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Estimation of the mean and variance components in some random effects models with composited samples

A method is presented for estimating the mean, variance components , and their variances in a two-way crossed classification where the experimental units are composited. Estimates of the variance components are obtained using a procedure similar to Henderson's method I in which the sums of squares are computed using weighted or unweighted observations or cell means. Four estimation procedures were suggested and compared. The results show that any best estimation procedure depends on the relative sizes of the variance components. Conditions on the various design parameters were investigated to determine when exact tests of hypotheses on the variance components can be made. For all estimation procedures considered, the designs should be completely balanced if exact F-tests are required. Using some of the results from the two-way crossed classification, estimates of the mean, variance components and their variances in 2-and 3-stage nested designs were derived. These nested designs allowed for the experimental units to be composited in all but the last stage. Four estimation procedures were proposed and evaluated. For both the 2-and 3-staqe nested designs any best estimation procedure depends on the relative sizes of the variance components.