Generalized staggered nested designs for variance components estimation

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.     

Statistical tests for random effects in staggered nested designs

In this paper, we present certain statistical tests under a staggered nested design set-up, for the hypotheses that certain variance components are zero. To do so, the particular variance-covariance, structure induced by the staggering is exploited and certain results of multivariate analysis are used. In most problems, the test statistics can be easily computed. An example is provided for illustration and some power computations for comparison of test statistics are shown.     

Fast and robust estimators of variance components in the nested error model

Usual fitting methods for the nested error linear regression model are known to be very sensitive to the effect of even a single outlier. Robust approaches for the unbalanced nested error model with proved robustness and efficiency properties, such as M-estimators, are typically obtained through iterative algorithms. These algorithms are often computationally intensive and require robust estimates of the same parameters to start the algorithms, but so far no robust starting values have been proposed for this model. This paper proposes computationally fast robust estimators for the variance components under an unbalanced nested error model, based on a simple robustification of the fitting-of-constants method or Henderson method III. These estimators can be used as starting values for other iterative methods. Our simulations show that they are highly robust to various types of contamination of different magnitude.     

Estimation of variance components in an unbalanced one-way classification

In the unbalanced one-way random effects model the weighted least squares approach with estimated weights is used to develop a relatively simple estimator of variance components. As the number of classes increases, the proposed estimator is seen not only to be best asymptotically normal but also to be asymptotically equivalent to the maximum likelihood estimator.     

General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

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.     

Estimation of variance components in linear mixed measurement error models

In this paper, we consider a linear mixed model with measurement errors in fixed effects. We find the corrected score function estimators for the variance components. An iterative algorithm is proposed for estimating the parameters. The computations on each iteration of this algorithm are those associated with computing estimates of fixed and random effects for given values of the variance components. We also derive the consistency of the estimators under regularity conditions. The simulation study shows that for relatively small sample size the corrected estimators perform very well. Finally, an example of real data is given for illustration.     

Additional comparisons of designs to estimate variance components in a two-way classification model

In this paper, the research of Muse and Anderson is extended to include additional comparisons of designs, featuring planned unbalance, for the estimation of variance components in a two-way cross classification model. Their results are extended to Include the following: (i) a small sample study of the original off-diagonal (OD) design and (ii) an asymptotic maximum likelihood investigation of three modifica-tions of the balanced diagonal rectangles (BD) design and one modification of the 01) design to permit the estimation of row, column, interaction and error variance components. Also a general iterative least.     

A survey of nested designs

A nest with parameters (r,k,λ)→(r′,k′,λ′) is a BIBD on (b,v,r,k,λ) where each block has a distinguished sublock of cardinality k, the sublocks forming a (b,v,r,k,λ)-design.These designs are ‘nested’ in the sense of W.T. Federer (1972), who recommended the use of these designs for the sequential addition of periods in marketing experiments in order to retain Youden design properties as rows are added. Note that for a Youden design, the b columns and v treatments are in an SBIBD arrangement with parameters v=b, k=r, and λ.     

Random balanced resampling: A new method for estimating variance components in unbalanced designs

Variance components in factorial designs with balanced data are commonly estimated by equating mean squares to expected mean squares. For unbalanced data, the usual extensions of this approach are the Henderson methods, which require formulas that are rather involved. Alternatively, maximum likelihood estimation based on normality has been proposed. Although the algorithm for maximum likelihood is computationally complex, programs exist in some statistical packages. This article introduces a simpler method, that of creating a balanced data set by resampling from the original one. Revised formulas for expected mean squares are presented for the two-way case; they are easily generalized to larger factorial designs. The results of a number of simulation studies indicate that, in certain types of designs, the proposed method has performance advantages over both the Henderson Method I and maximum likelihood estimators.     

Equineighboured balanced nested row-column designs

This paper presents equineighboured balanced nested row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. These are two-dimensional designs with the property that all pairs of treatments are neighbours equally frequently at all levels in both rows and columns. Methods of construction are given both for designs based on Latin squares and those where pq≤v. Cyclic equineighboured designs are defined and tabulated for 5≤v≤10, p≤3, q≤5, where r=bpq/v is the number of replications of each treatment.     

On nested block designs geometry

Nested block designs and block designs properties such as orthogonality, orthogonal block structure and general balance are examined using the concept of a commutative quadratic subspace and standard properties of orthogonal projectors. In this geometrical context conditions for existence of the best linear unbiased estimators of treatment contrasts are also discussed.     

Testing for autocorreiation in nested analysis of variance

This paper considers the problem in which repeated measures are taken on each subject in an experimental design and the model errors follow a first order autoregressive process. The appropriate form of the Durbin-Watson test statistic for autocorrelated errors is presented. Critical values for the test statistic are reported.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

Estimation of variance components in the mixed effects models: A comparison between analysis of variance and spectral decomposition

The mixed effects models with two variance components are often used to analyze longitudinal data. For these models, we compare two approaches to estimating the variance components, the analysis of variance approach and the spectral decomposition approach. We establish a necessary and sufficient condition for the two approaches to yield identical estimates, and some sufficient conditions for the superiority of one approach over the other, under the mean squared error criterion. Applications of the methods to circular models and longitudinal data are discussed. Furthermore, simulation results indicate that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.     

Minimum variance rectangular designs for U-statistics

We consider a certain class of rectangular designs for incomplete U-statistics based on Latin squares and show it to be optimal with respect to the minimal variance criterion. We also show it to be asymptotically efficient when compared with the corresponding complete statistics, as well as uniformly more efficient than the random subset selection. We provide the necessary and sufficient conditions for the existence of our design and give some examples of applications.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.