A construction for generalized hadamard matrices

We prove that if p^r and p^r ? 1 are both prime powers then there is a generalized Hadamard matrix of order p^r(p^r ? 1) with elements from the elementary abelian group Z_p x?x Z_p. This result was motivated by results of Rajkundia on BIBD's. This result is then used to produce p^r ? 1 mutually orthogonal F-squares F(p^r(p^r ? 1); p^r ? 1).     

Arrays of strength s on two symbols

This paper gives necessary and sufficient conditions on σ, s, t and on μ, s, t for an array with s+t rows to have strength s and weight σ, or to be balanced and have strength s and weight μ. If a balanced array can exist, the conditions provide a construction. The solutions for t=1,2 are also given in an alternate form useful for the study of trim arrays. The balanced solution for t=1 is more detailed than that known so far, and permits one to determine whether or not a solution exists in possibly fewer steps.     

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.     

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.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Four mols of order 10 with a hole of order 2

A transversal design with hole T[6;10]?T[6;2] is constructed from a separable group divisible design GD[5,1,8;48].     

On a robustness property of PBIBD

In this paper we study a robustness property of partially balanced incomplete block designs based on association schemes with m classes (PBIBD(m)) against the unavailability of data in the sense that, when any t (a positive integer) observations are unavailable the design remains connected w.r.t. treatment. We characterize the robustness property of PBIBD(m) completely for m=2 and partially for m=3.     

Projective (2n,n,λ,1)-designs

A projective (2n,n,λ,1)-design is a set $\text{D}$ of n element subsets (called blocks) of a 2n-element set V having the properties that each element of V is a member of λ blocks and every two blocks have a non-empty intersection. This paper establishes existence and non-existence results for various projective (2n,n,λ,1)-designs and their subdesigns.     

Construction of (M,S)-optimal design for block size 3

(M,S)-optimal designs are constructed for block size three when the number of treatments is of the form 6t + 3.     

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.     

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).     

Generalized Bhaskar Rao designs of block size three

We show that the necessary conditions

$\text{λ≡0 (}\text{mod}\text{|G|)}$

,

$\text{λ(υ?1)≡0 (}\text{mod}\text{2)}$

,

$\text{λυ(υ?1)≡}\text{0 (}\text{mod}\text{6)}\text{for}\text{|G|}\text{odd}\text{,}\text{0 (}\text{mod}\text{24)}\text{for}\text{|G|}\text{even}$

, are sufficient for the existence of a generalized Bhaskar Rao design GBRD(υ,b,r,3,λ;G) for the elementary abelian group G, of each order |G|.     

Saturated designs for multivariate cubic regression

This paper deals with the problem of finding saturated designs for multivariate cubic regression on a cube which are nearly D-optimal. A finite class of designs is presented for the k dimensional cube having the property that the sequence of the best designs in this class for each k is asymptotically efficient as k increases. A method for constructing good designs in this class is discussed and the construction is carried out for 1?k?8. These numerical results are presented in the last section of the paper.     

Serial P-values

When a collection of hypotheses is to be tested it is necessary to maintain a bound on the simultaneous Type I error rate. Serial P-values are used to define a serial test that does provide such a bound. Moreover, serial P-values are meaningful in the context of multiple tests, with or without the ‘rejection-confirmation’ decisions. The method is particularly suited to the analysis of unbalanced data, especially contingency tables.     

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.     

Some sufficient conditions for establishing (M,S)-optimality

Two sufficient conditions are given for an incomplete block design to be (M,S- optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.     

Some d-optimal weighing designs for n≡ (mod 4)

A number of D-optimal weighing designs are constructed with the help of block matrices. The D-optimal designs (n,k,s)=(19,13,10), (19,14,7), (19,14,8), (19,15,7), (19,15,8), (19,17,6), (19,18,6), (23,16,8), (23,17,8), (23,18,8), (4n?1,2n+3,(3n+4)/2), (4n?1,2n+4,n+3), (4n?1,2n+4,n+2) where n≡0 mod 4 and a skew H_n exists, (31,24,8), (31,25,8) and many others are constructed. A computer routine leading to locally D-optimal designs is presented.     

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.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.