Two-level Fractional Factorial Designs Research Articles

Screening the Experimental Domain for the Microwave-Assisted Extraction of Secoisolariciresinol Diglucoside from Flaxseed Prior to Optimization Procedures

This paper presents the first part of a study that aims at developing an optimized microwave-assisted method for extracting the lignan secoisolariciresinol diglucoside from defatted flaxseed meal. Two-level fractional factorial designs were used for screening the following factors: the microwave power level (30–360 W), the time of residence in the microwave cavity (1–25 min), the concentration of NaOH (0.25–1 M), and the mode of microwave power application (power on 30 and 60 s/min). The experimental domain had to be adjusted after each screening in order to focus the future optimization study within the factors' ranges likely to contain the optimal extraction conditions.

Orthogonal and nearly orthogonal designs for computer experiments

We introduce a method for constructing a rich class of designs that are suitable for use in computer experiments. The designs include Latin hypercube designs and two-level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs. The basic construction method is simple, building a series of larger designs based on a given small design. If the base design is orthogonal, the resulting designs are orthogonal; likewise, if the base design is nearly orthogonal, the resulting designs are nearly orthogonal. We present two generalizations of our basic construction method. The first generalization improves the projection properties of the basic method; the second generalization gives rise to designs that have smaller correlations. Sample constructions are presented and properties of these designs are discussed.

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.

A Class of Partially Replicated Two Level Fractional Factorial Designs

Resolution III fractional factorial designs permit the uncorrelated estimation of all main effects under the assumption of no interactions among the factors and are often used in the initial phases of an experimental program to screen factors. Even in these experimental situations, it may be desirable to include partial replication of the design, thereby allowing for the unbiased estimation of the experimental error while maintaining the orthogonality of the main effects. It is often then possible to assess the no-interactions assumption by comparing the sample of confounded interactions with the pure error estimate. In this paper, a new class of partially replicated two-level fractional factorial designs based on Hadamard matrices is proposed. A method of constructing these designs is given and the relative performance of the designs is evaluated by comparing them with competing fractional factorial designs.

Optimal Foldover Plans for Two-Level Fractional Factorial Split-Plot Designs

We consider the construction of foldovers of two-level fractional factorial split-plot designs. As a follow-up strategy, the foldover technique is useful when the objective is to systematically de-alias effects of interest after the initial fractional factorial split-plot design has been run. Due to the sequential nature in which the runs of the initial and foldover designs are conducted, the resulting combined design (the design consisting of both the initial design and its foldover) is a blocked fractional factorial split-plot design. Using the minimum aberration criterion for blocked fractional factorial split-plot designs, in conjunction with other optimality criteria, we provide a catalog of optimal foldover plans for initial minimum aberration fractional factorial split-plot designs consisting of 16 and 32 runs.

An isomorphism check for two-level fractional factorial designs

Two fractional factorial designs are isomorphic if one can be obtained from the other by reordering the treatment combinations, relabelling the factor levels and relabelling the factors. By defining a word-pattern matrix, we are able to create a new isomorphism check which is much faster than existing checks for certain situations. We combine this with a new, extremely fast, sufficient condition for non-isomorphism to avoid checking certain cases. We then create a faster search algorithm by combining the Bingham and Sitter [1999. Minimum aberration fractional factorial split-plot designs. Technometrics 41, 62–70] search algorithm, the isomorphism check algorithm of Clark and Dean [2001. Equivalence of fractional factorial designs. Statist. Sinica 11, 537–547] with our proposed isomorphism check. The algorithm is used to extend the known set of existing non-isomorphic 128-run two-level regular designs with resolution ⩾ 4 to situations with 12, 13, 14, 15 and 16 factors, 256- and 512-run designs with resolution ⩾ 5 and ⩽ 17 factors and 1024-run even designs with resolution ⩾ 6 and ⩽ 18 factors.

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2 n - p fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.

SELECTION OF BLOCKED TWO-LEVEL FRACTIONAL FACTORIAL DESIGNS FOR AGRICULTURAL EXPERIMENTS

Blocked two-level fractional factorial designs are a very useful tool for efficient data collection in agricultural and other scientific research. In most experiments, in addition to the main effects, some two-factor interactions are also meaningful and need to be estimated. We propose a method for efficiently selecting blocked two-level fractional factorial designs when some of the two-factor interactions are non-negligible. We then present some results for a design with only 8 or 16 runs to illustrate how to use this method.

A complementary design theory for doubling

Chen and Cheng [Ann. Statist. 34 (2006) 546--558] discussed the method of doubling for constructing two-level fractional factorial designs. They showed that for $9N/32\le n\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $5N/16$ factors which is constructed by repeatedly doubling the $2^{5-1}$ design defined by $I=ABCDE$. This paper develops a general complementary design theory for doubling. For any design obtained by repeated doubling, general identities are established to link the wordlength patterns of each pair of complementary projection designs. A rule is developed for choosing minimum aberration projection designs from the maximal design with $5N/16$ factors. It is further shown that for $17N/64\le n\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $N$ runs and $5N/16$ factors.

Partial duplication in two-level fractional factorial designs

Most fractional factorial designs have no replicated points and thus do not provide an estimate for pure error. The construction methods for orthogonal main-effect plan in the literature usually do not produce designs with duplicate points. However, it is possible to combine four fractions to provide a set of duplicate points without sacrificing the orthogonality of main effects. This paper proposes two techniques of this idea to produce designs with replicate points in two-level fractional factorial designs.

Connections Between the Resolutions of General Two-level Factorial Designs

Indicator functions are new tools for studying two-level fractional factorial designs. This article discusses some properties of indicator functions. Using indicator functions, we study the connection between general two-level factorial designs of generalized resolutions.

Classification of Three-word Indicator Functions of Two-level Factorial Designs

Indicator functions have been in the literature for several years, and yet only a few of their properties have been examined. In this paper, we study some properties of indicator functions of two-level fractional factorial designs. For example, we show that there is no indicator function with only two words, and also classify all indicator functions with only three words. The results imply that there is no valuable non-regular design with only three or less words in its indicator function.

Results for two-level fractional factorial designs of resolution IV or more

The main theorem of this paper shows that foldover designs are the only (regular or nonregular) two-level factorial designs of resolution IV (strength 3) or more for n runs and n / 3 ⩽ m ⩽ n / 2 factors. This theorem is a generalization of a coding theory result of Davydov and Tombak [1990. Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems Inform. Transmission 25, 265–275] which, under translation, effectively states that foldover (or even) designs are the only regular two-level factorial designs of resolution IV or more for n runs and 5 n / 16 ⩽ m ⩽ n / 2 factors. This paper also contains other theorems including an alternative proof of Davydov and Tombak's result.

Partially replicated two‐level fractional factorial designs

AbstractIn a two‐level factorial experiment, the authors consider designs with partial duplication which permit estimation of the constant term, all main effects and some specified two‐factor interactions, assuming that the other effects are negligible. They construct parallel‐flats designs with two identical parallel flats that meet prior specifications; they also consider classes of 3‐flat and 4‐flat designs. They show that the designs obtained can have a very simple covariance structure and high D‐efficiency. They give an algorithm from which they generate a series of practical designs with run sizes 12, 16, 24, and 32.

Construction of Two-Level Split-Lot Fractional Factorial Designs for Multistage Processes

Split-lot designs, also known as multiway split-unit designs, are useful in factorial experiments where there are multiple processing stages. Such experiments occur, for example, in the fabrication of integrated circuits in the semiconductor industry and in product assembly. The designs have a split-plot structure at each stage so that settings of the factors are applied to sublots of experimental runs rather than to individual runs. This allows experimental runs to be processed at much less expense than if the runs were completely randomized. In this article, split-lot designs are constructed for two, three, and four processing stages. The designs have minimum aberration under the split-lot structure and minimize in various senses the confounding of main effects and two-factor interactions with the sublots at each stage.

Irregular Four-level Response Surface Designs

Four-level response surface designs based on regular two-level fractional factorial designs were introduced by Edmondson (1991). Here, the methods are extended to include designs based on irregular two-level fractional factorials. These designs allow orthogonal blocking and require fewer experimental units than the regular designs.

Fractional factorial designs for two-step production processes

In this paper a method for constructing efficient two-level fractional factorial designs for the analysis of two-step production processes is described. First a design for analyzing the first step of the production process is chosen. The chosen design is then extended to one for analyzing both steps. The design is extended in a such a way that the number of different combinations for the first step is small.

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3n−k fractional factorial designs.

Classification of efficient two-level fractional factorial designs of resolution IV or more

Two-level fractional factorial designs that are efficient in terms of aberration or other aliasing properties are classified into four types of designs of resolution IV or more: the half-fraction designs, the even designs, the five-column designs and the join designs. The designs are shown to have concise grid representations which provide simple interpretations of their aliasing structure. New efficient 128-run designs are presented and blocking of the designs is considered.

Minimum G2-aberration properties of two-level foldover designs

This paper provides theoretical results on the construction of two-level fractional factorial designs with minimum G 2-aberration. Attention focuses on foldover designs which are shown to have minimum G 2-aberration across the whole class of orthogonal designs for n=24 runs and any m⩽ n/2 factors. Minimum G 2-aberration foldover designs are also provided for n=32, 48 and 64 runs.