Variance Balanced Block Designs Research Articles

Graph decomposition methods for variance balanced block designs with correlated errors

Block designs have an important historical footing in the design of experiments. Let θ be the treatment mean vector in the statistical model with fixed block effects. It is desirable to have the covariance of the least squares estimator of the treatment effects, Cov(θ̂), be a completely symmetric matrix. When this occurs, we have a variance balanced design (VBD). It is known that a pairwise balanced design can be converted into a VBD by choosing blocks with a multiplicity inversely proportional to the block size. This assumes that experimental errors are uncorrelated. However, this need not be the case, especially in ecological experiments where blocks have spatial correlation or laboratory settings with temporal correlation. This paper proposes the use of graph decompositions as a generalization of balanced incomplete block designs for the construction of VBDs in the presence of correlated errors. A general result is obtained to construct VBDs for various error correlation structures. Several applications are given to illustrate the relationship between graph decompositions and VBDs.

Robustness of Variance Balanced Block Designs

The robustness of block designs against missing observations is revisited. It has been shown that A-efficiency criterion is not an appropriate measure to judge the efficiency of the residual design. As an alternate to this, E-efficiency criterion is defined. A lower bound of this criterion for the loss of any t observations in binary variance balanced block design is obtained. Balanced incomplete block designs (BIBD) that are robust as per E-efficiency criterion are identified.

Robustness of block designs on the basis of pairwise treatment comparisons against missing data

Robustness of incomplete block designs against the unavailability of data has been investigated in the literature in terms of average variance of all possible pairwise treatment comparisons in the design. But if we examine the robustness of a design for the loss of observation(s) on the basis of pairwise treatment comparisons, loss of information of some of treatment comparisons may be more than that the loss of information on the basis of average variance of the residual design. A design that is robust on the basis of overall efficiency may not be robust when the efficiency is worked out on the basis of paired treatment comparisons. We, therefore, investigate all the estimates of individual pairwise treatment contrasts for the loss of any number of observation(s) in a block for balanced incomplete block design and variance balanced block designs. Designs that are robust on the basis of average variance but not on the basis of pairwise treatment contrasts have been identified.

Some new construction methods of variance balanced block designs with repeated blocks

Some new construction methods of the variance balanced block designs with repeated blocks are given. They are based on the specialized product of incidence matrices of the balanced incomplete block designs.

The analysis of mixed and random effects models for equireplicate variance-balanced block designs

Equirephcate variance-balanced block designs possess many desirable properties and include several important subclasses of designs, such as balanced incomplete block designs. However, while attention has been paid to the derivation of properties and the construction of these designs, methods of analyzing experiments using these designs have not been developed completely. Thus, we consider models appropriate to these designs, construct test statistics for basic hypotheses and derive the distributions for these statistics. Four additive models that include all of the possible combinations of fixed and random effects are examined. The analyses are generalizations of the analyses for balanced incomplete block designs, and include the intra-block, intra-treatment and inter-block analyses, Estimators of fixed treatment effects and their distributions also are given.

Resistance of block designs

Abstract Hedayat and John (1974) introduced resistant and susceptible balanced incomplete block designs. The results were essentially on locally/globally resistant balanced incomplete block designs of degree one. This paper extends the concept of resistance to general variance balanced block designs as well as universally optimal block designs with unequal block sizes. Some methods of constructing locally resistant designs of degree one have been given.

A Relation Between Pairwise Balanced and Variance Balanced Block Designs

Abstract We show that the problems of constructing pairwise balanced block designs and variance balanced block designs are equivalent. This provides incentive to study only the direct construction for the simplest of these designs—namely, pairwise balanced designs.

A Relation between Pairwise Balanced and Variance Balanced Block Designs

Abstract We show that the problems of constructing pairwise balanced block designs and variance balanced block designs are equivalent. This provides incentive to study only the direct construction for the simplest of these designs—namely, pairwise balanced designs.

THE ANALYSIS OF THE ADDITIVE MIXED MODEL FOR CLASSES OF NON ORTHOGONAL DESIGNS

Tests for fixed and random effects can be difficult to derive for nonorthogonal designs with mixed models. However, extensions of the intrablock and inter-block analyses of Balanced Incomplete Block Designs can often be obtained. Here we derive the extensions for the broad class of Group Divisible Designs. Decompositions of the design space are used to develop exact tests for fixed and random effects in the additive mixed model with random block effects. Conditions on the design which permit the standard use of the intra-block and inter-block test statistics are given. Important subclasses of Group Divisible Designs include Equireplicate Variance Balanced Block Designs and Group Divisible Partially Balanced Incomplete Block Designs with Two Associate Classes. These two subclasses are also examined. An example from the literature of an experiment on fruit trees is used to illustrate the methods.

On Fisher's Inequality for Variance Balanced Block Designs

A cbaraeteriz1tion of all the connected variance balanced block designs, for which the Fisher's inequality b⩾ v holds, is given.

Conditions for the robustness of block designs against the unavailability of data

According to the concept of Ghosh (1982), a binary and connected block design D, with the smallest treatment replication r [ ν] , is said to be maximally robust against the unavailability of data and with respect to the estimability of treatment contrasts, if a design D #, obtained from D by deleting any r [ ν] −1 blocks, remains connected irrespective of the choice of the blocks deleted. Three sufficient conditions for the maximal robustness of a block design are derived. Their applications are shown in the context of certain variance-balanced block designs constructed by Kageyama (1976) and variance-and-efficiency-balanced block designs listed by Gupta and Jones (1983).

A Note on a Parametric Relation of Variance Balanced Block Designs

The object of this communication is to prove the important parametric relation [Formula: see text] and hence [Formula: see text] for all connected Variance Balanced (VB) block designs, excepting the cquireplicatcd orthogonal designs which arc trivially VB.

Some bounds for partially balanced block designs

Bounds on eigenvalues of theC-matrix for a partially balanced block (PBB) design are given together with some bounds on the number of blocks. Furthermore, a certain equiblock-sized PBB design is characterized. These results contain, as special cases, the known results for variance-balanced block designs and so on.

Characterization of equireplicated variance-balanced block designs

Characterization of equireplicated variance-balanced block designs

Some methods for constructing efficiency-balanced block designs

Abstract Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.

On a class of variance balanced block designs

On a class of variance balanced block designs