QUANTITATIVE INHERITANCE 205 



The variance of the estimate of the regression coefficient should, in my opinion, be 

 computed in the usual way but from the weighted sum of squares. Thus 



°* I w - 2 12 (wx 2 ) b \ 



The error variance on which this is based is the weighted mean of the squared deviations 

 of family means from the regression line. Kempthorne and Tandon, 702 however, 

 say that the appropriate error variance is the variance (of Y) within families. 



Sib analyses. — Although the correlation between sibs is usually less efficient than 

 the offspring-parent regression as a method of estimating heritability in laboratory 

 mammals, there are occasions when it has to be used. If the measurement of the 

 character requires the death of the animals before they can breed, then the regression 

 method is obviously inapplicable, and the correlation between sibs is the only method 

 that can be used. 



The correlation of full sibs does not provide a reliable estimate of heritability 

 because it may be augmented by nonadditive genetic variance and often also by non- 

 genetic causes of resemblance. A sib analysis designed to estimate heritability should 

 preferably, therefore, be based on the correlation between paternal half sibs. To provide 

 half-sib data, each of a number of males is mated to several females and the progeny 

 of each male constitutes a half-sib family. Ideally, if the estimation of heritability is 

 the sole object, only one offspring of each female should be measured, so that there are 

 no full sibs within the half-sib families. The problem of design is then a fairly simple 

 one. The total number of progeny that can be measured will be fixed by the amount 

 of space available or by the amount of effort that can be expended on the measuring. 

 The question then is : how large should the families be ? Where does the optimum lie 

 between the extremes of having many small families and few large families ? It can be 

 demonstrated 1061 that the sampling variance of the correlation coefficient will be mini- 

 mal when n = \jt approximately, n being the number of offspring per family and t the 

 phenotypic correlation. So, again, it is found that the optimal design cannot be 

 determined precisely without prior knowledge of the correlation to be estimated. 

 But, by guessing whether the heritability of the character is likely to be high or low, 

 a useful guide to the design can be obtained. Since the correlation between half 

 sibs will be one quarter of the heritability (/ = \h 2 ), the optimal design will have 

 n — 4jh z . Therefore if the heritability is 20 per cent, 20 offspring per family should 

 be planned, and if it is 40 per cent, 10 offspring per family should be planned. The 

 precision of the estimate falls off much more rapidly when the families are smaller than 

 the optimum than when they are larger. Therefore it is better to err on the side 

 of having too many offspring per family than too few, and, in the absence of any 

 knowledge of what the heritability is likely to be, it would seem reasonable to plan 

 on having families of about 20 half sibs. 



To mate each male to 20 females and to measure only one offspring from each 

 female, as the optimal design requires, is, however, not a convenient design to carry 



