RANDOMLY BREEDING POPULATIONS AND CORRELATIONS 



progenies. The values of these statistics also depend on D and H, 

 thus: — 



Variance of biparental means jD + i^H + E^ 



Mean variance of biparentals ^D + i^eH + E^ 



Covariance of F2 and biparentals jD 



They may be used to supplement or to replace the statistics from F3 

 families in the estimation of D and H. 



When we distinguish between the biparental families obtained 

 by the intercrossing of different pairs of individuals from Fg, we 

 break the heritable variance down into two parts: the variance of 

 the means, which measures variation between families, and the mean 

 variance, which measures variation within families. The total herit- 

 able variation of this generation is the sum of these two parts, viz. 

 (iD+ r6H)+ (iD+ i%H) =|D+ JH. It is exactly the same 

 as the heritable variance of F2 itself. And if we breed still another 

 generation of biparental progenies, the pairs of parents being taken 

 at random, without distinction of family, from the first biparental 

 generation, we fmd that its heritable variance is once again ^D + jH. 

 This variance is, in fact, characteristic of the random mating system 

 used. The variances will be the same from generation to generation 

 provided that the system of random mating is continued, and pro- 

 vided also, of course, that the environmental variation, and with it 

 the E component, is constant. The covariance of parent and offspring 

 is also constantly jD under this system of mating, with no E 

 component appearing in it. 



In the particular case under discussion we commenced with the Fg 

 of a cross between two true-breeding lines, so that the frequencies 

 of the two allelomorphs of each gene by which the parents differed, 

 A-a, B-b, etc., must be equal. The variance and the parent-offspring 

 covariance still retain their values of ^D+ 4H4- E and jD even, 

 however, where the frequencies of yl and a, B and b, etc., are not 

 equal in the group of individuals. The variation formulae are 

 therefore characteristic of all randomly breeding groups. 



The effect of variation in the gene frequencies appears in a different 

 way, viz. in the contributions which the genes make to D and H. 

 With equal gene frequencies and no linkage we have D = S(d-) 

 and H = S (h^) ; but when allelomorph A has the frequency u^, 



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