Balanced Arrays Research Articles

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.

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

The norm ‖A‖ = {tr(A′A)} 1 2 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 ‖ A‖ will be given for a balanced fractional 2 m factorial design of resolution 2 l + 1 which obtained from a simple array with parameters ( m; λ 0, λ 1,…, λ m ). This array is identical with a balanced array of strength m, m constraints and index set { λ 0, λ 1,…, λ m }. In the class of the designs of resolution V ( l = 2) obtained from S-arrays, ones which minimize ‖ A‖ 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.

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

A connection between a balanced fractional 2 m factorial design of resolution 2 l + 1 and a balanced array of strength 2 l with index set { μ 0, μ 1,…, μ 2 l } 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 l 0l 1l 2 }.

D-Optimal Fractions of Three-Level Factorial Designs

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.

D-Optimal Fractions of Three-Level Factorial Designs

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.

On computer generation of balanced arrays

This paper provides an algebraic (and hence computing) procedure for generation of balanced arrays having two symbols, m rows, specified minimum and maximum column weights, arbitrary strength t≦ m, and index set parameters μ t 1, μ t 2,…, μ t t . μ t 0 is unspecified, and calculated as part of the algorithm, although the procedure for specifying it is straightforward and can be used if desired. Array generation is herein reduced to finding integral solutions to a linear programming problem. It is shown that the integral solutions of the system of equations comprise all balanced arrays with the given set of parameters. A computing algorithm is provided which constructs the system of equations to be solved; it has been interfaced with a standard linear programming package to provide some preliminary results. Additional algorithms whose development should result in substantial decreases in computing costs are discussed.

Contributions to balanced fractional $2\spm$ factorial designs derived from balanced arrays of strength $2l$

Contributions to balanced fractional $2\spm$ factorial designs derived from balanced arrays of strength $2l$

On the construction of balanced and orthogonal arrays

AbstractTwo 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.

Balanced Fractional $2^m$ Factorial Designs of Even Resolution Obtained from Balanced Arrays of Strength $2l$ with Index $\mu_l = 0$

Consider a balanced array $T$ of strength 2$l,$ size $N, m$ constraints and index set $\{\mu_0, \mu_1,\cdots, \mu_{2l}\}$ with $\mu_l = 0$. Under some conditions $T$ yields a design of even resolution ($2l$, say) with $N$ assemblies such that all the effects involving up to $(l - 1)$-factor interactions are estimable provided $(l + 1)$-factor and higher order interactions are assumed negligible and that the covariance matrix of their estimates is invariant under any permutation of $m$ factors. The alias structure of the effects of $l$-factor interactions is explicitly given. Such an array $T$ is called an $S$-type balanced fractional $2^m$ factorial design of resolution $2l$. Necessary conditions for the existence of the design $T$ are given. For any given $N$, there are in general a large number of possible $S$-type balanced fractional $2^m$ factorial designs of resolution $2l$. Finally a criterion for comparing these designs is given.

Note on the construction of partially balanced arrays

Thep-symbol partially balanced arrays of strengtht, wherep=3 and 4, are given by a method due to A. Dey, A. C. Kulshreshtha and G. M. Saha [4].

Balanced arrays of strength 2l and balanced fractional 2 m factorial designs

A connection between a balanced fractional 2 m factorial design of resolutionV and a balanced array of strength 4 with index set {μ 0,μ 0,μ 1,μ 2,μ 3,μ 4} has been established by Srivastava [3]. The purpose of this paper is to generalize his results by investigating the combinatorial property of a fractionT and the algebraic structure of the information matrix of the fractional design. Main results are: A necessary and sufficient condition for a fractional 2 m factorial designT of resolution 2l+1 to be balanced is thatT is a balanced array of strength 2l with index set {μ 0,μ 1,μ 2, ⋯,μ 21} provided the information matrixM is nonsingular.

Contributions to the Theory and Construction of Balanced Arrays

Balanced arrays were introduced and first studied by I. M. Chakravarti and called partially balanced arrays. J. N. Srivastava and D. V. Chopra have recently made major contributions to the theory and construction of these arrays. They suggested dropping the adjective "partially" and we have followed their lead. Whereas their work has been concerned mainly with arrays of strength four with two symbols, we are here concerned primarily with arrays of strength two with two symbols. However, when the proofs can be carried out as easily for any strength and any number of symbols, we do state the theorems in full generality. We are concerned here with finding bounds on the maximum possible number of rows and with the problem of constructing balanced arrays for given sets of parameters. Analogous to the problem of constructing other combinatorial configurations, we investigated whether some schemes, i.e. some subsets of columns of balanced arrays, could be extended to full balanced arrays. It is shown, analogously to orthogonal arrays, that balanced arrays of even strength, say $2u$ are extendable to arrays of strength $2u + 1$. A new technique of construction of balanced arrays with the maximum number of constraints is also described. It is shown that BIB designs with $\lambda = 1$ can be utilized for constructing balanced arrays with the number of symbols equal to the block size of the BIB design. For completeness we include an example of the analysis of a partially balanced array of strength two with two symbols when it is used as a fractional factorial design. We exhibit in this way an explicit method of estimating main effects when higher ordered interactions are assumed to be negligible.

Optimal balanced 27 fractional factorial designs of resolutionV, withN≦42

In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).

Optimal balanced 27fractional factorial designs of resolution V, 49 ≤ N ≤55

In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).

Three symbol partially balanced arrays

Two methods of construction of partially balanced arrays of strength two and three are presented.

Some general existence conditions for balanced arrays of strength t and 2 symbols

This paper gives necessary and sufficient conditions for the existence of a balanced array ( B-array) (“partially balanced” array, in the terminology of Chakravarti [7]) of 2 symbols, strength t, and m rows (with m ⩽ t + 2). Necessary existence conditions, for all m, are also derived in the form of many classes of diophantine equations.

Amino Acid Retention and Balance in the Young Rat Fed Varying Levels of Casein

Amino acid balance and retention efficiency were determined in rats as a function of the level of dietary protein (casein). Twelve diets were prepared ranging in total protein from 4.8 to 39.1%. It was anticipated that the most limiting amino acids would show the greatest retention efficiency as long as protein synthesis was the dominant influence on utilization. The expected pattern was observed for the first three limiting amino acids but not valine and isoleucine. Protein efficiency ratio (PER) and individual amino acid retention efficiencies declined rapidly below 8% dietary protein despite continued weight gain. The decline in PER was accompanied by a change in the relative proportions of several amino acids in the carcass gain. It was proposed that the latter change resulted from a shift in the rate of protein synthesis favoring proteins essential for maintenance of life at the expense of more expandable proteins such as hair or muscle. The data are essentially in agreement with a previous study employing lactalbumin as the protein source and presenting a better balanced array of amino acids.

Balanced confounding of factorial experiments

The paper discusses the use of pseudo-factors for the construction of balanced factorial arrangements. The method is developed for s×q designs and then more generally for s×q1×q2×…,×qn designs, where s is necessarily a prime power. Usually blocks of st plots are wanted. The procedure is to replace each factor after the first by one or more pseudo-factors at s levels, and to form a parent design that confounds sT in blocks of st for a suitably large T. An arbitrary correspondence is then established between the combinations of levels of pseudo-fators and the qu, levels of factor Qu. This gives a set of initial blocks for the required design, and a series of permutations then introduces balance. Chief emphasis is placed on designs for two factors. The special case of s×q designs, with s>q blocks of s plots, is related to the use of balanced incomplete block designs in constructing asymmetrical factorial designs. The relation with partially balanced arrays of strength two is also discussed. Methods for reducing the number of blocks first produced are examined, as are the effects of choice of correspondence of factor levels and choice of permutations on the number of replicates, the loss of information on main effects, and resolvability. Blocks of fewer than st plots m considered.

Derivation of intermodulation outputs of a pair of general nonlinear elements in a balanced mixer array

Derivation of intermodulation outputs of a pair of general nonlinear elements in a balanced mixer array

Orthogonal and partially balanced arrays and their application in design of experiments

Orthogonal and partially balanced arrays and their application in design of experiments