SPECIFIC AND GENERAL COMBINING ABILITY 355 



which sample observation. As an illustration, .Vi might be associated with bi, 

 the population mean. Then Xn would have the value 1 in each observation; 

 Xii might denote the inbreeding coefficient of the dam. 



Now comes the really crucial part of the model. 'J'he ^'s are regarded as 

 having some multivariate distribution with means zero and variance-co- 

 variance matrix, 



Also the d's and ^''s are regarded as having a joint distribution with covari- 

 ances aekyi- The way in which this problem differs from the ordinary estima- 

 tion problem in statistics is that here we wish to estimate the values of indi- 

 vidual ^'s which are regarded as a sample from some specified population. 



Selection for Additive Effects in the Normal Distribution 

 What is the ''best" way to estimate the 0's? Suppose that they represent 

 additive genetic values of individuals and that any linear function of the y's 

 is normally distributed. Lush (1948) has shown that, subject to the normality 

 assumptions, improvement in additive genetic merit of a population through 

 selection by truncation of the estimates (indexes) of additive genetic values 

 is maximized by choosing that index which has maximum correlation with 

 additive genetic value. This principle has been used in the index method of 

 selection by Fairfield Smith (1936), Hazel (1943), and others. These workers 

 have shown that the index can be found in a straightforward manner pro- 

 vided certain variances and covariances and all of the 6's, the fixed elements 

 of the model, are known. 



The values of Kei which maximize ree where 6 — KeiWi + . . . + KbnWn 

 are the solution to the set of simultaneous equations (1). The w's are the 

 ^''s corrected for the fixed elements of the model such as the population mean 

 (not the sample mean). Thus Wi = yi — biXn — ■ ■ . —b,,Xpi. 



ei y^y-i ' B2 y.^ ' ' e^ y.,y\ J/.." 



(1) 



K„,(T + K„„a -|- . . . + K„..(T-^ = o-„ „ 



d\ y^yx ^' y^'Ux *-^ l^X yN'' 



Selection when Form of Distribution is Unspecified 

 and b's Are Unknown 

 Maximization of ree is a satisfactory solution to the problem of selection 

 for additive genetic values under the normality assumption and the as- 

 sumption of known b's,. Is a comparable solution available when nothing is 

 known of the distribution or of the b's? So far as I am aware there is not. 



