multistage selection for genetic gain by orthogonal ...equation 4 is the multitrait equivalence of...

Copyright 0 1991 by the Genetics Society of America

Multistage Selection for Genetic Gain by Orthogonal Transformation

Shizhong Xu and W. M. Muir Department of Animal Sciences, Purdue University, West Lafayette, Indiana 47907

Manuscript received April 29, 199 1 Accepted for publication August 1, 199 1

ABSTRACT An exact transformed culling method for any number of traits or stages of selection with explicit

solution for multistage selection is described in this paper. This procedure does not need numerical integration and is suitable for obtaining either desired genetic gains for a variable proportion selected or optimum aggregate breeding value for a fixed total proportion selected. The procedure has similar properties to multistage selection index and, as such, genetic gains from use of the procedure may exceed ordinary independent culling level selection. The relative efflciencies of transformed to conventional independent culling ranged from 87% to over 300%. These results suggest that for most situations one can chose a multistage selection scheme, either conventional or transformed culling, which will have an efficiency close to that of selection index. After considering cost savings associated with multistage selection, there are many situations in which economic returns from use of independent culling, either conventional or transformed, will exceed that of selection index.

F OR the same overall selected proportion, expected genetic gain from index selection is never

smaller than that from independent culling (HAZEL and LUSH 1942). However, for the same fixed re- sources and selection intensity, total economic returns may be greater for the latter. This results because the independent culling method sets minimum standards for each trait, as such, in contrast to index selection, not all individuals need to be measured for each trait. Enhancements from that aspect of independent culling level selection may be gained in several ways.

First, traits to be measured may be obtained at different ages, thus allowing early culling of some individuals. Initial culling at an early age has two advantages: 1) An increased selection intensity on traits measured at the earlier age. Due to subsequent growth, space required per individual is greatest at the age at which the plant or animal is finally proc- essed. With fixed facilities the limiting factor determining final population size is space required per mature individual. Thus, with the same facilities, a greater number of individuals can be reared at an earlier age than later allowing a higher selection intensity on those traits. With index selection, the same individuals must be measured for each trait, hence the number measured per mature individual is the same as that for an immature. For example, on an acre of land several thousand tree seedlings can be measured for rate of growth in the first few years. At maturity, the same acre may support fewer than a hundred trees. Similar problems are encountered in fish, poul- try, beef, and swine breeding. 2) Savings of feed and facilities. If some individuals can be culled before final traits are measured, savings will be realized in terms

Genetics 129 963-974 (November, 1991)

of feed, labor, and facilities (SAXTON 1989). Examples of such traits are weaning weights in swine and beef cattle breeding.

Second, costs of measuring traits may be greatly different (NAMKOONG 1970). Even if traits to be selected can be measured contemporaneously, costs associated with measuring various traits may vary greatly. As such, economic savings can be realized by initially culling on the least expensive traits. For example, swine may be selected based on rate of gain and percent lean. Accurate assessment of percent lean may require sophisticated equipment at a relatively high cost to operate. An initial culling based on rate of gain would greatly reduce overall costs.

The practical problem with independent culling is the difficulty in choosing proper truncation points for traits in order to obtain desired genetic gains or to maximize aggregate breeding value. Independent culling for two traits, or similarly, two-stage selection, has been well developed (YOUNG and WEILER 196 1 ; YOUNG 1964; WILLIAMS and WEILER 1964; SMITH and QUAAS 1982; BROWN 1967; NAMKOONG 1970; COTTERILL and JAMES 198 1). Recently, SAXTON (1 989) and DUCROCQ and COLLEAU (1 989) developed computer programs to calculate optimum truncation points for up to seven stages.

With the advent of those computer programs, po- tential use of independent culling may increase. Un- fortunately, computer time needed to obtain solutions for more than four stages is impractical on most computers. SAXTON (1989) noted that it would take approximately 1 1 CPU hours for five traits on an IBM 3084 mainframe. The corresponding time on a per- sonal computer (386 CPU, 22 mhz) would be several

964 S. Xu and

days. Furthermore, these programs are usually based on a nonlinear optimization procedure which may converge to a local maximum rather than the global maximum (SAXTON 1989), so maximum genetic gains are not guaranteed.

An approximate procedure, without multivariate integration, for n-stages of selection for m traits with explicit solutions has been proposed (MUIR and Xu 1991). But, the usefulness of their procedure was generally limited to cases in which phenotypic correlations among stages were small. An exact method for independent culling without use of multivariate numerical integration has not been previously reported.

The efficiency of independent culling can be increased by combining information across stages. HAN- SON and BRIM (1963) selected the sum of the first and second traits in the second stage of selection on soybean yields. The same idea was extended to multiple stage index selections by YOUNG (1964) and CUN- NINGHAM (1975). Multistage index selection is more efficient than ordinary independent culling because more information is used in subsequent stages of selection. But, multivariate integration is still needed to determine truncation points for each subsequent index.

An exact transformed culling procedure for any number of traits or stages of selection with explicit solution is described in this paper. This procedure does not need numerical integration and is suitable for obtaining either desired genetic gains for a variable proportion selected or optimum aggregate breeding value for a fixed total proportion selected. In addition, it will be shown that in certain situations the expected genetic gain from use of this new procedure will exceed that of conventional independent culling selection.

THEORETICAL CONSIDERATIONS AND DERIVATIONS

Notation and definitions:

n

k P G X

W

g AG AH

T

ti

number of traits measured or stages of selec-

number of traits of genetic importance n x n phenotype variance-covariance matrix k x n genetic variance-covariance matrix n X 1 vector of phenotypic records, expressed

as deviations from the population means k X 1 vector of economic weights k x 1 vector of breeding values k X 1 vector of genetic gains Gain in aggregate economic value,

n X n Cholesky decomposition of P, i e . , T'T

The ith column vector of T"

tion

AH = w' AG

= P

W. M. Muir

AX n X 1 vector of selection differentials for X bg.x k X n matrix containing regression coefficients

o f g o n X

normal variables obtained by linear orthogonal transformation of X

AZ n X 1 vector of standardized selection differentials of transformed variables

i overall selection intensity I, n X n identity matrix

Z n X 1 vector of independent standardized

P total proportion selected 9 n X 1 vector of proportions selected for indi-

U n X 1 vector of truncation points vidual variables

Multivariate selection responses: Using the notation of FALCONER (I 98 l) and assuming infinite population size, for selection based on a single trait, R = AG, S = AX and

R = h2S (1)

This formula can be extended to the case of multiple trait selection responses. The multivariate linear relationship between AG and AX can be expressed as:

AG = b,.,AX, (2) where bg.% has the form of:

bg.% = cov(g,X)[Var(X)]" = GP". (3)

Since cov(g,X) is defined as G and Var (X) is P,

AG = GP"AX. (4)

Equation 4 is the multitrait equivalence of (1). There- fore, AX and AG are respectively referred to as the multivariate selection differentials and responses. Sim- ilarly, GP" is referred to as the multivariate heritability matrix.

In the case of single trait selection, retrospective selection differentials can be determined from selection responses as:

S = h-'R.

Similarly, the retrospective vector of selection differentials can be determined from the vector of genetic gains for multitrait selection. If traits of genetic importance are the same as those selected, the relationship between AX and AG is simply

AX = PG"AG. (5)

But, in the general case, traits for which genetic improvement is desired may not be the same as those selected. In that case G" may not exist and there is no unique solution for AX. But, the best choice among all possible solutions for AX is that which minimizes overall selection intensity. Overall selection intensity for multitrait selection is defined by:

Multistage Selection 965

i = (AX'P-~AX)" (6)

which is a simple generalization of selection intensity for a single trait. For example, in the case of single trait selection, P" = a-' so that (6) simplifies to i = (SU-~S)"' = S/u. The solution for AX can be found subject to constraint (4) by minimizing (6), or equiva- lently, for ease of computation, minimizing 'h i2 . The equation to be minimized is:

Q = %(AX'P"AX) + X'(GP"AX - AG) (7)

where X is an k x 1 vector of Lagrange multipliers. Setting partial derivatives of (7) with respect to AX and X to zeros, the following equations are obtained:

[" GP" 0 p-lG] [:"I =[;J (8)

or

and

GP"AX = AG. (ab)

Constants X are found by multiplying both sides of (sa) by G, then substituting (8b) into (8a) and solving,

X = -(GP-'G')-'AG. (84

The vector AX is obtained by substituting (8c) into ( 8 4

AX = G'(GP"G')"AG. (9)

Standardization of the original traits: The next step is to find truncation points corresponding to X. In order to avoid numerical integration, it is necessary to transform the original traits into a new set of uncorrelated variables so that culling can be con- ducted on the transformed variables. TAI (1 989) suggested a transformation for index selection in which canonical variables were substituted for traits. Unfor- tunately, this type of transformation is not appropriate for sequential culling because one would have to wait until all traits are measured before the data could be transformed. As such, independent culling would lose its major advantage. In this paper, a new kind of linear transformation is proposed which has the same property of orthogonality but does not require information on all traits before it can be applied and hence pre- serves the advantage of independent culling.

Since P is positive definite, a decomposition of P is possible. One such decomposition is the Chole- sky decomposition (MARTIN, PETERS and WILKINSON 1965; MIHRAM 1972). The Cholesky decomposition matrix, T, is an upper triangular matrix with the

property of

T'T = P. (1 0) MARTIN PETERS and WILKINSON (1965) provided a procedure to calculate T. Also, this decomposition is commonly available in computer software packages such as SAS (1985). Since the inverse of a triangular matrix remains triangular, the inverse can be used as a transformation matrix to convert the original traits into a new set of uncorrelated normal variables, Z, assuming that the original traits were multivariate normal:

Z = (T')"X. (1 1)

Notice that

Var(Z) = (T')-' Var(X)T" = (T')"PT"

= (T')-~(T'T)T-' = I,.

The usefulness of this transformation can be ascer- tained by further examination of the Z matrix. First, define tq as elements of (T')-' then

Thus

Z =

tlIX1

t 2 l X I + t22X2

Each Zi represents an index comprised of all traits measured up to that stage. The overall procedure is, therefore, a form of multistage index selection. As such, genetic gains from use of this procedure may exceed ordinary independent culling level selection since additional information is being added at each stage as suggested by YOUNG (1 964) and CUNNINGHAM (1975). The procedure will here after be referred to as transformed culling.

Use of transformed culling for desired gains: The procedure is particularly applicable to situations in which genetic gains to accomplish a breeder's objec- tive are known. Equation 11 implies that

AZ = (T')-'AX. (12)

Let AG be a vector of desired genetic gains. After substituting (1 2) into (9), selection intensities necessary to achieve those gains are found as:

966 S. Xu and W. M. Muir

AZ = (T’)”G’(GP”G’)”AG. (1 3)

From AZ, a vector of truncation points, u, can be obtained by use of tables, such as given by FALCONER (1 98 l), relating selection intensity, A Z ; , with proportion selected, q i , and truncation points of the normal distribution, ui, or computationally as:

= [ q;(2*)1/2 ] EXP(-Yzu?)

where u, is the truncation point, ui = PROBIT(1 - q,), q i is the proportion selected at the ith stage, q, = 1 - @(ui),

1 q u i ) = - (2T)1/2 1. EXP(-fiZ2) dZ

and PROBIT ( ) is the inverse function of the standard normal distribution available in computer software packages such as SAS (1 985).

Optimum transformed culling: Although the Z’s in (1 1) represent sequential indices, they have not been optimized with respect to aggregate economic gain. But, with selection index the covariance between aggregate economic value ( H ) and the index (I) is always positive. Therefore, an optimum index is one which maximizes that covariance. In contrast, the covariance between H and Zi is

COV(H,Z;) = cOv(w’g,&’X) = w ’ G ~ (15)

which can be positive or negative. In the event of a negative covariance, Z, is selected in the negative direction; an optimum index would therefore be one which minimizes the covariance, i.e., gives a large negative value. In order to standardize the procedure such that the covariance is always maximized, the sign of ti is reversed if the covariance is negative. The relationship between AH and AZ is:

AH = w’AG

= w ’ ~ ~ - l ~ ~

= w ’ ~ ~ - l ~ ’ ~ ~

= w ’ ~ ~ - l ~ ~ n

= w‘ GCAZ, i= 1

Because (1 6) gives an explicit expression relating AH to the truncation point of each transformed variable, the optimum AH can be found by maximizing AH with respect to u given the constraint I Iy= , q, = p , i e . , the product of proportions selected of all stages

must equal a predetermined total proportion selected, p . T o maximize AH with this restriction, the constraint must be incorporated into the computations for AZ. This result may be accomplished by expressing the last selection differential, AZ,, as a dependent function of the others. Set the proportion selected for the last variable as:

therefore

AZ, = exp(-%u:) qn(2T)1’2

where u, = PROBIT(1 - 4,). Optimum truncation points can now be found by taking partial derivatives of AH with respect to u and setting these equal to zero.

6 A H 6 A Z i 6AZ, 6u, “

6Ul - w’G4 [x] + w’Gh [x][z] =

where

6 AZ, ”

6U: - U i ( U i - u,)

and

(see APPENDIX A). Therefore,

6 A H 6Uj ” - w’GCAZi(AZ; - ~ i )

= w’GC(AZ; - u,) - w’Gh(AZ, - u,) = 0

or

w’GC(A2; - ui) = w’Gt(A2, - u,) = T (17)

where 7 is a constant chosen such that n n qi = p - (18)

t= 1

To solve for 7 and the optimal u, a system of n + 1 nonlinear equations is constructed based on relationships given by (1 7) and ( 1 8).


This set of nonlinear equations in u and 7 is solved iteratively using the multidimensional Newton’s method (DUCROCQ and COLLEAU 1989) as shown in APPENDIX B. A general computer program for any number of traits, written in SAS/IML (SAS 1985), is available from the authors. Because numerical multiple integration is not needed, the program can accom- modate any number of traits or stages. On an IBM 3090, only 0.5 CPU seconds were required to obtain optimum culling points for five traits.

General relationships: The general relationship between single and multivariate selection response is made clear by further examination of Equations 1 and 4. If S in ( 1 ) is standardized by u, then

R = h2S = h2ui.

Similarly, if T’(T’)-’ is inserted into (4) we have

AG = GP”T’(T’)”AX = GP”T’AZ

which can be found in (16), the corresponding relationship between single and multivariate response is now obvious. The terms AG, GP”, T’ and AZ are the respective multivariate equivalents of R , h2, u and i. Thus, GP” is referred to as the multivariate heritability matrix. This matrix may be very informative in describing the dynamics of selection either in nat- ural or domestic populations. Similarly, with single trait selection, a selection differential can be standardized to find the corresponding truncation point from a standard normal table. Our procedure is a generalization to multiple traits whereby the multivariate selection differentials are standardized from which truncation points can be obtained from tables. But, this result should be distinguished from the usual z score. A z score of variable X is obtained by z = ( X - p)/u. Corresponding z scores of two correlated variables are still correlated. In contrast, our multivariate standardization procedure provides uncorrelated z scores.

EXAMPLES AND COMPARISONS

Two examples will be given to illustrate and nu- merically evaluate the procedure. An arbitrary set of parameters is given by:

P = PO 8 20 8 10 31 G = [5 3 5 3 ‘1. 3 10 30 4 5 20

The linear transformation matrix is

[ 0.3162 0

0.0295 -0.1 121 0.2006 O I (T’)-’ = -0.2169 0.2712 0

Desired genetic gains: Assume that all three traits are to be selected but only the first two are of economic importance, G will reduce to

G =[I :].

Let the vector of desired genetic gains be

AG = [0.8 1.01’.

The vector of standardized selection differentials, determined by (13), is

AZ = [0.4810 0.5422 0.30391’.

Corresponding proportions selected for each stage were obtained from FALCONER’S ( 1 98 1) Table A as

q = [0.708 0.666 0.8281’

with truncation points of:

u = [-0.5476 - 0.4289 - 0.94631’.

The overall proportion selected is p = 0.708 X 0.666 X 0.828 z 39%.

Optimum transformed culling: Let w = [ l 2]’, and the overall proportion selected be 20%, the corresponding selection intensity is i = 1.3998. From (16)

AH = 3.4785A21 + 1 .1389Az2 + 1.675623

using the multidimensional Newton’s method (APPEN- DIX B) optimal truncation points were:

u = [0.6651 - 1.7392 - 0.9307]’,

leading to

q = [0.2533 0.9585 0.82371’

AZ = [1.2634 0.0919 0.31461’

and the corresponding maximum aggregate genetic gain is

AH = 5.0264.

Note that the direction of selection for all three variables is positive.

Relative comparisons: The relative efficiency of transformed culling to index selection and conventional culling can be determined by the ratio of their corresponding A H ’ S . However, both relative costs and selection efficiencies must be considered before decid- ing on a selection program since cost of measuring traits will be different with each procedure. Also, comparisons of AH with conventional culling are only possible for specific cases, using numerical integration, because explicit solutions do not exist. Further, numerical evaluation is limited to a small number of stages because the computer time requirement for higher dimensional multivariate integration is tremen- dous (Saxton 1989). Comparisons between both culling procedures and index selection, for two, three and four traits were partially based on examples given by MUIR and Xu ( 1 99 1). Comparisons were based on the


TABLE 1

The expected total economic gain ( A H ) for two traits using transformed and conventional independent culling, and index selection for differing orders of selection, proportions selected (p ) , economic weights (W), genetic (rg), and phenotypic (rp) correlations, assuming that each trait has a unit phenotypic variance

Method of selection Relative efticiencv

Independent cullin& ~~

Case Index

P rP r, Trait h' W T C (S) RTIS RTIC Rcls

1

2

3

4

5

6

7

8

9

10

11

12

13

0.01

0.01

0.01

0.0 1

0.0 1

0.01

0.01

0.01

0.5

0.5

0.5

0.5

0.5

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.8

0.8

0.8

0.8

0.2

0.2

0.2

0.2

0.8

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

0.2 1

1 0.2

1 0.2

0.2 1

0.2 1

1 0.2

1 0.2

0.2 1

0.2 1

1 0.2

1 0.2

0.2 1

1 0.2

2.70

2.71

0.83

0.83

2.77

2.86

1.46

1.50

0.8 1

0.81

0.23

0.24

0.42

2.71

2.71

0.84

0.83

2.72

2.85

1.50

1.48

0.81

0.81

0.24

0.23

0.44

2.72

2.72

0.89

0.89

2.90

2.90

1.55

1.55

0.8 1

0.81

0.26

0.26

0.46

99.3

99.6

93.3

93.2

95.6

98.6

94.2

96.8

100.0

100.0

88.5

82.3

91.3

99.6

100.0

98.8

100.0

101.8

100.0

97.3

101.4

100.0

100.0

95.8

104.3

95.4

99.6

99.6

94.4

93.2

93.8

98.3

96.8

95.5

100.0

100.0

92.3

88.5

95.6


Method of selection Relative efficiency”

Independent culling

Case P rP rg Trait h‘ W T c (S) rr1.5 Ri-IC; RC, Index

14

15

16

17

18

19

20

21

22

23

24

25

26

27

0.5

0.5

0.5

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.5

0.5

0.5

0.2

0.2

0.2

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.2

0.2

0.2

0.2

0.8

0.8

0.8

0.8

0.2

0.2

0.2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

1 0.2

0.2 1

0.2 1

0.2 1

1 0.2

0.2 1

1 0.2

1 0.2

0.2 1

0.2 1

1 0.2

1 0.2

0.2 1

0.2 1

1 0.2

1 0.2

0.45

0.8 1

0.86

4.06

3.80

0.72

0.77

3.39

3.28

1.59

1.58

1.22

1.05

0.44

0.85

0.86

2.71

2.70

0.77

0.77

2.86

2.84

1.48

1.48

0.81

0.81

0.20 0.23

0.46

0.87

0.87

4.08

4.08

0.77

0.77

3.51

3.51

1.68

1.68

1.22

1.22

0.23

97.8

93.1

98.8

99.5

93.1

93.5

100.0

96.6

93.4

94.6

94.0

100.0

86.1

86.9

102.3

95.3

100.0

149.8

140.7

93.5

100.0

118.5

115.5

107.4

106.7

150.6

129.6

87.0

95.6

97.7

98.8

66.4

66.2

100.0

100.0

81.5

80.9

88.1

88.1

66.4

66.4

100.0


TABLE 1-Continued

The expected total economic gain (AH) for two traits using transformed and conventional independent culling, and index selection for differing orders of selection, proportions selected (p) , economic weights (W), genetic (r8), and phenotypic (rP) correlations, assuming

that each trait has a unit phenotypic variance

Method of selection

Independent cullingb

Case P TP rg Trait h2 W T C

Relative efficiency

Index (S) RT/s RT/c Rc/s

28 0.5 0.8 0.2 2 1 0.2 1 0.2 1

0.23 0.23 0.23 100.0 100.0 100.0 29 0.5 0.8 0.8

1 0.2 0.2 2 1 1

1.00 0.86 1.05 95.2 116.3 81.9 30 0.5 0.8 0.8

2 1 1 1 0.2 0.2

0.92 0.86 1.05 87.6 107.0 81.9 31 0.5 0.8 0.8

1 0.2 1 2 1 0.2

0.46 0.44 0.50 92.0 104.5 88.0 32 0.5 0.8 0.8

2 1 0.2 1 0.2 1

0.45 0.44 0.50 90.0 102.3 88.0

RTIS = 100 X AH(Transformed culling)/AH(Index), RTlC = 100 X AH(Transformed culling)/AH(Conventional culling), RcIs = 100 X AH(Conventiona1 culling)/AH(Index).

* T = transformed culling, C = conventional culling.

relative efficiencies of transformed culling to selection index (RT/s), transformed culling to conventional culling ( R T I c ) , and conventional culling to selection index ( R c ~ s ) . For two trait selection a wide range of heritabilities, economic weights, genetic correlations, phenotypic correlations, overall selection intensities, and orders of selection were examined. Due to the vast number of possible combinations with three- and four- trait selection, those comparisons were limited to cases of equal heritabilities of 0.5 for each trait, equal phenotypic correlations, and equal genetic correlations among traits. These are given in Tables 1 , 2 and 3.

Results show that the relative efficiencies of transformed culling to conventional (RT/C) ranged from 87% to over 300%. In general, the relative efficiency of transformed culling to conventional culling was inversely related to the relative efficiency of conventional culling to selection index (Rc/s), see Figure 1 which is plotted by using data from the three tables. On average, from Figure 1, the transformed culling procedure was superior to conventional culling where conventional culling was notably inferior to selection index (Rcls < 89%). Those cases usually involved high phenotypic correlations, dissimilar products of economic weights and heritabilities, and generally im-

proved with selection intensity or number of traits selected.

DISCUSSION

The remarkable aspect of this procedure is that for any number of traits, optimum culling points are obtainable with unique explicitly determined solutions without use of numerical multiple integration. By use of this equation it is possible to further examine the theoretical merits of multistage selection.

The usually greater efficiency of transformed culling is because 1) selecting a linear combination of the original traits may have a higher efficiency, like selection index. 2) With conventional independent culling level selection, the direction of selection for a trait is usually determined by the sign of its economic weight to give the maximum gain (SAXTON 1989). This may not be justified because the direction of selection for a trait also depends on heritabilities and genetic correlations. In the case of two traits, for instance, if hyrA(xyJhx is greater than one, indirect selection for y will give a greater response for x than direct selection of x (FALCONER 198 1). Thus, the direction of selection should not be determined by the sign of economic weight itself. In contrast, with transformed culling, the direction of selection for a standardized normal

Multistage Selection 97 1

TABLE 2

The expected total economic gain ( A J Y ) for three traits using transformed and conventional independent culling, and index selection for differing proportions selected (p), genetic correlations (re), phenotypic correlations (r,) and economic weights (W), assuming that

each trait has a unit phenotypic variance

Economic weight trait

Case" TP 1.4 P 1 2 3

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.2 0.2 0.2 0.2

-0.2 -0.2 -0.2 -0.2

0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2

0.8 0.2

-0.2 0.2

-0.2 -0.8

0.8 0.2 0.8 0.8 0.2 0.8 0.2 0.8 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.10 0.10 0.10 0.10 0.50 0.50 0.50 0.50

1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1

1 1 1 1 1 1 0.1 1 0.1 1 0.1 1 1 -1 1 1 1 1 0.1 1 1 -1 1 1 1 1 0.1 1 0.1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1

~

T

3.56 1.86 0.81 1.96 2.78 4.06 1.44 2.42 4.50 1.66 1.90 6.89 3.7 1 2.55 1.63 2.30 0.93 1.53 1.20 1.04 0.41 0.65 0.50

~

Method of selection

Independent culling' Relative efficiencyb

~

~

3.61 3.72 1.94 2.00 0.83 0.86 1.35 2.00 1.30 2.80 1.22 4.19 1.39 1.58 2.50 2.73 4.64 5.07 1.71 1.87 1.87 2.15 6.30 7.75 3.39 4.17 2.3 1 2.85 1.47 1.71 2.37 2.45 0.87 1.04 1.61 1.79 1.17 1.42 1.05 1.1 1 0.39 0.47 0.70 0.82 0.49 0.64

95.7 93.0 94.1 98.0 99.2 96.9 91.1 88.6 88.8 88.8 88.4 88.9 89.0 89.5 95.3 93.9 89.4 85.5 84.5 93.7 87.2 79.3 78.1

98.6 95.9 97.6

145.2 227.7 332.8 103.6 96.8 97.0 97.1

101.6 109.3 109.4 110.4 110.9 97.0

106.9 95.0

102.6 99.1

105.1 92.8

102.0

97.0 97.0 96.5 67.5 46.4 29.1 88.0 91.6 91.5 91.4 87.0 81.3 81.3 81.1 86.0 96.7 83.6 89.9 82.4 94.6 83.0 85.4 76.6

a For all traits, heritability = 0.5, genetic and phenotypic correlations are equal for all traits. * RT/S = 100 X AH(transformed culling)/AH(index), RTlc; = 100 X AH(transformed culling)/AH(conventional culling), and Rcls = 100 X

' T = transformed culling, C = conventional culling. AH(conventiona1 culling)/AH(index).

TABLE 3

The expected total economic gain (A?Z) for 4 traits using transformed and conventional independent culling, and index selection for differing proportions selected (p), genetic correlations (ra), phenotypic correlations (r,) and economic weights (W), assuming that each

trait has a unit phenotypic variance

Method of selection

Economic weight trait Independent

culling' Relative efficiencyb

Case" TP Ta P 1 2 3 4 T C Index ( S ) R7/(; Rc/,

1 0.8 0.8 0.10 1 1 1 1 3.01 3.12 3.24 92.9 96.5 96.3 2 0.8 0.8 0.10 1 1 1 -1 1.59 1.66 1.75 90.8 95.8 94.9 3 0.8 0.8 0.10 1 -1 1 1 1.61 1.66 1.75 92.0 97.0 94.9 4 0.8 0.8 0.10 -1 1 1 1 1.47 1.66 1.75 84.0 88.6 94.9 5 0.8 0.8 0.10 1 1 - 1 -1 0.70 0.43 0.78 89.7 162.8 55.1 6 0.8 0.8 0.50 1 1 1 1 1.36 1.36 1.47 92.5 100.0 92.5 7 0.8 0.8 0.50 1 1 1 -1 0.72 0.75 8 0.8 0.8 0.50 1 1 -1 -1 0.30 0.13

0.80 90.0 96.0 93.8 0.36 83.3 230.1 36.1

10 0.2 0.2 0.50 1 1 -1 -1 0.53 0.48 0.71 74.6 110.4 67.6 1.57 80.9 105.8 76.4

11 0.8 -0.2 0.10 0 0.1 1 10 19.59 8.58 19.76 99.1 228.3 43.4

9 0.2 0.2 0.10 1 -1 -1 1.27 1.20 1

For all traits, heritability = 0.5, genetic and phenotypic correlations are equal for all traits.

AH(conventiona1 culling)/AH(index). RT/S = 100 X M(transformed cuhg)/AH(index), RT/c: = 100 X AH(transformed culling)/AH(conventional culling), and R , : ~ ~ = 100 x

' T = transformed culling, C = conventional culling.

972

Y = Rt/c (%)

S. Xu and W. M. Muir

h

Y = 528 - 9.02X+ 0.047X2

FIGURE 1 .-Relationship between relative efficiency of transformed to conventional culling (Rtlc) and the relative efficiency of conventional culling to selection index ( R c l s ) .

20 30 40 50 60 70 80

X = R d . (“A)

variable is determined by the sign of the coefficient of the selection differential which is a function of a set of genetic and phenotypic parameters as well as economic weights.

The main reason for the sometimes lower efficiency of transformed culling over conventional is due to the restriction of orthogonality placed on the solution space. As such, the comparison is similar to that of a restricted selection index, ie., the responses will always be less than an unrestricted index (KEMPTHORNE and NORDSKOG 1959). Thus, in situations where gain from conventional culling is close to that of index selection, gain from the restricted index, or transformed culling, will be less than either. In situations where gains from selection index greatly exceed that of conventional culling, gains from transformed culling will remain less than that of the index but are now superior to that of conventional culling. Thus the relative efficiency of transformed culling to conventional culling is contingent on the relative efficiency

90 100

of selection index to conventional culling. YOUNG (1 961), as summarized by TURNER and

YOUNG (1 969), made extensive comparisons between conventional culling and index selection and con- cluded that:, 1) The superiority of index selection over conventional culling increases with selection intensity and 2) the number of traits under selection, but 3) decreases with increasing differences in relative importance among traits and is maximum when the traits considered are equally important. 4) The relative efficiency of index over independent culling is highly affected by the phenotypic correlation between traits when the traits are of equal importance, the relative efficiency of the index being highest when the phenotypic correlation is low or negative. 5) The effect of genetic correlations is apparent only when the traits are of unequal importance.

With respect to comparisons between conventional culling and index selection, our results support YOUNG’S (1 96 1) conclusions regarding selection inten-


sity, number of traits selected and effect of genetic correlations, but generally disagrees with YOUNG’S (1 96 1) other conclusions. These discrepancies are most easily discerned from cases examined with 3- trait selection given in Table 2. Comparison of cases 2 us. 4, 3 vs. 5 , 9 vs, 10, 12 vs. 14, and 13 vs. 15 all show that, in contrast to YOUNG’S (1 96 1) third conclusion, the efficiency of index selection is greatest with dissimilar economic weights. Also, in contrast to YOUNG’S (1 96 1) fourth conclusion, from comparison of cases 1 vs. 9, 2 vs. 8, and 1 vs. 12, the efficiency of index selection was always greater with high positive phenotypic correlations and the magnitude of the phenotypic correlation had a relatively small effect on the efficiencies when the traits were of equal importance.

The interesting aspect of these results is that for any situation or set of genetic parameters, one can always choose a multi-stage selection scheme, either conventional or transformed culling, which will have an efficiency close to that of selection index. After considering cost savings associated with independent culling selection, there are probably many situations in which the economic returns from use of independent culling, either conventional or transformed, will exceed that of selection index.

The procedure for determining optimum proportions selected was based on maximizing genetic gain. However, the advantage of multi-stage selection is cost savings associated with early culling. An obvious extension can be made to maximize the gain to cost ratio or profit. The procedure can also be extended to the case of multiple stage selection with more than one trait selected at each stage as discussed by YOUNG (1 964) and COTTERILL and JAMES (1 98 1). A generalization of this procedure, including the costs of measuring traits, is given by X u and MUIR (1 99 1).

We wish to acknowledge ALAN SCHINCKEL, T . STEWART and T . G. MARTIN for helpful comments on preliminary versions and comments of anonymous reviewers whose suggestions were very helpful. Journal Paper No. 12804 of the Purdue University Agri- cultural Experiment Station.

LITERATURE CITED

BROWN, G. H., 1967 The use of correlated variables for preliminary culling. Biometrics 23: 551-562.

COTTERILL, P. P., and J. W. JAMES, 1981 Optimizing two-stage independent culling selection in tree and animal breeding. Theor. Appl. Genet. 59: 67-72.

CUNNINGHAM, E. P., 1975 Multi-stage index selection. Theor. Appl. Genet. 4 6 55-61.

DUCROCQ, V., and J. J. COLLEAU, 1989 Optimum truncation points for independent culling level selection on a multivariate normal distribution, with an application to dairy cattle selection. Genet. Sel. Evol. 21: 185-198.

FALCONER, D. S., 198 1 Introduction to Quantitative Genetics, Ed. 2. Longman, London.

HANSON, W. D., and C. A. BRIM, 1963 Optimum allocation of

test materials for two-stage testing with an application to evaluation of soybean lines. Crop Sci., 3: 43-49.

HAZEL, L. N., and J. L. LUSH, 1942 The efficiency of three methods of selection. J. Hered. 33: 393-399.

KEMPTHORNE, O., and A. W. NORDSKOG, 1959 Restricted selection indices. Biometrics 15: 10-19.

MARTIN, R. S., G. PETERS and J. H. WILKINSON, 1965 Symmetric decomposition of a positive definite matrix. Num. Math. 7: 362-383.

MIHRAM, G. A,, 1972 Simulation: Statistical Foundations and Meth- odology. Academic Press, New York.

MUIR, W. M., and S. Xu, 1991 An approximate method for optimum independent culling level selection for n stages of selection with explicit solutions. Theor. Appl. Genet. (in press).

NAMKOONG, G., 1970 Optimum allocation of selection intensity in two stages of truncation selections. Biometrics 26: 465-476.

SAS, 1985 SASIIML Users Guide, Version 5 Edition. SAS Institute Inc., Cary, NC.

SAXTON, A. M., 1989 INDCULL Version 3.0: Independent culling for two or more traits. J. Hered. 8 0 166-167.

SMITH, S. P., and R. L. QUAAS, 1982 Optimal truncation points for independent culling level selection involving two traits. Biometrics 38: 975-980.

TAI, G. C. C., 1989 A proposal to improve the efficiency of index selection by ‘rounding.’ Theor. Appl. Genet. 78: 798-800.

TURNER, H. N., and S. S. Y. YOUNG, 1969 Quantitative Genetics in Sheep Breeding. Cornell University Press, Ithaca, N.Y.

WILLIAMS, J. M., and H. WEILER, 1964 Further charts for the means of truncated normal bivariate distributions. Aust. J. Statist. 6: 1 17-1 29.

XU, S., and W. M. MUIR, 1991 Selection index updating. Theor. Appl. Genet. (in press).

YOUNG, S. S. Y., 1961 A further examination of the relative efficiency of three methods of selection for genetic gains under less restricted conditions. Genet. Res. 2: 106-121.

YOUNG, S. S. Y., 1964 Multi-stage selection for genetic gain. Heredity 1 9 131-143.

YOUNG, S. S. Y., and H. WEILER, 1961 Selection for two correlated traits by independent culling levels. J. Genet. 57: 329- 338.

Communicating editor: B. S. WEIR

APPENDIX A

Derivatives:

Since A Z j = EXP(--1/2u?)

(27r)’/2[1 - qui ) ]

6aZi [ E X P ( ” ~ / ~ U : ) - -uj 6Uj (27r)1/*[1 - q u i ) ] 1

= -ujaZ, + AZl‘


APPENDIX B

Since

U , = PROBIT(1 - qn)

and

" 6un G[PROBIT(l - qn)] 69, 6U: 69, 6U;

- -

where

G[PROBIT( 1 - qn)] ( 2 7 p 69, EXP( PROBI BIT^( 1 - 9,)) 1

and,

Therefore

sf; (3)

Since5 = w'G&(AZi - ui) - 7

- sf; = w'G&[;; 6 A z i - 11 6Ul

Since

Multidimensional Newton's methods: To obtain solutions at the t + 1 iteration, solutions at the tth iteration are adjusted by the inverse of the matrix of partial derivatives evaluated at the tth iteration, i e . , u = d t ) and 7 = T(*), times the vector of differences also evaluated at the tth iteration, i . e . ,

[ : I 1 ) = [I] - su s7 -c3 6 f 6f

6r 6r 6u 67 "

where (see APPENDIX A):

"=o sf; 6Uj

6r - = 0. 67

The final solutions,

is obtained when,

evaluated at the tth iteration is sufficiently small. Convergence is rapid if starting values d o ) and d o ) , are not too far from their final solutions. By trial and error, we found that

?(O) = w'Gb'(i - C) @'Pb)'/'

is a good starting value for 7 , where c is the truncation point corresponding to the total proportion selected p , and b=P"G'w. Starting values of uo are obtained after Substituting d o ) into (1 7).

multistage selection for genetic gain by orthogonal ...equation 4 is the multitrait equivalence of...

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