Variance Balanced Designs Research Articles

Universally optimal designs for three interference models under circulant correlation

In many areas such as agriculture, forestry, and medicine the interference models arise when the response of a treatment is affected by the other treatments in neighbour experimental plots. In this paper, the interference models are considered with correlated errors according to a circulant correlation structure, and the structure of information matrices are obtained. For these models, we determine some sufficient conditions for variance balanced designs in the class of circular block designs under circulant correlation structure, and specify the information matrices for variance balanced designs. The universally optimal binary designs are obtained in this class based on Kushner's method and its generalization by Kunert and Martin. We present some methods for construction of the universally optimal designs.

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.

Construction of variance balanced designs using incidence matrix

Construction of variance balanced designs using incidence matrix

On the Construction of Variance Balanced Designs

On the Construction of Variance Balanced Designs

On a Class of Variance Balanced Designs

This paper discusses the method of construction of Variance balanced designs using (i) factorial designs and (ii) some incidence matrix. Further the discussion also involves the efficiency criteria of Variance balanced designs. We have also carried out some properties of Variance balanced designs. This study has been supported by suitable examples.

Variance balanced incomplete block designs

Variance balanced incomplete block designs

On Variance Balanced Designs

On Variance Balanced Designs

Nonbinary variance-balanced designs–part I, optimality

Let D(v,b,k) be a class of designs with ν treatments and b blocks, each of size k. We show that (i) if a variance-balanced design exists in D(v, b, k), then it is E-optimal. Let Δ(ν, b, k) ⊂ D(v, b, k) containing all the variance-balanced designs. Then we prove that (ii) a design d ∈′ Δ(ν, b,k) is ER-optimal in Δ(ν, b,k) if its replication numbers are as uniform as possible, (iii) a design d in Δ(ν, b,k) is AR-optimal and DR-optimal in Δ(ν, b, k) if its replication numbers are all equal except one, and (iv) a design d in Δ(ν, b, k) has maximum average efficiency in Δ(ν, b,k) if its replications numbers are all equal but one. Average efficiencies of designs in three classes are computed and tabulated at the end.

Some Construction Methods of Optimum Chemical Balance Weighing Designs III

Methods of constructing the optimum chemical balance weighing designs from symmetric balanced incomplete block designs are proposed with illustration. As a by-product pairwise efficiency and variance balanced designs are also obtained.

Some Construction Methods of A-Optimum Chemical Balance Weighing Designs

Some new construction methods of the optimum chemical balance weighing designs and pairwise efficiency and variance balanced designs are proposed, which are based on the incidence matrices of the known symmetric balanced incomplete block designs. Also the conditions under which the constructed chemical balance weighing designs become A-optimal are also been given.

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.

Universally optimal balanced changeover designs with first residuals

Considering the presence of first order residual effects of treatments, a family of variance balanced changeover designs has been presented and universal optimality of the designs is established. The designs use only v experimental units and (v − 1)/2 periods for v = 4t + 3 prime or prime power number of treatments; t being a positive integer. A special feature of the proposed designs is that ‘in the order of presentation of treatments to experimental units over periods, each treatment is once immediately preceded by only half of the other treatments and is immediately followed once by the remaining half of the treatments’. This characteristic results in reducing the size of the variance balanced designs considerably.

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.

Variance balanced circular designs involving sequences of treatments with first and second residuals

Designs involving sequences of treatments (also called changeover, crossover, or repeated measurements designs) balanced for first and second residuals available in literature are usually large even for a moderate number of treatments. Besides, most of these designs require number of periods at least equal to the number of treatments. In this paper a class of variance balanced circular designs for prime or prime power number of treatments, v (= mp + 1; m, a positive integer) using p ( 4, < v) periods and only mv experimental units has been proposed. A simple method of analysis of these designs is given and the efficiency factors relative to the orthogonal designs have been tabulated for v 31 and p 4, 12.

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.

Constructions of Nested Pairwise Efficiency and Variance Balanced Designs

ABSTRACT Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.

Variance-balanced designs under interference for dependent observations

It is shown that variance-balanced designs can be obtained from Type I orthogonal arrays for many general models with two kinds of treatment effects, including ones for interference, with general dependence structures. These designs can be used to obtain optimal and efficient designs. Some examples and design comparisons are given.

Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.

Variance-balanced change-over designs for dependent observations

We show that variance-balanced change-over designs can be obtained from orthogonal arrays for many dependence structures. Some examples and design comparisons are given.

Multiple Comparisons in the General Linear Model

Whereas multiple comparisons computations in a one-way model are well understood, multiple comparisons computations in a general linear model (GLM) are not. For models with the so-called “one-way structure,” no new technique is needed beyond proper substitution of terms. Examples of designs that guarantee a one-way structure include variance balanced designs and orthogonal designs. For models without a one-way structure, more sophisticated computational techniques are needed. Approximations based on the probabilistic inequalities of Bonferroni, Šidák, and Slepian are too conservative. Even the second-order Hunter—Worsley inequality is rather conservative. The so-called factor analytic approximation is quite accurate for multiple comparison with a control (MCC) and multiple comparison with the best (MCB), but conditions for it to be conservative are not known. This article describes a highly accurate, deterministic, conservative approximation that is applicable to a popular class of general linear models, and a fast, stochastic, conservative approximation that is generally applicable.