Copyright 0 1996 by the Genetics Society of America
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. Account was 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 ( C 8 ) can be approximated as
where N, is the effective population size, = Ne& is the genetic variancewithinfamilies 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 withrespect 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.
G
ENETIC response to one cycle of selection for a quantitative character is solely a function of selection accuracy (the correlation between the selection criterion and breeding values of individuals), selection intensity and additive genetic variance in the population (FALCONER and MACKAY 1996). However, for long term response to selection in a finite population, other factors such as genetic drift, gametic linkage disequilib rium, mutationalvariance and effective population size are all variables depending on character, population structure and selection strategy, which should be incorporated to predict thecumulative response to selection. While long term response is also a function of the distribution of the effects and frequencies of individual loci, such information is not available, s o it is necessary to consider only known, albeit restricted, parameters. ROBERTSON (1960) developed the theory of selection limits under mass selection. He did not consider, however, that additive variance is reduced by selection due to gametic linkage disequilibrium, the “Bulmer effect” (PEARSON 1903; BULMER1971). To predict long term response, the effects of gametic linkage disequilibrium and genetic drift on additive variance need to be considered simultaneously. A selection index (LUSH1947) and animal model best linear unbiased prediction (BLUP) (HENDERSON Curresponding authur: William G. Hill, Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd., Edinburgh EH9 3JT, United Kingdom. E-mail:
[email protected] Genetics 144: 1961-1974 (December, 1996)
1975) using all information available are formally the methods to achieve the maximum response for one generation of selection. Assuming an infinite population, WRAY and HILL (1989) computed the asymptotic response (the constant rate of response) after accounting for the Bulmer effect for various index selection schemes. DEKKERS (1992) and VILLANUEVA et aZ. (1993) extended the theory to BLUP selection. These works, however, did not consider the reduction in selection response due to the finite size ofthe populations (HILL 1985), which can be accounted for by means of the effective population size. In populations under selection, the effective size can be predicted depending on selection methodsand mating schemes. ROBERTSON (1961), WRAY and THOMPSON (1990), WOOLLIAMS et aZ. (1993), and SANTIAGO and CABALLERO (1995) developed methods to predict the rate of inbreeding when a population is undergoing mass selection. WRAYet al. (1994) extended the theory to handle the situation of index selection. For a review of the principles of the above methods see CABALLERO (1994). Selection intensity is also a factor that depends on population size. For the same proportion selected, the selection intensity is reduced when there are a small number of families and by the increased correlation between individuals’estimated breeding values, particularlywith selection using family information (HILL 1976; RAWLINGS 1976; MEUWISSEN1990). Different methods of selection (mass selection, index
1962
M. Wei, A. Caballero and W. G. Hill
selection, etc.) differ in the accuracywith which individual breeding values are estimated. A high selection accuracy gives a high response in the short term but because of coselection of family members it also gives a high rate of inbreeding, so that it reduces the selection response in thelongrun (ROBERTSON 1960, 1961). Thus methodsachieving maximum short term response do not necessarily optimize long term response. Some simulation studies of long term selection response have beenconducted (e.&, BELONSKY and KENNEDY 1988; VEWER et al. 1993) that give some insight into the comparison of short and longterm selection response. Spontaneous mutation has been found to be an important source of newvariation (CLAETON and ROBERTSON 1955; FRANKHAM 1980; LYNCH1988). HILL (1982) developed the theory for genetic response from new mutations, but the model including both initial segregating and mutational variances to predict selection response has not been developed, except under simple assumptions (HILL1985; KEIGHTLEYand HILL 1992). The aim of this study is to develop the analytical theory to predict cumulative response to selection under various selection schemes (mass, family, within-family, index and BLUP selection) taking into account genetic drift, gametic linkage disequilibrium, initial and mutational variances, inbreeding depression, common environmental variance and reducedselection intensity due to correlation of family members. The main objective is not to give the most accurate predictions of response to selection but to incorporate all the factors that affect short, medium and long-term response in relatively simple equations as functions of known parameters, whichallow the effect of such factors and parameter values to be evaluated. Results are discussed in relation to previous theory and the results of long term selection experiments. THEORY
Modeland selectionschemesassumed: A single quantitative character is considered that is controlled by an infinitesimal model of gene effects ( i e . , the character is controlled by genes at many unlinked loci, each of small additive effect). A closed and finite population with a nested mating structure (full-sib families nested within half-sib families), discrete generations, random mating, constant population structure and sizeisassumed. Each full-sib family is assumed to have a constant number of individuals (n), n/2 of each sex. At generation t, the total phenotypic variance (a&) consists of additive genetic variance (&), environmental variance (a:) and common environmental variance between full-sib family members ( a : ) . The residual variance (a:) is defined as a: = a: 0%and is assumed to be constant over generations. The constant input of new genetic variance per generation due to mutation is assumed to be aL. Subscripts for generationsare
+
omitted for simplicity if a formula or parameter applies for any generation. In this study, mass selection refers to truncation selection on individual phenotypic values. Family selection is defined as the selection of the best families based on phenotypic family means. Within-family selection is the selection of the best progeny within each family based on individual phenotype. Index selection refers to the selection of individuals based on an index of individual, full-sib and/or half-sibfamily information, optimally weighted every generation (FALCONER and MACKAY 1996). BLUP selection is carried out by the selection of individuals with highest breeding values estimated by the “pseudo-animal model BLUP”, i e . , an index of individual, full- and half-sib information plus estimated breeding values of the dam, sire and mates of the sire (WRAY and HILL 1989).Pseudo-BLUP is almost identical to true BLUP when there are no fixed effects other than the overall mean (WRAY and GODDARD 1994), and we will simply call it BLUP henceforth. There are equations available to approximate effective population size under theselection schemes underlined above (WRAYet al. 1994). In this paper, however, we use values obtained by simulation, for simplicity. For a description of the simulation procedure and examples of predictions using the equation of WRAY et al. (1994), see CABALLERO et al. (1996b). The reduced selection intensity due to the correlationbetween index of family members is calculated by using the formulae of HILL (1976) and MEUWISSEN(1990) to correct for finite populations. Prediction of genetic variance: Based on the infinitesimal model, the additive genetic variance at generation t (&) can be predicted if gametic linkage disequilibrium is ignored. Itis then equal to the genic variance (&), Le., ait = a : , = aio [ ( l - 1/(2N,)lt = aioe-1’/2N~, where a:o is the initial genetic variance and Ne is the effective population size. When gametic linkage disequilibrium is accounted for, the prediction of ail is not straightforward. However, the within-family additive variance (aiwl) is not affected by the linkage disequilibrium (BULMER 1971) and therefore can still be predicted, i e . , aiw = uiuoe-t/2Ne,so wewill use it as the reference point in all predictions. We now add to this last equation the contributionof mutation. The withinfamily additive variance due to mutations accumulated until generationtisequal to NgL(1 Lie., half the total additive variance, given by CLA~TONand ROBERTSON (1955) and HILL (1982)l. Thus, the total within-family genetic variance at generation t is o A2W l
= aiwm
+ ( o A2W O
- a~wm)e-t/2~,
(1)
where 2
CAW^ =
Ne&
(2)
is the steady state within-familyadditive genetic variance, regardless of the value of ~ W O .
Selection Response
1963
Prediction of response: The response to selection at generation t(&) is
R,=
(3)
ZPptffAt, '
where i is the selection intensity and pt is the accuracy of selection at generation t. The cumulative response to selection until generation t (CR,) is CR, =
R, r=l
=
s'
(4)
ipraArdT.
0
R, and CR, can, therefore, beevaluated if pt and ait are expressed in terms of the within-family additive variance at generation t, which is predicted from (1).Equations 3 and 4 can be tedious to derive in terms of aiw,except under simple assumptions, and numerical solutions can be obtainedusing a mathematical package (e.g., MAPLE; see HECK1993). However, a first order Taylor series approximation can always be used providing t/N, is sufficiently small (say, t/N, < 2),
where & = ipoaAo is the selection response at generation 0 and the derivative
P=
d log R d log aiw
(log indicates natural logarithm) gives the relative rate of change inresponse ( R ) and aiwNo subscript generation is given in the derivative, because it can be evaluated at any generation, but here we evaluate it using i.e., the genetic parameters at the reference generation, generation zero. Substituting (1) into theabove expression,
or approximately
assuming t/N, < 1. The cumulative response to selection until generation t (CR,) can be obtained by integrating functions (5a) or (5b) over generations,
X [ t - 2N,(1 -
or
e-"'y)]
I
, (6a)
Note that, if mutation is ignored (a" = 0 ) , then from (2) aiwm = 0 in (6, a and b). The general equations (Equation 6, a and b) allow the cumulative response to selection tobe approximated for a range of selection methods and parameter values. We now describe how gametic linkage disequilibrium is specifically accounted for. In an infinite p o p ulation, a limiting value of genetic variance is closely approached after the Bulmer effect occurs in a few cycles of directional selection. The response (variance) predicted with this limiting value was called the asymp totic response (variance) by WRAYand HILL (1989). In this study, the response (variance) is continuously increased by mutation and erased by drift and, therefore, it does not necessarily asymptote. We shall still call it the asymptotic response (variance) for simplicity, 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 of response would not be significantly influenced because the asymptotic response is closely approached within a few generations. We assume that the variance within families at generation zero (aim),which is the result of previous selection, mutation and genetic drift, is known. Then we calculate aioas the asymptotic variance assuming thatthe initial genetic variance was 2aio.Analogously at any generation t we predict the genetic variance within families (aiwt)accounting for mutation and drift using (1), and ait as the asymptotic variance assuming that the initial genetic variance was 2aiW. Explicit formulae to calculate the asymptotic variance are derived in APPENDICES A.1 (Equation A l ) and A.2 (Equation A3) for mass and family selection, respectively. Those for index and BLUP were obtained 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 ai, account forinitial asymptotic variance, genetic drift, mutation and Bulmer effect for a further t generations. To find solutions for the derivative p = d log R / d log o ~ needed W in (6, a and b), this can be expressed as
d log R d log a: P=p1P2=dlopdloga~w' where PI indicates the relative proportional rate of change of R and 02. This can be derived for mass selec-
M. Wei, A. Caballero and W. G. Hill
1964
tion by setting h2 = &‘(ai i&ap to give
p 1 -
+
a:) and R = ihoA =
d log R d ( b A ) ai 1 - 1 - - h2. dlog hoA 2
da;
(7)
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 additive variance and, therefore, pw = pwl.For index and BLUP selection, derivatives of ,8 are more complicated (APPENDICES B.3 and B.4). Note that, for BLUP selection (subscript B ) , p B = Pel, because the response including the Bulmer effectis proportional to that without including it (DEKKERS1992; see APPENDIX B.4). RESULTS
Characteristics of equations and validation: In the above section we have derived explicit equations to predict cumulative response in terms of known parameters of the population. To check the validity of these equations, we can compare them with more exact recurrence calculations. In these, gametic linkage disequilibrium is accounted for (see, e.g., VERRIER et al. 1991) and we include mutational variance ( a : ) . Let aist and a h n be t the variance between half-sib families and fullsib familieswithin sires at generation t , respectively, and a :, the genic variance at generation t , i.e., the additive variance which would result in the absence of selection. Thus
a:,
2
= anl-l(l-
1/2Ne) + ,a :
(8a)
where pst-l is the accuracy of selection of sires at generation t - 1; Ns is the number of sires selected; (1 l/Ns) is used to correct for the finite population size (KEIGHTLEY and HILL1987; VERRIERet al. 1993), assuming sampling of parents without replacement; ks = is( is - xS), where is is the selection intensity and xs is the standardized truncation point; and corresponding quantities apply for dams (subscript D ) . Note that, in Expressions 8, the mutational variance (a&) initially arises within families(Equation 8b) but it is later redistributed among and within families (Equation 8a). In the absence of selection a i , = a:,, and for t a,a : = -P 2NpL and oiwm”* Nea:, as expected. -+
12
0
0.5
1
1.5
2
GenerationlNe
FIGURE1.-Cumulative response (in upunits)to mass selection predicted by Formulas 6a (---) and 6b ( - - -) compared with that from recurrent equations (Equation 8) ( - ). Parameters used were as follows: 20 males, 40 females, six progeny scored per couple, N, = 46, $ = 0.2 and uk = 0.
The accuracy of selection ( p s and pul) was obtained with the usual procedure (see FALCONER and MACKAY 1996, pp. 243-244). Response to selection every generation was calculated from (3) using the asymptotic variance as the initial value (aio). The cumulative response (C&) predicted by ( 6 , a and b) is compared in Figure 1 with that obtained by the recurrentequations (Equation 8). Comparisons are shown only for mass selection, for simplicity, but the results are similar for otherselection methods. Formula 6a fits well until about 2N, generations, after which it starts to overestimate response; (6b) fits well up to N, generations, after which response starts to be substantially underestimated. The role of the different parameters determining cumulative 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 ~ ~ J a i ~ ) t ~ / ( 4 NG, )t ]so that CR, = t&. With increasing t , however, the roles of N, and p become more important. The value of p determines therelative rate of change in response and variance, i.e., how much the response is reduced due to the loss of variance by genetic drift and the Bulmer effect. As the response is proportional to the accuracy of selection ( p ) (Equation 3), p will be small if the accuracy is fairly insensitive to changes in additive variance (or heritability), and it will be large if the accuracy is highly dependent on theheritability. In general, p lies between 0.5 and 1, being smaller with a larger p, h2, and/or familysize (n).This is shown in Figure 2, A-F, where the values of p and p are plotted against h2 for different selection schemes and population parameters. In general, 0 is small for selection
Selection Response
A
’ 0.8
0.9
Ea
.-cg .-P
4 0.4
0 0.7
h 0
1 !Kl
0.6
0 0
0.8
1
0.6
0.2
0.5
0 0
0.2
0.6 Heritability
0.4
0.8
1
0
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0.8
1
0.8
1
0.8
1
Heritability
E 1 B
1 0.9
0.8
6
.-$! .-2!
0.6
Ea
1
d
0
3
0.8
U
0.4
0.7
0.6
0.2
0.5
0
0
0.2
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0
1
0.2
0.4
0.6
Heritability
Heritability
c 1 0.9
0.8
0.9
0.7
.-$!
0.6
m
5
0
0.8
U
m
0.5
.->
1
2 0.4
0 0.7
0.3 0.2
0.6
0.1
4
0
0.5 0
0.2
0.4
0.6
Heritability
0.8
1
0
1
0.2
0.4
0.6
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. Caballero and
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 valuesof 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. For a given heritability, an increase in common environmental variance ( a : ) implies a reduction in a: and thus an increase in the selection accuracy for withinfamily selection, but a decrease for family, index and BLUP selection (cf: Figure 2, B and C). Accordingly, the value of p increases with increasing a : for family, index and BLUP selection, but decreases for withinfamily selection, especially with high h2 ($ Figure 2, E and F). Role of mutational variance:The role of mutational variance (a") in determining the cumulative response to selection is clearly shown in Formula 6b. If aiwo> aiwm,the term 1 - aiwm/aiwo is positive and selection response will become slower every generation, otherwise it will accelerate. Estimates of 0%from various experiments (LYNCH 1988) are in the range from 0.0001 to 0.005 of, depending on the character, species and population, although most estimates come from Drosophila. Estimates for body weight in mice range from 0.001 to 0.006 a: (CABALLERO et al. 1995). Figure 3 shows cumulative responses to mass and BLUP selection (using Equation 6b) for no mutation and two extreme values of aL (= 0.001 a; and 0.01 a;). Mutation can substantially increase the long term selection response. The cumulative response to mass selection surpasses that to BLUP selection earlier when mutation is present, because selection schemes with smaller accuracy but larger Ne (e.g., mass selection) exploit the mutational variation more efficiently than selection schemes with larger accuracy but smaller Ne (e.g., BLUP). The reason is that the long termresponse will eventually be determined by the variance atthe steady state, i.e., ai, = 2Np" (cf: Equation 2 ) . Different selection methods (say j and k ) can be compared in terms of the ratio of long term responses at the limit [ R , from (3), assuming the same selection intensity], 1 Re =
PmjffAmj
h
P m kPfmf Ak m J Nk e k '
k
- pm& jq -
(9)
where pmis the accuracy of selection at the steady state. In general, a selection method with low Ne is predicted
W. G. Hill
10
0
20 Generation
30
40
FIGURE3.-Cumulative response (in apunits) to mass selection (---) and BLUP selection (-) in case of different mutational variance: ak = 0 (lines without symbols); a" = 0.001 a$ (lineswith X); and 0%= 0.01 a$ (lineswith A). Parameters used were as follows:20 males, 40 females, 12 progeny scored per couple, @ = 0.2.
to give a lower asymptotic response than another with larger Ne, even if the first has a higher accuracy of selection for a given value of genetic variance. This is because the accuracy will eventually depend on the steady-state variance (ab) that is also a function of Ne. Thus, if Ne is low the accuracy will eventually be low at the steady state. Compare, for example, mass selection (subscript m ) and within-family selection (subscript w ) . In this case, (Nm/Nm)1/2 2 & in general, because N, = 2N irrespective of selection, while N, is usually smaller than N , particularly with high h2 and intense selection (see, e.g., SANTIAGOand CABALLERO 1995). The accuracies at the steady state are pmw
=
JYLPL/~(N~PL + (~NLPL)
and
+
pm, = 2 N , a ~ / d ( 2 N m a ~ a:) (2N,aL).
Thus, again because N, 2 2N,, it follows that pmw/pm, 2 l/&. Therefore, substituting these inequalities into (9), ReW/km 2 1, and a larger response in the long term is predicted for within-family selection than mass selection, even though the accuracy for within-family selection is about one half that for mass selection, for a given genetic variance. Short vs. long term selection response for different selection methods: Inbreeding depression needs to be taken into account in long term selection (FALCONER and MACKAY 1996). Assuming a linear relationship between inbreeding coefficient and genetic mean, thetotal reduction of genetic response at generation t is D(1 - e-'lzNe) = Dt/ ( 2 N e )where , D is the inbreedingdepression per unit of inbreeding (in am units). Thus from Equation 6b,
1967 '"
Selection Response
.
I
0
10
20
30 40 Generation
50
60
FIGURE 4.-Cumulative response (in apunits) to mass selection (---) and BLUP selection (-) with @ = 0.2 in case of different population sizes and different proportions selected, i.e., NS = 20 males, N D = 40 females, n = six progeny scored per couple (lines without symbols); Ns = 20, N D = 40, n = 12 = 40, ND = 80,n = 12 (lines with A). (lines with X); and
C&=&
[ ( t-P
2;)
1 - 7
-
t
-D-.
2N,
(10)
The more general Equation 10 can now be used to compare differentselection methods. For example, the generation when two different selection schemes (jand k) areexpectedto achieve the same cumulative response ( Tmss)is
/ *
4N* 4% defining a = 1 - a:wm/a:wo, for simplicity. Selection intensity affects thelong term response both directly and indirectly through its influence on Ne and theBulmer effect, particularly for intenseselection, low h2 and index or BLUP (HILL 1985; MEUWISSEN 1990). For example, with h2 = 0.2,20 sires and 40 dams selected every generation out of 240 progeny, i = 1.296 for aninfinite population. If computed fora finite p o p ulation, i = 1.283for 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
0
10
20
30
40
Generation
FIGURE 5.-Cumulative response (in cpunits) to mass selecwhen common environtion (---) and BLUP selection (-) mental variance (a;) is assumed to be 0 (lines without A) and 0.1 a; (lines with A ) . Parameters used were as follows: 20 males, 40 females, 12 progeny scored per couple, @ = 0.2.
are compared in Figure 4 where no inbreeding depression is assumed. The cumulative response to mass selection surpasses that to BLUP earlier with a small population size or a high selection intensity. If inbreeding depression were considered, mass selection would catch up BLUP even earlier, as Ne for BLUP is smaller and, therefore, it is more affected by inbreeding depression than is mass selection. The effect of common environmental variance (a:) 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 selection 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 ) . Comparison of equations with previousresults: If the Bulmer effect is ignored, Formulas 3 and 4 can easily be developed for illustration and as a reference for comparison to previous results. Let us first consider mass selection, for which the accuracy of selection at generation tisp f = ht, the square rootof the heritability. Ignoring the Bulmer effect, the total genetic variance at generation tis azf= 2 0 ; ~from ( l ) ,and theresponse from (3) is
1968
M. Wei, A. Caballero and
W. G. Hill
TABLE 1 Parameters used to predict cumulative response tofamily and within-family selection, which substitutefor the corresponding parametersin Formulas 3-7 for mass selection
Mass selection
Within-family selection
+
where a& = ai, af is the steady-state phenotypic variance. The cumulative response to selection until generation t can be obtained by integrating (12),
X
tanh"
(am apPm
gpl)cJPm -
2
UP-
where tanh" is the hyperbolic arc tangent function. If there is no mutation, aL = aim= 0, and (13) reduces to
CR, = 4Nei(am- a&).
(14)
Predictions for the cumulative response to family and within-family selection (ignoring theBulmer effect) can also be obtained from the same formulae (Equations 12-14), but replacing Ne, i and other corresponding parameters as listed in Table 1. Expressions for withinfamily selection in the table are taken from HILLet al. (1996). ROBERTSON (1960) derived a similar formula to predict cumulative response to mass selection without taking account of the Bulmer effect, i e . , CR, = 2Ne& (1 giving a limit of C& = 2Ne& = 2Neia~0/am. In this formula, however, a& is assumed to be a constant over generations, while in our derivation of (14), both a i t and a& are assumed to change, and only a: remains constant.Hence apm= a, and 2 noting that aio = oPo - a: = (apo a,)(apo- aT), the selection limit from (14) is C& = 4Ne2bio/ (apo a?). Thus, the ratio of ROBERTSON'S limit to the limit derived in this study is
+
+
This ratio is near to one for small g, but if this is large, ROBERTSON'S formula predicts a substantially smaller response (down to one-half for = 1). If the population is initially isogenic (aio= 0 ) , the cumulative response from mutational variance alone can be obtained from (13), substituting a& = &
Family selection
+
(1 - e - ' l n N e ) of. The resulting equation differs from that derived by HILL (1982), i e . , CR, = 2Nei(aL/ap) [ t - 2N,(1 - e - t / 2 N e ) ] ,whichalso does not allow for 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 -2% at generation80, but with a value of 0%five times larger, the difference is 3% at generation 20, and 10% at generation 80. A formula to predict cumulative response accounting for both initial and mutational variance has not been formally derived previously. KEIGHTLEY and HILL (1992) used an approximation by adding two independent terms from the formulae of ROBERTSON (1960) and HILL (1982), but this inappropriately applies two different values of selection accuracy to the same selection process. Equation 13, by contrast, uses a value of selection accuracy applicable to the initial and mutational variation together, so if evaluated with aio = 0 it would not simply be the difference between the results of Equations 13 (aiof 0, & # 0) and 14 (DL= 0). DISCUSSION
We have derived expressions to predict selection response as a functionof known parameters in the population, including a number of factors. These expressions are useful in the assessment ofthe impact of such factors and parameter values on the selection limits, the comparison of the response with different selection methods, and the interpretation of selection experiments. We discuss now some of the possible usesof the equations. Selection limits and interpretation of results from long term selection experiments: WEBER and DIGGINS (1990) reviewed responses in long term selection experiments and found that the cumulative response predicted by ROBERTSON'S formula was always larger than that achieved after 50 generations of selection at various population sizes. One possible reason for this discrepancy would be the fact that ROBERTSON'S formula ignores theBulmer effect. Generally, the predictedcumulative response from (6a) is -5-10% smaller than
1969
Selection Response
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 selected (average)
Achieved CR,
Robertson's, no Bulmer (UL = 0 )
76 88
22.4 28.8
24.9 27.6
Equation 14, no Bulmer (UL = 0 )
Equation 6a, Bulmer ( U t = 0)
Equation 6a, Bulmer (UL = 0.002 u;)
26.0 28.9
23.1 25.8
26.7
29.2
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'S equation, and would give a better fit to the examples given by WEBERand DIGGINS(1990). Let us consider the results of YOO'S(1980a) long term selection experiment to illustrate these effects. In Table 2 are shown the observed responses in this experimentandthe predictions without accounting for Bulmer effect and mutation (Equation 14), accounting for Bulmer effect but no mutation (Equation 6a with o', = O), accounting for Bulmer effect and mutation (Equation 6awith o', = 0.002a;), and ROBERTSON'S (1960) prediction. We note that ROBERTSON'S prediction is slightly smaller than the corresponding prediction by (14), as shown by (15). The inclusion of the Bulmer effect (Equation 6a with g', = 0 ) reduces considerably the predictions, while the account for mutation (Equation 6a with o& > 0 ) increases them again. For the first group of three lines selected for 76 or so generations, the formula excluding mutation but accounting for the Bulmer effect (Equation 6a with o& = 0 ) fits bestto the experimentalresults. For the second group of three lines selected for 88 or so generations, predictions with and without accounting for both mutation and the Bulmer effect (Equation 6a with a& > 0 and Equation 14, respectively) are closest to observations. Interestingly, mutations of large effect were detected in two lines of the second group (Yo0 1980b). However, not too muchemphasis can be given to these comparisons, because of the restrictions in the prediction model (see below). Based on an infinitesimal model without mutation, ROBERTSON'S (1960) theory predicts a selection limit of 2NeR0.Such a limit is very unlikely to be achieved, but even if the infinitesimal model were true, it would be very difficult to be observed experimentally. First, the limit cannot be reached in a real experiment unless population size and generation interval are very small. Further, populations of small size are difficult to maintain for many generations because of reproduction and fitness problems (FALCONER and MACKAY 1996). Finally, mutational variance cannot be ignored in thelong term, and additive genetic variance would never be exhausted in theory under the model used here.
DEMPFLE(1974) comparedthe selection limits to within-family ( w ) and mass selection ( m )considering the effects of genetic drift and Bulmer effect, and concluded that within-family selection achieves a higher selection limit. We can generalize this result as follows. Assume first, for simplicity, that a& = 0 , a: = 0 and ignore the Bulmer effect. Using (14), C L - 4Nk,jk(apmo - 0,) C L 4N,i,(ffm - 0,)
"
[z]
1 - (1 -
=
(1 -
/&2/2)1/2
-
h2)1/2
(1 -
h2)1/2.
Because N, = 2Nm and, if n is sufficiently large, i, = i, the above expression reduces to
C L -= C L
1 - (1 2[(1 -
h2/2)1/2
h2)1/2
- (1 - h y ]
5
1.
The ratio has a maximum of 1 for h2 0 , equals about 0.9 for h2 = 0.5, and a minimum of about 0.7 for h2 = 1. If the Bulmer effect were included, the ratio would be even smaller because this reduces effective size and genetic variance for mass selection, but not for withinfamily selection, so that N, 2 2N, in most cases. Analogously, an increase in common environmental variance (a:) decreases effective size for mass selection but not for within-family selection, so CR,,/ CR,, becomes even smaller with 0% > 0. Finally, the long term response from mutation will also be smaller for mass selection than for within-family selection, as was explained before. The above argument refers to the theoretical limit to selection. In a feasible time span, however, within-family selection willgive smaller responses than massselection, as it utilizes only about half of the total genetic variance. For this reason, the use of such a scheme in breeding practice is not useful except, perhaps,for breed conservation purposes. Comparison of short and long term response for different selection methods: Equation 14 nicely illustrates the antagonistic relation between effective size and se+
1970
M.Wei. A. Caballero and
lection accuracy in the cumulative response. This latter is a function of two factors, Ne and (gA)- oR).On the one hand, the magnitude of the value of om - optfor a given tis mainly determined by the selection accuracy, which depends on the particular selection scheme: a scheme of higher accuracy generally results in a larger reduction of variance and, therefore, in a larger value of om - C T ~On . the other hand, the larger Ne, the larger is the cumulative response resulting from the same reduction in variance. Because a selection method of higher accuracy gives a smaller Nebut alarger reduction of additive variance (i.e., larger om - oR),the two factors have an antagonistic effect on the response. It is then expected that the quicker the genetic variance is reduced, theless efficientlythis is exploited to change thepopulationmean.Therefore, selection schemes with higher accuracy achieve less response in the long term, andit seems to beimpossible to manipulateaccuracy and effective size simultaneously to achieve maximum cumulative response for both the short and long term. WRAY and HILL (1989) concluded, ignoring genetic drift, that the ranking of breeding schemes is not greatly altered when compared by predicted responses after one-generation rather than asymptotic responses, and DEKKERS(1992) showed something similar for BLUP selection. Even if genetic drift is considered, the predicted response after one generation of selection may be still sufficient to rank breeding schemes in the short term (say, up to five generations), because the accuracy of selection is still the major determinant of response. Simulation studies on long term selection response in finite populations have been conducted by many authors, with conclusions depending on population size and time span. For example, BELONSKY and KENNEDY (1988) concluded that BLUP achieves larger response than mass selection for 10 generationsof selection, because the increased accuracy of BLUP selection more than counterbalances the reduction in effectivesize. VERRIER et al. (1993) used different population sizes and showed that mass selection achieves larger response than BLUP if population size is small. QUINTON et al. (1992) showed that mass selection can yield higher response than BLUP selection when the comparison is made at the same level of inbreeding, i.e., when the population size is assumed to be different for the two schemes, so that more intensemass selection yields the same inbreeding rate as BLUP selection. As shown by (11) and illustrated by Figures 3-5, the time for the ranking in response of selection methods to change depends mainly on effective size,but also on other parameters such as ,&r i, &, etc. Some systematic methods to control inbreeding for long term selection have been proposed (e.g., TORO and PEREZ-ENCISO 1990; VILLANUEVA et al. 1994), for example, the useof a biased upward heritability in
W. G. Hill
BLUP evaluation gives lessweight to family information, reducing the rate of inbreeding more than selection accuracy (GRUNDY et al. 1994). WRAYand GODDARD (1994) and BRISBANEand GIBSON (1995) have proposed an iterative selection algorithm to maximize genetic response in a certain time horizon by balancing response and inbreeding, i.e., selecting individuals based on their estimated breeding values and relationship to other individuals. These methods enable BLUP selection schemes to achieve more response for agiven timehorizon, but mass selection will eventually produce a larger response because of its larger effective size. Limitations of the study: The infinitesimal model is unrealistic because there can only be a finite number of loci that control the character and the gene effects may not be all small.Change of variance due to change of gene frequencies as well as dominance, linkage and epistatic effects are not accountedfor under this model. Thus, an infinitesimal model is likely to be adequate forpredictingshort or medium term selection response, but to become less valid as the numberof generations increases. Approximate formulae giveninthis paper apply up to about Ne generations for which the infinitesimal model assumptions may still hold. Inthemodel,aconstant environmental variance (a:) is assumed, but its pattern of change is actually unknown. A phenomenon observed from selection experiments (e.g., BUNGER and HERRENDOWER 1994; HEATHet al. 1995) is that it increases with selection. However, little is known of the magnitude of this increment and how it relates to genetic response or the reduction of additive variance. In this study, mutational variance is assumed to be constant, and all mutants are neutral with respect to fitness and of small effect (CLAWONand ROBERTSON 1955; HILL1982; LYNCH and HILL1986). Mutants with large effects on quantitative traits and deleterious pleiotropic effects on fitness certainly occur, however ( U T TER 1966; HILL and KEIGHTLFY 1988; CABALLERO and KEIGHTLEY 1994). The fate of mutations of large effect under artificial selection with different selection methods has been investigated by CABALLERO et al. (1996b). Natural selection on phenotype, ie., stabilizing selection, may also play a role, so that the actual response is smaller than predicted here (UTTER 1966; NICHOLAS and ROBERTSON 1980; ZENG and HILL 1986). The model of mutation assumed in this paper is the random walk mutation model (e.g., LYNCHand HILL 1986). A house-of-cards mutation model ( C O C K E M and TACHIDA 1987; ZENG and COCKERHAM 1993) was not considered but is not likely to affect the results of this paper as the difference between the random walk and house-of-cards mutation model is expected to be large only in a time scale far beyond that considered in this paper (see ZENG and COCKERHAM 1993). Finally, random mating is assumed in the analysis,
Response
Selection
but in practice mating can be controlled to reduce the rate of inbreeding andto maintain more genetic variation (e.g.,TOROand PEREZ-ENCISO 1990; CABALLEROet al. 1996a). CHEAVALET (1994) has developed a theory for selection response assuming a finite number of unlinked loci, and HILLand RASBASH (1986) proposed a noninfinitesimal model allowing for different distributions of gene effects and frequencies, although nonadditive effects and linkage are not accounted for. As more information on distribution of gene effects and gene frequencies and on collateral effects on fitness for quantitative traits become available, more realistic models can be applied. In the meantime,however, an infinitesimal model continues to bea useful reference point to interpret selection experiments and to predict selection response. We are grateful to B. VILLANUEVA and particularly J. DEKKERS for useful comments on the manuscript and to the Biotechnology and Biological Sciences Research Council for financial support.
LITERATURE CITED BELONSKY, G. M., and B.W. KENNEDY, 1988 Selection on individual phenotype and best linear unbiased predictor of breeding value in a closed swine herd. J. Anim. Sci. 66: 1124-1131. BRISBANE, J. R., and J. P. GIBSON, 1995 Balancing selection response and rate of inbreeding by including genetic relationships in selection decisions. Theor. Appl. Genet. 91: 412-431. BULMER, M. G., 1971 The effect selection on genetic variability. Am. Nat. 105: 201-211. BUNGER,L.,and G. HERRENDOER,1994 Analysisof a long-term selection experiment with an exponential model.J. Anim. Breed. Genet. 111: 1-13. CABALLERO,A,,1994 Developments in the prediction ofeffective population size. Heredity 7 3 657-679. CABALLERO, A., and P.D. KEIGHTLEY, 1994 A pleiotropic nonadditive model of variation in quantitative traits. Genetics 138: 883900. CABALLERO, A,, P.D. KEIGHTLEY and W. G. HILL,1995 Accumulation of mutations affecting body weight in inbred mouse lines. Genet. Res. 65: 145-149. CABALLERO, A., E. SANTIAGO and M. A. TORO,1996a Systemsof mating to reduce inbreeding in selected populations. Anim. Sci. 62: 431-442. CABALLERO, A., M. WE1 and W. G. HILL, 1996b Survival rates of mutant genes under artificial selection using individualand family information. J. Genet. (in press). CHEVALET, C., 1994 An approximate theory of selection assuming a finite number of quantitative trait loci. Genet. Sel.Evol. 26: 379-400. CLAWON, G., and A. ROBERTSON, 1955 Mutation and quantitative variation. Am. Nat. 89: 151-158. COCKERHAM, C.C., and H. TACHIDA, 1987 Evolution and maintenance of quantitative genetic variation by mutations. Proc. Natl. Acad. Sci. USA 8 4 6205-6209. DEMPFLE, L., 1974 A note on increasing the limit of selection through selection within families. Genet. Res. 24: 127-135. DEKKERS, J. C. M., 1992 Asymptotic response to selection on best linear unbiased predictors of breeding values. Anim. Prod. 5 4 351-360. FALCONER, D. S., and T. F. C. MACKAY, 1996 Introduction to Quantitative Genetics, Ed. 4. Longman, London. FRANKHAM, R., 1980 Origin of genetic variation in selection lines, pp. 56-68 in Selection ExperZments in Laboratory and Domestic Animals, edited by A. ROBERTSON. Commonwealth Agricultural Bureaux, Slough. GOMEZ-RAYA,L., and E.B. BURNSIDE, 1990 Linkage disequilibrium effects on genetic variance, heritability and response after repeated cycles of selection. Theor. Appl. Genet. 79: 568-574.
1971 GRUNDY, B., A. C A B A L L EE. RO SANTIAGO , and W.G. HILL,1994 A note on using biased parameter values and non-random mating to reduce rates of inbreeding in selection programmes. Anim. Prod. 5 9 465-468. HEATH, S. C., G. BULFIELD, R THOMPSON and P. D. KEIGHTLEY, 1995 Rates ofchange of genetic parameters of body weightin selected mouse lines. Genet. Res. 6 6 19-25. HECK,A., 1993 Introduction to Maple. Springer-Verlag, NewYork. HENDERSON, C. R., 1975 Best linear unbiased estimation and prediction under a selection model. Biometrics 31: 423-447. HENDERSON, C. R, 1982 Best linear unbiased prediction in populations that have undergone selection, pp. 191-201 in Proceedings of the World Congress of She@ and Beef Cattle Breeding, New Zealand, Vol. 1, edited by R. A. BARTON and W. C. SMITH.Dunmore Press, Palmerston North, New Zealand. HILL,W. G., 1976 Order statistics of correlated variables and implications in genetic selection programmes. Biometrics 3 2 889902. HILL,W. G., 1982 Predictions of response to artificial selection from new mutations. Genet. Res. 4 0 255-278. HILL,W. G., 1985 Effects of population size on response to short and long term selection.J. Anim. Breed. Genet. 102: 161-173. HILL,W. G., and P. D. KEIGHTLEY, 1988 Interaction between molecular and quantitative genetics, pp. 41 -55 in Advances in Animal edited Breeding-Symposium in Honour of Professor R D. POLITIEK, by S. KORVER, H. A.M. VAN DER STEEN, J. A. M. ARENDONK, H. Pudoc. WagenBAKKER, E. W. BRASCAMPand J. DOMMERHOLT. ingen, Netherlands. HILL,W. G., and J. &BASH, 1985 Modelsof long term artificial selection in finite population. Genet. Res. 4 8 41-50. HILL,W. G., A. CABALLERO and L. DEMPFLE, 1996 Prediction of response to selection within families.Genet. Sel. Evol. (in press). KEIGHTLEY, P.D., and W. G. HILL,1987 Directional selection and variation in finite populations. Genetics 117: 573-582. KEIGHTLEY, P. D., and W.G. HILL,1992 Quantitative genetic variation in bodysizeofmice from new mutations. Genetics 131: 693-700. LATTER, B. D.H., 1966 Selection for a threshold character in Drw sophila. 2. Homeostatic behaviour on relaxation of selection. Genet. Res. 8: 205-218. LYNCH, M., 1988 The rate of polygenic mutation. Genet. Res. 51: 137-148. LYNCH,M., and W. G. HILL,1986 Phenotypic evolution by neutral mutation. Evolution 4 0 915-935. LUSH,J. L., 1947 Family merit and individual merit as bases for selection. Am. Nat. 81: 241-261, 362-379. MEUWISSEN,T. H. E., 1990 Reduction of selection differentials in finite populations with a nested full-halfsibfamily structure. Biometrics 47: 195-203. NICHOLAS, F.W., and A. ROBERTSON, 1980 The conflict between natural and artificial selection in finite populations. Theor. Appl. Genet. 56: 57-64. PEARSON,IC, 1903 Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs. Phil. Trans. R. SOC.Lond. A 200 1 66. PRESS,W. H., S. A. TEUKOLSKY, W. T. VETTERLING and B. P. FLANNERY, 1992 Numerical RecipesinC,Ed. 2. Cambridge University Press, Cambridge. RAWLINGS, J. O., 1976 Order statisticsfor a special classof unequally correlated multinomal variates. Biometrics 3 2 875-887. ROBERTSON, A., 1960 A theory of limits in artificial selection. Proc. R. SOC.Lond. B. 153: 234-249. ROBERTSON, A., 1961 Inbreeding in artificial programmes. Genet. Res. 2: 189-194. QUINTON, M.,C. SMITHand M.E. GODDARD, 1992 Comparison of selection methods at the same level of inbreeding. J. Anim. Sci. 70: 1060-1067. SANTIAGO, E., and A. CABALLERO, 1995 Effective size of populations under selection. Genetics 139: 1013-1030. TORO,M. A., and M. PEREZ-ENCISO, 1990 Optimization of selection response under restricted inbreeding. Genet. Sel. Evol. 22: 93107. VERRIER, E., J.J. COLLEAU and J. L. FOULLEY, 1991 Methods for predicting response to selection in small populations under additive genetic models; a review. Livest. Prod. Sci. 2 9 93-114. VERRIER, E., J.J. COLLEAU and J. L. FOULLEY, 1993 Long-term effects
M. Wei, A. Caballero and
1972
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 using univariate and multivariate bestlinear unbiased predictors. Anim. Prod. 57: 1-13. VIIMNUEVA,B., J. A. WOOLLIAMS and G. SI", 1994 Strategies for controlling rates of inbreeding in adult MOET nucleus schemes for beef cattle. Genet. Sel. Evol. 2 6 517-535. WEBER,K E., and L. T. DIGGINS,1990 Increased selection response in larger populations. 11. Selection for ethanol vapor resistance in Dmsophila melanogaster at two population sizes. Genetics 125 585-597. WOOLLIAMS, J. A., N. R. WRAYand R. THOMPSON, 1993 Prediction of long term contributions and inbreeding in populations undergoing mass selection. Genet. Res. 6 2 231-242. WRAY,N. R., and M. E. GODDARD, 1994 Increasing long-termresponse to selection. Genet. Sel. Evol. 2 6 431-451. WRAY,N. R., and W. G. HILL,1989 Asymptoticrates of response from index selection. Anim. Prod. 49: 217-227. WRAY, N. R., and R. THOMPSON, 1990 Prediction of rates of inbreeding in selected populations. Genet. Res. 55: 41-54. WRAY,N. R., J. A. WOOLLIAMS and R. THOMPSON, 1994 Prediction of rates of inbreeding in populations undergoing index selection. Theor. Appl. Genet. 87: 878-892. YOO, B. H., 1980a Long-term selection for a quantitative character in large replicate populations of Drosophila melanogaster. 1. Response to selection. Genet. Res. 35: 1-17. YOO, B. H., 1980b Long-term selection for a quantitative character in large replicate populations of Drosophila melanogaster.2. Lethals and visible mutants with large effects. Genet. Res. 35: 19-31. ZENG, Z. B., and C.C. COCKERHAM, 1993 Mutationmodels and quantitative genetic variation. Genetics 133: 729-736. 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.
W. G. Hill
+ h2,I2 + 47dh:(1
2 1/2 - ho))
where @ = uio/ (aio + a:). Equation A1 reduces to and BURNSIDE(1990) when that given by GOMEZ-RAYA N7and ND+ w, i.e., yrnl= 1 and y d = ( k ~ s kD)/2. Family selection: The additive variance offamily means at the limit (05~) can be derived as a function of initial additive variance of family means (a:p) and k, ( i e . , k for family selection). Analogously as before, from (8) the recurrent equation at the limit is
+
ai = 0.25 (1 - l / N . ) ( l
- k,p;)a;
+ 0.25 (1 - l/ND) (1 - k&)a: + 0:0/2, where k, = kD = k . under family selection. p; = a&/(aia$) is the limiting value of squared accuracy 0% is the under family selection, where a$ = aif phenotypic variance of family means at the limit. The recurrent equation can be rearranged as ai = a i o / [ (2 yf) + kfyp;] where yf = 1 - (l/Ns + 1/N1)/2. Substituting p; gives
+
[r,(k,- 1) + 2 1 4 + r p d ,
Communicating editor: Z-B. ZENC
APPENDIX A
Derivation of additive genetic variance at the limit under mass selection and family selection: The expressions derived in this appendix are given as a function of ai and aio. To apply them for any generation t, replace a : and aioby a i t 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 population. For a finite population, we still assume that the within-family variance (aio/2) does not change over generations, but consider a correction in the betweenfamily variance for the number of sires and dams selected (N, and No) due to the sampling of parents with replacement (KEIGHTLEY and HILL1987;VERRIERet al. 1993). Thus, from (8)
ai
=
0.25(1 - l/N.) (1 - k.&)ai
+ 0.25 (1 - l/ND)(l
-
k,&)d
+ d0/2,
where p: and p; are the squared limiting accuracy of selection in siresand dams, respectively, and are equal to each other and to the limiting heritability h2 = aV(ai + a:). Rearranging the equation a i = aio/(yml+ y,h2), where Yrnl = 1 (I/& + 1/ND)/2, and ym2 = [(I - 1/Ns)k.$ f (1 - 1/ND)kD1/2, giving
u2
-
Af-
1 2[yXk,-
1) + 21
where
APPENDIX B
Rate of change in selection responsewith respect to rate of change in genetic variance for different selection schemes: All the derivations in this appendix are evaluated at generation 0, which is taken as a reference generation. The subscript 0 , however, hasbeen omitted from all parameters, for simplicity. Mass selection: The derivative pm= d log R,,,/d log aiwcan be expressed as Pmlpm2 = ( d log R J d log ai) X ( d log o i / d log aiw). From (7), pml= 1 - h 2 / 2 . Substituting (1) into (Al) and deriving
+ [h2 + (yml+ 2y,)(l- h2)l +([yrnl(l- h2)+ h2I2+ 4y,h2(1 - h2))l/' yrnl(l- h2) - h2 + {[yrnl(l - h2) + h2I2 ' + 4y,h2(1 - h2))l/'
-1 p,=1-h2
(A4)
1973
Selection Response
where ymland ym2are given in Mass selection of APPENDIX A. The function Pm2increases with increasing hz, and limh' + 1 Pm2= 1. Equation A4 also applies to a full-sib ( y m l= 1 1/[2ND] and ym2= [ l - l/ND]kD) or half-sibfamily structure ( y m l= 1 1/ [2Ns], and ym2 = [1 - 1/Nslks). Family selection: Analogously to the above, Pp = (1 - h;/2), where hj is the limiting value of heritability of family means, and
+
+
(1 - n ) y f - 2
-
n+ 1
- 2Yf2]
(A5) where yf, yfl and yf2 are given in Family selection of APPENDIX A. The function Pf2 decreases with increasing h; and family size, and limh2 + 1 Pf2 = 1. Index selection: First, we just derive an expression (Pn)without accounting for the Bulmer effect. The index consists of individual performance (expressed as deviation from full-sib family mean), full-sib mean (as deviation from half-sib familymean) and half-sib family mean. The phenotypic variance-covariance matrix is diagonal with the following elements: PI1
=
PIZ = PI,
=
[ ( n- l)/nI(aS
+ ai/2),
[(m - l ) / ( n m ) l [ d + {a: + na2c
+
[n(m + 1) + 2Iai/4)/(nm),
wn = [(n - 1 ) / ( 2 n ) I g i , =
wn =
[(n + 2)(m - 1)/(4nm)]ai,
[2
+ n(m + 1)]oi/(4nm).
The variance of the index is a : = Z dj/po. Thus, Pn = d log Rl/d log aiw is
+
+
+
+
+
PBl
= [(n - 1 ) / n ] ( d
[(m - l ) / ( n m ) I [ d
+ d/2),
+4
pB2
=
pB3
+ ( n + 2)ai/4] - [1/4 - 1/4m]p2ai, = [a: + n a: + ( n + nm + 2)ai/4]/nm - [1/4 + 1/4m]p2ui,
d + (n + 2)d/41,
where n is the full-sib family size, and m is the numberof dams mated toa sire. The vector of covariance between genetic values and index information has the following elements:
WE
reduces to mass selection and (A6) reduces to (7). For the extreme case where n UJ and a : = 0, Pn 0.5h2 (1 - h z ) / ( 2 - h'). In general, the value of Pn lies between 0.5 and 1, and decreases with increasing familysize ( n and m) and/or h2. When the Bulmer effect is accounted for it is, however, very tedious to express ai as a function of aiw, a: and k, and to obtain its derivative. The formula for PI accounting for the Bulmer effect is not presented, and a numerical solution has been used (PRESSet al. 1992). BLUP selection: BLUP selection is approximated by selection based onanindex (pseudo-animal model BLUP) consisting of individual record (I), full-sib mean (FS), half-sib mean (HS), estimated breeding values of thedam (AD), themean of the estimated breeding value of all dams mated to the sire ( A D , ) and the estimated breeding value of the sire (A,y) (WMY and HILL 1989). To facilitate the derivation, the index is set up using independent linear functions of the records so the variance-covariance matrix is diagonal, where the diagonals are for (1- m), (n- HS - &/2 Au/2), (HS - As/2 - Au/2), and (AD/2 As/2). These diagonal elements are
p,
= p2ai/2.
Here, p2 is the squared accuracyofBLUP selection. The vector of covariancesbetween breeding values and information sources has elements WBl
=
[(n- 1 ) / ( 2 n ) ] d ,
WBZ
=
[(n+ 2)(m - l ) / ( n m ) l a i - [1/4 - 1/4m]p2ai,
WBS
=
[(n+ 2 + nm)/(4nm)]ai
wB4
= p2&2.
-
[1/4
The variance of the index is aiI = Z . . . , 4. Let 6 = d p2/d ai, then
2prs -
+
2
+ n(m + 1)
$3 ~3
"1
. (A6)
When half-sib information is not available ( m = l ) , then the second term in these equations ( wR, p R ) drops out; and if there are onlyhalf-sibs ( n = l ) , the first term drops out.Similarly, if n = 1 and m = 1, the index
+ 1/4m]p2ai,
wij/ps, for j
= 1,
1974
M. Wei, A. Caballero and W. G. Hill
HENDERSON (1982) found that under animal model BLUP the variance of prediction error is determined only by the amountof information on self and relatives and is not affected by selection. Hence, DEKKERS (1992) showed that theratio of asymptotic to original response is equal to a constant, [2/ (2 ks k D ) ]1’2. Therefore, in this case p B= pBl= d log R/d log cTiwand,therefore,
+ +
m = 1, and the second term in w and p in (A7) and (A8) is dropped. Further,when the pseudo-BLUP index includes only individual information and parental estimatedbreeding values, Le., the case that DEKKERS (1992) considered to provide lower limit to accuracy of = 0; BLUP, wBJ = (1 - p2/2)& WM = p2&2, 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,
