QUANTITATIVE INHERITANCE 201 



Offspring-parent regression. — The chief problem of experimental design concerns the 

 number of offspring from each parent or pair of parents that should be measured. When 

 the most efficient design has been decided, the number of animals that must be measured 

 to attain a given degree of precision can be determined. The size of an experiment is 

 limited either by the amount of breeding or rearing space available or by the number of 

 individuals that can be measured. The problem is to decide how to divide the space 

 or effort among the offspring of different parents. Either few offspring from each of 

 many parents or many offspring from each of few parents can be reared and measured. 

 The solution of this problem comes from a consideration of the expected standard error 

 of the estimate of heritability that will be obtained. Approximate formulae for the 

 variance of the estimate are as follows: 335 



By regression : 



„ . \ + (n — l)t , r . 



on one parent, op = 4 ^ (b) 



on mean of both parents ofe = 2 ^^ (7) 



In these formulae n is the number of offspring measured per parent (equation 6) or pair 

 of parents (equation 7), TV is the number of parents or pairs of parents, and / is the pheno- 

 typic correlation between the offspring of the same parent or pair of parents. In 

 work with laboratory animals the offspring are likely to be full sibs. In this case 

 the phenotypic correlation, t, will be approximately equal to half the heritability, or 

 greater than this if there are also nongenetic causes of resemblance between the offspring 

 of the same parents. The solution of the problem of design depends on the circum- 

 stances that limit the size of the experiment. The simplest solution is when the 

 limiting factor is the total number of offspring that can be reared or measured. The 

 total number of offspring is nN, and, if this is fixed, then the denominators of both the 

 above formulae are fixed. The variance of the estimate is then, in both cases, minimal 

 when n = 1 . This means that the most efficient design has only one offspring measured 

 from each parent or pair of parents. The standard error of the estimate of the herita- 

 bility then becomes: 



By regression : 



on one parent op — V2/N (8) 



on mean of both parents a h ^—2jvN. (9) 



These relationships are also shown by the graphs in figure 34. Graph (b) refers to the 

 regression on one parent and graph (a) to the regression on the mean of both parents. 

 The horizontal axis shows the total number of offspring measured (that is, N) and the 

 vertical axis the corresponding standard error expected. It is again evident that 

 precise estimates of the heritability cannot be obtained without the measurement of 

 large numbers of individuals. For example, to attain a standard error of 0.10 it is 



