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A method of computing restricted best linear unbiased prediction of breeding values for some animals in a population

Genetic Diversity Department, National Institute of Agrobiological Sciences, Tsukuba 305-8602, Japan

	Abstract

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Restricted BLUP (R-BLUP) is derived by imposing restrictions directly within a multiple-trait mixed model. As a result, the R-BLUP procedure requires the solution of high-order simultaneous equations. If restrictions are imposed on breeding values for only some animals in a population, calculations become more complex. A new procedure for computing the R-BLUP of breeding values was derived when constraints were imposed on the additive genetic values of only some animals in a population. Rules for including records when proportional constraints are imposed were developed based on the traits that are recorded for an animal. The technique was better than the previous method in both memory requirement and central processing unit time.

Key Words: Genetic Evaluation • Mixed Linear Model • Restricted Best Linear Unbiased Prediction

	Introduction

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Restricted BLUP (R-BLUP) was derived by imposing restrictions on multiple-trait BLUP (Quaas and Henderson, 1976). It is an efficient method for antagonistic selection. For example, in cattle breeding, increased weaning weight is desired, whereas birth weight should be kept at the current level to avoid calving difficulty. This is a situation typical to antagonistic selection because the genetic correlation between the birth and weaning weights are generally quite high.

The original method for calculating R-BLUP, imposing the same restrictions on all animals, has been improved (Quaas and Henderson, 1976; Itoh and Iwaisaki, 1990, Satoh, 1998). These methods have also been used in several studies based on field data or computer simulation (e.g., Satoh et al., 1998; Díaz et al., 1999; Ieiri et al., 2003). However, the breeding values of all animals need not be subjected to the same restrictions. For example, if selection criteria in each subline, such as sire and dam lines, from the same population are different, then different constraints should be imposed. Another example is that restrictions imposed on breeding values of dead animals are meaningless because the constraints should generally apply only to breeding values of animals available for selection. However, if restrictions are imposed on breeding values for only some animals in a population, calculations become more complex, requiring a matrix whose elements are coefficients of additive relationships between all animals and animals with restricted breeding values, and the second power of the matrix for writing R-BLUP equations.

The objectives of the present paper were to derive a procedure for computing R-BLUP of breeding values and to discuss its application when constraints were imposed on the additive genetic values of only some animals in a population.

	Materials and Methods

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Theoretical Background
An additive genetic mixed model for q traits is assumed. The model for the ith trait is written as follows:

where y_i is a vector of observations for the ith trait; b_i is a vector of unknown fixed effects; X_i is a known incidence matrix relating elements of b_i to y_i; u_i is a vector of unknown random additive genetic effects for n animals; Z_i is a known incidence matrix relating elements of u_i to y_i; and e_i is a vector of random errors. Let q_k be the number of traits in the kth animal; k = 1, 2, ..., n; and 0 q_k q. The model for all traits is written as:

where records are ordered by animals within traits. Assume that u and e are multivariates normally distributed with E(u) = E(e) = 0, var(u) = G, var(e) = R, and cov(u, e') = 0; G = G₀A, where G₀ is a q x q additive genetic variance-covariance matrix for the q traits; A is an n x n additive relationship matrix for the n animals; and R is an n. x n. (n. = _kq_k) error variance-covariance matrix for the q traits for the n animals.

Let the set of restrictions on u be C'u. If the same constraints are imposed on the additive genetic values of all animals, C = C₀I_n, where C₀ is a q x r matrix with full columns rank. The number of columns of C₀, r, depends on the number and type of constraints imposed: no change and/or proportional change. This will be illustrated subsequently. Kempthorne and Nordskog (1959) defined C₀ for no-change constraints. For example, if the restriction is no change for the second trait, C₀ might be:

Here r (= z) is the number of traits constrained. For the case of proportional constraints (2 p q traits), define:

[1]

where k'₁ = [c₁ c₂ ... c_{p – 1}] and c_i is a predetermined proportional change for Traits 1, 2, ..., p (Mallard, 1972; Itoh and Yamada, 1987). Note that r = p – 1. Furthermore, if constraints include both no change and proportional change, the form of C₀ = [C_z | C_p ], and so r = z + p – 1. The R-BLUP of u, û, is obtained by solving the following equations (Quaas and Henderson, 1976):

[2]

where is a vector of some solution to b and is a vector of Lagrange multipliers. Equation [2] involves G = G₀ A; in contrast to its inverse, it is very time consuming to compute with large numbers of animals. Quaas and Henderson (1976) represented Eq. [2] as [3]:

[3]

where P = G₀C₀ I_n. Equation [3] is simpler because it does not involve A.

Furthermore, Eq. [3] is represented in Eq. [4] as:

[4]

where

[5]

where subscripts Z, R, and N correspond to z characters with no change, p characters with proportional constraints, and q – z – p characters without constraints, respectively; k' = [k₁'/c_p 1]; and K₀ is a q x (q – r) matrix. Thus û_Z = 0 and û_R = kû_P (Satoh, 1998). Equations [3] and [4] are represented more simply when an animal model without repeated records is used (see Quaas and Henderson, 1976; Satoh, 1998).

Constraints for Some Animals in a Population
If the constraints are imposed on the additive genetic values of some animals in a population, then

[6]

where J_m is an identity matrix with columns pertaining to n – m animals without constraints deleted. We can apply different restrictions to different animals (e.g., different constraints for different groups from the same base population):

where C_j (j = 1, 2, ..., s) is a matrix for constraints of jth group and m_j is the number of animals with constraints in jth group. Thus, if the constraints are imposed on breeding values for only some animals in the population, Eq. [2] cannot be represented as in Eq. [3] or [4]. As an example, for an animal model with one record in each trait, no missing observations, and the same restriction of breeding values for some of the animals, the cell in the third row and third column of the coefficient matrix in Eq. [2] is as follows:

[7]

where R₀ is a q x q error variance-covariance matrix. Because Eq. [2] includes computation of elements of AJ_m and the product of J_m'A and AJ_m, it is quite difficult to write R-BLUP equations. Furthermore, with repeated records, partially missing observations, and two or more constraints, Eq. [7] is more complex. If Eq. [2] is denoted as W = v, an equivalent set of equations for and û is:

where

where t is the number of columns of X. Then Eq. [2] is represented as in Eq. [8]

[8]

Equation [8] is simpler than Eq. [2] because it has fewer nonzero elements and does not include the second power of a submatrix of A in the cell in the third row and third column of the coefficient matrix. However, after solving Eq. [8], we need AJ_m to obtain û from û + GC and .

Now let û + GC = w₁ and = w₂. Then,

[9]

Equation [9] does not include A. It can be rewritten as

[10]

where û_i, w_1i, and w_2i are subvectors for the ith trait of û, w₁, and w₂, respectively, and p_ij is the ijth element of G₀C₀. The coefficient matrix in Eq. [10] is alwaysA^–1. Therefore, Eq. [10] can be solved easily and separately for each trait after solving Eq. [8]. Furthermore, from Eq. [6] and the third row in Eq. [8], C'û = (C₀' J_m')û = 0. Then, to obtain û relating only to animals with restrictions (û₀), û_0Z = 0 (constraints with no change in some traits) and û_0R = k û_0p (constraints with proportional changes), where subscripts Z, R, and p are as in Eq. [5].

Computational Remarks
Linear Dependencies Among Fixed Equations.
Quaas and Henderson (1976) showed three types of linear dependencies among the fixed-effect equations of an R-BLUP procedure in which the same restrictions were imposed on the predicted breeding values of all animals. When the constraints were imposed on breeding values for only some animals in a population, the rules for eliminating the linear dependencies among the fixed effect equations were similar to those of Quaas and Henderson (1976) and are summarized as follows: If the classes in the fixed effects include only animals with constraints, the rules are similar to those of Quaas and Henderson (1976); otherwise, the rules are similar to those of multiple-trait BLUP (Henderson and Quaas, 1976).

If some records in the restricted traits are missing, some columns of X may be identical to other columns of X. In extreme cases, this occurs not only in columns of X between traits, but also in those within traits. Hence, we need to investigate which columns of X are identical to each other before constructing R-BLUP equations.

Rules for Ignoring Records of Observations.
Quaas and Henderson (1976) indicated that if q_k r, all the records of any animals missing q_k – r observations are ignored in R-BLUP; exceptions to this can occur if some of the restricted traits are genetically uncorrelated with some of the unrestricted traits. However, they did not consider constraints with proportional change. The rules for ignoring records of observations are summarized below.

First assume that the observations are ordered traits within animals. Let Z_k be a known incidence submatrix on the diagonal of Z, relating elements of breeding values to observations for the kth animal. If Z_kG₀C₀ = 0, this means no restriction for the kth animal (Case 2 in Appendix). If the elements of the jth row of G₀C₀ are all zero, constraints are not imposed on the trait corresponding to the jth row of G₀C₀. The trait should be treated as an unrestricted trait. Restricted traits may be divided into three types. First, all records of the kth animal are ignored if Z_kG₀C₀ has full row rank (Case 1 in the Appendix). If a linear combination of the columns of Z_kG₀C₀ can be 0, linearly dependent columns of Z_kG₀C₀ are deleted (Case 5 in the Appendix). Next, if a vector exists with only one nonzero element corresponding to the jth trait from a linear combination of columns of Z_kG₀C₀, the records of the jth trait of the kth animal are ignored (Case 3 in the Appendix). Otherwise, the records of the kth animal are always used for R-BLUP (Case 4 in the Appendix).

For example, G₀ is assumed to be

If proportional changes in Traits 1, 2, and 4 are wanted on the basis of proportional constraints in the ratio 2:1:1, then C₀ is expressed as:

If there are only observations on the first trait, and then no restrictions for the first trait because Z_kG₀C₀ = 0 (Case 2 in the Appendix). If there are only observations of any one trait except the first trait or the observations of the first and another trait are missing, the records are ignored because Z_kG₀C₀ has full row rank (Case 1 in the Appendix). If the observations of only one trait other than the first trait are missing, records other than the first trait are ignored because a vector exists with only one nonzero element corresponding to the second, third, or fourth trait from a linear combination of columns of Z_kG₀C₀ (Case 3 in the Appendix). If the observations of any two traits except the first trait are missing, the records other than the first trait are ignored (Cases 3 and 5 in the Appendix). If no observations or only those on the first trait are missing, the records are used (Case 4 in the Appendix).

	Results

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Numerical Example
Data from a Monte Carlo computer simulation were used to compare two methods for calculating R-BLUP of breeding values for some animals in a population. Method I uses Eq. [2] and Method II uses Eq. [8] and [9]; Method III, which imposes the same restrictions on the predicted breeding values of all animals by using Eq. [4], was also used for reference.

For computer simulation, the base population (G₀) of 150 males and 1,500 females was mated at random to produce 3,000 males and 3,000 females. Six separate generations (G₀ to G₅) and two traits were simulated. One hundred fifty males and 1,500 females in G₁ through G₄ were selected by using a selection index with zero-gain constraints. For simplicity, the animals were assumed to have a record in each trait, and two fixed effects (sex and generation) and two random effects (additive direct genotype and error) were assumed. Breeding values of a trait of only the animals in the last generation were restricted to be zero. The total number of animals was 31,650 and the number of restricted animals was 6,000. A preconditioned conjugate gradient based iterative method using iteration on data was used to solve each R-BLUP equation. The convergence criterion was based on relative adjusted right-hand sides: ||b – Ax||/||b|| < 10^–9, where Ax and b are left- and right-hand sides of R-BLUP equations, respectively, and ||b|| represents the norm of b. All computations were performed on a personal workstation (3.06 GHz, 1.5 gigabytes RAM).

Table 1 shows the number of nonzero elements in the upper triangular matrix on the left-hand side of the R-BLUP equations and central processing unit time for each Method. The number of nonzero elements in Method II was reduced to 8.6% compared with Method I. Central processing unit time for Method II was about 41 times faster than that for Method I.

	Discussion

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The method R-BLUP can be used for selection with two practical constraints: 1) zero change in one or a few traits and 2) proportional change for several or all traits. Itoh and Iwaisaki (1990) found that a canonical transformation technique could be applied to R-BLUP to decrease the number of equations for an animal model. However, the method has a limitation in that models must be identical for all traits and there cannot be any partially missing observations. Satoh (1998) showed a simple method for computing R-BLUP of breeding values; this procedure works, in particular, without repeated records. These computations for formulating BLUP equations are relatively simpler than those of Quaas and Henderson (1976); however, restrictions of breeding values should essentially be imposed only on candidates for evaluation.

The calculation of R-BLUP imposing the same restrictions on all animals is easier (Method III). However, if one wants to apply it only to candidates for evaluation, it is difficult to write R-BLUP equations because the need to include matrix A. In this case, however, Eq. [8] and [9] were available to compute R-BLUP. Method II gave much better performance than Method I in both memory requirement and central processing unit time, even if computing time was affected by the algorithm and programming to some extent. The technique in the present study can be easily applied to animal models with more than two random effects.

	Implications

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Restricted best linear unbiased prediction is available for selection when constraints include no change in some traits or predetermined relative changes for several or all traits. Restricted best linear unbiased prediction of breeding values imposing the same restrictions on all animals has been used because the method for calculation is easy. However, restrictions of breeding values essentially should be imposed only on candidates for evaluation. A new procedure for computing restricted best linear unbiased prediction of breeding values is presented when constraints are imposed on the additive genetic values of only some animals in a population. The technique is better than the previous method in both memory requirement and central processing unit time.

	Appendix

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Eliminating from Eq. [3], we get for and û:

where S = R^–1 – R^–1ZP(P'Z'R^–1ZP)^–P'Z'R^–1 (Quaas and Henderson, 1976). Now consider that the observations are ordered traits within animals; then S is a block diagonal matrix,

where S_k is a q x q matrix peculiar to the kth animal and

[11]

where R_k^–1 is the inverse of the error variance-covariance matrix for the q traits for the kth animal, and Z_k is a known incidence matrix relating elements of breeding values to observations for the kth animal. We use the following four lemmas in order to show the property of the columns of S_k.

Lemma 1.
When vectors x₀ and x₁ are linearly independent (LIN), for A nonsingular, vectors y₀ = Ax₀ and y₁ = Ax₁ are LIN. The proof of this lemma is that if y₀ and y₁ are linearly dependent, then for some constant c 0, Ax₀ = cAx₁. Because A^–1 exists, x₀ = cx₁, but this contradicts the assumption that x₀ and x₁ are LIN. Therefore y₀ and y₁ are LIN.

Lemma 2.
When X has full low rank, then AX(X'AX)^–X'A = A. The proof of this lemma is that when X has full low rank, XX' is non-singular (Searle, 1971). Pre- and post-multiplying X'AX(X'AX)^–X'AX = X'AX by (XX')^–1X and X'(XX')^–1, respectively, gives AX(X'AX)^–X'A = A.

Lemma 3.
For A nonsingular and positive definite (p.d.), A^–1 is p.d. The proof of this lemma is that any p.d. matrix can be written as A = UDU', where U is the matrix of eigenvectors and D is a diagonal matrix of eigenvalues (all of which are positive). Then the inverse is U(D^–1)U' because UU' = I, and still diagonals of (D^–1) are positive, so the inverse is p.d.

Lemma 4.
For P 0 and X 0, the rows and columns of P = I – X(X'X)^–X' are orthogonal to and LIN of columns of X. The proof of this lemma is that X(X'X)^–X' is symmetric (Searle, 1971) and P = I – X(X'X)^–X' is symmetric. Because (X'X)^–X' is a generalized inverse of X, PX = X – X(X'X)^–X'X = 0, the rows and columns of P are orthogonal to the columns of X. Assume that the columns of P depend on those of X. Then P = XA for some A0, and P = PP = PXA = 0. Therefore the assumption is false, and so the columns of P are LIN of those of X. Because P is symmetric, the rows of P are LIN of those of X.

Because the columns of C₀ are LIN and G₀ is nonsingular, the columns of G₀C₀ are LIN by Lemma 1. We can classify Z_kG₀C₀ into the following five cases: Z_kG₀C₀ has full row rank (Case 1); Z_kG₀C₀ = 0 (Case 2); a linear combination of the column vectors of Z_kG₀C₀ has only one nonzero element (Case 3); a linear combination of the column vectors of Z_kG₀C₀ has more than one nonzero element (Case 4); a linear combination of the column vectors of Z_kG₀C₀ is 0 (Case 5).

Case 1
For Case 1 if Z_kG₀C₀ has full row rank, using Lemma 2 leads to

Therefore, S_k = 0.

Cases 2 to 5
Cases 2 through 5 apply to situations when Z_kG₀C₀ does not have full row rank.

Case 2.
If Z_kG₀C₀ = 0, then R_k^–1 is p.d. from Lemma 3 because R_k is p.d. Hence, C₀'G₀Z_k'R_k^–1Z_kG₀C₀ = 0 only when Z_kG₀C₀ = 0. This means that the submatrices of the kth animal of the third row and the third column in Eq. [3] are all zero. Therefore S_k = R_k^–1.

Case 3.
If a linear combination of the column vectors of Z_kG₀C₀ has only one nonzero element then R_k^–1 can be written as L_k'L_k for a nonsingular L_k because R_k^–1 is p.d. From [11], S_k = L_k'[I_k – L_kZ_kG₀C₀(C₀'G₀Z_k'L_k'L_kZ_kG₀C₀)^–C₀'G₀Z_k'L_k'] L_k.

Let W_k = I_k – L_kZ_kG₀C₀(C₀'G₀Z_k' L_k'L_kZ_kG₀C₀)^–C₀'G₀Z_k'L_k'. Because the rows of W_k are orthogonal to the columns of L_kZ_kG₀C₀ by Lemma 4, for any vector t0, the rows of W_k are orthogonal to L_kZ_kG₀C₀t. Hence, W_kL_kZ_kG₀C₀t = 0; then the rows of W_kL_k are also orthogonal to Z_kG₀C₀t. This means that the jth column vector of W_kL_k is 0 if only the jth element in a linear combination of the columns of Z_kG₀C₀ is zero. Therefore, the jth column vector of S_k = L_k'W_kL_k is 0. Then the records of the jth trait of the kth animal are ignored.

Case 4.
If a linear combination of the column vectors of Z_kG₀C₀ has more than one nonzero element, and if L_kZ_kG₀C₀t has more than one nonzero element, then each row of W_k has more than one nonzero element because columns of W_k are orthogonal to L_kZ_kG₀C₀t. Therefore, the column vectors of S_k relating traits with records are not 0. Then the records of the kth animal are always used for R-BLUP.

Case 5.
If a linear combination of the column vectors of Z_kG₀C₀ is 0, then some linearly dependent columns of Z_kG₀C₀ exist, because for some vector t 0, Z_kG₀C₀t = 0. Because the rows of W_kL_k are orthogonal to the columns of Z_kG₀C₀, they are also orthogonal to the linear combinations of columns that eliminate linearly dependent columns of Z_kG₀C₀. Then if the linear combinations of columns that eliminate linearly dependent columns of Z_kG₀C₀ have only one nonzero element corresponding to the jth trait, the records of the trait are ignored (refer to Case 3); otherwise, the records are always used (refer to Case 4).

¹ Correspondence: Kannondai 2-1-2, Tsukuba-shi 305-8602 (phone: 81-29-838-7041; fax: 81-29-838-7408; e-mail: hereford{at}affrc.go.jp).

Received for publication September 29, 2003. Accepted for publication March 1, 2004.

	Literature Cited

Top Abstract Introduction Materials and Methods Results Discussion Implications Appendix Literature Cited

 


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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Satoh, M. Search for Related Content PubMed PubMed Citation Articles by Satoh, M.