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Multistage selection with fixed proportions and selection indices based on covariates of the target variable is studied. Assuming a multivariate normal distribution before the selection, expressions are presented for the expectation and the variance of the target variable in the retained subpopulation. As the numerical evaluation for finite populations requires lengthy computations, some approximations using methods for infinite populations are proposed. Numerical illustrations are given for selections in up to three stages.

SUMMARY Tallis [1962] formulated and solved the problem of constrtucting a selectioin index whose use will produce changes in the means of the genotypic values of various traits that are proportional to ultimately desired changes. Several meaningftul variations of this problem are described, and their soluitions are presented in a common framework. A common problemn in animal and plant breeding is that of selecting animals or plants for breeding purposes from a group of candidates. The selections are based on a vector

Previous formulations of expected population size under phenotypic selection and index selection are extended to include tests containing mixtures of full-sib and half-sib families generated by regular mating plans. Projections of effective size are presented for a single cycle of selection from tests conducted with factorial and hierarchical mating designs, and with fractions of diallel plans. Consideration of variable family sizes enables similar projections for selection from polycross tests.

The selection index criterion for aggregate genetic value defined as H = v* g*, where g* is a vector of additive genetic values of the several economically important traits and v* is a vector of relative economic values is, as is well known, I = (P-'G*v*)'x, where P is the phenotypic covariance matrix of the data vector, x, and G* is the genetic covariance matrix of the economically importatn traits. When each trait has a direct and a maternal additive genetic component the selection index criterion becomes I = (P-'A'Gv)' x, where G is the covariance matrix of the direct and maternal genetic components of all the traits, (twice the order of G*), v is the vector of economic values of the direct and maternal components of all the traits (twice the length of v*), and A is a particular matrix consisting of zeros and additive relationships of the animal being evaluated with its relatives having records and with the dams of those relatives. The correlated response for the kth component (direct or maternal) of thejth trait is Agi = Agjk = v'G'AP-1(A'G)j/0-1 times the standardized selection differential, where (A'G)i is the ith column of A'G.

SUMMARY The nature of the index selection procedure when sample estimates are used in place of the true parameter values is considered. Approximate equations are developed for the low order moments of the joint distribution of the expected progress from selection for a particular calculated index and the predicted progress from this index. These equations involve the variances and covariances among the estimates used for index construction. Formulae are developed for these variances and covariances for paternal half-sib estimation. This joint distribution is also studied by a simulated sampling procedure. Numerical results for the two phases of the study are tabulated and studied for verification of the validity of the approximate equations and for assessment of the influence of the errors of estimation upon index selection procedures when estimates are obtained from paternal half-sib analyses of variance and covariance.