SPECIFIC AND GENERAL COMBINING ABILITY 367 



biixn.. +^-1.1.) +^2(a-2i.. +:t2.i.) +^i(«i. + ".i + ff^cr*^) 



= 3'!.. +>'.!. 



and similarly for the other gi equations. 



^l-Vl.l. + ^2-V2.1. + gl«.l+2^ gi«.l+ Wi (;/.!+ Cr!/ffl,)+ 2 5i,M,i = J.l. 



and similarly for the other ;«; equations. 



^(■Vn2. +-Vi21.) +6.2(.V212. +-V22I.) + (gl+ g2) ("12 + W21) +/«1«21 



+ W2H12+ ■Jl2(«12+»21 + ae/(rJ = yi2. +3'21. 



and similarly for the other Sij equations. 



These equations are not particularly difficult to solve, for each Sij can be 

 expressed as a function of ya., y;,-., 5i, 62, Qi, Qj, and mj. Utilizing this relation- 

 ship the equations can be reduced to a set involving none of the ,Sij. Also an 

 iterative solution is usually easy because of the relatively large diagonal co- 

 efficients. Once the estimates of gi, nij, and s^ are obtained it is a simple mat- 

 ter to evaluate the lines and crosses. The estimate of the value of a line as the 

 male parent in topcrosses is Qi, and the estimate of its average value as the 

 female parent is ^i + ifu. The value of a single cross is estimated simply as 

 9 1 + di + '^ij + •^u- It is appropriate to add the estimates in this manner be- 

 cause they have the desirable property of invariance. 



If solution of the large set of simultaneous equations required for most ef- 

 ficient appraisal of lines is considered too burdensome, certain approximate 

 solutions can be employed. An approximation suggested by the common 

 practice in construction of selection indexes is the choosing of certain infor- 

 mation most pertinent to the particular line or cross to be appraised. For ex- 

 ample, the estimate of g, might be based entirely on y^.. and y.,., each cor- 

 rected for the b's as best can be done with the information available regarding 

 their values. As a further simplification it might be assumed that the gi are 

 uncorrelated and have common variance o-|. Similarly w, might be estimated 

 entirely from y,. . and y.,. . These approximate solutions are 



g. =C ,a +C><7 

 fh ■ =C ,(j -\-C„(j , 

 where the C's are the solution to 



