SPECIFIC AND GENERAL COMBINING ABILITY 365 



shall confine ourselves to discussion of the analysis of single crosses. Applica- 

 tion of these principles to multij)le cross data involves no new principles. 



Let us consider first what type of model might be reasonable for a single 

 cross. It is not too difficult to suppose that the value of a particular observa- 

 tion on a single cross is the sum of the general combining ability of the male 

 line, the general combining ability of the female line, a maternal effect coming 

 from the line used as the female, a specific effect due to dominance and 

 epistasis and peculiar to the particular cross, non-random environmental ef- 

 fects, and a multitude of random errors such as Mendelian sampling and the 

 environment peculiar to the particular progeny on whom the record is taken. 

 More complicated models could of course be proposed, but the one which we 

 have just described would seem to account for the major sources of variation 

 among crosses. Furthermore it is amenable to mathematical treatment. Put- 

 ting the above description in a mathematical model we have 



Jiik = ^i-^iu'^- + boXi^jk + g, + gj + W; -f s,j + eijk , 



where yuk is the observation on the ^th progeny of a cross between the ith 

 line used as a male parent and thejth line as a female parent, the 6's and a;'s 

 are related to the mean and other non-random environmental factors as de- 

 scribed in the model for the topcross test, giigj) is the general combining abili- 

 ty of the ith(yth) line, nij is an effect in addition to the additive genetic value 

 which is common to all progeny of they'th line used as a female parent, 5,, is 

 an effect over and above the additive genetic and maternal effects and which 

 is common to all progeny of the cross of the ith line by thej'th line or of the 

 yth line by the ith line, and Cijk is a random error associated with the particu- 

 lar observation. 



In this model the gi are regarded as having some multivariate distribution 

 with means zero and variance-covariance matrix, 



11 '.,-., II- 



The wy, Sij, and e^j are all regarded as independently distributed with means 

 zero and variances o-^, o",, and al, respectively. It is of course conceivable that 

 the variances of the m, and sa and the covariances between them vary with 

 the inbreeding and relationships of the lines. Also gi and ntj may be correlat- 

 ed. In the absence of any real knowledge concerning such covariances we 

 shall ignore them for our present purposes. If, however, something is known 

 about these covariances, the estimation procedure can be modified to take 

 them into account. The procedure should also be modified if the lines are not 

 homozygous and each parent has more than one progeny. 



A single cross test can supply answers to the following questions with re- 

 spect to the lines tested: 



1. What are the best estimates of the relative values of the tested lines 



