Asymmetric Factorials Research Articles

Projection properties of some orthogonal arrays

In factor screening experiments, one generally starts with a large pool of potentially important factors. However, often only a few of these are really active. Under this assumption of effect sparsity, while choosing a design for factor screening, it is important to consider projections of the design on to smaller subsets of factors and examine whether the projected designs allow estimability of some interactions along with the main effects. While the projectivity properties of symmetric 2-level and a few 3-level fractional factorial designs represented by orthogonal arrays have been studied in the literature, similar studies in respect of asymmetric or, mixed level factorials seems to be lacking. In this paper, we initiate work in this direction by providing designs with good projectivity properties for asymmetric factorials of the type t × 2 m based on orthogonal arrays. We also note that the results of Cheng (1995) regarding the projectivity of symmetric two-symbol orthogonal arrays do not necessarily extend to arrays with more than two symbols.

Uniformity in factorial designs with mixed levels

The uniformity can be utilized as a measure for comparing factorial designs. Fang and Mukerjee (Biometrika 87 (2000) 193–198) and Fang et al. (in: K.T. Fang, F.J. Hickernell, H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002) found links among uniformity in terms of some non-uniformity measures, orthogonality and aberration for regular symmetric factorials. In this paper we extend their results to asymmetric factorials by considering a so-called wrap-around L 2-discrepancy to evaluate the uniformity of factorials. Furthermore, a lower bound of wrap-around L 2-discrepancy is obtained for asymmetric factorials and two new ways of construction of factorial designs with mixed levels are proposed.

Level changes and trend resistance on replacement in asymmetric orthogonal arrays

When level changes and trend-resistant effects are considered in factorial experiments, the order of experimental runs is essential. Under such considerations, it is helpful to use orthogonal arrays to arrange experimental runs and their orders simultaneously. To use this method, one needs to derive the level changes and trend resistance in the columns of the arrays to gain an appropriate order. Most research on this topic has focused on the case of symmetric factorials or only on the construction of trend-resistant run orders of asymmetric factorials. Recently, Wang (Statist. Sinica, 6 (1996) 471–479) derived the level changes and trend resistance in the columns of asymmetric orthogonal arrays for 2 m 4 n factorial plans. In this article, we extend Wang's results to general asymmetric orthogonal arrays.

Use of asymmetrical factorials for generating designs for quadratic mixture model

Mclean and Anderson (1966), Snee (1975) and Nigam et al. (1983) have used symmetric factorials for constructing designs for fitting the quadratic mixture models, when the mixture components are constrained by upper and lower bounds. In this paper, we have used asymmetrical factorial plans for constructing these designs.

On the construction of orthogonal factorial designs of resolution 11

A new method of construction of orthogonal resolution IV designs for symmetrical and asymmetrical factorials has been presented. Many new series of orthogonal factorial designs of resolution IV can be obtained by the above general method.

Search designs for searching for one among the two-and three-factor interaction effects in the general symmetric and asymmetric factorials

This paper describes the construction of search designs which permit the estimation of the general mean and main-effects, and allow the search for and estimation of one possibly unknown non-zero effect among the two-and three-factor interactions in the general symmetric and asymmetric factorial set-up.

Shorter Communication: Some Orthogonal Main-Effect Plans for Asymmetrical Factorials

Hadamard matrices are used to construct some saturated asymmetrical main-effect plans of types 4 s × 2 t and n × 4 s × 2 t . For Hadamard numbers t and n with t ≤ n and n ≥ 4, I show that there exists a saturated orthogonal 4 t−1 × 2 nt−3t+2 main-effect plan with nt runs. If t and n are Hadamard numbers such that t, n ≥ 4 and t/2 is also a Hadamard number, then there exists a saturated orthogonal n × 4 h−1 × 2 n(t−1)−3(h−1) main-effect plan with nt runs, where h = min(t, n).

Search designs for general asymmetric factorials

This paper is in connection with search designs which permit the estimation of the general mean and main-effects, and allow the search and estimation of the possibly unknown non-zero effect among the two-factor interactions in the general asymmetric factorial set-up. This paper describes the construction of search designs in the above set-up.

Mixed orthogonal arrays for variance estimation with unequal numbers of primary selections per stratum

For estimating the variance of nonlinear statistics in large-scale complex surveys, the method of balanced repeated replications has received special attention. The method at present is useful in stratified clustered designs with equal number of primary selections from each stratum. This paper extends the method to general sample designs with arbitrary number of selections from each stratum. It is shown that mixed orthogonal arrays of strength two, or equivalently, equal frequency orthogonal main-effect plans for asymmetrical factorials give a set of balanced subsamples useful in variance estimation.

Some search designs for symmetric and asymmetric factorials

This paper describes the construction of search designs involving a larger number of factors from those involving smaller numers of factors. Some search designs for symmetric factorials have been proposed. These results may be utilized for the construction of search designs for the series s r × w m− r where s and w are any positive integers.

Orthogonal Resolution-IV Designs for Some Asymmetrical Factorials

Orthogonal resolution-IV designs for 42 · 22n–4, 43 · 24n–12, and 82 · 24n–8 experiments are constructed, where n is a multiple of four. The plans for 42 · 22n–4 and 82 · 24n–8 experiments are minimal as well.

Orthogonal Resolution-IV Designs for Some Asymmetrical Factorials

Orthogonal resolution-IV designs for 42 · 22n–4, 43 · 24n–12, and 82 · 24n–8 experiments are constructed, where n is a multiple of four. The plans for 42 · 22n–4 and 82 · 24n–8 experiments are minimal as well.

Orthogonal Main-Effect Plans for Asymmetrical Factorials

A technique for constructing orthogonal main-effect plans from resolvable orthogonal arrays is described. This technique is applied to obtain two series of orthogonal main-effect plans for asymmetrical factorials of the type t · sm . Both series of plans are saturated. A number of new plans can be derived using these series.

On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.

Remarks on extension of the anderson-federer (0,1)-matrix procedure for main effects to effects of a higher degree

The theory of (0,1)-matrices plays a fundamental role in many combinatorial investigations. This is especially true in factorial designs where it is desirable to transform, if possible, the design matrix to a (0,1) matrix so that results on (0,1)-matrices can be utilized in solving construction (e.g. optimal designs) and analysis problems. In this connection this paper shows, firstly, that the Anderson-Federer (0,1)-matrix procedure for main effect plans in the general asymmetrical factorial does not extend in a natural way to include plans of a higher degree. Secondly, a transformation is provided, in the case of the symmetrical factorial, which converts design matrices defined relative to a suitable class of observation vectors and admissible sets of effects of degree at most two which include the mean into (0,1)-matrices. Further a transformation is given which converts any design matrix relative to an admissible set of effects into a (-l,0,l)-matrix.

On non-orthogonal main effect plans for asymmetrical factorials

Following the lines of Raktoe and Federer [19] a unified approach for constructing main effect plans in any\(k_1^{n_1 } \times k_2^{n_2 } \times \cdots \times k_r^{n_r } \) factorials whereki's are the numbers of equispaced levels of each of theni factors, andki's are not necessarily primes or prime powers and need not satisty any relations among themselves, is presented. The method consists of, first, dividing the totality of treatment combinations, omitting, of course, some, if necessary, in to pairs such that the differences within the pairs are clear of ‘even’ effects and the sums are clear of ‘odd’ effects, and then, depending on the number of error d.f. wanted, selecting a suitable sub-set of these pairs which lead to the solution of the estimates of main effects. A general class of non-orthogonal main effect plans for 2m×2n factorials is proposed. Information matrices and their inverses for such plans are worked out. An example followed by discussions and comparison statements is presented.

A Calculus for Factorial Arrangements

This paper introduces a special calculus for the analysis of factorial experiments. The calculus applies to the general case of asymmetric factorial experiments and is not restricted to symmetric factorials as is the current theory which relies on the theory of finite projective geometry. The concise notation and operations of this calculus point up the relationship of treatment combinations to interactions and the effect of patterns of arrangements on the distribution of relevant quantities. One aim is to carry out complex manipulations and operations with relative ease. The calculus enables many large order arithmetic operations, necessary for analyzing factorial designs, to be partly carried out by logical operations. This should be of importance in programming the analysis of factorial designs on high speed computers. The principal new results of this paper, aside from the new notation and operations, are (i) the further development of a theory of confounding for asymmetrical factorials (Section 4) and (ii) a new approach to the calculation of polynomial regression (Section 5). In particular, the use of the calculus enables one to write the inverse matrix of the normal equations for a polynomial model as a partitioned matrix. As a result it only requires inverting matrices of smaller order.

Symmetrical and Asymmetrical Fractional Factorial Plans

Symmetrical and Asymmetrical Fractional Factorial Plans