504 



R. E. COMSTOCK AND H. F. ROBINSON 



be a sampling deviation between the actual and expected values of the prog- 

 enies. The variation among expected values is the variation due to genetic 

 differences among parents, and hence that to be considered in evaluating o-^ 

 and al^l. c7^„ is the progeny variance due to genetic differences among the vari- 

 able parents, i.e., the variance of the pair means indicated in the next to the 

 last column of Table 30.5. a;„i, the progeny variance from interaction of geno- 

 types of the variable and homozygous parents, is one-half the variance of the 

 pair differences indicated in the final column.- 



TABLE 30.5 

 POPULATION SAMPLED IN EXPERIMENT III 



* The one chosen from F2 or later generation of the cross between lines 

 A and B. 



s symbolizes an infinitely large number. 



X's are expected genetic values of progenies, subscripts indicate parentage 

 of individual progenies. 



Xi = mean of Xai and Xbi [where the subscript / identifies the variable 

 parent, e.g., Xi = (.Vqi + A'6i)/2]. 



Di = Xai — Xbi- 



Now note that a pair mean or difference is the sum of contributions from 

 individual loci. Let 



Xij be the contribution of the 7th locus to the ith pair mean, and 

 d ij be the contribution of the 7th locus to the /th pair difference. 

 Then 



.Y,- = .^',1 -|- .v,2 -\- . . . Xij^ 



Di = d,i + dr2-\- ■ ■ ■ d,jy 



where .Vis the number of contributing loci. Then the variances of pair means 

 and differences must be as follows:^ 



<t\. ( = a- ) = > cr- . -|- 2 7 (T ., 



A m ' ^^ I] ' ^^^ X]k 



] i, k 



D lilt ^^^ d] ^^ djk 



(1) 



(2) 



i.k 



2. It is well known and easily verified that in the analysis of variance of any 2 X .^ 

 table, the interaction variance is one-half that of the pair differences. 



3. Since the variance of the sum of any number of variables is the sum of the variances 

 of those variables plus twice the sum of all covariances among them. 



