Selection Index for Attaining Breeding Goals. Y. Yamada. Generally, a selection index estimates the value of an individual

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1 127 Selection Index for Attaining Breeding Goals Y. Yamada Kyoto University Kyoto, Japan Generally, a selection index estimates the value of an individual for an aggregate genotype which is expressed in financial terms in a linear form of additive genetic merits for component traits weighted by their relative economic values. The first paper oll selection index was given by Smith (1936) who applied directly the discriminant function of Fisher (1936) to multi-traits selection in plant populations. Hazel (1943) developed essentially the same index to apply in animal populations using path coefficients devised by Wright (1921). Later, selection indexes have been extended ill various directions. The most important development is the restricted selection index by Kempthorne and Nordskog (1959). Subsequently, some modifications of this index were proposed, e.g., the index with inequality constraints by Rao (1962), the index for optimal genotype by Tallis (1962), and the index with proportionality constraints by Harville (1975). Selection index for desired genetic gains was derived by Pesek and Baker (1969) in plant, Yamada et ai.(1975), and Rouvier (1977) in animals. These were independently developed from each other. Nevertheless, the unique point of Yamada et a1.(1975) from others is that neither the economic weights nor the underlying aggregate genotype is considered in the derivation, which is essential in the others. For the past few years I have been studying the characteristics

2 128 of these selection indexes proposed by various authors, and the results were published elsewhere (Itoh and Yamada, 1986, 1987, 1988). Therefore, the details will not be given here. The objectives of this paper are to clarify the difference of our selection indexes from others and to provide the usefulness of our recent methods under various conditions in animal and poultry breeding practice, although very brief explanations of various selection indexes will be given for understanding theoretical basis of each index. I. Smith-Hazel's selection index The breeding value of an individual is defined as H = a'g = algl an_ 7 (I) where a = a vector of relative economic weights g = a vector for additive genetic values The selection index is defined as I = b'p = blpl +.., + bnp n (2) where b = a vector of weight coefficients P = a vector of phenotypic values expressed as the deviation from their respective means The variance and covariance of I and H are a 2I = Vat(1) = b' Pb, a _ = Vat(H)= a' Fa, ahi = Cov(H,l) = b'ga where P = Vat(p), F = Vat(g), G = Cov(p,g), which are assumed to be known. Assuming I and H be distributed as bivariate normal, then from the definition of regression, the genetic change in H due to selection for I is expressed to be:

3 129 AH = bhi AI = irh[c H where bhi= regression of H on [, AI = selection differential i = standardized selection differential, rhi= accuracy of selection, Maximizing AH, we obtain 2 Pb = [ Ga (3) aih The value _ / OiH does not affect the relative size of the 2 : 1 Then, we obtain elements in b, thus we set I / CIH ' Pb = Ga (4) Consequently, the vector of appropriate weights is obtained by b = P-IGa (5) Another derivation is also possible. Now consider the expected squared difference between [ and H as: E(I-H) 2 = b'pb - 2b'Ga + a'fa (6) This is sometimes devoted by Var(I-H). Using 2/ / _IH = i, it can be shown that (Henderson, 1963): E(f-H) 2 = all(i-r/h) (7) This indicates that minimizing E(I-H) 2 is equivalent to maximizing rhi. Thus, E(I-H) 9 _b - = 2Pb - 2Ga = O Pb = Ga. (8) II. Restricted selection index of Kempthrone and Nordskog (1959) Kempthorne and Nordskog (1959) proposed the selection index which gives no change in some linear functions and traits. Consider the

4 130 linear functions of g given by C'g, where C is a constant m x r matrix and its columns are linearly independent. If C'g is desired not to be changed by selection for an index, then the index should satisfy the condition: Cov(C'g,I) = C'G'b = 0 (9) So, the index is to be chosen such that rhi is maximum (or E(I-H) 2 e minimum) subject to the constraint that C'G'b = O. The index weight vector of this index is given by bkn = [I - p-1g C(C'G'p-1G C)-Ic'G']p-1Ga (10) This index is called the restricted selection index. A simple example of this index is the case where the r traits themselves, instead of their iinear functions, are considered not to be changed. Let C : [Irx r O] and G = [G 1 G2 ] and further a'=[a i a½] in which the subscripts 1 and 2 are for the constrained and unconstrained traits, respectively. The index weight vector becomes bk _ = [I - p-igi(gip-igi)-igilp-iga = [I - p-igi(gip-igi)-igi]p-i[giai+ G2a 2] = [I - p-igi(gip-1g1)-igi]p-ig2a2 (11) The result indicates that the relative economic u eights of the constrained traits have no effect on the derived selection index. Abpllanalp, Ogasawara and Hill (1963) developed a simple method to restrict the genetic change in a component trait, using (9) as the contraint to solve normal equations. biuch simpler technique to obtain the restricted selection index as that of Kemptho_-ne and Nordskog (1959) was presented by Abe, Yamada and Nishida (1969) and Cunningham, Moen and Gjedrem (1970). Their method is to solve the following equations directly:

5 131 where k is a vector of Lagrange multipliers. These equations can be obtained by minimizing Var(l-H) subject to the equations (9). Ill. Difficulties of assessing relative economic values I. Assessing relative economic values is essential for constructing selection index, and very accurate economic analyses are required to assess economic importance of each trait, which will be incorporated in the index. To attain reliable estimates a large volume of data is required. 2. In the manifestation of these traits various stages of animal's life are involved. But not all of these traits are included in the index. We must select only a few but very important traits. Nevertheless, there is no objective standard to choose these traits. 3. Economic situations may change with time and location. They may change easily with external factors and the change in the population mean by selection. Therefore, even though an accurate economic assessment is made, inaccurate relative weights may likely result after selection. 4. There are several traits such that one can hardly assess their importance, i.e., body conformation. 5. Desirability views of breeders, producers, marketing dealers and consumers are often quite different. Breeder's view for economically optimal genotype is often different by locality. 6. It is very logical and desirable to improve a population mean linearly toward the economically optimal level. But economical

6 132 environments tend to shift with time and the balance between demand and supply. In the past few years a good number of papers on the evaluation of selection goals or breeding objectives have been published (James, 1978, 1982; Melton et al., 1979; Smith, 1983; Elsen et al, 1986; Brascamp et al., 1985; Shultz, 1986; Nordskog, 1986; Smith et al., 1986 and so on). Among these authors, Elsen et ai.(1985) made twenty remarks on economical evaluation of selection goals and concluded that the selection goal is to maximize decider's profit. Nevertheless, breeding objectives based on any profit function can not be free from these difficulties mentioned above. On the contrary, my own attitude for the breeding goal is very simple and straightforward. Breeding in practice must be undertaken to accomplish a definite goal in each population. The goal depends on the level of performance in each trait and thus it may vary in each population. Breeder's immediate concerns are to level up certain traits which are decidedly inferior to his competitor's, providing that other traits showing superiority or equivalency remain the same levels. If breeder's view is negative or otherwise it takes many generations to catch up the level of his competitor's flock, he should replace the stock or give up his business before bankruptcy. N. Selection index for attaining breeding goals (Yamada, Yokouchi, Nishida, 1975; Itoh and Yamada, 1986) I. Selection for definite goals Breeding goals are definite as fixed values and k is a vector of desired genetic gains in m traits due to one generation of selection

7 133 for I which is composed of n traits: 49 = C v(9'i) _ i Cov(9 I) - i G'b (13) a} _I- a---7, a[ Let e be a positive scalar and the reciprocal of e is equal to the number of generations to attain the goal, then b should satisfy the following condition: i G'b = ek (14) Let O = i/_ I, then G'b = k (15) When n = m, we obtain, b = (G )-ik (16) which has a unique solution. However for n > m, no unique solution exists. Thus, the optimal choice is to choose such a b which makes dg maximal. For doing so, as Ag is inversely proportional to gi' we should choose b such that g_ = b'pb is minimal subject to the condition (15). In order to obtain the solution in more generalized form, we express the problem in the following way, G'b = k b'pb _ minimum To obtain b, we solve the following equations, (0] = (17) 6' 0 X k Then, we obtain for n > m, b = P-IG(8'P-18)-Ik (18) If we set k be the vector of the difference of the means between the present population and the breeding goal, tlle number of generations of selection to attain the goal is equal to ai/i.

8 134 Equation (18) was obtained by Harville (1975), Rouvier (1977), Essl (1981), Tallis (1985) from different approaches. Equation (16) is identical to the solution by Pesek and Baker (1969), who derived the equation based on equation (5). As it was mentioned earlier along with equation (11), once constraints are assigned to the genetic changes in certain traits, the relative economic weights of the constrained traits have no effect on solution. 2. Merits of this index (I) Breeding goal should be attained at tile minimal number of generations of selection, which is equal to (7i/i. (2) Accurate economical analysis is not required. (3) As the breeding goal is similar to "ideal type", it is easy to understand intuitively. (4) The goal does not change from generation to generation. (5) Breeders can choose the goal by their own decision. V. Selection indexes for desired gains with inequality contraints Despite of the usefulness of our index, there are some cases where it is difficult or unnecessary to determine definite breeding goals of certain traits precisely. It may be more convenient and more efficient to improve the population for such a flexible goal than for a strictly fixed goal. Furthermore, there are cases where one does not know or does not want to specify the breeding goals for certain traits but simply wants to avoid their deterioration. All inequality constraint forcing the genetic changes for such traits to be merely non-negative seems much more adequate than the arbitrary choice of a

9 135 direction of genetic improvement. Itoh and Yamada (1988) extended the selection index for proportional desired gains with inequality constraints in more general forms. However, the principles are essentially the same as of Itoh and Yamada (1988). Partition the traits into two groups, and let the desired gains of the first group be equal to k I and those of the second group be equal to k 2. For example, consider only two traits for simplicity. If the genetic gain for trait 2 is desired to be at least twice the genetic gain for trait l, then kl= l as a reference and k2= 2. If the genetic change for traits 2 is desired to be simply non-negative, we set kl= I, and k2> O. Let gl and g2 be the vectors of additive genetic values for the first and second groups, and GI= Cov(P,gl), G2= Cov(P,g2), then b should satisfy the following conditions, based on the same argument to obtain (15): Gib = k I and G½b _ k 2 If no unique solution exists, the optimal b is such that G{b = k 1 G½b Z k2 (19) b'pb _ minimum This is a problem of quadratic programming subject to inequality constraints. Next, consider such a case where k I be the vector of the desired genetic gains in the first group and that of the second group be equal to or less than k3 and g3 be the vector of additive genetic values of the second group traits, G3= Cov(P,g3). Then we solve

10 136 G{b = k 1 G_b _< k 3 (20) b'pb _ minimum We can extend the problem further. Namely, k I is the vector of the desired relative gains in the first group and the desired gains of the second group of the traits are between k 4 and k5. And g4 be the vector of the additive genetic values of the second group of the traits, and G4= Cov(P,g4). Then the optimal b will be obtained from: Gib = k I F k 4.< G4b.< k 5 (21) b'pb _ minimum It is also possible to combine some constraints mentioned above in the following way: G_b = k 1 G½b _> k2 G_b s k 3 (22) k 4 -< G4b <. k 5 b'pb _ minimum Discussion The most crucial issue in the application of selection index in animal breeding practice is how to define, the.breeding. goals. Some people recommend using profit equations for deriving economic weights(brascamp et al. 1985). Others emphasize the importance of biological productivity rather than economic one(maijala, 1976). Both

11 137 views could be legitimate ill appropriate circumstances. One should not hesitate to adopt a profit function as far as it represents explicitly the profit of the industry concerned. But, there is no assurance for the fitness of this function for a long time span, even though we ignore the time lag between the price in the market and the cost in the farm. Profit may fluctuate from time to time so that this function should be re-evaluated in every generation. Analyses, however, are very rare. The arguments on the derivation of economic weights so far published (Smith et ai.,1986; Brascamp et al., 1985) were more or less theoretical or otherwise sophistical and thus they do not have much impact upon practical implication. On the other hand, the decision of the breeding goals by breeders was often criticized as not objective. In fact, objective choice of these differential weighting of each trait by breeders is difficult. But, for the experienced breeders it is easy to evaluate his own flocks or populations in relation to his competitors' products on the basis of Random Sample Tests, i.e., how much improvement should be made in trait A and trait B so as to exceed his competitors. This can be made from the breeder's experience based on his accumulated data rather than a profit function. If a breeder has no such capability, he will face bankruptcy sooner or later anyway. I can decide very definitely the breeding goal for each strain in my own breeding farm, in terms of the difference in each trait between the desired level and the present population mean. Our selection index which was developed to attain these objectives simultaneously will tell how many generations are needed to accomplish the goal in theory. If the number of generations with a given intensity of selection is too long, some additional traits can be taken into consideration as concomitant

12 138 traits. This may reduce the number of generations to some extent. If such does not affect the number, the breeder should give up this population and select other strains for further improvement. Selection index for attaining breeding goals can be applied either to a short-term program or a long-term one, depending on the amount to be improved. The generational fluctuation of the population means can be incorporated into k in every generation without changing ultimate goals. The number of traits included in the breeding objective must be fairly large. As far as their economic importance is recognized, these traits should be incorporated into k. Nordskog (1986) listed the number of type and production traits in e_-tvpe== _ chickens. On the contrary, the number of traits included in the index should be small, preferably be less than five. Theoretically the more number the better. Considering the error of the measurement and the bias from multinormal distributions of variance and covariance estimates, the number should be limited. If the number of the traits in the breeding objective exceed that in the index, i.e., m > n, solution is not easily obtained. Because m- dimensional breeding goal is approximated by n-dimensional index. The best choice would be the index which has the minimal variance (a{). I References Abe,T., Yamada,Y., Nishida,A. (1969) On the method constructing restricted selection index. Jpn. Poult. Sci. 6: Abplanap,H., Ogasawara, F.K. and Asmundson,V.S. (1963) Influence of selection for body weight at different ages on growth of turkeys. Brit. Poult. Sci. 4: Brascamp,E.W., Smith,C., and Guy,D.R. (1984) Derivation of economic

13 139 weights from profit equations. Antm. Prod. Sci. 40: Cunningham,E.P., Moen,R.A., and Gjedrem,T. (1970) Restriction of selection indexes. Biometrics 26: Elsen,d.M., Bibe,B., Landais,E. and Ricordeau,G. (1986) Twenty remarks on economic evaluation of selection goals. Proceedings of 3rd World Congress on Genetics Applied to Livestock Production. Lincoln, Nebraska, 12: Essl,A. (1981) Index selection with proportional restriction: Another view point. Z. Ti_rzuchti_. Zuchtungsbiol. 98: Fisher,R.A. (1936) The use of measurements in texonomic problems. Ann. Eugen. 7: Harville,D.A. (1975) Index selection with proportionality constraints. Biometrics 31: Hazel,L.N. (1943) The genetic basis for constructing selection indexes. Genetics 28: Henderson,C.R. (1963) Selection index and expected genetic advance. Statistical Genetics and Plant Breeding. ed. W.D.Hanson, W.D.Handon, and H.F.Robinson. Natl. Acad. Sci. Nat. Res. Council Publ. 982: Itoh,Y. and Yamada,Y. (1986) Re-examination of selection index for desired gains. G6n6t. $61. Evol. 18: Itoh,Y. and Yamada,Y. (1987) Comparisons of selection indices achieving predetermined proportional gain_.'gene. Sel. Evol. 19: Itoh,Y. and Yamada,Y. (1988) Selection indices for desired relative genetic gains with inequality constraints. Theor. Appl. Genet. 75: James,J.W. (1978) Index selection for both current and future generation gains. Anim. Prod. 26:111-I18. James,J.W. (1982) Economic aspect of developing breeding objectives: general considerations. Future Developments in the Genetic Improvement of Animal. Academic Press Australia. pp Kempthorne,O. Nordskog,A.W. (1959) Restricted selection indices. Biometrics 15: Maijala,K. (1976) General aspects in defining breeding goals in farm animals. Act. Agric. Scand. 36: Melton,B.E., Heady,E.O., Willham,R.L. (1979) Estimation of ecomomic values for selection indices. Anim. Prod. 28: Nordskog,A.W. (1986) Economic evaluation of breeding objectives in layer-type chickens. Proceedings of 3rd World Congress on Genetics Applied to Livestock Production. Lincoln, Nebraska, 10:

14 140 Pesek,J., Baker,R.J. (1969) Desired improvement in relation to selection indices. Can. J. Plant Sci. 49: Rao,C.R. (1962) Problems of selection with restrictions. J. Roy. Statist. Soc. Set. B 24: Rao,C.R. (1965) Problem of selection involving programming techniques. Proceedings of the IB>I Scientific Computing Symposium on Statistics, White Plains, New York, IBM Data Proceeding Division, Rouvier,R. (1977) Mise au point sur le modele classique d'estimation de la valuer genetique. Ann. G6n4t. Sel. Anim.'9_ Shultz,F.T. (1986) Formulation of breeding objectives for poultry meat production. Proceedings of 3rd World Congress on Genetics Applied to Livestock Production. Lincoln, Nebraska, 10: Smith,C., James,J.W. and Brascamp,E.W. (1986) On the derivation of economic weights in livestock improvement. Anim. Prod. 43: Smith,H.F. (1936) A discriminant function for plant selection. Ann. Eugenics 7: Tallis,G.M. (1962) A selection index for optimum genotype. Biometrics 18: Tallis,G.M. (1985) Constrained selection. Jpn. J. Genet., 60: Corrigendum and addendum, Jpn. J. Genet. 61: Wright,S. (1921) Correlation and causation. J. Agric. Res. 20: Yamada,Y., Yokouchi,K., Nishida,A. (1975) Selection index when genetic gains of individual traits are of primary concern. Jpn. J. Genet. 50:33-41.

15 141 APPENDICES 4 Numerical Examples 1 Figure

16 142 >< i.,-,.].,,,o i CO /%. f..d r_ x... -:._ "_: -._

17 The number of traits is The number of constraints is 5 List of constraints 1 delta g(1) = k 2 delta g(2) < -0.i000 k 3 delta g(3) > k 4 delta g(3) < k 5 delta g(4) < k Phenotypic variance covariance matrix , Genotypic variance covariance matrix Satisfies the Equality constraints No Variance inequalities? i No * No * * No * * No * * * No * * No * * *? No * * No * * * Yes * * * Yes * * * * Yes * * * Yes * * * * Optimal index is No.ll Optimal index weights Variance of the index Expected genetic gains

18 The number of traits is 4 The number of constraints is List of constraints 1 delta g(1) = k 2 delta g(2) = k 3 delta g(3) = k 4 delta g(4) = " k Phenotypic variance covariance matrix Genotypic variance covariance matrix i Satisfies the Equality constraints No Variance inequalities? Yes * * * * Optimal index is No. 1 Optimal index weights Variance of the index Expected genetic gains

19 The number of traits is 4 The number of constraints is List of constraints 1 delta g(1) = k 2 delta g(2) = k 3 delta g(3) > k 4 delta g(4) = k Phenotypic variance covariance matrix i Genotypic variance covariance matrix Satisfies the Equality constraints No Variance inequalities? Yes * * Yes * * * Optimal index is No. 1 Optimal index weights Variance of the index Expected genetic gains

20 The number of traits is 4 The number of constraints is List of constraints 1 delta g(1) = k 2 delta g(2) = k 3 delta g(3) > k 4 delta g(4) > k Phenotypic variance covariance matrix , Genotypic variance covariance matrix , S'atisfies the Equality constraints No Variance inequalities? Yes * * Yes * * * Yes * * * Yes * * * * Optimal index is No. 1 Optimal index weights Variance of the index Expected genetic gains

21 147 Question: G. Herbert Should the breeder not be concerned with moving as quickly as possible from his current position on the profit surface to the next higher iso-profit contour as rapidly as possible rather than with changing traits by (fixed) specific amounts in the minimum amount of time? Response: Y. Yamada I disagree with your viewpoint, because the profit surface changes with the balance between the demand and the supply as well as individual taste. The latter changes substantially depending on people's socio-economic situations. Therefore, there is no guarantee that the same profit equations can be applied to two successive generations. You may face a catastrophy if the balance of demand/supply were broken. Breeders prefer to improve their animals for attaining a more realistic goal rather than uncertain economic weights. Question: J.P. Gibson You say that restricted indexes can be used in many cases because economic weights are uncertain. However, every selection index has an implied or retrospective set of underlying economic weights, so that the economic weights have been predetermined without ever seeing what they are. Would it not be better to use your best estimates or even guesses at the economic weights in a standard index than to avoid the issue by using a restricted index? Response : Y. Yamada Your argument that every selection index has a retrospective set of underlying economic weights depends on the assumption that the relationship of Pb=Ga exists a priori rather than... posterlorl so that you can estlmate a-g Pb. On the contrary, my logic starts from the fact that the breeder concerns primarily the ratio of correlated genetic gains of the index, i.e., _G:,I/mG, I, in which I is a selection index. Surely, one can'deriv_'a set of retrospective economic weights by setting _GI.I/_GjI = _ /_,. But, I do not, "o understand why you thlnk it is better to than to avoid the _w issue by using mg_.i/ag]. I. In my notation, a stands for retrospective sets.

22 148 Question: Alan Ensley The commercial product is a cross. Component lines under selection are pure with variable contributing merits on a given trait. Comment on how your procedure takes this into account, please. Response: Y. Yamada According to Drs. Hutt and Cole of Cornell, the performance of a hybrid or a cross between two purebreds was nearly parallel to the genetic improvement obtained in two purebreds. Thus, assuming the desired improvement in the crossbred berg, then the breeding goals of two strains would be PI + A C and P_ +_ C, respectively. Appropriate modification can be made, based on the information gathered from preliminary combining tests.