Restricted best linear unbiased prediction using canonical transformation

neous equations if there are many traits and a large number of animals to be evaluated. In this paper, a canonical transformation technique through which new independent traits are introduced is presented. Thus only equations of relatively low order for each transformed trait have to be solved. Furthermore, it is shown that the number of independent transformed traits is reduced by the number of restrictions imposed. The technique is applicable when a multiple-trait animal model is assumed.


INTRODUCTION .
Kempthorne and Nordskog (1959) proposed the restricted selection index which is a modification of the usual selection index for predicting genetic merits. Selection based on this index can change population means of some traits, but holds some linear functions of other traits unchanged. Quaas and Henderson (1976a) extended the restricted selection index and proposed the restricted best linear unbiased prediction (restricted BLUP) procedure which could include observations with unknown means, missing observations and related animals. They suggested that it provides a useful selection criterion, for example, for altering the growth curve of beef cattle in a favorable manner (Quaas and Henderson, 1976b). However, this method requires the solution of high order simultaneous equations if there are many traits and a large number of individuals to be evaluated.
Of course, this difficulty holds true for the ordinary multiple-trait BLUP evaluation (Henderson and Quaas, 1976). Such a computational difficulty can be overcome in several ways. One of them is an application of the canonical transformation technique (Thompson, 1977;Lee, 1979;Arnason, 1982) through which new independent traits are introduced, and consequently only mixed model equations of relatively smaller order for each trait need to be solved. The objective of this paper is to discuss the application of canonical transformation to the restricted BLUP. THEORY A multiple-trait animal model with the number of traits expressed as q is assumed.
The model for the i-th trait is written as: where: y i is a vector of observations for the i-th trait, X o is an incidence matrix relating fixed effects to observations, ! is a vector of unknown fixed effects, Z o is an incidence matrix relating u i to observations, u i is a vector of unknown additive genotypic values of animals and e i is a vector of errors.
It is assumed that X o and Z o are the same for all traits. Only the records of individuals who have records on all the traits or who have no record on any trait are used, but the records of individuals whose records are partially missing on some traits are not used. The number of individuals to be evaluated is denoted by p, the number of individuals with records by n and the number of the columns of X o by f.
When the records are ordered by individuals within traits, the model for all the traits is written as: where 1! denotes the identity matrix of order q x q and * denotes the direct product operation. It is assumed that where A is the numerator relationship matrix among individuals to be evaluated, Go and R o are the additive genotypic and error variance-covariance matrices among traits, and I!, is the identity matrix of order n x n.
Consider the restricted BLUP proposed by Quaas and Henderson (1976a). In that method, a linear predictor b'y is used which is uncorrelated with some linear function of u, say C'u. This is expressed algebraically as Cov(b'y, C'u) = b'ZGC = O. If the same constraints are imposed on the additive genotypic values of all animals, then C is expressed as C = C o * In where C o is a matrix of order q x r and the same as that used by Kempthorne and Nordskog (1959). The columns of C o are assumed to be linearly independent. Subject to this additional constraint, the best linear unbiased predictor can be derived. Such a predictor of u, denoted by u, is obtained by solving the following equations: Eliminating t by absorption gives: where Note that: in animal models. Using this, S can be rewritten as: where Note that So has rank q &mdash; r.
Because Go is positive definite and So is positive semidefinite, there exists a non-singular matrix Q such that Q'Go 1 Q = Iq and Q'S oQ = D where D is a diagonal matrix whose diagonal elements, denoted by a l _> ... > Aq-r (> 0) and Aq-r + l = ... = Aq(= 0), are the roots of the equation: Such a matrix Q and a = 's are easily obtainable through general program packages.
Premultiplying 1 Q/ 0 *If Q/ 0 * IP where I and Ip are identity matrices of order L ! !*!pJ p f x f and p x p, equations (2) can be modified as: or where Because D is diagonal, equations (5) can be subdivided into q independent equation systems which have different forms depending on Aj . When A i is nonzero (i = 1 to q &mdash; r), the equations for the i-th transformed trait become: These equations can be solved with a computing program for single trait BLUP.
On the other hand, when a i is zero (i = q &mdash; r + 1 to q), the equations are reduced to: thus in every case ui = 0 and íi: is indefinite.
From the facts stated above, the restricted BLUP can be computed easily as follows without directly solving equations (1) of high order. First, transform the observed records by (6). Then compute Gg by solving equations (7) for each of the first q &mdash; r transformed traits, and set ui = 0 for the remaining r traits. Finally, obtain u by the inverse transformation: The solution u derived in this way is identical to the solution u given by solving equations (1). However, the fixed effects are not estimable because some !s's are indefinite.

NUMERICAL EXAMPLE
The following example including 2 traits, birth weight and weaning weight, illustrates the use of the method outlined above: There are 4 animals to be evaluated, but animal no 4 has no record. The numerator relationship matrix among them is: The genetic and error variance-covariance matrices are: The fixed effects in the model are only common means. Thus the matrices and vectors included in the model are expressed as: Suppose that it is desirable to improve weaning weight but to keep birth weight unchanged, then C o is: First, the direct solution of restricted BLUP will be shown. Equations (1) become (10). Solving these equations gives: Thus the predicted additive genetic values are: Next, the procedure using the canonical transformation will be shown. From (4), So becomes: Using a program package, the following matrices can be computed from So and G-1: 0 1 Observations transformed by (6) are: yg is used in the next step, but y2 will not be used any more. Equations (7) for the first transformed trait where 'B1 = 0.278 80 become: The solution is: thus: As to the second transformed trait, set u2 = O. Thus û . is obtained. Finally, transforming û . by (9) gives the predicted additive genetic values which are identical to (11) and (12).

DISCUSSION
It has been shown that a canonical transformation technique is applicable to the restricted BLUP evaluation when an animal model is assumed. It is not necessary to solve the independent equation systems for all the transformed traits. The number of equation systems to be solved becomes q &mdash; r, the number of traits minus the number of constraints. Furthermore, a standard computing program for single trait BLUP is applicable to the restricted BLUP evaluation, and no special program is needed. Therefore, the computational task to obtain the solution is much reduced.
However, this method is applicable only to animal models. When the model used is not an animal model, equality (3) does not hold. Consequently, no simple formula can be obtained after that. This method also has another limitation: it assumes that the models are the same for all traits and there are no partially missing observations. Hence, the method is not applicable to, for example, multiple lactation records in dairy cows where the number of lactations is not constant for all cows generally. It was shown that there is no estimable function of the fixed effects in general.