QUANTITATIVE INHERITANCE 203 



the offspring of the same parent. The second simple course of action is to take the 

 mean of each family and to regress this on the parental value. This gives too little 

 weight to the larger families, but is reasonably satisfactory when there is a strong 

 resemblance between offspring of the same parents, either for genetic or for nongenetic 

 reasons. A slightly less simple, but much more satisfactory, procedure is as follows. 

 The families are divided into groups according to the number in the family, and the 

 regression of family means on parental values is computed separately within each group. 

 The variance of each estimate is computed in the usual way, the degrees of freedom 

 being based on the number of families in the group. A weighted average of the 

 regression coefficients is then taken, the weight being the reciprocal of the variance of 

 each estimate. The variance of this average regression is the reciprocal of the sum 

 of the weights. The chief disadvantage of this method is that one degree of freedom 

 is lost for each subdivision of the data, a sacrifice to simplicity that may not willingly 

 be made when much effort has gone into the collection of the data. It may therefore 

 seem worthwhile to use a fully satisfactory, but more complicated, method of combining 

 the data from families of different size. 



The method that makes the best use of all the data depends on combining the sums 

 of squares and products from families of different size according to a weighting factor 

 appropriate to the family size. The derivation of the weighting factors is explained by 

 Kempthorne and Tandon 702 and Reeve. 1046 The principle is that families of different 

 size are weighted in proportion to the reciprocal of the variance of the estimate of re- 

 gression that would be obtained from families all of that particular size. It has already 

 been pointed out, in connection with planning, that the proportionate effect of the 

 number in the families on the precision of the estimate is governed by the phenotypic 

 correlation, t, between members of families. But it also depends, to a lesser extent, 

 on the value of the regression being estimated, and, although this factor can be omitted 

 in connection with planning, it must be brought in if the best use is to be made of the 

 data. The first step in computing the weighting factors is therefore to determine the 

 phenotypic correlation, /, between members of families, from an analysis of variance 

 of the offspring, within and between families. Then one must make a rough estimate 

 of the regression, b, which is to be estimated. This need not be very precise, and it 

 can be obtained from a graphical representation of the data in a scatter diagram of the 

 values of parents and the means of their offspring. Approximate values of the corre- 

 lation, t, and the regression, b, having been obtained, the weighting factors can most 

 easily be calculated in two steps. First compute the quantity T, as follows : 

 when the regression is to be made on a single parent 



T = (t - b*)l(\ - t), 

 and when the regression is to be made on the mean of both parents 



T=(t-±b*)l(l -t). 

 Then the weighting factor, w n , appropriate to families of n offspring is 



w n = (n + nT)l(\ + nT). 



