. While in the short term (13) agrees well with HILL'Sformula, the difference becomes significant in the long term, especially when aL and/or Ne are large. For example, with a" = 0.001 a& and N, = 100, the difference in predicted cumulative response is only (am - gpl)cJPm -2% at generation 80, but with a value of 0%five times X tanh" 2 apPm - UP- larger, the difference is 3% at generation 20, and 10% at generation 80. where tanh" is the hyperbolic arc tangent function. If A formula to predict cumulative response accounting there is no mutation, aL = aim= 0, and (13) reduces for both initial and mutational variance has not been to formally derived previously. KEIGHTLEY and HILL (1992) used an approximation by adding two indepen- CR, = 4Nei(am- a&). (14) dent terms from the formulae of ROBERTSON(1960) Predictions for the cumulative response to family and and HILL (1982), but this inappropriately applies two within-family selection (ignoring theBulmer effect) can different values of selection accuracy to the same selec- also be obtained from the same formulae (Equations tion process. Equation 13, by contrast, uses a value of 12-14), but replacing Ne, i and other corresponding selection accuracy applicable to the initial and muta- parameters as listed in Table 1. Expressions for within- tional variation together, so if evaluated with aio = 0 it family selection in the table are taken from HILLet al. would not simply be the difference between the results (1996). of Equations 13 (aiof 0, & # 0) and 14 (DL= 0). ROBERTSON(1960) derived a similar formula to predict cumulative response to mass selection without DISCUSSION taking account of the Bulmer effect, ie., CR, = 2Ne& We have derived expressions to predict selection re- (1 - giving a limit of C& = 2Ne& = sponse as a function of known parameters in the popu- 2Neia~0/am.In this formula, however, is assumed a& lation, including a number of factors. These expres- to be a constant over generations, while in our deriva- sions are useful in the assessment ofthe impact of such tion of (14), both ait and are assumed to change, a& factors and parameter values on the selection limits, and only remains constant.Hence apm= a, and a: the comparison of the response with different selection noting that = oPo2 - = - aT), aio a: (apo+ a,) (apo methods, and the interpretation of selection experi- the selection limit from (14) is C& = 4Ne2bio/ ments. We discuss now some of the possible uses of the Thus, the ratio of ROBERTSON'Slimit to the (apo+ a?). equations. limit derived in this study is Selection limits and interpretation of results from long term selection experiments: WEBERand DIGGINS (1990) reviewed responses in long term selection experiments and found that the cumulative response pre- This ratio is near to one for small g, but if this is large, dicted by ROBERTSON'Sformula was always larger than ROBERTSON'Sformula predicts a substantially smaller that achieved after 50 generations of selection at various response (down to one-half for = 1). population sizes. One possible reason for this discrep- If the population is initially isogenic (aio= 0),the ancy would be the fact that ROBERTSON'Sformula ig- cumulative response from mutational variance alone nores theBulmer effect. Generally, the predictedcumu- can be obtained from (13), substituting a& = & lative response from (6a) is -5-10% smaller than Selection Response 1969

TABLE 2 Predicted cumulative response to mass selection (CRJcompared with that achieved in the long term selection experiment of YOO

Predicted C& by different formulae

~~~~ Generations Robertson's, Equation 14, Equation 6a, Equation 6a, selected Achieved no Bulmer no Bulmer Bulmer Bulmer (average) CR, (UL = 0) (UL = 0) (Ut = 0) (UL = 0.002 u;) 76 22.4 24.9 26.0 23.1 26.7 88 28.8 27.6 29.2 28.9 25.8 In YOO'S(1980) experiment, six replicate lines were selected to increase abdominal bristle number in Drosophila (three lines selected for an average of 76 generations, another three lines for 88 generations). Every generation, 50 pairs of parents were selected at an intensity of 20% (Nc= 60). In the base population, = 0.5 and @ = 0.2.

ROBERTSON'Sequation, and would give a better fit to DEMPFLE(1974) comparedthe selection limits to the examples given by WEBERand DIGGINS(1990). within-family (w) and mass selection (m)considering Let us consider the results of YOO'S(1980a) long the effects of genetic drift and Bulmer effect, and con- term selection experiment to illustrate these effects. In cluded that within-family selection achieves a higher Table 2 are shown the observed responses in this experi- selection limit. We can generalize this result as follows. mentand the predictions without accounting for Assume first, for simplicity, that a& = 0, a: = 0 and Bulmer effect and mutation (Equation 14),accounting ignore the Bulmer effect. Using (14), for Bulmer effect but no mutation (Equation 6a with "CL 4Nk,jk(apmo - 0,) o', = O), accounting for Bulmer effect and mutation - (Equation 6awith o', = 0.002a;), and ROBERTSON'S CL 4N,i,(ffm - 0,) (1960) prediction. We note that ROBERTSON'Spredic- 1 - (1 - h2)1/2 tion is slightly smaller than the corresponding predic- = (1 - /&2/2)1/2 - (1 - h2)1/2. tion by (14), as shown by (15). The inclusion of the [z] Bulmer effect (Equation 6a with g', = 0) reduces con- siderably the predictions, while the account for muta- Because N, = 2Nm and, if n is sufficiently large, i, = tion (Equation 6a with o& > 0) increases them again. i,, the above expression reduces to For the first group of three lines selected for 76 or so CL 1 - (1 - h2)1/2 generations, the formula excluding mutation but ac- -= 5 1. CL 2[(1 - h2/2)1/2 - (1 - hy] counting for the Bulmer effect (Equation 6a with o&

= 0) fits bestto the experimentalresults. For the second The ratio has a maximum of 1 for h2 + 0, equals about group of three lines selected for 88 or so generations, 0.9 for h2 = 0.5, and a minimum of about 0.7 for h2 = predictions with and without accounting for bothmuta- 1. If the Bulmer effect were included, the ratio would tion and the Bulmer effect (Equation 6a with a& > 0 be even smaller because this reduces effective size and and Equation 14, respectively) are closest to observa- genetic variance for mass selection, but not for within- tions. Interestingly, mutations of large effect were de- family selection, so that N, 2 2N, in most cases. Analo- tected in two lines of the second group (Yo0 1980b). gously, an increase in common environmental variance However, not too muchemphasis can be given to these (a:) decreases effective size for mass selection but not comparisons, because of the restrictions in the predic- for within-family selection, so CR,,/ CR,, becomes tion model (see below). even smaller with 0% > 0. Finally, the long term re- Based on an infinitesimal model without mutation, sponse from mutation will also be smaller for mass selec- ROBERTSON'S(1960) theory predicts a selection limit of tion than for within-family selection, as was explained 2NeR0.Such a limit is very unlikely to be achieved, but before. even if the infinitesimal model were true, it would be The above argument refers to the theoretical limit to very difficult to be observed experimentally. First, the selection. In a feasible time span, however, within-family limit cannot be reached in a real experiment unless selection willgive smaller responses than mass selec- population size and generation interval are very small. tion, as it utilizes only about half of the total genetic Further, populations of small size are difficult to main- variance. For this reason, the use of such a scheme in tain for many generations because of reproduction and breeding practice is not useful except, perhaps,for fitness problems (FALCONERand MACKAY 1996). Finally, breed conservation purposes. mutational variance cannot be ignored in thelong Comparison of short and long term response for dif- term, and additive genetic variance would never be ex- ferent selection methods: Equation 14 nicely illustrates hausted in theory under the model used here. the antagonistic relation between effective size and se- 1970 M.Wei. A. Caballeroand W. G. Hill lection accuracy in the cumulative response. This latter BLUP evaluation gives lessweight to family information, is a function of two factors, Ne and (gA)- oR).On the reducing the rate of inbreeding more than selection one hand, the magnitude of the value of om - optfor accuracy (GRUNDYet al. 1994). WRAYand GODDARD a given tis mainly determined by the selection accuracy, (1994) and BRISBANEand GIBSON(1995) have proposed which depends on the particular selection scheme: a an iterative selection algorithm to maximize genetic re- scheme of higher accuracy generally results in a larger sponse in a certain time horizon by balancing response reduction of variance and, therefore, in a larger value and inbreeding, i.e., selecting individuals based on their of om - CT~.On the other hand, the larger Ne, the estimated breeding values and relationship to other in- larger is the cumulative response resulting from the dividuals. Thesemethods enable BLUP selection same reduction in variance. Because a selection method schemes to achieve more response for agiven time hori- of higher accuracy gives a smaller Ne but alarger reduc- zon, but mass selection will eventually produce a larger tion of additive variance (i.e., larger om - oR),the two response because of its larger effective size. factors have an antagonistic effect on the response. It Limitations of the study: The infinitesimal model is is then expected that the quicker the genetic variance unrealistic because there can only be a finite number is reduced, theless efficientlythis is exploited to change of loci that control the character and the gene effects thepopulation mean. Therefore, selection schemes may not be all small.Change of variance due to change with higher accuracy achieve less response in the long of gene frequencies as well as dominance, linkage and term, andit seems to beimpossible to manipulateaccu- epistatic effects are not accountedfor under this model. racy and effective size simultaneously to achieve maxi- Thus, an infinitesimal model is likely to be adequate mum cumulative response for both the short and long forpredicting short or medium term selection re- term. sponse, but to become less valid asthe numberof gener- WRAYand HILL (1989) concluded, ignoring genetic ations increases. Approximate formulae given in this drift, that theranking of breeding schemes is not greatly paper apply up to about Ne generations for which the altered when compared by predicted responses after infinitesimal model assumptions may still hold. one-generation rather than asymptotic responses, and Inthe model, a constant environmental variance DEKKERS(1992) showed something similar for BLUP (a:) is assumed, but its pattern of change is actually selection. Even if genetic drift is considered, the pre- unknown. A phenomenon observed from selection ex- dicted response after one generation of selection may periments (e.g., BUNGER and HERRENDOWER1994; be still sufficient to rank breeding schemes in the short HEATHet al. 1995) is that it increases with selection. term (say, up to five generations), because the accuracy However, little is known of the magnitude of this incre- of selection is still the major determinant of response. ment and how it relates to genetic response or the re- Simulation studies on long term selection response duction of additive variance. in finite populations have been conducted by many In this study, mutational variance is assumed to be authors, with conclusions depending on populationsize constant, and all mutants are neutral with respect to and time span. For example, BELONSKYand KENNEDY fitness and of small effect (CLAWONand ROBERTSON (1988) concluded that BLUP achieves larger response 1955; HILL1982; LYNCHand HILL1986). Mutants with than mass selection for 10 generationsof selection, be- large effects on quantitative traits and deleterious pleio- cause the increased accuracy of BLUP selection more tropic effects on fitness certainly occur, however (UT- than counterbalances the reduction in effectivesize. TER 1966; HILL and KEIGHTLFY 1988; CABALLERO and VERRIER et al. (1993) used different population sizes KEIGHTLEY 1994). The fate of mutations of large effect and showed that mass selection achieves larger response under artificial selection with different selection meth- than BLUP if population size is small. QUINTONet al. ods has been investigated by CABALLERO et al. (1996b). (1992) showed that mass selection can yield higher re- Natural selection on phenotype, ie., stabilizing selec- sponse than BLUP selection when the comparison is tion, may also play a role, so that the actual response made at the same level of inbreeding, i.e., when the is smaller than predicted here (UTTER1966; NICHOLAS population size is assumed to be different for the two and ROBERTSON 1980;ZENG and HILL 1986). schemes, so that more intensemass selection yields the The model of mutation assumed in this paper is the same inbreeding rate as BLUP selection. As shown by random walk mutation model (e.g., LYNCHand HILL (11) and illustrated by Figures 3-5, the time for the 1986). A house-of-cards mutation model (COCKEM ranking in response of selection methods to change and TACHIDA1987; ZENG and COCKERHAM1993) was depends mainly on effective size, but also on other pa- not considered but is not likely to affect the results of rameters such as r&, i, &, etc. this paper as the difference between the random walk Some systematic methods to control inbreeding for and house-of-cards mutation model is expected to be long term selection have been proposed (e.g., TORO large only in a time scale far beyond that considered and PEREZ-ENCISO 1990;VILLANUEVA et al. 1994), for in this paper (see ZENG and COCKERHAM 1993). example, the useof a biased upward heritability in Finally, random mating is assumed in the analysis, Selection Response Selection 1971 but in practice mating can be controlled to reduce the GRUNDY,B., A. CABALLERO,E. SANTIAGO and W.G. HILL,1994 A note on using biased parameter values and non-random mating rate of inbreeding andto maintain more genetic varia- to reduce rates of inbreeding in selection programmes. Anim. tion (e.g.,TORO and PEREZ-ENCISO1990; CABALLEROet Prod. 59 465-468. HEATH,S. C., G. BULFIELD,R THOMPSONand P. D. KEIGHTLEY,1995 al. 1996a). Rates of change of genetic parameters of body weightin selected CHEAVALET(1994) has developed a theory for selec- mouse lines. Genet. Res. 66 19-25. tion response assuming a finite number of unlinked HECK,A., 1993 Introduction to Maple. Springer-Verlag, New York. HENDERSON,C. R., 1975 Best linear unbiased estimation and predic- loci, and HILLand RASBASH (1986) proposed a nonin- tion under a selection model. Biometrics 31: 423-447. finitesimal model allowing for different distributions HENDERSON,C. R, 1982 Best linear unbiased prediction in popula- of gene effects and frequencies, although nonadditive tions that have undergone selection, pp. 191-201 in Proceedings of the World Congress of She@ and Beef Cattle Breeding, New Zealand, effects and linkage are not accounted for. As more in- Vol. 1, edited by R. A. BARTONand W. C. SMITH.Dunmore Press, formation on distribution of gene effects and gene fre- Palmerston North, New Zealand. quencies and on collateral effects on fitness for quanti- HILL,W. G., 1976 Order statistics of correlated variables and impli- cations in genetic selection programmes. Biometrics 32 889- tative traits become available, more realistic models can 902. be applied. In the meantime,however, an infinitesimal HILL,W. G., 1982 Predictions of response to artificial selection from new mutations. Genet. Res. 40 255-278. model continues to bea useful reference point to inter- HILL,W. G., 1985 Effects of population size on response to short pret selection experiments and to predict selection re- and long term selection. J. Anim. Breed. Genet. 102: 161-173. sponse. HILL,W. G.,and P. D. KEIGHTLEY,1988 Interaction between molec- ular and quantitative genetics, pp. 41 -55 in Advances in Animal We are grateful to B. VILLANUEVAand particularly J. DEKKERSfor Breeding-Symposium in Honour of Professor R D. POLITIEK,edited useful comments on the manuscript and to the Biotechnology and by S. KORVER, H. A.M. VAN DER STEEN,J. A. M. ARENDONK,H. Biological Sciences Research Council for financial support. BAKKER,E. W. BRASCAMPand J. DOMMERHOLT.Pudoc. Wagen- ingen, Netherlands. HILL,W. G., and J. &BASH, 1985 Models of long term artificial LITERATURE CITED selection in finite population. Genet. Res. 48 41-50. HILL,W. G., A. CABALLERO and L. DEMPFLE,1996 Prediction of BELONSKY,G. M., and B. W. KENNEDY,1988 Selection on individual response to selection within families.Genet. Sel. Evol. (in press). phenotype and best linear unbiased predictor of breeding value KEIGHTLEY,P. D., and W. G. HILL,1987 Directional selection and in a closed swine herd. J. Anim. Sci. 66: 1124-1131. variation in finite populations. Genetics 117: 573-582. BRISBANE,J. R., and J. P. GIBSON,1995 Balancing selection response KEIGHTLEY,P. D., and W. G. HILL,1992 Quantitative genetic varia- and rate of inbreeding by including genetic relationships in se- tion in bodysize of mice from new mutations. Genetics 131: lection decisions. Theor. Appl. Genet. 91: 412-431. 693-700. BULMER,M. G., 1971 The effect selection on genetic variability. Am. LATTER,B. D. H., 1966 Selection for a threshold character in Drw Nat. 105: 201-211. sophila. 2. Homeostatic behaviour on relaxation of selection. BUNGER,L., and G. HERRENDOER, 1994 Analysisof a long-term Genet. Res. 8: 205-218. selection experiment with an exponential model. J. Anim. Breed. LYNCH,M., 1988 The rate of polygenic mutation. Genet. Res. 51: Genet. 111: 1-13. 137-148. CABALLERO,A,, 1994 Developments in the prediction ofeffective LYNCH,M., and W. G. HILL,1986 Phenotypic evolution by neutral population size. Heredity 657-679. 73 mutation. Evolution 40 915-935. CABALLERO,A., and P.D. KEIGHTLEY,1994 A pleiotropic nonaddi- LUSH,J. L., 1947 Family merit and individual merit bases for tive model of variation in quantitative traits. Genetics 138: 883- as selection. Am. Nat. 81: 241-261, 362-379. 900. MEUWISSEN,T. H. E., 1990 Reduction of selection differentials in CABALLERO,A,, P.D. KEIGHTLEY and W. G. HILL,1995 Accumula- tion of mutations affecting body weight in inbred mouse lines. finite populations with a nested full-half sib family structure. 47: Genet. Res. 65: 145-149. Biometrics 195-203. NICHOLAS, W., and A. ROBERTSON,1980 The conflict between CABALLERO, A., E. SANTIAGOand M. A. TORO,1996a Systemsof F. mating to reduce inbreeding in selected populations. Anim. Sci. natural and artificial selection in finite populations. Theor. Appl. 62: 431-442. Genet. 56: 57-64. CABALLERO, A., M. WE1 and W. G. HILL, 1996b Survival rates of PEARSON,IC, 1903 Mathematical contributions to the theory of evo- mutant genes under artificial selection using individualand fam- lution. XI. On the influence of natural selection on the variability ily information. J. Genet. (in press). and correlation of organs. Phil. Trans. R. SOC.Lond. A 200 1 - CHEVALET,C., 1994 An approximate theory of selection assuming 66. a finite number of quantitative trait loci. Genet. Sel. Evol. 26: PRESS, W. H., S. A. TEUKOLSKY, W.T. VETTERLINGand B. P. FLANNERY, 379-400. 1992 Numerical Recipesin C, Ed. 2. Cambridge University CLAWON, G., and A. ROBERTSON,1955 Mutation and quantitative Press, Cambridge. variation. Am. Nat. 89: 151-158. RAWLINGS,J. O., 1976 Order statistics for a special classof unequally COCKERHAM,C.C., and H. TACHIDA,1987 Evolution and mainte- correlated multinomal variates. Biometrics 32 875-887. nance of quantitative genetic variation by mutations. Proc. Natl. ROBERTSON,A., 1960 A theory of limits in artificial selection. Proc. Acad. Sci. USA 84 6205-6209. R. SOC.Lond. B. 153: 234-249. DEMPFLE,L., 1974 A note on increasing the limit of selection ROBERTSON,A., 1961 Inbreeding in artificial programmes. Genet. through selection within families. Genet. Res. 24: 127-135. Res. 2: 189-194. DEKKERS,J. C. M., 1992 Asymptotic response to selection on best QUINTON,M., C. SMITHand M.E. GODDARD,1992 Comparison of linear unbiased predictors of breeding values. Anim. Prod. 54 selection methods at the same level of inbreeding. J. Anim. Sci. 351-360. 70: 1060-1067. FALCONER,D. S., and T. F. C. MACKAY, 1996 Introduction to Quantita- SANTIAGO,E., and A. CABALLERO,1995 Effective size of populations tive Genetics, Ed. 4. Longman, London. under selection. Genetics 139: 1013-1030. FRANKHAM,R., 1980 Origin of genetic variation in selection lines, TORO,M. A., and M. PEREZ-ENCISO,1990 Optimization of selection pp. 56-68 in Selection ExperZments in Laboratory and Domestic Ani- response under restricted inbreeding. Genet. Sel. Evol. 22: 93- mals, edited by A. ROBERTSON. CommonwealthAgricultural Bu- 107. reaux, Slough. VERRIER,E., J. J. COLLEAUand J. L. FOULLEY,1991 Methods for pre- GOMEZ-RAYA, L.,and E.B. BURNSIDE,1990 Linkage disequilibrium dicting response to selection in small populations under additive effects on genetic variance, heritability and response after re- genetic models; a review. Livest. Prod. Sci. 29 93-114. peated cycles of selection. Theor. Appl. Genet. 79: 568-574. VERRIER,E., J.J. COLLEAUand J. L. FOULLEY,1993 Long-term effects 1972 M. Wei, A. Caballeroand W. G. Hill

of selection based on the animal model BLUP in a finite population. Theor. Appl. Genet. 87: 446-454. VILLANUEVA,B., N. R. WRAY and R. THOMPSON,1993 Prediction of asymptotic rates of response from selection on multiple traits 2 1/2 using univariate and multivariate bestlinear unbiased predictors. + h2,I2 + 47dh:(1 - ho)) Anim. Prod. 57: 1-13. VIIMNUEVA,B., J. A. WOOLLIAMSand G. SI", 1994 Strategies for where @ = uio/ (aio + a:). Equation A1 reduces to controlling rates of inbreeding in adult MOET nucleus schemes that given by GOMEZ-RAYAand BURNSIDE(1990) when for beef cattle. Genet. Sel. Evol. 26 517-535. WEBER,K E., and L. T. DIGGINS,1990 Increased selection response N7and ND + w, i.e., yrnl= 1 and yd = (k~s+ kD)/2. in larger populations. 11. Selection for ethanol vapor resistance Family selection: The additive variance offamily in Dmsophila melanogaster at two population sizes. Genetics 125 means at the limit can be derived a function 585-597. (05~) as WOOLLIAMS,J. A., N. R. WRAYand R. THOMPSON,1993 Prediction of initial additive variance of family means (a:p) and of long term contributions and inbreeding in populations under- k, (ie., k for family selection). Analogously as before, going mass selection. Genet. Res. 62 231-242. from (8) the recurrent equation at the limit is WRAY,N. R., and M. E. GODDARD, 1994 Increasing long-termresponse to selection. Genet. Sel. Evol. 26 431-451. WRAY,N. R., and W. G. HILL, 1989 Asymptoticrates of response ai = 0.25 (1 - l/N.)(l - k,p;)a; from index selection. Anim. Prod. 49: 217-227. WRAY,N. R., and R. THOMPSON,1990 Prediction of rates of inbreed- + 0.25 (1 - l/ND) (1 - k&)a: + 0:0/2, ing in selected populations. Genet. Res. 55: 41-54. WRAY,N. R., J. A. WOOLLIAMSand R. THOMPSON,1994 Prediction of where k, = kD = k. under family selection. p; = rates of inbreeding in populations undergoing index selection. a&/(aia$) is the limiting value of squared accuracy Theor. Appl. Genet. 87: 878-892. YOO, B. H., 1980a Long-term selection for a quantitative character under family selection, where a$ = aif + 0% is the in large replicate populations of Drosophila melanogaster. 1. Re- phenotypic variance of family means at the limit. The sponse to selection. Genet. Res. 35: 1-17. recurrent equation can be rearranged as ai = aio/ [ (2 - YOO, B. H., 1980b Long-term selection for a quantitative character in large replicate populations of Drosophila melanogaster. 2. Lethals yf) + kfyp;] where yf = 1 - (l/Ns + 1/N1)/2. Substitut- and visible mutants with large effects. Genet. Res. 35: 19-31. ing p; gives ZENG, Z. B., and C.C. COCKERHAM,1993 Mutationmodels and quantitative genetic variation. Genetics 133: 729-736. [r,(k,- 1) + 214+ rpd, ZENG, Z. B., and W. G. HILL, 1986 The selection limit due to the conflict between truncation and stabilising selection with mutation. Genetics 114 1313-1328. Communicating editor: Z-B. ZENC

APPENDIX A 1 Derivation of additive genetic variance at the limit u2 - Af- 2[yXk,- 1) 21 under mass selection and family selection: The expres- + sions derived in this appendix are given as a function of ai and aio.To apply them for any generation t, where replace a: and aioby ait and 2aiwt,respectively. Mass selection: GOMEZ-RAYA and BURNSIDE(1990) gave a formula to calculate the genetic variance at the limit (ai) for mass selection assuming an infinite popu- APPENDIX B lation. For a finite population, we still assume that the within-family variance (aio/2) does not change over Rate of change in selection responsewith respect to generations, but consider a correction in the between- rate of change in genetic variance for different selec- family variance for the number of sires and dams se- tion schemes: All the derivations in this appendix are lected (N, and No) due to the sampling of parents with evaluated at generation 0, which is taken as a reference replacement (KEIGHTLEY and HILL1987; VERRIER et al. generation. The subscript 0,however, hasbeen omitted 1993). Thus, from (8) from all parameters, for simplicity. Mass selection: The derivative pm= d log R,,,/d log ai = 0.25(1 - l/N.) (1 - k.&)ai aiw can be expressed as Pmlpm2= (d log RJd log ai) X (d log oi/d log aiw).From (7), pml = 1 - h2/2. + 0.25 (1 - l/ND)(l - k,&)d + d0/2, Substituting (1) into (Al) and deriving where p: and p; are the squared limiting accuracyof selec- -1 + [h2+ (yml+ 2y,)(l- h2)l tion in sires and dams, respectively, and are equal to each +([yrnl(l- h2)+ h2I2+ 4y,h2(1 - h2))l/' = p,=1-h2 other and to the limiting heritability h2 aV(ai + a:). yrnl(l- h2) - h2 + {[yrnl(l- h2) + h2I2 ' Rearranging the equation ai = aio/(yml + y,h2), where + 4y,h2(1 - h2))l/' Yrnl = 1 (I/& + 1/ND)/2, and ym2 = [(I - 1/Ns)k.$ f (1 - 1/ND)kD1/2, giving (A4) Selection Response 1973 where ymland ym2are given in Mass selection of APPEN- reduces to mass selection and (A6) reduces to (7). For DIX A. The function Pm2increases with increasing hz, the extreme case where n + UJ and a: = 0, Pn + 0.5h2 and limh' + 1 Pm2= 1. Equation A4 also applies to a + (1 - hz)/(2- h'). full-sib (yml= 1 + 1/[2ND] and ym2= [l - l/ND]kD) In general, the value of Pn lies between 0.5 and 1, or half-sib family structure (yml= 1 + 1/ [2Ns], and and decreases with increasing familysize (n and m) ym2 = [1 - 1/Nslks). and/or h2. Family selection: Analogously to the above, Pp = (1 When the Bulmer effect is accounted for it is, how- - h;/2), where hj is the limiting value of heritability of ever, very tedious to express ai as a function of aiw, family means, and a: and k, and to obtain its derivative. The formula for PI accounting for the Bulmer effect is not presented, - (1 - n)yf- 2 and a numerical solution has been used (PRESSet al. n+ 1 - 2Yf2] 1992). BLUP selection: BLUP selection is approximated by selection based onan index (pseudo-animal model BLUP) consisting of individual record (I), full-sib mean (FS), half-sib mean (HS), estimated breeding values of (A5) thedam (AD), themean of the estimated breeding where yf, yfl and yf2 are given in Family selection of value of all dams mated to the sire (AD,) and the esti- APPENDIX A. The function Pf2decreases with increasing mated breeding value of the sire (A,y) (WMY and HILL h; and family size, and limh2 + 1 Pf2 = 1. 1989). To facilitate the derivation, the index is set up Index selection: First, we just derive an expression using independent linear functions of the records so (Pn)without accounting for the Bulmer effect. The the variance-covariance matrix is diagonal, where the index consists of individual performance (expressed as diagonals are for (1- m),(n - HS - &/2 + Au/2), deviation from full-sib family mean), full-sib mean (as (HS - As/2 - Au/2), and (AD/2 + As/2). These diago- deviation from half-sib familymean) and half-sib family nal elements are mean. The phenotypic variance-covariance matrix is di- PBl = [(n - 1)/n](d + d/2), agonal with the following elements: pB2 = [(m - l)/(nm)I[d + 4 PI1 = [(n- l)/nI(aS + ai/2), + (n + 2)ai/4] - [1/4 - 1/4m]p2ai, PIZ = [(m - l)/(nm)l[d + d + (n + 2)d/41, pB3 = [a: + na: + (n + nm + 2)ai/4]/nm PI, = {a: + na2c + [n(m + 1) + 2Iai/4)/(nm), - [1/4 + 1/4m]p2ui, where n is the full-sib family size,and m is the numberof dams mated toa sire. The vector of covariance between p, = p2ai/2. genetic values and index information has the following Here, p2 is the squared accuracy of BLUP selection. elements: The vector of covariancesbetween breeding values and information sources has elements wn = [(n - 1)/(2n)Igi, WBl = [(n- 1)/(2n)]d, WE = [(n + 2)(m - 1)/(4nm)]ai, WBZ = [(n+ 2)(m - l)/(nm)lai - [1/4 - 1/4m]p2ai, wn = [2 + n(m + 1)]oi/(4nm). WBS = [(n+ 2 + nm)/(4nm)]ai - [1/4 + 1/4m]p2ai, The variance of the index is a: = Z dj/po. Thus, Pn = d log Rl/d log aiw is wB4 = p2&2. The variance of the index is aiI = Z wij/ps,for j = 1, . . . , 4. Let 6 = d p2/d ai, then

2 + n(m + 1) 2prs - + $3 "1 . (A6) ~3

When half-sib information is not available (m = l), then the second termin these equations ( wR, pR)drops out; and if there are onlyhalf-sibs (n = l), the first term drops out.Similarly, if n = 1 and m = 1, the index 1974 M. Wei, A. Caballero and W. G. Hill

HENDERSON(1982) found that under animal model m = 1, and the second term in w and p in (A7) and BLUP the variance of prediction error is determined (A8) is dropped. Further,when the pseudo-BLUP index only by the amountof information on self and relatives includes only individual information and parental esti- and is not affected by selection. Hence, DEKKERS(1992) matedbreeding values, Le., the case that DEKKERS showed that theratio of asymptotic to original response (1992) considered to provide lower limit to accuracy of is equal to a constant, [2/ (2 + ks + kD)]1’2. Therefore, BLUP, wBJ = (1 - p2/2)& WM = p2&2, = 0; - in this case pB= pBl= d log R/d log cTiwand,therefore, p2a;/2, pH = p2a?/2, and the first two terms in w and p are dropped. Then

and

with 4 from (A7). If half-sib information is not available,