149. Note: On Optimum Family Size in Selection ProgrammesAuthor(s): Alan RobertsonSource: Biometrics, Vol. 16, No. 2 (Jun., 1960), pp. 296-298Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2527559 .

Accessed: 28/06/2014 09:50

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access toBiometrics.

http://www.jstor.org

This content downloaded from 193.105.245.35 on Sat, 28 Jun 2014 09:50:30 AMAll use subject to JSTOR Terms and Conditions

296 BIOMETRICS, JUNE 1960

occupies approximately 800 drum locations, and its running time is 20 minutes.

REFERENCES

Emik, L. 0. and Terrill, C. E. [1949]. Systematic procedures for calculating inbreeding coefficients. Jour. Hered. 40, 51-55.

Robinson, H. F. and Comstock, R. E. [1955]. Analysis of genetic variability in corn with reference to probable effects of selection. Cold Spring Harbor Symp. Quant. Biol. 20, 127-136.

Wright, S. [1922]. Coefficients of inbreeding and relationship. Amer. Nat. 56, 330-338.

149 NOTE: On Optimum Family Size in Selection Programmes

ALAN ROBERTSON

A.R.C. Unit of Animal Genetics, West Mains Road, Edinburgh, Scotland.

In a recent publication (Robertson, [1957]) the problem of the optimum structure of a breeding programme using progeny-testing or family selection was discussed in general terms. It was shown that by the introduction of a factor K (the number of animals N whose per- formance could be tested divided by the number of groups which would be selected 8) a general solution could be given.

When the groups are of half-sibs and it is known that there are no non-genetic between-group variations, it can be shown that the expected improvement is a function solely of the proportion of groups selected p and K/a, where a = (4 - h2)/h2, h2 being the heritability of the measurement on which selection is based. We have for the expected genetic superiority of chosen sires

AG =Z 4 P P r+ (a/K)V9

where z is the ordinate of the unit normal curve at the point at which the area cut off is p. The determination of the structure for optimum improvement is then a problem of maximizing AG in terms of p. At the maximum, it was found that

K _1 2px-z a 2p z-px

where x is now the abscissa of the normal curve at the point of cut-off. It was thus possible to calculate the optimum value of p, and conse- quently the maximum value of AG, for different values of K/a.

This content downloaded from 193.105.245.35 on Sat, 28 Jun 2014 09:50:30 AMAll use subject to JSTOR Terms and Conditions

QUERIES AND NOTES 297

It was found that, as expected, AG increases as K/a increases but also, rather surprisingly, that above a value of K/a of about 8, a given proportional increase in K/a produced a constant absolute increase in AG. This relationship remained at an empirical level in the original paper. It is the function of the present paper to investigate further the nature of this relationship and to use it to consider further some of the problems of population structure. p is then determined by the above relationship

K 1 2px-z a 2pz-px

Substituting in the expression for AG, we have that at the optimum

AG = [(z2px-z) ]-

The algebraic expression of the empirical relationship would be

AG = A + B log, (K/a).

If this held true, then we should have

d(AG)/d(K/a) = B(a/K).

We can in fact find the value of the differential as a function of p, by evaluating d(AG)/dp and d(K/a)/dp. After much algebra we find that

d(AG) _ a z-px 2px-z d(K/a) K p Z

= (a/K)f(p).

If we plot f(p) against p, we find that it is zero when p = 0.27 (when 2px - z = 0), rises to a maximum when p is in the neighborhood of 0.07, and then declines slowly to zero as p approaches zero. But over a very wide range of p, f (p) changes very little. Thus, between p = 0.16 to p = 0.006, corresponding to K/a values from 3 to 500, f(p) lies between 0.285 and 0.325. It follows then that the apparent linearity of the plot of AG against log K/a is an algebraic accident in that the linearity holds merely over a wide range of K/a. This does not however alter the fact that over the range of K/a from 3 to 500, we may with reasonable accuracy write the superiority of the sires of the chosen groups when the structure is optimum as

AG = oq[O.5 + Q.3 log, (K/a)].

This content downloaded from 193.105.245.35 on Sat, 28 Jun 2014 09:50:30 AMAll use subject to JSTOR Terms and Conditions

298 BIOMETRICS, JUNE 1960

There is an odd consequence of this relationship. The number of groups chosen each generation S is generally fixed by the inbreeding depression that it is thought the population could stand. Can the optimum value of S be determined a priori? For given values of N and h2, any increase in S reduces AG when the structure is optimum, but also reduces the total amount of inbreeding depression AP. The best value of S is that which gives the greatest value of AG - AP.

Let us consider a programme of half-sib family selection in which we select the progeny groups of S sires in each generation. The increase in homozygosis each generation is 1/8S. If D is the inbreeding depres- sion for each unit of inbreeding, then the inbreeding depression each generation will be D/8S. As we deal with the selection of half-sib groups, then we have

AG = 'o-,[0.5 + 0.3 log. (K/Sa)].

At the optimum, we will have d(AG - AP)/dS = 0, or -0.15(of/S) + (D/882) = 0, giving for the optimum value of S,

5 D S= 6 o-

In a progeny-testing programme, in which the greater part of the improvement comes from breeding young males for testing by proven sires from which males would be bred each generation, the inbreeding depression would be reduced by a factor of 4 and the genetic improve- ment per generation only by a factor of two, so that the optimum value would be twice that given above. Perhaps the reader should be warned against a too literal interpretation of the above formulae. Inbreeding depression generally affects many other aspects of an animal besides those to which selection is immediately directed and in any practical application the value of D would have to be modified accordingly. However, it is interesting and rather surprising that the optimum number of families selected is independent of the total amount of test- ing facilities. This will of course lose its relevance if the population is to be used mainly for crossing purposes so that the purebred perform- ance is not of great importance.

REFERENCE

Robertson, A. [1957]. Optimum Group Size in Progeny Testing and Family Selec- tion. Biometrics 13, 442-450.

This content downloaded from 193.105.245.35 on Sat, 28 Jun 2014 09:50:30 AMAll use subject to JSTOR Terms and Conditions