Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

A class of new pbib designs

Two series of PBIB designs, one with three associate classes and theother with four associate classes ars developed. Efficiency factors for two designs are computed.     

Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

A note on generalized binary proper efficiency-balanced block designs

Some methods of construction of generalized binary proper efficiency-balanced block (GB-EB) designs with two different replications are derived and the solution of the unknown designs given by Das and Kageyama (1991) are obtained.     

Some Methods of Construction of Gd-Rc and Rectangular-rc Designs

A number of methods of construction of GD-RC and RectanguIar-RC designs, based upon the series A of BIB-RC designs, using the method of Kronecker product, are developed.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

m-Associate PBIB designs using Youden-m squares

In this paper, we have proposed a type of arrangement that we call Youden-m square and is similar to the usual Youden square but generates PBIB designs instead of BIB designs when its columns are taken as blocks. We have also discussed its construction methodologies, introduced two new m-associate class association schemes, and also constructed some series of Youden-m square type PBIB designs.     

On the construction of hypercubic design

Hypercubic design was introduced by Shah (1958) and Kusumoto(1965) in which the t=v^m treatments are represented by n-plets (x₁, x₂, ..., x_m),where 1≤ x₁, x₂, ..., x_m ≤v, and two treatments are said to be i-th associates if they differ in exactly i components. This paper deals with the construction of some hypercubic designs and gives their application to confounding plans for symmetrical factorial experiments. The proposed methods prove to be quite flexible in terms of choice of possible block sizes and are easy to use.     

On d- and e- minimax optimal designs for estimating the axial slopes of a second-order response surface over hypercubic regions

Design of experiments for estimating the slopes of a response surface is considered. Design criteria analogous to the traditional ones but based upon the variance-covariance matrix of the estimated slopes along factor axes are proposed. Optimal designs under the proposed criteria are derived for second-order polynomial regression over hypercubic regions. Best de¬signs within some commonly used classes of designs are also obtained and their efficiencies are investigated.     

The interplay of optimality and combinatorics in experimental design

The design of statistical experiments, as developed by R. A. Fisher and his followers, often used combinatorial structures that yielded simple calculation of estimates and/or symmetric variances and covariances. Examples are block designs with balance, regression experiments with equally spaced observations, etc. More recently, considerations of optimality (choosing a design that achieves most accurate inference in some sense) have sometimes justified the traditional designs, but have sometimes led to new combinatorial investigations. Illustrations are given.     

On Some Two- and Three-dimensional D-minimax Designs for Estimating Slopes of a Third-order Response Surface

Design of experiments is considered for the situation where estimation of the slopes of a response surface is the main interest. Under the D-minimax criterion, the objective is to minimize the generalized variance of the estimated axial slopes at a point maximized over all points in the region of interest in the factor space. For the third-order model over spherical regions, the D-minimax designs are derived in two and three dimensions. The efficiencies of some two- and three-dimensional designs available in the literature are also investigated.     

THE USE OF A CLASS OF FOLDOVER DESIGNS AS SEARCH DESIGNS

It is shown that members of a class of two-level nonorthogonal resolution IV designs with n factors are strongly resolvable search designs when k, the maximum number of two-factor interactions thought possible, equals one; weakly resolvable when k = 2 except when the number of factors is 6; and may not be weakly resolvable when k≥ 3.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also presented.     

On The Optimal Choice of Cube and Star Replications in Restricted Second-Order Designs

This article examines the central composite design in which some of the experimental runs are replicated. Three different classes of N-point designs are compared using the criterion of Schur's ordering under orthogonality, rotatablity, and slope- rotatablity conditions. The response surface designs with the star portion replicated seem to have more potential than others under orthogonality condition, while the designs with the cube portion replicated is preferable to the designs with their star portion or only the center point replicated under rotatable and slope-rotatable conditions.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

An investigation of first-order and second-order designs for extrapolation outside a hypersphere

Methods for selecting a first-order or second-order rotatable response surface design when both variance and bias error exist are applied to a situation in which it is desired to extrapolate the fitted model in all directions outside of a sphere within which all the experiments are to be made. The extrapolation region is a spherical shell.     

Partially efficiency-balanced designs-II

efficiency balanced (£R) and partially efficiency balanced (PEB) designs were studied hy Calinski (1971) and Puri and Higam (1971, 1977a). In tais paper we have studied the dual desinas of two classes of PEB designs. It is shown that the dual of any symmetrical PCS design with HN'=N'H is again a PEB design. Also the dual of a PEB(S) with μ₁ = 0 μ₂ = 0and is shown to the again a PEB(S). The relations between PEB designs and linked and partially linked blocks have heen established. As a conseguence, the analysis of linked Mock and partially linkei block designs can he sinrlififid. Sore useful classes of supplencntcd block designs having simple analysis have also been discussed.     

Foldover Conference Designs for Screening Experiments

A conference matrix is a square matrix C with zeros on the diagonal and ±1s off the diagonal, such that C ^T C = CC ^T  = (n ? 1)I, where I is the identity matrix. Conference matrices are an important class of combinatorial designs due to their many applications in several fields of science, including statistical-experimental designs, telecommunications, elliptic geometry, and more. In this article, conference matrices and their full foldover design are combined together to obtain an alternative method for screening active factors in complicated problems. This method provides a model-independent estimate of the set of active factors and also gives a linearity test for the underlying model.