A note on partially balanced ternary designs

Two new methods of constructing a series of partially balanced ternary designs are presented. One from a BIB design and a PBIB design, and the second from a PBIB design alone, obtained by method of differences in both the cases.     

On the incomplete block designs derivable from the association schemes

Blackwelder (1969) has given two methods of constructing balanced incomplete block (BIB) designs from the association matrices of association schemes with two and three associate classes. In this note these two methods are incorporated in a general method, and the existence of a series of BIB designs is shown by the generalized method. In addition, a remark about partially balanced incomplete block (PBIB) designs with respect to the method is made.     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

Balanced nested designs and balanced arrays

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4.     

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.     

On construction of partially balanced n-ary designs

A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT designs and PB n-ary designs from a PBIB design alone based on NC_m-scheme as well as from a group divisible PBIB design with smaller number of blocks and moderate block sizes.     

Edge-coloured designs with block size four

Summary The existence of edge-coloured block designs with block size four is studied for all nonisomorphic colourings of the edges of aK ₄. There are 25 nonisomorphic edge-colouredK ₄'s; for each, we examine the existence of edge-coloured designs with the minimum possible index. Uniform cases lead to block designs, perpendicular arrays, nested Steiner triple systems, idempotent Schroeder quasigroups, and other combinatorial objects.     

A construction of balanced arrays of strengtht and some related incomplete block designs

Summary Saha [6] has shown the equivalence between a ‘tactical system’ (or at-design) and a 2-symbol balanced array (BA) of strengtht. The implicit method of construction of BA in that paper has been generalized herein to that of ans-symbol BA of strengtht. Some BIB and PBIB designs are also constructed from these arrays. Majindar [2], Vanstone [8] and Saha [6] have all shown that the existence of a symmetrical BIBD forv treatments implies the existence of six more BIBD's forv treatments in (v/2) blocks. An analogue of this result has been obtained for a large class of PBIB designs in this paper.     

Balanced arrays of strength two from block designs

Some constructions of balanced arrays of strength two are provided by use of rectangular designs, group divisible designs, and nested balanced incomplete block designs. Some series of such arrays are also presented as well as orthogonal arrays, with illustrations. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 303–312, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10016     

Sets pooling designs

Pooling desings have previously been used for the efficient identification of distinguished elements of a finite setU. Group testing underlies these designs: For any , a binary result is obtainable, indicating whether or not the number of distinguished elements included inS is zero. The current generalization of pooling designs will enable the efficient identification of distinguished subsets of a finite setU. In this case, for any , a binary result is obtainable, indicating whether or not the number of distinguished subsets included inS is zero. Such designs are called sets pooling designs, comprising standard pooling designs in the special case where all the distinguished subsets are elements. The new designs are similar to the standard designs but are subject to new constraints because the set of subsets included inS is its power set. To illustrate the feasibility of constructing sets pooling designs, random, non-adaptive designs are investigated for the special case where all distinguished subsets have the same size. An optimum probability for including an object in a pool is approximated as a function of the size and number of distinguished subsets, adopting the criterion of minimizing the average number of non-distinguished subsets whose status would not be resolved by the pooling design. Deterministic and adaptive designs are also described.This work was supported by the US Department of Energy under contract W-7405-ENG-36, through a Laboratory Directed Research and Development Grant at Los Alamos National Laboratory.     

A robust and powerful rank test of treatment effects in balanced incomplete block designs

This paper is concerned primarily with a Monte Carlo investigation into the small sample robustness and power of an aligned rank transformation statistic. The aligned rank transformation statistic (ART) is compared to the classical F-test and to Durbin's (1951) rank test for its ability to detect treatment effects in balanced incomplete block (BIB) designs.     

The number oft-wise balanced designs

We prove that the number oft-wise balanced designs of ordern is asymptotically , provided that blocks of sizet are permitted. In the process, we prove that the number oft-profiles (multisets of block sizes) is bounded below by and above by for constants c₂>c₁>0.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

Construction of BIB designs with various support sizes—with special emphasis for v = 8 and k = 4

This paper deals with methods of constructing new BIB designs from known designs. First, we provide a method of finding trades on a given BIB design so that new designs with the same parameters ν, b, k can be constructed. Second, we point out that the method of combining designs mentioned in [9] is particularly useful for constructing designs based on ν = 2k, k even, when designs based on ν = 2k ? 1 are available. By utilizing the above methods with the aid of Table I in [4], we have constructed a table of 53 BIB designs based on V = 8 and k = 4 with support sizes $\text{14 ? b 1 (}\text{8}\text{4}\text{)=70}$ except for $\text{b}{}^1\text{= 15, 16, 17, 19}$. BIB designs with ν = 8, k = 4 and $\text{b}{}^1\text{< 14}$ are shown to be nonexistent.     

A relation between BIB designs and chemical balance weighing designs

The paper gives certain new construction method for optimum chemical balance weighing designs. It utilizes a relation between the incidence matrices of a set of BIB designs and the design matrix of a chemical balance weighing design.     

Pooling, lattice square, and union jack designs

Simplified pooling designs employ rows, columns, and principal diagonals from square and rectangular plates. The requirement that every two samples be tested together in exactly one pool leads to a novel combinatorial configuration: The union jack design. Existence of union jack designs is settled affirmatively whenever the ordern is a prime andn3 (mod 4).     

Pairwise and variance balanced incomplete block designs

Summary The purpose of this paper is three-fold. The first purpose is to compile and to systematize published and dispersed results on two aspects of balancing in incomplete block designs, i.e., pairwise balance and variance balance. This was done in order to establish the status of these two concepts of balance in published literature and to put them in a form which is useful for further work in this area. Also, the results in this form are necessary for the development of the remainder of the paper. The second purpose of this paper is to present a method of constructing unequal replicate and/or unequal block size experiment designs for which the variance balance property is achieved. The method of construction involves the union of blocks from two or more block designs and the augmentation of some of the blocks with additional treatments; the method is denoted asunionizing block designs. A straight-forward extension of the method would produce a partially balanced block design with unequal replicate and/or unequal block designs. The enlargement of the concept and availability of variance blanced block designs to accommodate unequal replication and/or unequal block sizes is important to the researcher, the teacher, and the experimenter needing such designs. For example, an animal nutritionist or a psychologist is no longer required to have constant litter or family sizes for the blocks and may have unequal replication on the treatments for those treatments with insufficient material and still attain the goal of equal variances on all normalized treatment contracts. The third purpose of the paper is to apply the unionizing block designs method to construct a family of unequal replicate and unequal block size variance balanced designs. Some comments are given on the extension of the unionizing block designs method to construct other families of variance balanced or partially balanced block designs. This investigation was supported in part by PHS Research Grant No. R01-GM-05900 from General Medical Services.     

Connectedness of PBIB designs having asymmetrical association schemes

Summary A necessary and sufficient condition for the connectedness ofm-associate partially balanced incomplete block (PBIB) designs having an asymmetrical association scheme is given, only in terms of design parameters, without inner structure parameters of designs. Supported in part by Grant 321-6066-58530013 (Japan).     

Nested sampling for estimating spatial semivariograms compared to other designs

In spatial studies, use is commonly made of nested sampling plans. By applying such plans, one takes observations according to a hierarchical scheme, with decreasing distances between observations. As observed by Miesch (1975), the cumulative sum of variance components provided by the nested sampling plan may be used in some situations to obtain semivariogram values. In this article, proofs are given for both balanced and unbalanced designs. Different estimation procedures for obtaining semivariogram values are compared with each other. The paper is illustrated with two numerical examples, one on actual soil pH data and one on simulated random fields. Mean squared pair differences are shown to be inferior to expected mean squares and restricted maximum likelihood for variance component estimation and several other spatial sampling plans may be superior to the nested sampling plan for estimating the spatial semivariogram.