Journal ArticleDOI
General formulae for expectations, variances and covariances of the mean squares for staggered nested designs
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In this article, the authors derived the canonical form for the arbitrary m-factors in a staggered nested design and applied it to obtain the expectation, variances and covariances of the mean squares.Abstract:
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 combinati...read more
Citations
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A bibliography on variance components an introduction and an update: 1984-2002
Hardeo Sahai,Anwer Khurshid +1 more
TL;DR: In particular, the study of variance through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications as discussed by the authors.
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Generalized staggered nested designs for variance components estimation
TL;DR: In this article, a generalized version of the staggered nested design is proposed, which has a simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom.
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Randomization-based models for multitiered experiments: I. A chain of randomizations
R. A. Bailey,Chris Brien +1 more
TL;DR: In this article, the authors derive randomization-based models for experiments with a chain of randomizations and derive formulae for the estimators of treatment effects, their standard errors and expected mean squares in the analysis of variance.
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Crossing balanced and stair nested designs
TL;DR: In this article, the algebraic structure of the balanced, the stair and the staggered nested designs is studied and the cross between balanced and unbalanced nested models is investigated. But the results of the inference for models with unbalanced nesting are not as good as those with balanced nesting.
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Comparison of designs for the three-fold nested random model
TL;DR: In this paper, the authors compared three designs, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model, based on the quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components.
References
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Book
Testing statistical hypotheses
TL;DR: The general decision problem, the Probability Background, Uniformly Most Powerful Tests, Unbiasedness, Theory and First Applications, and UNbiasedness: Applications to Normal Distributions, Invariance, Linear Hypotheses as discussed by the authors.
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Variance component estimation for unbalanced hierarchical classifications
TL;DR: A systematic method for computing the coefficients of the variance components in the expected mean squares of a hierarchical analysis of variance is described, together with a simple method of pooling if the outermost hierarchy is analysed separately for each of its categories.
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The Analysis of Variance of the S-Stage Hierarchal Classification
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Statistical tests for random effects in staggered nested designs
TL;DR: In this paper, the particular variance-covariance, structure induced by the staggering is exploited and certain results of multivariate analysis are used to compute the test statistics for the hypotheses that certain variance components are zero.