Three Stage Nested Designs for Estimating Variance Components

Based on a completely random three-stage nested model, five fundamental sampling structures are defined. From the five fundamental structures, 61 designs were enumerated such that each design contained no more than three fundamental structures and a multiple of twelve third-stage samples such that the designs would permit the ANOVA estimation of all three variance components. The table of designs contains in addition to the design structure, a design code for classifying the designs as well as a set of four coefficients that simpliiy the computation of the expected mean squares for the design. General formulas for the variances and covariances of the ANOVA estimators are presented in an appendix. Three criteria derived from the covariance matrix of the variance component estimators are proposed and then used to compare the designs as well as determine the optimum design for 49 diierent variance component configurations and 10 diierent sample sizes.