368 C. R. HENDERSON 



The variances and covariances needed in this approximate solution can be 

 computed easily from a^, a^, a;, and a^. Approximate values of §ij can then 

 be obtained by substituting the approximate bi, hi, Qi, and rhj in equations (9) . 



ESTIMATION OF VARIANCES OF GENERAL, MATERNAL, 

 AND SPECIFIC EFFECTS 



As mentioned earlier, one might take as the additive genetic variance and 

 covariance among the lines the theoretical values based on relationships 

 among the lines, degree of inbreeding among the lines, and the genetic vari- 

 ance in the original population from which the lines came. It is necessary even 

 then to estimate al^, af, and a;. It is well known that methods for estimating 

 variance components are in a much less advanced stage than estimation of 

 individual fixed effects. It is seldom possible to obtain maximum likelihood 

 estimates. Consequently many different methods might be used, and the 

 relative efficiencies of alternative procedures are not known. 



We shall consider as desirable criteria of estimation procedures for vari- 

 ance components ease of computation and unbiasedness. If the single cross 

 experiment is a balanced one, that is if there are the same number of observa- 

 tions on each of the possible crosses, it is not difficult to work out the least 

 squares sums of squares for various tests of hypotheses, regarding the line 

 and cross line characteristics as fixed. Then assuming that there are no co- 

 variances between the various effects and interactions, one can obtain the ex- 

 pectations of the least squares sum of squares under the assumption that the 

 effects and interactions have a distribution (Henderson, 1948). In case the 

 experiment is not a balanced one, it is still possible to obtain least squares 

 tests of hypotheses and to find expectations of the resulting sums of squares- 

 This, however, is ordinarily an extremely laborious procedure (Henderson, 

 1950). 



A much easier procedure is available. It probably gives estimates with 

 larger sampling variance, although that is not really known, and gives almost 

 exactly the same results in the balanced experiments as does the least squares 

 procedure. This involves computing various sums of squares ignoring all cri- 

 teria of classification except one, taking expectations of these various sums of 

 squares, and solving the resulting set of simultaneous equations. The latter 

 procedure will now be illustrated for single cross data in which we wish to 

 obtain estimates of the variances pertaining to general combining ability, 

 maternal ability, specific effects, and error. It will be assumed that the only 

 fixed element in the model is /x. Now let us compute certain sums of squares 

 and their expectations. These are set out below. 



Total: i^(X;2: Vy?,,)=>/..(M^ + 2a^ + ai + a^4-ap 



