Balanced arrays of strength 4 and balanced fractional 3m factorial designs

A connection between a balanced fractional 2^m factorial design of resolution 2l + 1 and a balanced array of strength 2l with index set {μ₀, μ₁,…, μ_2l} was established by Yamamoto, Shirakura and Kuwada (1975). The main purpose of this paper is to give a connection between a balanced fractional 3^m factorial design of resolution V and a balanced array of strength 4, size N, m constraints, 3 levels and index set {λ_l0l1l2}.     

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.     

Search designs for 2m factorial experiments

Theorems 5, 6 and 10, and Tables 1–2 in Ghosh (1981) are corrected. These are concerned with search designs which permit the estimation of the general mean and main effects, and allow the search and estimation of one possibly unknown nonzero effect among the two- and three-factor interactions in 2^m factorial experiments. Some new results are presented.     

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.     

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).     

On the robustness of the optimum balanced 2m factorial designs of resolution v (given by Srivastava and Chopra) in the presence of outliers

In this paper we study the sensitivity of the optimum balanced resolution V plans for 2^m factorials, to outliers, using the measure suggested by Box and Drapper (1975). We observe that the designs are robust, i.e., have low sensitivity.     

Construction of some classes of balanced cross-over designs from terraces

5 and 6 have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.     

On some new search designs for 2m factorial experiments

In this paper, we obtain search designs with reasonably small number of treatments which permit the estimation of the general mean and main effects and search of one more unknown possible nonzero effect among two and three factor interactions in 2^m factorial experiments, 3 ? m ? 8.     

Some bounds on balanced block designs

Bounds on the latest root of the C-matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.     

Weakly resolvable IV.3 search designs for the pn factorial experiment

A series of weakly resolvable search designs for the pⁿ factorial experiment is given for which the mean and all main effects are estimable in the presence of any number of two-factor interactions and for which any combination of three or fewer pairs of factors that interact may be detected. The designs have N = p(p–1)n+p runs except in one case where additional runs are required for detection and one case where $\text{(p?1)}\text{2}$ additional runs are needed to estimate all (p–1)² degrees of freedom for each pair of detected interactions. The detection procedure is simple enough that computations can be carried out with hand calculations.     

A construction for generalized hadamard matrices GH(4q, EA(q))

We give a construction for a generalized Hadamard matrix GH(4q, EA(q)) as a 4 × 4 matrix of q × q blocks, for q an odd prime power other than 3 or 5. Each block is a GH(q, EA(q)) and certain combinations of 4 blocks form GH(2q, EA(q)) matrices. Hence a GH(4q, EA(q)) matrix exists for every prime power q.     

Minimum number of runs for two-level factorial search designs

A lower bound is given for the number of experimental runs required in search designs for two-level factorial models.     

D-optimal block designs with at most six varieties

For given positive integers v, b, and k (all of them ≥2) a block design is a k × b array of the variety labels 1,…,v with blocks as columns. For the usual one-way heterogeneity model in standard form the problem is studied of finding a D-optimal block design for estimating the variety contrasts, when no balanced block design (BBD) exists. The paper presents solutions to this problem for v≤6. The results on D-optimality are derived from a graph-theoretic context. Block designs can be considered as multigraphs, and a block design is D-optimal iff its multigraph has greatest complexity (=number of spanning trees).     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and c_i(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[¹¹¹⁰₀₁₂₁], c₁=[¹₀], c₂=[⁰₂].) Let T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) is a vector of variables.(Thus, EG(4, 3) consists of the 3⁴=81 points of the form (t₁,t₂,t₃,t₄), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=c_i is s^n?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T₁ and T₂ are 2-flats consisting of 3^4?2=9 points each. The flats T₁,T₂,…,T_f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T₁ are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T₂ consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sⁿ symmetric factorial experiment obtained by taking T₁,T₂,…,T_f together. (Thus, in the example, 3⁴=81 treatments of the 3⁴ factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T₁ and T₂ enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X._j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′._jX._j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′._jX._j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X._j into the X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X._j and X′._jX._j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.     

On invariance and randomization under factor permutation in fractional factorial designs

This paper establishes the spectrum invariance of the information matrix under an arbitrary subgroup Г of the group Δ of factor permutations. In addition, it provides the randomized unbiased estimation of a linear parametric function under the composition Ω°Г, where Ω is the group of level permutations. These two results are achieved for the most practical partitioning of the whole parametric vector using the concepts of Г-closed and admissibility of a parametric subvector. Applications are given with an explicit illustration using the minimal resolution III design setting for the 2³ factorial.     

Generalized Bhaskar Rao designs

Generalized Bhaskar Rao designs with non-zero elements from an abelian group G are constructed. In particular this paper shows that the necessary conditions are sufficient for the existence of generalized Bhaskar Rao designs with k=3 for the following groups: ?G? is odd, G=Z^r₂, and G=Z^r₂×H where 3? ?H? and r?1. It also constructs generalized Bhaskar Rao designs with υ=k, which is equivalent to υ rows of a generalized Hadamard matrix of order n where υ?n.     

Complete designs with blocks of maximal multiplicity

The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size $\text{b}{}^1\text{= (}{}^{\mathrm{v}}{}_3\text{) ? 4(v ? 2)}$. Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (^v₃) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.     

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.     

A construction of repeated measurements designs with balance for residual effects

Families of Repeated Measurements Designs balanced for residual effects are constructed (whenever the divisibility conditions allow), under the assumption that the number of periods is less than the number of treatments and that each treatment precedes each other treatment once. These designs are then shown to be connected for both residual and direct treatment effects.     

Some families of group divisible designs

Generalizing methods of constructions of Hadamard group divisible designs due to Bush (1979), some new families of semi-regular or regular group divisible designs are produced. Furthermore, new nonisomorphic solutions for some known group divisible designs are given, together with useful group divisible designs not listed in Clatworthy (1973).