The Estimation of Heritability with Unbalanced Data: I. Observations Available on Parents and Offspring

Hill and Nicholas (1974) and Thompson (1976) have recently discussed the design of experiments to estimate heritability when data are available on parents and offspring. They show the advantage of maximum likelihood (ML) methods over regression and sib covariance methods. They consider the efficiency of various balanced hierarchical designs, that is designs in which the same number of dams are mated to each sire and the same number of offspring are measured from each dam. In practice this balance is seldom attained and in this paper we consider estimation of heritability for unbalanced data. In balanced designs it is useful to divide the data into five parts due to differences within dams, between dams within sires, between sires with offspring, between animals with no offspring and between groups of animals with and without offspring (Thompson 1976). This partition suggests estimators in the unbalanced case but the weighting to be given to each full-sib and half-sib mean has yet to be decided. The weighting of the means has been discussed previously, in the context of using only the parent-offspring covariances, for example by Kempthorne and Tandon (1953) and Ollivier (1974), and in the context of using the sib covariances alone, for example Robertson (1962). The weighting to be given to the means is not obvious when we wish to use both the parent-offspririg and sib covariances efficiently and we use instead an ML procedure. We try to take advantage of the special structure of the data, noting that the formulation for the unbalanced case is completely different from the formulation for the balanced case. In Section 2 we discuss a model for the unbalanced case and give a suitable form for the variance-covariance structure. In Section 3 we use this structure to indicate how the parameters might be estimated. In Section 4 we indicate the modifications necessary when different models are assumed. The method we develop can be useful in the more general problem of estimating variance components and we discuss this in Section 4.