Progeny Testing with Auxiliary Traits

Progeny Testing with Auxiliary TraitsAuthor(s): Prem NarainSource: Biometrics, Vol. 41, No. 4 (Dec., 1985), pp. 895-907Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2530962 .

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BIOMETRICS 41, 895-907 December 1985

Progeny Testing with Auxiliary Traits*

Prem Narain Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi, India

SUMMARY

The problem of determining the breeding worth of a male on the basis of the phenotypic values of his female progeny is discussed. The use of one or more auxiliary traits in conjunction with the main trait for progeny testing seems to have an edge over the conventional method in which no auxiliary traits are used. A general expression for the accuracy of selection based on the progeny test is derived and a generalised sire index is proposed. Detailed numerical investigation with one auxiliary trait reveals that the accuracy of the progeny test in such a case is always increased. The maximum gain in accuracy is found when the phenotypic and additive genetic correlations between the main and the auxiliary traits are of opposite signs. The number of progeny required to attain a pre-assigned value of accuracy is determined for several cases. It is found that the use of auxiliary trait reduces this number resulting in decreased cost of the progeny testing programme. The effect of the number of auxiliary traits on the gain in accuracy is also studied under some simplified situations.

1. Introduction

For sex-limited traits such as milk production in dairy cattle, the breeding value of a sire is assessed with the help of the phenotypic values of his female progeny for the given trait. The correlation between the two tends to unity as the number of progenies is made infinitely large. This forms the basis for progeny testing and sire evaluation in a livestock breeding programme. In this connection, Narain (1979, 1980) proposed a new sire index for milk production corrected for an auxiliary trait, such as age at first calving, first calving interval, or length of first lactation. This index is based on the phenotypic values of the progeny of the sire for the main trait expressed as deviations from its expected value predicted with the help of another correlated auxiliary trait. It seems desirable to use one or more auxiliary traits in conjunction with the basic trait for the progeny testing of a sire, since the genes affecting the main trait may also have pleiotropic effects so that the segregation of these very genes may also produce variation in a number of other traits of the individual which are ancillary to the main trait. Such effects tend to cause correlation between the concerned traits both at the genotypic as well as phenotypic levels. The breeding value of the sire for the basic trait is then strictly dependent on the phenotypic values of his progeny for the basic trait as well as other correlated auxiliary traits. We examine in this paper the accuracy of progeny testing from this angle. It may be noted that if, because of circumstances of physiological and/or managemental nature, it is preferable not to use the main trait itself as selection criterion, but to use only the auxiliary traits either singly or in optimum combination as a selection index, the procedure known as indirect selection has already been studied by Binet (1965) and Searle (1965, 1978).

* Presented in the Session on "Animal Genetics and Breeding" during the XV International Congress of Genetics (December 12-2 1, 1983) at New Delhi, India.

Key words: Accuracy of selection.;..Auxiliary traits; Genetic improvement; Progeny-testing.

89-5



896 Biometrics, December 1985

2. Statistical Model

Consider k auxiliary traits xi (i = 1, ..., k) related to the main trait y. Let the breeding values of a sire for the characters xi and y, expressed as deviations from the population means, be denoted by G(xi) and G(y), respectively, for i = 1, . . . , k. Also, let the phenotypic values be standardised to have unit variances so that the heritabilities of the traits, h2 (i = 1, ..., k) and ho (for trait y), are the same as the respective additive genetic variances. Further, let the average of the phenotypic values of n progenies of the sire for the characters xi and y be denoted by D(x1) and D(y), respectively (i = 1, ..., k). We denote the phenotypic and additive genetic correlations between y and xi by R pi and RI respectively, for i = 1, . . ., k, and phenotypic and additive genetic correlations between xi and xj by R ~, and R 4j, respectively, for i $ j = 1, ..., k. It is assumed that there is no common environment among paternal half-sibs. We can then set up the relationship between the breeding value of the sire and the average phenotypic values of his progenies for the (k + 1) characters as follows:

k

E[G(y)] = boD(y) + z biD(x1), (1) 1=1

where E stands for the expected value and bo, bl, . . ., bk are coefficients to be determined such that the multiple correlation coefficient between the breeding value of the sire and its expectation on the basis of the performance of the progenies for (k + 1) characters is maximised. The resulting normal equations are given by:

k

boV[D(y)] + E bicov[D(y), D(xi)] = cov[D(y), G(y)] i=l1

k

bocov[1D(y), 1D(xi)] + biV[D(xi)] + Z bjcov[1D(xi), D(xj)] (2) j=1 (joi)

= cov[D(x1), G(y)], i= 1, ..., k,

where, fori= 1, ...,kandj= 1,..., k(j54i),

V[D(y)] = I[1 +(n 1)ho];

V[D(xi)] = -[I +( 4)h2]

cov[D(y), D(xi)] n [R?i ( 4 ) Gihj;

cov[D(y), G(y)] =h (3)

(n 1) cov[D(xi), D(xj)] = n IRP, + ( 4 ) i]

cov[D(x1), G(y)] = 'Cih C1 =~~~2

Ci = R`I{hjho).

Expressing the normal equations in matrix notation yields

(n + ao)bo + [(1 + ao)r T + (n - I)CT]b = 2n; [(1 + ao)ro + (n - l)c]bo + [(1 + ao)R + (n - l)H]b = 2nc; (4)



Progeny Testing with Auxiliary Traits 897

where, with superscript T denoting the transpose of a vector or matrix,

bT = (bl, bk);

rT = (Ro, .. .,

CT= (Cl, . . , ;

R =((R

H=((R g hi hj1ho) H =~~2 ao= (4 - ho)lh 0

With n = 1, these equations reduce to similar ones given by Narain and Mishra [1975, equations (5)], except for a factor of 1 due to parent-offspring relationship utilised in a progeny testing scheme. Solving the equations, we get

bo = 2n(1 - aTQ c)/(a TQ-1(5) b = 2nQ-l[(ac - - cTQ-I# + ##TQ-Ic]/(a _ TQ-)

where

=( + ao)ro + (n - I)c;

Q = [(1 + ao)R + (n - 1)H] = (nI + A)H; (6)

A = (4RH1 - h2)1h2;

a = (n + ao).

3. Accuracy of Progeny Test

The accuracy of the progeny test measured in terms of the maximised multiple correlation coefficient (h * ) between G(y) and E[G(y)] is then given by the expression

h*2 -=h2 1 - (n + ao)dl + 2(1 + ao)d3- (1 + ao)2(di d2- d2)1 pr prL1 -(n+aO)dj +2(1 +ao)d3-(I +aO)2d2/(n+ao)]' (7)

where

hpr= (n/a) = n/(n + ao);

d = cTQ-1c; (8) d2= (c - ro)TQ(c-ro)

d3 = c Q(c - ro)

It is interesting to note that for given values of n, ho, h2 , RR, Rg-, and Rpi, the expression (7) is a function of C, or Rgi. In fact, it is a scalar function of vector variable c at fixed values of other parameters. We therefore take the vector derivative of this function with respect to c and equate it to the null vector. This gives, on simplification, the solution c = ro with positive second vector derivative at this solution. This shows that the accuracy is minimal when c = ro, i.e., the corresponding elements of the two vectors are equal in magnitude and possess the same sign. When c = ro, i.e., when Rpi = Rgihi/ho, d2 and d3 reduce to 0 and hp*r = hpr, indicating that the gain in accuracy due to the use of auxiliary traits is 0. In fact, we can express (7) in terms of the gain in accuracy due to the use of




auxiliary traits as

A= (hpr - hpr)/hpr

=(1 + ao)2 ({(n + ao)d2 + [1 -(n + ao)d]d2}/{(n + ao)[I - (n + ao)dl + (1 + ao)d3] - (1 + ao)[(l + ao)d2 - (n + ao)d3]j). (9)

It is found that AA is the same when either C1 and R pi are both positive or both negative for i = 1, . .. , k. It is also the same when both these sets of parameters have opposite signs, i.e., Ci positive and Rpi negative give the same results as C1 negative and Rpi positive. But AA with opposite signs for c and ro is necessarily greater than AA with the same signs for c and ro. Thus, the accuracy is expected to increase as the vector of differences (c - ro) increases in magnitude and would have maximum effect when c and ro are of opposite signs. This aspect is numerically studied in detail, for k = 1, in Section 5.

For the limiting case when n tends to be infinitely large, we see that

Q = H-1(nI + A)-'

= I-V1[I _ (I/n +Al)l] (10) n n _

tends to 0 so that di, d2, and d3 also tend to 0, and, since h~r tends to 1, so does hpr. In fact, with very large number of progenies available for a sire, we expect to know with complete accuracy the breeding values for each of the main and auxiliary traits. The index will then help very little as we would be virtually selecting on the basis of main trait only. This was also noted by Robertson (1961) while discussing selection for several characters. On the other extreme, if we take n = 1,

Q = (I + A)H = 4R/ho (11)

and hpr reduces to

h (2 ) (cTR-ic + Er)2 (12) (r 4 ) 2

where

EP = (1 -cTR-'ro)(1 - rTR-'ro)-f/2 (13)

is the efficiency of the phenotypic index relative to individual selection given by Narain and Mishra (1975), which for a single trait, reduces to the expression given in Searle (1965). It was also shown in Narain and Mishra (1975) that Ep is related to Es, the efficiency of selection index based on several characters, by

p= S- CTR-c. (14)

Since accuracy is ho times efficiency, accuracy of the progeny test reduces to half of the accuracy of selection index, as expected in view of the parent-progeny relationship.

When the genetic correlation coefficients Rgi and RM. are all 0, c = O H = 0, Q = (1 + ao)R, giving di = d3= 0 and d2 = rTR-'ro/(l + ao) and h*2 reduces to

= n/(n + aO*), (15)

where a* = (4 - ho2)/ho2. Here ho2 = h/(l rR-'ro) is the heritability of the main trait y corrected for auxiliary traits x,, ..., Xk which are correlated with the main trait at the




environmental level only as given in Narain and Mishra (1975). This heritability was also used in Narain (1979) for proposing a new sire index for milk production corrected for an auxiliary trait such as age at first calving or first calving interval. This means that when all the auxiliary traits are related to the main trait only at the environmental level, the accuracy of a progeny test is of the same form as that without any auxiliary traits but the main trait y is replaced by the phenotypic index developed in Narain and Mishra (1975).

4. A Generalised Sire Index

Substituting the optimum values of bo, b1, . .. , bk given by (5) in (1), we get the generalised index for the breeding value of sire, as

nW1D( ) -(I _ +Q-IC) a0 r C) - A I (cr- rocT)]Ql'd(x)} (16) SI 2n~~~k D~~~y k #TQIC! [(ro - ) #Q(r

where Wk = (1 _ 6TQ-I c)/(a -#TQ-Ia)

dT(X) = (D(x1), ... , D(xk)). (17)

Again, we note that when c = ro, the index reduces to

/2n - SI = I JD(y), (18)

\f + a0o

the usual sire index (based on daughters' average, making allowance for finite number of daughters), as if there were no auxiliary traits. For n -- oo, Q' tends to 0 as we saw earlier and the index reduces to twice the simple daughter average for trait y, as it should. For n = 1, we get

h 2 1 r D(y) (ro - c)T - rTR'(crT - rocT)] x (19)

the usual selection index in which the auxiliary traits xi, . .. , xk are used merely as an aid to more efficient selection for improvement in the main trait, the relative economic values of x,, . .. , xk all being taken as 0.

When the genetic correlation coefficients Rgi and Rgj are all 0, c = 0, H = 0 so that =(+ ao)ro,

2n SI = (n + *) [D(y) - rOR'd(x)], (20)

which is a generalised form of the index proposed in Narain (1979) for k auxiliary traits, all correlated with the main trait as well as amongst themselves, but only at the environmental level.

5. Numerical Results

It is apparent from the theory presented in the preceding sections that a study of the behaviour of the accuracy of the progeny test or that of the generalised sire index, with variation in the various parameters, is practically impossible unless we introduce some simplifications. In the first instance, we assume that there is only a single auxiliary trait, i.e., k = 1. Thereafter we consider the effect of the number of auxiliary traits by assuming simplified situations.




5.1 Single Auxiliary Trait

Since k = 1 the vectors ro and c become the scalars R1 and C, = Ro h1I/ho respectively, whereas matrices R and H become the scalars 1 and hMhI, respectively. The matrix Q then becomes the scalar (hl/h')(n + a,) with a, = (4 - h,)/h 2 and ft is now the scalar (1 + ao)RoP + (n - 1)CI. On simplification, the expression for h~? reduces to

hr - r [K -(C, -Rol)2(1 + ao)/(n + ao)1 (21)

where

K = (Cl -Ro)2 + (1-Ro) + (n - 1)(1 - Ro)/(1 + ai). (22)

This expression tallies, barring differences in notation, with (23) of Searle (1978), which contains a misprint of g instead of N?i4 in the denominator. It can be easily seen that h*r2 (k = 1) is always greater than or equal to h2r since the expression within the bracket is necessarily greater than or equal to unity. This shows that, with a single auxiliary trait, the accuracy of the progeny test is always increased an expected result also obtained by Searle (1978). This, however, would also be the case for any number of auxiliary traits since the accuracy, being a multiple correlation coefficient, will not decrease with more terms added to the model.

The corresponding sire index is given by

SI(k=l) = 2nWi[D(y) - (R0 - 1

D(xi)] (23) (I - RoICI) + (n - 1)(h, - hoC )/4

where

I= [(1 + ao)(1 - RoICI) + (n - 1)(1 - C2) - (n - 1)(1 - h 2/h2)] ix [(1 +ao)2(1 -RP) + 2(1 + ao)(n- 1)(1 -RPCj) + (n-1)2(1-Ct)

- (n + ao)(n - 1)(1 - h i/ho)f. (24) We notice that, even with k = 1, the accuracy of the progeny test is a function of as

many as five parameters, ho, h2, R pi, Rg1, and n. We therefore fix four of these parameters and study the accuracy as a function of the fifth parameter.

For getting numerical results, values of n were chosen to be 1, 5, 10, 20, 30, 50, and 100, and h2 was chosen to be 0.01, 0.10, 0.20, and 0.50. The heritability of the main character ho was set to 0.01, 0.05, 0.10, 0.15, 0.20, 0.30, 0.50, and 0.80. A large number of combinations of Ro, and Rg, lying between -0.80 and +0.80 were taken for obtaining the accuracy of the progeny test. It was found that positive and negative values of RoP or Rg produce the same effect because of their occurrence as square terms. We have, however, chosen a few specific combinations of parameters to illustrate the general pattern on the behaviour of the accuracy of the progeny test as the parameters vary.

First, we fix n = 20, h2 = .5, and ho = .1, and examine the accuracy at five different values of correlations Ro, and R91, namely, -0.8, -0.5, 0.0, 0.5, and 0.8. This gives rise to 25 combinations, the results for which are presented in Table 1.

As expected, there is a rotational symmetry in the table, positive and negative combinations of the two correlations producing the same value. We have distinct values for 13 combinations out of 25 given in the table, including the one corresponding to the combination (0, 0) which gives the lowest value of 0.58. The accuracy is greater when correlations have opposite signs than when they have the same signs. The highest accuracy occurs for (?0.8, TO.8). However, the same accuracy of 0.72 for (?0.5, TO.5) can be produced at higher values of Ro1 but with same sign as (+0.5, +O.8). At a given value of




Table 1 Accuracy of progeny test with one auxiliary trait for different combinations of (Rg1, R91) when n =

20, hi = 0.5, andho = 0.1 DP 01VO

MO 1 -0.8 -0.5 0.0 0.5 0.8 -0.8 0.70 0.72 0.77 0.84 0.89 -0.5 0.59 0.61 0.65 0.72 0.77 0.0 0.62 0.60 0.58 0.60 0.62 0.5 0.77 0.72 0.65 0.61 0.59 0.8 0.89 0.84 0.77 0.72 0.70

Rgo, the variation in accuracy over different values of Ro is less compared to that over different values of Rg1 at a given value of Rol. In fact, the accuracies given in the row for

goI= 0 are very similar but those given in the column for Ro1 = 0 vary considerably. This shows that genetic correlation affects the accuracy more than the phenotypic correlation. In general, compared to the combination (0, 0) for no auxiliary traits, the accuracy values are found to be highest for correlations with opposite signs at each value of n, ho, and h . This is true for several combinations of (Rol, Rg1), the results of all of which are not presented here but can be obtained from the author on request. It was also noted in all cases that the presence of genetic correlation is more significant in increasing the accuracy than that of phenotypic correlation except at higher values of n or ho when values for all combinations are close to each other. As such, a combination such as (?0.5, ?0.5) gives results similar to those for (0, ?0.5).

The effect of the numerical difference 6 = (C, - RPI) on the accuracy was further examined. As seen from the theory presented in Section 4, the gain in accuracy is minimal, being 0, when C, = R or 6 = 0. As 6 increases, either in positive or negative direction, the accuracy increases. For different combinations of (RoI, R91) producing the same value of 6, the accuracies are close to each other, the value being, however, maximum for numerically highest value of Rp1. Here we present results in Figure 1 for some typical values of the parameters to illustrate the effect of 6 on the accuracy.

For h2 = ho = h2 so that (C, - RP1) = (Rg, - RPI) = 6 and Ro, = +0.8, the figure shows different curves depicting variation in accuracy as 6 increases or decreases. There are eight curves for four values of n = 1, 10, 30, and 100, each at two values of h2 = 0.1 and 0.2. In each case, the accuracy increases with increase in 6 from 0 to 1.60 as well as decrease in 6 from 0 to - 1.60, the curves for negative 6 being mirror images of those for positive 6. As n increases from 1 to 100, the rate of increase, however, decreases. At a given value of n, the accuracy is always more for the higher value of h2 = 0.2 compared to those for h2 = 0.1. The figure clearly brings home the point that the gain in accuracy is more when Rp, and Rol are of opposite signs than when they are of the same sign at each value of n and h2. The range of 6 between 0 and ? 1.6 consists of two halves, 0 to ?0.8 and ?0.8 to ? 1.6. At 6 = 0.8, R Iis 0, below it is negative, and above it is positive with reverse signs for negative 6. The rates of increase in accuracy are more and almost linear in the ranges of 6 from 0.8 to 1.6 and from -0.8 to - 1.6, corresponding to opposite signs of the two correlations, than those in the ranges from 0 to 0.8 and from 0 to -0.8, corresponding to the same signs, respectively.

We now fix n = 20 and h I = 0.2 and plot the accuracy as a function of ho for combinations of (RoP, R91) as (+0.5, ?0.5), (0.0, ?0.5), and (?0.5, 0.0) as in Figure 2. For comparison purposes, we plot also the curve for the combination (0.0, 0.0).

It is apparent that the accuracy increases sharply for smaller values of ho less than or equal to about 0.25 in all four cases. The most important point is that accuracy with an



902 Biometrics, December 1985 1 00 Rl= +0 8 Rol= -0 8

070 --

0*75 \ X -X

0~~~~~~~~~~3

10

LLJ 02 5 -

0

OL U

< = (C1- Ro1)=( Ro1 Ro)

Figure 1. Accuracy of the progeny test with one auxiliary trait as a function of 3 = (C, - Rg1)- (Ro - Rol) for n = 1, 10, 30, 100, and h2 = h2= h2= 0.1, 0.2 when Rol = ?0.8.

auxiliary trait is always more than that without it for all values of h2. The two curves for the combinations (0.0, ?0.5) and (?0.5, 0.0) intersect each other at h2 = 0.2. For h2 < 0.2, the accuracy for (0.0, ?0.5) is more than that for (+0.5, 0), but for h4> 0.2 this is reversed. The accuracy of the progeny test, when correlations are of opposite signs, is found to be always more than the other cases, the gain being substantial at lower values of hT.

The most significant finding about the accuracy of the progeny test is shown in Figure 3, wherein it is plotted against the group size n for fixed values of h2 = 0.2 and h2 = 0.2. The three curves pertain to combinations of (Rol, Ro1) as (+0.5, +0.5), (?0.5, 0.0), and

(+0.5 acuayfr+0.505)s.oeta ha o ?., u o h2>02tisi eesd

Clearly, the curve with correlations of opposite sign is the highest and that with correlations of same sign is the lowest. As expected, the accuracy increases sharply as we go from n = 1 to n = 30, beyond which the curves taper off at higher values of n, tending, in the limit, to unity in all the cases. The message is that for group size between 10 and 30, it pays substantially to progeny test sires, taking into account an auxiliary trait, provided the genetic and phenotypic correlations are of opposite signs.

The above result on the behaviour of the accuracy with variation in progeny group size clearly brings out the need for knowing the number of progeny which would provide a certain degree of accuracy. This can be achieved by solving the expression of h* for n. Such a solution would give, prior to conducting a progeny test programme, the number of progeny required in order that hpr shall be a pre-assigned value, say 0. This gives a quadratic




1.00

w 0 7 5 -

z U n 20

0 2 p g

LL 0 -5 0 ~ ~I 0-2 0.0 0.0

> .l II: 0 2 -o05 05

U / 0 5 -05

z / -------- III 0~~~~0 2 oo0 + 0.5

0 ........... ....- IV 0 2 t. 0 5 ? ?

0 025 050 080 100

2

Figure 2. Accuracy of the progeny test with one auxiliary trait as a function of heritability (ho) of main trait for (RoI, Rg1) = (+0.5, ?0.5), (0.0, ?0.5), and (?0.5, 0.0) when n = 20 and h2 = 0.2.

equation in n given by

Dn+ En + F= 0, (25)

where

D=(1-_02)( - Rol);

E = (1- 02)(1 + a,)[(C1 - Rp )2 + (1 - Rp)I _-[1 -(1 - ao)02](1 - R ); (26)

F =021(l + a,)(C,- RoPI)2- ao[(l + al)(1 - R p2 (I - R )] }.

Of the two roots of this equation, G + H. where

)2+ (1-R)I o2\I 2\1

1 J4[(ci _ Rol (_ / PIR4-2 ' 2 h 2 I2 I' 2 h 1 01) 0[ 4-h \1h1 L - R p)2 - -PI -4_-_ ho

2 ~ ~ ~ ~~ 6 '02 (C2

-h2h2 1I _ 021 (1 -R0,2 (28) the positive root is to be taken. As we can see, the root is a function of four parameters, ho, h2, RPP, and R21, plus the predetermined value of hp* It may be seen that when




075 2

0~~~~~~~~~~~~~~~~ w~~~~~~~~~~~~~~~~

p , F i/ ./ ~~~~~~~~~~Ro Ro 01 01

u / - I: 0 5 - 0 5 s ~ ~~~~~~

~~~~~~~~~~~ 11 / 05 00 o 0 5 0

.5

LL -0.II: -05 0 5 0 005 -025

U

U U' 0.25

0 10 30 50 70 90 100

n

Figure 3. Accuracy of the progeny test with one auxiliary trait as a function of group size (n) for (RoP, Rg1) = (+0.5, +0.5), (?0.5, 0.0), (+0.5, ?0.5) when h2 = 0.2 and h2 = 0.2.

C1 = Ro,

or when Ro,

and Rg,

are 0, the values of n reduce to ao02/(1 - 02), as tabulated in Searle (1964) for no auxiliary trait.

For illustration, we present in Table 2 the values of progeny group size n when h2 = 0.2, for two values of h2 = 0.1 and 0.3, varying combinations of (Rt1, Ro1) as (0, 0), (?0.5, 0), (0, ?0.5), (?0.5, ?0.5), and (TO.5, ?0.5). The combination (0, 0) is taken for comparing

Table 2 Number of progeny requiredfor a pre-assigned value of accuracy when hM = 0.2

ho= 0.1 ho= 0.3

RoP0 0.0 ?0.5 0.0 ?0.5 T0.5 0.0 ?0.5 0.0 ?0.5 +0.5 R90 I 0.0 0.0 ?0.5 ?0.5 ?0.5 0.0 0.0 ?0.5 ?0.5 ?0.5

0.50 13 11 9 13 5 4 3 3 4 2 0.55 17 14 12 16 6 5 4 5 5 2 0.60 22 19 16 21 9 7 5 6 7 3 0.65 28 25 22 28 12 9 7 8 9 4

hp*r 0.70 37 34 30 37 18 12 10 10 12 6 0.75 50 46 42 49 26 16 13 14 16 8 0.80 69 65 60 68 41 22 19 20 22 12 0.85 101 97 91 101 70 32 28 29 32 19 0.90 166 161 154 165 131 52 48 49 52 36 0.95 361 356 348 360 323 114 109 110 114 94




the results with those already given in Searle (1964) without the use of any auxiliary trait, whereas other combinations are typical of the situations when the use of auxiliary trait is found to be most effective as borne out by the results already presented for the accuracy of the progeny test.

It is obvious from the results, and very significant to note, that in all cases the use of auxiliary traits reduces the number of progeny required for a pre-assigned accuracy, a favourable situation which reduces the cost of the progeny testing programme. Further, this gain is substantial when RP 1 and R g1 are of opposite signs and around 0.5 in magnitude. For instance, for h* = 0.70, the number drops by about 50%, i.e., from 37 to 18 when ho = 0.1 and from 12 to 6 when ho = 0.3. However, as we move on to higher accuracy, the difference narrows down, but, since n increases substantially in such a case, the gain is still substantial. Insofar as combinations (?0.5, 0) and (0, ?0.5) are concerned, it seems for ho = 0.1, which is less than h2 = 0.2, a situation with genetic correlation is favourable to that with no genetic correlation. When ho = 0.3, which is greater than h 2 = 0.2, this trend is reversed, though the differences are now only marginal.

5.2 Effect of the Number (k) ofAuxiliary Traits

We now assume that the k auxiliary traits are uncorrelated both at the genetic as well as at the phenotypic levels and that their genetic variabilities are of the same order. This would mean

R =I

H = (h2/h )I (29)

Q [(n + a)h2/h ]I a (4 - h )1h ,

where h2 is the common value of the heritability of auxiliary traits. We further assume that all phenotypic (genetic) correlations between y and x's are equal to R P (R g). We then have

d= kRg /(n + a)

2= d(I - )(30)

d d ( I _ R op d= di(I -

where C= R g(h/ho). We can then express, in percentage, the gain in (accuracy)2 in units of (accuracy)2 without

the auxiliary trait, as

A x I 00 000kC--) 2 (31) L(n + ao)(n + a)] kLR + (n -)]

For n= 1, we get

1O0k(C - R P)2 / I100kR2 (32 = I-kR2 =1- kR" if C is also 0), (32)

which are comparable to those given in Narain and Mishra (1975). The numerical results are presented in Table 3 for n = 20, ho = h2 = 0.25, typical values in a progeny testing




Table 3 Effect of the number of auxiliary traits (k) on the gain in (accuracy)2 (per cent) for n = 20 and ha =

h2 0.25 R? +0.2 ?0.1 0.0 ?0.2 0.0 w0.2 Rg-+0.2 0.0 ?0.1 0.0 ?0.2 ?0.2

1 0.00 0.21 0.21 0.84 0.85 3.34 5 0.00 1.06 1.06 4.36 4.44 16.74

k 10 0.00 2.13 2.15 9.12 9.48 33.54 15 0.00 3.24 3.28 14.34 15.23 50.38 20 0.00 4.36 4.44 20.07 21.88 67.27

programme with dairy cattle. The combinations of Ro and Rg taken for illustration are (?0.1, 0), (0, ?0.1), (?0.2, 0), (0, ?0.2), (+O.2, ?0.2), and (?0.2, ?0.2). It is apparent-that the gain due to inclusion of auxiliary traits is substantial provided Ro and Rg are of opposite signs.

6. General Comments

The use of auxiliary traits in increasing the efficiency of selection based on progeny testing is of general applicability, although this article is written with dairy cattle in mind and a generalised sire index is proposed in this connection. For a sex-limited character such as milk production, the procedure involves the use of age at calving or calving interval as auxiliary traits, as indicated in Narain (1979, 1980) and Kumar and Narain (1980). In the latter reference, this method was compared with other existing methods of sire evaluation with the help of data on Indian Sahiwal cattle. Sire-indexing with calving interval as an auxiliary trait improved the efficiency by about 14%. However, the use of auxiliary trait is dependent on the availability of the estimates of phenotypic and genetic correlation coefficients, which are treated as known quantities in this paper. Since the estimates of correlation coefficients, particularly the genetic correlation coefficient, are often found to lie outside the range of -1 to + 1, the application of the technique gets restricted. Further, in this paper we assume the absence of common environment among paternal half-sibs. When such environment is a major factor, as for instance in poultry, the efficiency of the method gets reduced.

ACKNOWLEDGEMENTS

The author is grateful to Professor Walter R. Harvey, Department of Dairy Science, The Ohio State University, Columbus, Ohio 43210, U.S.A. for helpful discussions. The assis- tance rendered by the referees in improving the article is gratefully acknowledged.

RE'SUME La discussion porte sur 1'estimation de la valeur genetique des miles 'a partir des valeurs phenotypiques de leurs descendantes. L'utilisation d'un ou plusieurs caracteres secondaires en plus du caractere principal mesur6 sur les descendants semble apporter une amelioration par rapport a la methode classique qui n'utilise pas de caracteres secondaires. Un indice male generalise est propose ainsi qu'une formulation generale de la precision d'une methode de selection base sur le testage. Dans une application numerique detaillne oti un caractere secondaire est utilis6, il est montr& que la precision du testage est toujours augmentee dans un tel cas. Le gain maximum de precision est obtenu lorsque les correlations phenotypiques et genetiques entre le caractere principal et les caracteres secondaires sont de signes diffhrents. Le nombre de descendants necessaires pour atteindre une valeur pre-determinee de la precision est calcul6 dans dif~erents cas. L'utilisation de caracteres secondaires permet de reduire ce nombre, et donc de reduire les cotits d'un programme de selection. L'effet du nombre de caracteres secondaires sur le gain de precision est aussi etudie dans des situations plus simple.




REFERENCES

Binet, F. E. (1965). On the construction of an index for indirect selection. Biometrics 21, 291-299. Kumar, D. and Narain, P. (1980). Different methods of sire evaluation. Indian Journal of Dairy

Science 33, 468-472. Narain, P. (1979). A new sire index for milk production corrected for an auxiliary trait. Indian

Journal ofAnimal Genetics and Breeding 1, 20-22. Narain, P. (1980). Livestock breeding. In Methodology for Improvement of Data Base on Livestock

Resources, 93-116. New Delhi: APHCA-IASRI. Narain, P. and Mishra, A. K. (1975). Efficiency of selective breeding based on a phenotypic index.

Journal of Genetics 62, 69-76. Robertson, A. (1961). Selection for several characters. In Lectures of II International Summer School

on Scientific Problems of Breeding Systems and Breeding Plans of Domestic Animals, 213-224. Mariensee: Schriftenreihe des Max Planck-Instituts fur Tierzucht und Tierernah-rung.

Searle, S. R. (1964). Progeny tests of sire and son. Journal of Dairy Science 47, 414-420. Searle, S. R. (1965). The value of indirect selection-I. Mass selection. Biometrics 21, 682-707. Searle, S. R. (1978). The value of indirect selection-II. Progeny testing. Theoretical and Applied

Genetics 51, 289-296.

Received May 1984; revised March and August 1985.



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Progeny Testing with Auxiliary Traits

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