SPECIFIC AND GENERAL COMBINING ABILITY 363 



and that the dj are mde{)endently distributed with means zero and common 

 variance a;, the estimates of the b's and g's which are "best" by the criterion 

 used in this paper are the solution to the following equations: 



i>i 



(8) 



^i-vi;,. + b.x.^ + ip ("p+ <T?ff^^) + ^ ^pff^cr"' = >' 





Dots in the subscripts denote summation over that subscript, and a'' denotes 

 an element of 



cr 



The above procedure for appraising lines on the basis of topcrosses assumes 

 either that the lines are homozygous or that only one progeny is obtained 

 from each randomly chosen male. If these assumptions are not correct, the 

 procedure is moditied to take into account intra-line variances and covari- 

 ances and the number of progeny per male. 



What are the consequences of appraising lines on the basis of the arithmet- 

 ic average of their respective progeny as compared to the more efficient 

 method just described? First, the errors are larger than necessary. Second, 

 selection of some small fraction of tested lines will tend to include a dispropor- 

 tionately large number of the less well-tested lines. The more efficient meth- 

 od discounts the higher averages in accordance with the number of tested 

 progeny and the relative magnitudes of a^ and a;. 



What if the number of lines tested is large and certain lines are related? 

 This means that a large matrix, 



11%,, II. 



has to be inverted and then a large set of simultaneous equations solved. 

 What approximations might be employed in the interest of reducing compu- 

 tations? For one thing, we might ignore the covariances between the g's, 

 thereby reducing the inverse matrix to l/a^, in the diagonal elements and 

 in the off-diagonal elements. Also if we know /i and non-random environmen- 



