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An iterative procedure for deriving selection indexes with constant restrictions¹

Dairy and Swine Research and Development Centre, Agriculture and Agri-Food Canada, Lennoxville, Quebec, Canada J1M 1Z3

	Abstract

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The objective of this study was to present an iterative procedure for deriving selection indexes with constant restrictions. Constant restriction means that the genetic responses of the restricted traits are preset to actual amounts for a given selection intensity (). Results of this study show that an index with constant restriction alone or in combination with other types of restrictions possesses three distinctive characteristics: 1) the coefficient matrix of the index equations is not symmetric and is nonlinear; 2) the coefficient matrix contains unknown , indicating that the index coefficients (b) to be derived depends on the value of predetermined before selection; and 3) the coefficient matrix contains unknown b, thus requiring iterative methods to solve the index equations. As a result of these unique characteristics, the index coefficients, genetic responses of the index traits, and overall genetic gain in net merit change nonlinearly with varying levels of , which is in sharp contrast to both unrestricted and restricted indexes reported in the literature. The construction of a constant-restricted index requires predetermining the value of intended for a selection program to derive the corresponding b. An index with constant restrictions has no meaning unless it is associated with a specific value of . Numerical examples are given to illustrate the construction of the index with constant restrictions and to validate the theoretical development proposed. The derived equations have yielded an index that maximized the total merit and fulfilled constant restriction at the same time.

Key Words: Constant Restriction • Iterative Procedure • Restricted Index

	Introduction

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The principle of restricted selection index was first introduced by Kempthorne and Nordskog (1959). Since then, various types of restricted selection indexes have been developed. Brascamp (1984) gave a detailed review of different restricted indexes. Niebel and Van Vleck (1983) presented a summary of methods for restricted indexes in terms of the imposed restrictions and the procedure for solving the index weights. Lin (2005) presented a method to simultaneously impose both proportional and fixed restrictions on an index. He imposed a fixed restriction in terms of the SD of an index (_I) assuming the selection intensity of 1. The problem of expressing fixed restriction in terms of _I is that the amount of _I is unknown at the beginning of index construction. The percentage of genetic improvement desired, say 5%, can be expressed in terms of constant restriction that is equal to 5% of the population mean. Furthermore, constant restriction is needed to maintain the same genetic trend as that estimated based on historical data. The question arises as to how to impose constant restriction. Constant restriction means that the genetic responses of certain restricted traits are restricted to actual values after one cycle of selection (e.g., restrict the genetic response of body weight to 1.5 kg in pigs). The purpose of this study, therefore, is to demonstrate how to derive indexes with constant restrictions, while maximizing overall selection response in net merit.

	Materials and Methods

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Notations
The notations used here follow those of Lin (1978), who reviewed both the theory and application of various types of selection indexes. The index and net merit are defined as I = b'x and H = a'g, respectively, where x and g are (m x 1) and (n x 1) vectors of phenotypic and genetic values, and b and a are (m x 1) and (n x 1) vectors of index coefficients and economic values, respectively. It follows that _I² = b'Pb, _IH = b'Ga, and _H² = a'Fa, where P is a (m x m) phenotypic covariance matrix of x, G is a (m x n) genetic covariance matrix between x and g, and F is a (n x n) genetic covariance matrix of g. The genetic responses in component traits of H due to restricted or unrestricted index selection are = G'b(/_I), where is an (n x 1) vector and is selection intensity. The genetic response of the ith trait in H is _i = G_i 'b(/_I), where G_i' is the ith row of G'. The total genetic gain in H is H = a' = a'G'b(/_I).

Constant Restrictions
Let r be the number of traits subject to constant restriction (r n), be an (r x 1) vector of Lagrange multipliers with elements _i (i = 1,2,....r), c be an (r x 1) vector of predetermined amounts of genetic gains for the r restricted traits, and G₁ be an (m x r) submatrix of G that contains the columns of the r restricted traits. Let the breeding goal be to maximize H, subject to constant restriction of G₁'b/_I = c associated with a prespecified value of . This is equivalent to minimizing E(I – H)² subject to the constraint G₁'b/_I = c. The method of Lagrange multipliers gives the following function:

Differentiating the function f with respect to b and and equating the resulting partial derivatives to zeros results in the following equations:

[1]

[2]

Equations [1] and [2] can be jointly expressed as the following system of equations:

[3]

The coefficient 1/2 of the upper diagonal element in Eq. [3] has been dropped without affecting the values of b. Notably, the coefficient matrix of Eq. [3] contains the unknown solution b, and thus, the solution to Eq. [3] needs to be obtained by iteration. It is logical to set the unknown b contained inside the coefficient matrix equal to the unrestricted index weights (b = P^–1Ga) to start iteration. Because the coefficient matrix is not a function of , there is no need to assume the initial values for .

In Eq. [3], let and . In an animal breeding context, selection intensity and the elements of c are rather small constants. By definition, P is an (m x m) nonsingular matrix, and G₁, a submatrix of nonsingular matrix G, has full column rank r (i.e., r linearly independent columns). Consequently, A has full column rank r, and B has full row rank r, indicating that the coefficient matrix of Eq. [3] has full rank (m + r). Therefore, iteration on Eq. [3] would converge to yield a unique solution. At convergence, the solution to Eq. [3] is as follows:

It is worth noting that the predetermined constant restriction c should be biologically reasonable. As an extreme example, a predetermined constant based on 100% of genetic gain in BW after one cycle of selection would be biologically impracticable. It is advisable to compute the expected responses of the index traits to unrestricted index selection to serve as a baseline for setting the constant restriction. A predetermined constant should not deviate too much from this baseline. A failure to achieve convergence in Eq. [3] is a good indication that the predetermined constant is too extreme to be genetically feasible.

When c is a null vector (zero restriction), Eq. [3] reduces to

[4]

with solution b = [I – P^–1G₁(G₁ 'P^–1G₁) ^–1G₁']P^–1Ga, which is identical to the index with zero restriction (Kempthorne and Nordskog, 1959). Thus, zero restriction is a special case of constant restriction. The elements of vector c may contain negative, zero and/or positive values. Note that the quantity of in the coefficient matrix of Eq. [4] can be dropped without affecting the solution vector b. The deletion of from Eq. [4] leads to a system of index equations that is identical to that of Cunningham et al. (1970).

Simultaneous Imposition of Constant and Proportional Restrictions
In addition to subjecting r traits to constant restriction, we also may constrain the genetic responses of s traits to proportionality of say k, where k is a (s x 1) vector (r + s n). Let G₂ be an (m x s) submatrix of G that contains the columns of the s traits with proportional restriction. The Lagrange multipliers function to maximize H subject to constant restriction of c, and proportional restriction of k is as follows:

where is a (s x 1) vector of Lagrange multipliers and is a scalar to be determined a posteriori. Differentiating the function f with respect to b, , , and and equating the partial derivatives to zeros result in the following equations:

These four sets of equations can be expressed in matrix notation as follows:

[5]

Note that the coefficient 1/2 for the second and third elements of the first set of equations in Eq. [5] has been dropped without affecting the values of b. Because the coefficient matrix of Eq. [5] contains unknown and b, direct inversion is impossible. An iterative procedure is necessary to solve for b at a given in this case or in cases that combine constant with any other types of restrictions. Following the same reasoning given for Expression [3], it can be shown that the coefficient matrix of Eq. [5] has full rank (m + r + s + 1) and is invertible. Therefore, iteration on [5] is guaranteed to converge. Deleting the third and fourth sets of equations from Eq. [5] reduces to Eq. [3] for constructing an index with constant restriction alone, and deleting the second set of equations from Eq. [5] yields a system of equations for constructing an index with proportional restrictions (Lin, 2005). Deleting the second, third, and fourth sets of equations from Eq. [5] results in unrestricted index b = P^–1Ga. Therefore, the presented procedure generalizes the construction of an index with single or multiple restriction(s) or without restriction.

Numerical Example
Parameter Estimates.
The relative economic weights (a) and phenotypic (P) and genetic (G) covariance matrices among egg weight (EW; g), BW (kg), egg production (EP; %), and feed requirement (FR; %) were obtained from Itoh and Yamada (1987):

Both I and H contain four traits (m = n = 4). The relative economic weights of –6.3 for BW and –2.5 for FR indicate that a decrease in these two traits is economically desirable. In an unrestricted case, b = [1.183 –12.767 0.071 –0.447] ', and = [0.962 –0.012 0.259 –4.349] ' when = 1.

Constant Restrictions.
Let the selection goal be to maximize H subject to the restrictions that EW will ncrease by 1.1 g (_EW = 1.1g) and BW will decrease by 0.02 kg (_BW = –0.02 kg) after selection. It follows accordingly that

These parameters were substituted into Eq. [3] and the unrestricted index coefficients b = P^–1Ga were used as priors to start iteration.

Simultaneous Imposition of Constant and Proportional Restrictions.
Let the breeding goal be to increase EP by 0.5% and to restrict the genetic responses of EW and BW to be 10: –1 with Trait 4 (FR) unrestricted. Thus,

These parameters were substituted into Eq. [5] to start iteration using the unrestricted index coefficients b = P^–1Ga as priors.

Two different levels of selection intensity ( = 1 or 2) were investigated for each of the above two examples. The successive estimates of solutions b and genetic gains of the index traits during iteration were given in Tables 1 and 2 for a constant-restricted index (first example) and a constant- and proportion-restricted index (second example), respectively. Total genetic gain in net merit (H), the variance of the index (²_I ), and the covariance between I and H (_IH) were compared among the constant-restricted index, constant- and proportion-restricted index, proportion-restricted index, and unrestricted index in Table 3.

View this table: [in this window] [in a new window]   Table 1. Successive estimates of index coefficients bi and genetic gains i of index traits subject to constant restriction of increasing egg weight by 1.1 g (1 = 1.1g) and body weight by –0.02 kg (2 = –0.02 kg) during iterationa

View this table: [in this window] [in a new window]   Table 2. Estimates of index coefficients and genetic gains subject to constant restriction of egg production to 0.5 % (3 = 0.5) and proportional restriction of 10 g: –1 kg between egg weight (g) and body weight (kg; 1:2 = 10: –1) during iterationa

View this table: [in this window] [in a new window]   Table 3. Comparison of total genetic gain in net merit (H), variance of index (2I), and covariance between I and H (IH) between the restricted and unrestricted indexesa

	Results and Discussion

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Dependency of the Constant Restricted Indexes on Selection Intensity
Tables 1 and 2 show that index Eq. [3] and [5] require only five rounds of iteration to converge when the unrestricted index coefficients (b = P^–1Ga) were used as priors. The rapid convergence occurred mainly because the coefficient matrices of Eq. [3] and [5] have full rank, as shown above, and are of small dimension (the number of equations being the total number of both index traits and restricted traits). Generally, achieving severe restrictions at a low intensity of selection would take longer to converge. At convergence, the expected responses are ₁ = 1.1 g and ₂ = –0.02 kg for the index with constant restriction (Table 1) and ₁:₂ = 3: –1 (g : kg) and ₃ = 0.5% for the index with constant and proportional restrictions (Table 2), regardless of = 1 or 2. The expected responses of these restricted traits are exactly the same as the restrictions imposed beforehand, thus supporting the theoretical development of Eq. [3] and [5]. For a specific restriction within either Table 1 or 2, the index coefficients vary between = 1 and = 2, indicating that index coefficients with constant restrictions depend on the value of because, as shown above, the coefficient matrices of index Eq. [3] and [5] are the functions of . As a result, the expected responses of the unrestricted traits in the index change with varying value of , whereas those of constant-restricted traits are invariant to the change in (Tables 1 and 2). The dependency of constant-restricted index on stands in sharp contrast to the conventional unrestricted and restricted indexes, where index coefficients are computed independently of the value of . Eq. [4] designed for zero restriction (a degenerate case of constant restriction) are independent of , as solution b is invariant to the value of used.

Prediction of Genetic Responses to Index with Constant Restrictions
Table 3 showed that for a specific , the total genetic gain in net merit (H) is greater for the unrestricted index than for the restricted indexes, as expected, because the restrictions are achieved at the expense of H. For a specific , the variance of the index (_I² = b'Pb) is equal to the covariance between I and H (_IH = b'Ga) in both the restricted and the unrestricted cases. It follows that H = a' = a'G'b(/_I) = _I where . This fundamental prediction equation in quantitative genetics shows that the value of H increases with increasing value of by corresponding magnitude. That is, doubling the value of would double the value of H; however, this prediction equation holds true only when the estimation of b is independent of so that and _I are independent of each other. Obviously, this is not true under constant restrictions because of a nonlinear relationship between b and , as explained above. Thus, the H of constant-restriction indexes changes nonlinearly with . For example, the constant-and proportion-restricted index yielded H of 12.98 when = 1 and of 28.41 when = 2 (Table 3), showing that doubling the value of did not increase the value of H by twofold. Conventionally, different selection methods were compared assuming that whatever the value of is used, the value of H will change by corresponding magnitude. This assumption is obviously invalid when different selection methods compared involve indexes with constant restrictions due to a nonlinear relationship between H and . Niebel and Van Vleck (1983) maximized _I subject to fixed and proportional restrictions to derive a "single restricted index" for application to all animals of a population. Their index was developed independently of , implying that both b and _I are independent of , which apparently differs from the index with constant and proportional restrictions presented here.

An important consequence of the dependence of b on under constant restrictions is that the basic prediction equation should be expressed as H_i = a'G'b_i(/_Ii) = i_Ii, where H_i, b_i, and _Ii are total genetic gain, restricted index coefficients, and SD of the constant restricted index for a specific , respectively. Similarly, the expected responses of the index traits should be expressed as _i = G'b_i(/_Ii). An index with constant restrictions has no meaning unless it is associated with a specific . Before the start of a selection program, animal breeders should decide on the proportion of the population selected as parents to determine the corresponding value of for constructing an index with constant restrictions based on Eq. [3] or [5].

Quaas and Henderson (1976) presented the restricted BLUP by directly imposing restriction on multiple-trait, mixed-model equations. Itoh and Iwaisaki (1990) applied canonical transformation technique to obtain restricted BLUP when all traits share a common model and have complete information. Satoh (1998) presented a simplified version of the restricted BLUP to decrease the number of equations. There are no reports on "constant-restricted" BLUP in the literature. Because an index with constant restriction depends on and a national livestock genetic evaluation center has no control over , it would be difficult to achieve constant restriction on a national basis.

	Implications

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The proposed method provides a general framework for restricting the genetic responses of certain traits to actual amounts while maximizing the overall genetic gains in net merit. Index equations with constant restrictions are not symmetric, contain selection intensity, and require an iterative approach to solve. These unique properties put the constant-restricted index in a class by itself, apart from all other types of restricted indexes. The constant-restricted index depends on selection intensity, which is in contrast to conventional selection indexes that are developed independently of selection intensity. Animal breeders should determine realistic levels of both selection intensity and constant restriction before constructing a constant-restricted index. Extreme restrictions and/or selection intensity would make it difficult for the breeding population to sustain a constant population size.

	Footnotes

 
¹ Dairy and Swine Research and Development Centre Contribution No. 861.

² Correspondence: Dept. of Anim. and Poultry Sci., Univ. of Guelph, Guelph, Ontario, Canada N1G 2W1 (phone: 519-824-4120, ext. 53339; fax: 519-767-0573; e-mail: clin{at}uoguelph.ca).

Received for publication February 18, 2005. Accepted for publication June 13, 2005.

	Literature Cited

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Brascamp, E. W. 1984. Selection indices with constraints. Anim. Breed. Abstr. 52:645–654.

Cunningham, E. P., R. A. Moen, and T. Gjedrem. 1970. Restriction of selection indexes. Biometrics 26:67–74.[Medline]

Itoh, Y., and H. Iwaisaki. 1990. Restricted best linear unbiased prediction using canonical transformation. Genet. Sel. Evol. 22:339–347.

Itoh, Y., and Y. Yamada. 1987. Comparisons of selection indices achieving predetermined proportional gains. Genet. Sel. Evol. 19:69–82.

Kempthorne, O., and A. W. Nordskog. 1959. Restricted selection indices. Biometrics 15:10–19.

Lin, C. Y. 1978. Index selection for genetic improvement of quantitative characters. Theor. Appl. Genet. 52:49–56.

Lin, C. Y. 2005. A simultaneous procedure for deriving selection indexes with multiple restrictions. J. Anim. Sci. 83:531–536.[Abstract/Free Full Text]

Niebel, E., and L. D. Van Vleck. 1983. Optimal procedures for restricted selection indexes. Z. Tierzuchtg. Zuchtungsbiol. 100:9–26.

Quaas, R. L., and C. R. Henderson. 1976. Selection criteria for altering the growth curve. J. Anim. Sci. 43:221. (Abstr).

Satoh, M. 1998. A simple method of computing restricted best linear unbiased prediction of breeding values. Genet. Sel. Evol. 30:89–101.

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