An infinite class of PBIB designs

In this paper, an infinite class of partially balanced incomplete block (PBIB) designs of m+1 associate classes is constructed through the use of a series of row-orthogonal matrices known as partially balanced orthogonal designs (PBOD) of m-associate classes. For the purpose, a series of PBOD is obtained through a method described herein. An infinite class of regular GD designs is also reported.     

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.     

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.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

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.     

Generalized neighbor designs with block size 3

A generalized neighbor design relaxes the equality condition on the number of times two treatments as neighbors in the design. In this article, we have considered the construction of some classes of generalized neighbor designs with block size k=3 by using the method of cyclic shifts. The distinguishing feature of this construction method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. A catalog of generalized neighbor designs with block size k=3 is compiled for v∈{5,6,…,18} treatments and for different replications. We provide the reader with a simpler method of construction, and in general the catalog that gives an open choice to the experimenter for selecting any class of neighbor designs.     

Some Classes of Binary Block Neighbor Designs

Neighbor designs are useful to neutralize the neighbor effects. In this paper two classes of these designs are constructed in circular binary blocks of size 4, 8, …, 24. First class consists of six infinite series of nearest neighbor designs in which each pair of distinct treatments appears once as neighbors. Second class also deals with six infinite series of these designs in which each pair of distinct treatments appears twice as neighbors. A catalog of nearest neighbor designs is also compiled in circular binary blocks for odd number of treatments from 23 to 99.     

Patterned constructions of partially ballvced incomplete block design

Starting from the incidence matrix of a BIB design, we obtain PBIB designs with two or three associate classes following certain patterns for their block structures .     

Weighted Searching Probability for Classes of Equivalent Search Designs Comparison

Search design is searching and estimating for a few non zero effects in a large set of effects along with estimation of elements in a set of unknown parameters. In presence of noise, the probability of discrimination between the true non zero effect from an alternative one depends on the design and an unknown parameter, say ρ. We develop a new criterion for design comparison which is independent of ρ and for a family density weight function show that it discriminates and ranks the designs precisely. This criterion is invariance to the variable noise which may be present between designs due to noise factors. This allows us to extend the design comparison to classes of equivalent designs.     

On the construction of balanced and orthogonal arrays

Two series of three symbol balanced arrays of strength two are constructed. Using special classes of BIB designs, two classes of two symbol orthogonal arrays of strength three are constructed.     

Robust designs for diallel crosses against the missing of one block

Dey & Midha (1996) showed that some of the complete diallel crosses plans, obtained by using triangular partially balanced designs with two associate classes, are optimal. In this investigation, it is derived that these optimal designs for diallel crosses are robust also against the unavailability of one block.     

Some partially variance balanced structurally incomplete row-column designs

Earlier results by the authors are used to provide the intrablock analysis for row-column designs that have observations at nodes of the row-column lattice, the design being structurally incomplete when some nodes are empty. Construction, properties, and intrablock analyses of some special b× b row-column designs with b empty nodes taken along the principal diagonal of the lattice are developed. The designs discussed have m > 1 associate classes and are said to be partially variance balanced. The special designs fall in two classes and are shown to be nearly optimal in a specified class of designs. A small catalog of designs constructed is provided and they should be useful when empty nodes do not represent wasted experimental units, perhaps because the row and column assignments of treatments are sequenced.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

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.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

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.     

Comparison of Four New General Classes of Search Designs

A factor screening experiment identifies a few important factors from a large list of factors that potentially influence the response. If a list consists of m factors each at three levels, a design is a subset of all possible 3 ^m runs. This paper considers the problem of finding designs with small numbers of runs, using the search linear model introduced in Srivastava (1975). The paper presents four new general classes of these 'search designs', each with 2 m −1 runs, which permit, at most, two important factors out of m factors to be searched for and identified. The paper compares the designs for 4 ≤ m ≤ 10, using arithmetic and geometric means of the determinants, traces and maximum characteristic roots of particular matrices. Two of the designs are found to be superior in all six criteria studied. The four designs are identical for m = 3 and this design is an optimal design in the class of all search designs under the six criteria. The four designs are also identical for m = 4 under some row and column permutations.     

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.     

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.