Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.
Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. The most important features of the staggered nested design are that it has a very simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom. Based on the features, a class of unbalanced nested designs that is a generalization of the staggered nested design is proposed in this paper. Formulae for the estimation of variance components and their sums are provided. Comparing the variances of the estimators to the staggered nested designs, it is found that some of the generalized staggered nested designs are more efficient than the traditional staggered nested design in estimating some of the variance components and their sums. An example is provided for illustration.
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