354 C. R. HENDERSON 



be made of a sequential type of testing in which some lines are discarded on 

 the basis of a very preliminary and inaccurate test? 



4. What relative emphasis in selection should be placed on general as 

 compared to specific combining ability? 



5. How much inbreeding should be done in the making of lines? How fast 

 should the lines be made? 



Obviously a complete discussion of all these problems and their possible 

 solutions in the time at our disposal is impossible. Consequently we shall 

 discuss primarily the problem of estimating general combining abilities of 

 lines and individuals and of estimating the values of specific crosses among 

 lines, given a particular set of records. In addition, since estimates of the 

 variances play an important role in these selection methods, we shall discuss 

 briefly the problem of estimating variance components from the results of 

 line-cross tests. 



So far as estimation of general combining abilities of individuals is con- 

 cerned, the methods to be presented here are essentially those of the selection 

 index. It will be shown that no assumption of normality of distributions is 

 required; that joint estimates of general combining abilities and certain 

 parameters such as the population means, the yearly effect, the age and in- 

 breeding effect, can be obtained; and that certain short-cut computational 

 procedures are sometimes distinctly advantageous. An application of the 

 principles of the selection index to estimation of general combining abilities 

 of lines or families also will be presented. Finally it will be shown that appli- 

 cation of the selection index need not be restricted as it has been to selection 

 for additive effects, but can be applied equally well to joint selection for 

 specific effects and general combining ability. The selection index approach 

 to appraising crosses can, under some circumstances, be much more efficient 

 than selection based on the mean of the progeny of a particular cross. 



ESTIMATION PROBLEMS IN SELECTION 

 Before turning to selection for general and specific combining abilities let 

 us consider the type of estimation problem which is involved and some gen- 

 eral solutions to it. Later the manner in which the solutions can be applied 

 to our present problem will be discussed. Our estimation problem can be 

 stated in this way. We have a sample of N observations, yi, y2, • • • , ^a^, 

 from which we wish to estimate ^i, 62, . . . , dg. The y's are assumed to have 

 a multivariate distribution (precisely what distribution need not be specified 

 for the present) with means, biXu-\- ^2-V2i + • • • + bpXpi, and variance- 

 co variance matrix, 



Tiie b's are fixed parameters such as the population mean and the regres- 

 sion of y on age of the dam, and x is an observable parameter, the first sub- 

 script denoting with which b it is associated, the second subscript with 



