366 C. R. HENDERSON 



when used as the male parent in topcrosses on the population from which 

 the lines are regarded as a sample? 



2. What are the best estimates of the relative values of the tested lines as 

 female parents in crosses with males from the above population? 



3. What are the best estimates of the relative values of specific single 

 crosses among the tested lines? 



Suppose that tin progeny of the cross tth line of male by 7th line of female 

 are tested (nn can be zero for some crosses) . Now the easiest way to estimate 

 the value of the ith line as a male parent is simply to compute the mean of 

 all progeny of the line when used as the male parent. This simple procedure, 

 however, fails to take into account the distribution among lines of the mates 

 of males of the /th line, the covariances among the general combining abili- 

 ties of lines, the consequences of specific effects, the size of the error variance, 

 and the number of progeny tested. Furthermore, since the zth line is used also 

 as the female parent in certain crosses, something can be gained by employ- 

 ing the measurements on these progeny. Estimation by the general procedure 

 we have described takes into account all of these factors. Similarly the easiest 

 way to estimate the maternal ability of thejth line is to compute the mean 

 of all progeny out of females of the /th line, but the most efficient procedure 

 takes into account the same factors as are needed in efficient estimation of 

 general combining ability. Finally the easy way to appraise the value of a 

 particular cross is merely to find the mean of all progeny of the specific cross 

 (if that cross has been tested). This latter estimate is subject to large sam- 

 pling error since it would seldom be feasible to test many individuals of the 

 numerous possible crosses among even a few lines. The error of estimation can 

 be materially reduced by utilizing the fact that the true merit of a cross is a 

 function of the general combining abilities of two lines, the maternal ability 

 of the female line, and the specific effect peculiar to that cross and to its re- 

 ciprocal. The method to be described places the proper emphasis on estimates 

 of general and maternal abilities and on the progeny averages of the specific 

 cross and its reciprocal. The procedure also enables estimates to be made of 

 the value of a specific cross even though that particular cross has not been 

 tested. 



The major step in these efficient estimation procedures is the setting up 

 and solving of a set of simultaneous equations in the 6's, ^'s, ?n's, and s's. 

 These equations are as follows: 



^1 2 y^ 2 -VluA- + ^2 23 2 2 •^■l'J>^"2 0-i- + 2^ gi i^l'.. + -^'l. I. ) 



i j k i i k i ( 9 ) 



and similarly for the bo equation. 



i<; i J k 



