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A simultaneous procedure for deriving selection indexes with multiple restrictions¹

Dairy and Swine Research and Development Centre, Agriculture and Agri-Food Canada, Guelph, Ontario, Canada N1G 2W1

	Abstract

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The formulas given in literature for the construction of restricted indexes were designed only for the imposition of a single restriction (zero, fixed, or proportional). This study presents both the theory and the methods of a simultaneous procedure for constructing indexes with single or multiple restriction(s). Numerical examples are given to verify the theoretical development and to demonstrate the implementation of the procedure. The simultaneous procedure presented brings the construction of various restricted indexes into a simple computational scheme. In addition to the use of the proposed procedure to handle multiple traits, it can be used to modify the growth curve of meat animals or the lactation curve of dairy animals, which generally requires simultaneous imposition of different restrictions on different stages of the curves. A misconception in the literature is that the variance of an index (b'Pb) is not equal to the covariance between an index and its net merit (b'Ga) when the index is a restricted one. This study showed generally that b'Pb and b'Ga are equal in the restricted or unrestricted case only when elements of b represent the original solutions from the index equations and are not equal when elements of b are expressed in terms of proportions.

Key Words: Generalized Procedure • Multiple Restrictions • Selection Index

	Introduction

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Kempthorne and Nordskog (1959) were the first to introduce the procedure of computing a restricted index. Tallis (1962) developed an "optimal index" to achieve the proportionality of genetic responses in certain traits of the index. James (1968) presented a general formula to compute an index with zero, nonzero, or proportional restrictions. Cunningham et al. (1970) presented a simplified version of deriving an index with zero restriction. Harville (1975) maximized the correlation between the index and net merit to derive an index with proportional restriction. Niebel and Van Vleck (1982) developed a procedure to impose restrictions when animals have different sources of information. Niebel and Van Vleck (1983) also established a general theory for deriving a "single-restricted index" and a "multiple-restricted index" when more than one selection index is used in a population. Brascamp (1984) gave a detailed review of selection index with constraints. Lin (1985) presented a stepwise procedure for constructing selection indexes with various restrictions. Itoh and Yamada (1987) showed the equivalency of the restricted index methods of Kempthorne and Norkskog (1959), Harville (1975), and Tallis (1985). However, all the formulas reported in the literature for the construction of a restricted index deal only with the imposition of a single restriction and are not applicable to an index with more than one restriction. The objective of this study, therefore, was to develop a generalized procedure for the construction of an index with a single or multiple restriction(s).

	Materials and Methods

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Selection index and net merit are defined as I = b'x and H = a'g, respectively, where x is a vector of m known phenotypic values, g is a vector of n unknown genetic values, and a is a vector of n known economic values. By definition,

where P is a (m x m) phenotypic covariance matrix, 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 size of m could be equal to, greater than, or smaller than n. If m = n, then G and F are identical.

Let k be a (r x 1) vector containing the predetermined proportions of the genetic gains for the r traits restricted. We can partition a, g, G, and H into two parts corresponding to the restricted and the unrestricted traits, respectively. For example, H may be partitioned into H₁ = a₁ ' g₁ and H₂ = a₂ ' g₂ (H = H₁ + H₂) and G into G₁ and G₁, where subscripts "1" and "2" denote the restricted and the unrestricted parts, respectively. Thus, G₁ has order m x r for the restricted traits, and G₂ has order m x (n – r) for the unrestricted traits. The implicit assumption for imposing a single restriction is that both n and r must be greater than or equal to 2 for a proportional restriction, and r 1 for a zero or fixed restriction.

The genetic response in H to selection on I is H = rIH H, where rIH is the correlation between I and H, is the selection intensity, and H is the standard deviation of H. The (n x 1) vector of genetic gains for component traits of H is = G'b( /I), where I is the standard deviation of I. Vector may be partitioned into ₁ = G₁ ' b(i/I) for the restricted traits and ₂ = G₂ ' b(i/I) for the unrestricted traits. Restrictions can be imposed on ₁ or on G₁ ' b to derive the desired index. This forms the basis for deriving the restricted indexes. The method of Lagrange multipliers is widely used to find the maximum or minimum of a function when restrictions are imposed on the variables of the function.

Imposition of a Single Restriction on an Index
Two-Step Procedure.
Tallis (1985) presented a two-step procedure to derive an index subject to the restriction of genetic gains of some traits to a vector of proportionalities (k).

Step 1: Tallis minimized the squared difference between I and H subject to the constraint G1 ' b = k, where is a scalar to be determined a posteriori. He differentiated the following Lagrange multiplier function:

with respect to b and to obtain:

[1]

where b is expressed as a function of .

Step 2: Tallis differentiated f = Var(b'x – a'g) with respect to to obtain the solution :

[2]

Then, he substituted Eq. [2] into [1] to obtain the final restricted index:

[3]

Tallis (1962) derived a proportionally restricted index based on a Lagrange multiplier function without the scalar . But this restricted index did not move the means of the restricted traits to the maximum possible extent, as pointed out by Mallard (1972) and Harville (1975). Subsequently, Tallis (1985) incorporated to move the means of the restricted traits as far away as possible from the original means. Itoh and Yamada (1987) showed that the restricted index is valid when > 0 and invalid when 0, which may arise when corresponding elements of a and k disagree in sign.

Simultaneous Procedure.
Instead of the two-step procedure of Tallis (1985), we can differentiate the La-grange function f = b'Pb – 2b'Ga + a'Fa + '(G1 ' b - k) with respect to b, , and simultaneously and equate the partial derivatives to zero:

These three sets of equations can be written jointly in the following form:

[4]

Note that the coefficient of G₁ was dropped from Eq. [4] because it does not affect the solutions for b and , although it changes the solutions for , which are not needed for index construction. Equation [4] is simple to set up and solve for b. It can be verified through the inversion of a partitioned matrix that the solutions b and to Eq. [4] are identical to Eq. [3] and [2], respectively. The main difference is that the simultaneous procedure yields the solutions b and simultaneously, whereas Tallis (1985) obtained b_T as a function of in the first step followed by obtaining in the second step to derive the final solution b_T. Because the simultaneous procedure differentiates the Lagrange function f with respect to b, , and in a single step, it is flexible and can be augmented to accommodate more than a single restriction as will be illustrated in a later section.

Restriction of Complete Proportionality
The above development deals with a partial restriction where the number of traits restricted (r) is less than the number of traits in H (n). Complete proportionality refers to the imposition of proportional restriction on all traits of H (r = n). Pesek and Baker (1969) obtained the desired gains index with complete proportionality (G'b = k) as follows:

[5]

where G^–1 exists only when I and H contain the same number of traits (m = n). When m n, multiplying both sides of G'b = k results in GG'b = Gk and thus,

[6]

where (GG')^–1 exists when m < n (Lemma 9 of Searle, 1971). When m > n, (GG')^–1 does not exist. Brascamp (1984) presented a formula to deal with this case. Itoh and Yamada (1986) showed that Brascamp’s formula is equivalent to

[7]

which was obtained by minimizing Var(I – H) subject to the constraints G'b = k. Alternatively, we can minimize Var(I – H) subject to the constraints G'b = k to obtain,

[8]

When G is a square matrix, Eq. [3] reduces to Eq. [8], whereas Eq. [6] and [7] reduce to Eq. [5]. Because the fraction in Eq. [8] can be dropped, Eq. [5] to [8] are equivalent in terms of ranking the animals. Although economic values are not required to construct restricted indexes with complete proportionality, they are needed to compute H.

Simultaneous Imposition of Multiple Restrictions on an Index
All formulas reported for the construction of a restricted index in the literature are concerned only with a single restriction and do not apply to an index with multiple restrictions. In marked contrast, the simultaneous procedure developed here can be easily extended to derive an index with multiple restrictions. For example, we may want to maximize H subject to the simultaneous imposition of G₁b = k and G₃b = c, where vector c contains some predetermined constants (fixed restrictions), and G₃ is a submatrix of G corresponding to those traits with fixed restrictions. Zero restriction implies c = 0. The Lagrange multipliers function takes the form of f = b'Pb – 2b'Ga + a'Fa + '(G1 ' b – k) + (G'₃b–c). Differentiating f with respect to b, , , and , and equating the partial derivatives to zeros, results in the following set of index equations:

[9]

A restricted index based on b from Eq. [9] is expected to maximize H and satisfy both proportional and fixed restrictions.

Index Eq. [9] can be augmented to accommodate as many restrictions as desired. If n restrictions (i.e., n condition equations) are imposed simultaneously on an index, there are n such sets of Lagrange multipliers required to set up the index equations. As a further example, in addition to proportional and constant restrictions, we could superimpose a linear restriction such as q₁G_i + q₂G_j = v where q₁, q₂, and v are constants (i.e., a linear combination of genetic gains between traits i and j is equal to a presp ecified constant). The Lagrange multipliers function then becomes

Likewise, we can differentiate this function with respect to b, , , , and and equate the partial derivatives to zeros to obtain a system of equations for computing an index with triple restrictions (proportional, constant, and linear).

Numerical Example
Imposition of a Single Restriction.\
The phenotypic and genetic covariance matrices among rate of lay (RL), age at sexual maturity (SM), egg weight (EW), and BW were obtained from Akbar et al. (1984). Let the breeding goal be to maximize H subject to a proportional restriction of G_RL:G_SM:G_EW = 3: –1: 2. A proportional restriction would make sense only when an element of k has the same sign as the corresponding element of a (e.g., negative sign for SM in k and a). Both I = b'x and H = a'g contain four traits, with three of them restricted to proportional changes (i.e., m = n = 4 and r = 3).

The simultaneous procedure yields the following set of index equations:

The solution vector is:

Note that b'Pb = b'Ga = 1125.69.

This set of index equations is easy to set up and solve for b and simultaneously. Equations [2] and [3] based on the two-step procedure of Tallis (1985) give:

Here, b_T ' Pb_T = b_T ' Ga = 1,125.69, and when = 1,

This example shows that both procedures produced identical solutions for b and and satisfied the same restriction of G₁: G₂: G₃ = 3: –1: 2.

Simultaneous Imposition of Two Restrictions.
The same parameter estimates as above were used to maximize H subject to simultaneous imposition of two restrictions: 1) the ratio of genetic gains between RL (Trait 1) and SM (Trait 2) to be 3:–1; and 2) the genetic gain of EW (Trait 3) to be 0.5 of the standard deviation of the index (i.e., G₃ = 0.5/_I). The second restriction is equivalent to G₃ ' b = 0.5 under the assumption of i = 1. Trait 4 (BW) is unrestricted.

The index Eq. [9] of the general procedure becomes

Solutions to this set of index equations lead to

The derived index satisfies the restrictions of G₁: G₂ = 1.12577: –0.37526 = 3: –1 and G₃ = 0.5/I = 0.01439 standard deviation units as expected.

	Results and Discussion

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Comparison Between the Simultaneous Procedure and Existing Methods
Formulas presented in the literature for the construction of restricted indexes (e.g., Kempthorne and Nordskog, 1959; Harville, 1975; Tallis, 1985) have dealt solely with a single restriction and do not apply to the construction of indexes with more than one restriction. In contrast, the simultaneous procedure developed here accommodates both single and multiple restrictions. Numerical examples validated the theoretical development of the procedure and demonstrated its ease of implementation for constructing various restricted indexes. For example, deleting the fourth set of equations from the proposed system of Eq. [9] reduces to Eq. [4] for the construction of an index with proportional restrictions; deleting the second and third sets of equations from Eq. [9] yields a system of equations for constructing index with fixed restrictions; deleting the third and fourth sets of equations from Eq. [9] gives a system of equations for an index with zero restrictions; and deleting the second, third, and fourth sets of equations from Eq. [9] simplifies to the familiar equation Pb = Ga for the ordinary unrestricted index. Therefore, the simultaneous procedure is indeed a generalized procedure to generate a system of equations for the construction of ordinary unrestricted indexes and restricted indexes with one or more constraints. It integrates the theory and the methods for constructing unrestricted or restricted indexes into a simple computational scheme. Because the elements of vector k for proportional restrictions can be negative, zero, or positive, zero and/or negative restrictions can be treated as part of the set of proportional restrictions (a special case) rather than as separate restrictions. Lin (1985) treated proportional and zero restrictions as two different constraints when developing his stepwise procedure for constructing restricted indexes.

In addition to the use of the simultaneous procedure for imposing restrictions on different traits in an index, the proposed procedure could be applied to modify growth curves of meat animals or lactation curves of dairy animals. Random regression models permit the estimation of breeding value for each individual day of growth or lactation, and of genetic covariances between BW or yields at any two days. Multiple restrictions could be imposed on individual EBV for particular days to formulate a restricted index to modify growth or lactation curves. The proposed procedure with multiple restrictions offers a potential tool to serve this purpose.

Relationship Between b'Pb and b'Ga
There have been some misconceptions about the relationship between b'Pb and b'Ga. Essl (1981) reported that b'Pb = b'Ga is true only for unrestricted indexes but not for restricted indexes. This conclusion is misleading. For instance, the index with zero restriction is b = [I – P^–1G₁(G₁ ' P^–1G₁)^–1G₁ ' ]P^–1Ga. It follows that

This proves that b'Pb = b'Ga holds true with zero restriction. This equality is also true for the simultaneous procedure with multiple restrictions, the modified version of Kempthorne and Nordskog (Mallard, 1972) and Tallis (1985), but it is not true for the methods of Tallis (1962) and Harville (1975). Generally, if the original values of b are expressed on a proportional basis, then b'Pb is not equal to b'Ga, regardless of whether the index is restricted. For example, if the original vector of solutions to the index equations b = [6 2 4]' is reduced to b = [3 1 2]', the equality b'Pb = b'Ga holds true in the former but not in the latter, in spite of the fact that both indexes rank animals in the same order and yield the same H.

The restriction of indexes with complete proportionality based on Eq. [5] to [7] from the literature leads to b'Pb b'Ga, whereas Eq. [8] developed here results in b'Pb = b'Ga. An important consequence of b'Pb = b'Ga is that H = a'G'b( /I) reduces to H = I in both the restricted and the unrestricted case. Thus, H = a'G'b( /I is the correct formula to use, regardless of whether b'Pb is equal to b'Ga, but H = I is applicable only when b'Pb = b'Ga.

	Implications

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Formulas reported in the literature for computing restricted indexes apply only to a single restriction and are not applicable to multiple restrictions. The simultaneous procedure proposed here unified both the theory and the methods of constructing indexes with one or more restriction(s) into a simple computational scheme. This procedure offers animal breeders a selection strategy to alter the genetic responses of restricted traits by a specified amount in the desired direction that would be otherwise impossible with ordinary unrestricted indexes. In addition to the construction of restricted indexes comprising different traits, the proposed procedure provides a useful tool to modifying the growth curve of meat animals or the lactation curve of dairy animals, which usually requires the simultaneous imposition of different restrictions on different parts of the curves.

	Footnotes

 
¹ Dairy and Swine Research and Development Centre Contribution No. 834.

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

Received for publication August 20, 2004. Accepted for publication November 9, 2004.

	Literature Cited

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Akbar, M. K., C. Y. Lin, N. R. Gyles, J. S. Gavora, and C. J. Brown. 1984. Some aspects of selection indices with constraints. Poult. Sci. 63:1899–1905.

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]

Essl, A. 1981. Index selection with proportionality restriction: Another viewpoint. J. Anim. Breed. Genet. 98:125–131.

Harville, D. A. 1975. Index selection with proportionality constraints. Biometrics 31:223–225.[Medline]

Itoh, Y., and Y. Yamada. 1986. Re-examination of selection index for desired gains. Genet. Sel. Evol. 18:499–504.

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

James, J. W. 1968. Index selection with restrictions. Biometrics 24:1015–1018.

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

Lin, C. Y. 1985. A simple stepwise procedure of deriving selection index with restrictions. Theor. Appl. Genet. 70:147–150.

Mallard, J. 1972. The theory and computation of selection indices with constraints: A critical synthesis. Biometrics 28:713–735.

Niebel, E., and L. D. Van Vleck. 1982. Selection with restriction in cattle. J. Anim. Sci. 55:439–457.[Abstract/Free Full Text]

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

Pesek, J., and R. J. Baker. 1969. Desired improvement in relation to selection indices. Can. J. Plant Sci. 49:803–804.

Searle, S. R. 1971. Linear Models. John Wiley & Sons, Inc., New York.

Tallis, G. M. 1962. A selection index for optimum genotype. Biometrics 18:120–122.

Tallis, G. M. 1985. Constrained selection. Jap. J. Genet. 60:151–155. [Corrigendum and addendum. Jap. J. Genet. 61:181–184]

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