A note on the robustness of PBIBD(2)s against breakdown in the event of observation loss

Robustness against design breakdown following observation loss is investigated for Partially Balanced Incomplete Block Designs with two associate classes (PBIBD(2)s). New results are obtained which add to the body of knowledge on PBIBD(2)s. In particular, using an approach based on the E‐value of a design, all PBIBD(2)s with triangular and Latin square association schemes are established as having optimal block breakdown number. Furthermore, for group divisible designs not covered by existing results in the literature, a sufficient condition for optimal block breakdown number establishes that all members of some design sub‐classes have this property.     

A-optimal partially balanced fractional 2m1+m2 factorial designs of resolution V,with 4≤m1>+m2≤6

By use of the algebraic structure, we obtain an explicit expression for the characteristic polynomial of the information matrix of a partially balanced fractional 2^m₁+m₂ factorial design of resolution V derived from a partially balanced array. For 4≤m₁+m₂≤6, A-optimal designs considered here are also presented for reasonable number of assemblies.     

Klingenberg structures and partial designs I: Congruence relations and solutions

We consider generalizations of projective Klingenberg and projective Hjelmslev planes, mainly (b, c)-K-structures. These are triples (φ, Π, Π′) where Π and Π′ are incidence structures and φ : Π → Π′ is an epimorphism which satisfies certain lifting axioms for double flags. The congruence relations of such triples are investigated, leading to nice factorizations of φ. Two integer invariants are associated with each congruence relation, generalizing a theorem of Kleinfeld on projective Hjelmslev planes. These invariants are completely characterized for a special class of triples: the balanced, minimally uniform neighbor cohesive (1,1)-K-structures. We show that a balanced neighbor cohesive (1,1)-K-structure Π “over” a PBIBD Π′ is again a PBIBD and compute its invariants. Several methods are given for constructing symmetric “divisible” PBIBD's on arbitrarily many classes.     

Balanced fractional factorial designs of resolution 2l+1 for interesting effects orthogonal to some nuisance parameters: 2m1+m2 series

For 2^m₁+m₂ factorial designs, this paper investigates balanced fractional 2^m₁ factorial designs of resolution 2l+1 with some nuisance parameters concerning the second factors. They are derivable from partially balanced arrays and further permit estimation of the effects up to the l-factor interactions concerning the first factors orthogonally to the nuisance parameters.     

On the use of partial confounding for the construction of alternative regular two-level blocked fractional factorial designs

In this paper, we consider experimental situations in which a regular fractional factorial design is to be used to study the effects of m two-level factors using n=2^m−k experimental units arranged in 2^p blocks of size 2^m−k−p. In such situations, two-factor interactions are often confounded with blocks and complete information is lost on these two-factor interactions. Here we consider the use of the foldover technique in conjunction with combining designs having different blocking schemes to produce alternative partially confounded blocked fractional factorial designs that have more estimable two-factor interactions or a higher estimation capacity or both than their traditional counterparts.     

Model-robust supersaturated and partially supersaturated designs

Supersaturated designs are an increasingly popular tool for screening factors in the presence of effect sparsity. The advantage of this class of designs over resolution III factorial designs or Plackett–Burman designs is that n, the number of runs, can be substantially smaller than the number of factors, m. A limitation associated with most supersaturated designs produced thus far is that the capability of these designs for estimating g active effects has not been discussed. In addition to exploring this capability, we develop a new class of model-robust supersaturated designs that, for a given n and m, maximizes the number g   of active effects that can be estimated simultaneously. The capabilities of model-robust supersaturated designs for model discrimination are assessed using a model-discrimination criterion, the subspace angle. Finally, we introduce the class of partially supersaturated designs, intended for use when we require a specific subset of _m1$m_1$ core factors to be estimable, and the sparsity of effects principle applies to the remaining (m-_m1$m-m_1$) factors.     

New light on fractional 2m factorial designs

New fractional 2^m factorial designs obtained by assigning factors to fractions of m columns of new saturated two symbol orthogonal arrays which are not isomorphic to the usual ones are proposed. Contrary to the usual assignment, examples show that some main effects are not totally but partially confounded with several two-factor interactions. Moreover, the recovery of the former from such partial confounding is possible in some cases by eliminating the latter.     

Some new factor screening designs using the search linear model

Factor screening designs for searching two and three effective factors using the search linear model are discussed. The construction of such factor screening designs involved finding a fraction with small number of treatments of a 2^m factorial experiment having the property P_2t (no 2t columns are linearly dependent) for t=2 and 3. A ‘Packing Problem’ is introduced in this connection. A complete solution of the problem in one case and partial solutions for the other cases are presented. Many practically useful new designs are listed.     

Designs for balanced observation of plant competition

Symmetric designs, for exploring the effect of competition between two varieties or plant species planted on a triangular lattice of hill plots, are discussed and the class of m-fold symmetric Beehive designs, based on certain symmetry properties of a regular hexagon, is introduced. The designs considered by Martin (1973) and Veevers and Boffey (1975) belong to the family arising when m = 6. Optimal designs for m = 2 and 3 are presented and, although not balanced, improvements in the sense of being nearer to balanced are achieved.Taking an alternative approach, a simple technique for constructing balanced, essentially rectangular, designs of arbitrary size is developed, based upon a set of twelve symmetric elementary arrays which possess a remarkable self-building property. The experimenter is at liberty to choose a balanced design to suit restrictions on space and material or to meet his desired degree of replication whilst the actual planting technique requires only that complete rows of each variety be suitably juxtaposed.     

On the construction of m-associate PBIB designs I

The m-associate triangular association scheme has been discussed, and several series of partially balanced incomplete block (PBIB) designs with m-associate triangular association scheme have been obtained in Section 1. In Section 2, an m-associate triangular-group divisible association scheme $\text{(T}{}_{\mathrm{q}}\text{—}\text{GD}\text{m}\text{, 1<q<m)}$ has been introduced and several series of PBIB designs with m-associate triangular group divisible association scheme, from m-associate triangular PBIB designs have been constructed. Some numerical values to the three associate triangular designs, and three associate triangular group divisible designs in the range b, v≦100; r, k≦10, with their average efficiencies are given, respectively, in Tables I and II, in Section 3, where as usual v denotes the number of treatments, b the number of blocks, r the number of replications of each treatment, and k the block size.     

Symmetric Sheffer sequences and their applications to lattice path counting

A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks.     

Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

Some new classes of orthogonal Latin hypercube designs

Orthogonal Latin hypercube (OLH) is a good design choice in a polynomial function model for computer experiments, because it ensures uncorrelated estimation of linear effects when a first-order model is fitted. However, when a second-order model is adopted, an OLH also needs to satisfy the additional property that each column is orthogonal to the elementwise square of all columns and orthogonal to the elementwise product of every pair of columns. Such class of OLHs is called OLHs of order two while the former class just possessing two-dimensional orthogonality is called OLHs of order one. In this paper we present a general method for constructing OLHs of orders one and two for n=s^m$n=s^m$ runs, where s and m may be any positive integers greater than one, by rotating a grouped orthogonal array with a column-orthogonal rotation matrix. The Kronecker product and the stacking methods are revisited and combined to construct some new classes of OLHs of orders one and two with other flexible numbers of runs. Some useful OLHs of order one or two with larger factor-to-run ratio and moderate runs are tabulated and discussed.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).     

Optimality aspects of 3-concurrence most balanced designs

Takcuchi (1961,1963) established E-optimality of Group Divisible Designs (GDDs) with λ₂=λ₁+1. Much later, Cheng (1980) and Jacroux (1980,1983) demonstrated E-optimality property of the GDDs with n=2,λ₁=λ₂+1 or with m=2,λ₂=λ₁+2. The purpose of this paper is to provide a unified approach for identifying certain classes of designs as E-optimal. In the process, we come up with a complete characterization of all E-optimal designs attaining a specific bound for the smallest non-zero eigenvalue of the underlying C-matrices. This establishes E-optimality of a class of 3-concurrence most balanced designs with suitable intra- and inter-group balancing. We also discuss the MV-optimality aspect of such designs.     

Minimal 2-coverings of a finite affine space based on GF(2)

Let EG(m, 2) denote the m-dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q-covering (where q is an integer satisfying 1?q?m) of EG(m, 2) if and only if T has a nonempty intersection with every (m-q)-flat of EG(m, 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q=2. Let N denote the size of the set T. Given N, we study the maximal value of m.     

Main effect plan for 2m factorials which allow search and estimation of one unknown effect

In this paper, we discuss resolution III plans for 2^m factorial experiments which have an additional property. We relax the classical assumption that all the interactions are negligible by assuming that (at most) one of them may be nonnegligible. Which interaction is nonnegligible is unknown. We discuss designs which allow the search and estimation of this interaction, along with the estimation of the general mean and the main effects as in the classical resolution III designs.     

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

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.     

On some optimal fractional 2m factorial designs of resolution V

This paper presents the trace of the covariance matrix of the estimates of effects based on a fractional 2^m factorial (2^m-FF) design T of resolution V for the following two cases: One is the case where T is constructed by adding some restricted assemblies to an orthogonal array. The other is one where T is constructed by removing some restricted assemblies from an orthogonal array of index unity. In the class of 2^m-FF designs of resolution V considered here, optimal designs with respect to the trace criterion, i.e. A-optimal, are presented for m = 4, 5, and 6 and for a range of practical values of N (the total number of assemblies). Some of them are better than the corresponding A-optimal designs in the class of balanced fractional 2^m factorial designs of resolution V obtained by Srivastava and Chopra (1971b) in such a sense that the trace of the covariance matrix of the estimates is small.