362 C. R. HENDERSON 



records need to be considered; in the former case the genetic differences 

 among Hnes, the environmental variance, and the number of progeny. Also in 

 both cases consideration of the genetic covariances between individuals or 

 between lines increases the accuracy of the ranking. 



Assuming that the population mean is known and that it and non-random 

 environmental factors have been subtracted from the means of the progeny 

 of the various lines, the estimate of gi, the general combining ability of the 

 iih line, is 



£.=C,a- --{-... -\-C a - 



°i 1 BiVi P BiVp 



where yi, • ■ ■ , Jp are the corrected means for the p tested lines and the C's 

 are the solution to a set of equations with 



as coefificients in the left members and corrected y\, . . . , y? as the right mem- 

 bers. Computation of o-^^j, and o-oiSi requires good estimates of 



Ik II 



and of al. Assuming that the corrected mean of a particular topcross is 

 yi = gi'\- ^i, and that the errors are independent with common variance af 

 we have the following variances and covariances 



a- = a- -]-a-/n. cr- — a~ 



a-- = a (where i 7^ j) a- = a (i ^ j) 



ViVj aivj -' viOj Qiuj ■' 



Frequently good estimates of n and non-random environmental factors are 

 not available and consequently must be estimated from the topcross data. 

 For example, it is very likely that the environment is not the same from test 

 to test and must be taken into account if the data from several tests are to be 

 combined into a "best" index. In such cases the method of equations (5) and 

 (6) can be employed to distinct advantage unless 



is too difficult to compute. To illustrate this method as applied to topcross 

 data we shall assume that y^j, the record of the/th progeny of the ith line, 

 can be represented by 



yu = byXuj -\- hoX'iij + gi -\- e,j . 



bi and 62 are examples of fixed parameters, gi is the general combining ability 

 of the /th line, and d, is a random error. Assuming that the gi are distributed 

 with means zero and known variance-covariance matrix. 



