Three-level Fractional Factorial Designs Research Articles

An integrative framework for geometric and hidden projections in three-level fractional factorial designs

Geometric and hidden projection are two important properties to consider for three-level fractional factorials. The effective development of methods that can address tasks involving both these properties requires a unified design framework that integrates them. We highlight one integrative framework that is based on indicator functions for three-level designs under the linear-quadratic parameterization system. We demonstrate the broad scope and utility of this framework by using it to develop a new procedure for addressing a task comprised of three serial problems in geometric and hidden projection. The first in the series is to construct the sets of smallest follow-up runs to a projection of a three-level design that yield specified aliasing relations. Second is to immediately characterize the partial aliasing relations among factorial effects in the resulting augmented designs. The final problem is to identify the augmented designs that maximize popular optimality criteria with respect to the second-order model among all of the candidate augmentations. The application of our procedure is illustrated with sample projections of definitive screening designs onto four, five, and six factors. The integrative framework that we present can ultimately facilitate the systematic development of methods that advance comprehensive understanding of multiple properties for general three-level designs.

Some new results on Triple designs

The method of doubling has been used to construct two-level fractional factorial designs. Zhang (2016) extended the method of doubling to the method of tripling, which has been used to construct three-level fractional factorial designs. Various popular screening criteria, such as E(fNOD) criterion, generalized minimum aberration (GMA), minimum moment aberration (MMA), B criterion and uniformity criterion, have been proposed from different viewpoints to compare and construct designs. In this paper, the analytic connections between Triple designs constructed by tripling and initial designs under the above screening criteria are considered. These connections are suitable to general original three-level fractional factorial designs, whether regular or nonregular. Furthermore, the connections provide theoretical basis for constructing a kind of optimal three-level designs with large sizes by tripling successively.

A fresh look at effect aliasing and interactions: some new wine in old bottles

Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the “de-aliasing” of aliased effects is deemed to be impossible. We argue that this “impossibility” can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme’s) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme’s. A historical recollection is given on how these ideas were discovered.

Allocation of the equipment path in a multi-stage manufacturing process

The allocation of equipment in a multi-stage process is discussed in this article. In most of the multi-stage manufacturing processes, multiple equipment are operated to minimize the waiting times between stages. Thus, the allocation of the equipment path becomes an issue in choosing the equipment for the next stage. In solving the allocation problem of the multi-stage process, it is assumed that main effects and two-way interaction effects for the two adjacent stages are significant. The efficient allocation problem for the multi-stage process for a given historical data is solved by the general linear model approach, and then the predicted responses are ordered to choose the subsequently optimal equipment paths. The effectiveness of the proposed allocation strategy is evaluated in terms of the probabilities for detecting all true effects and detecting optimal equipment path for three cases of precisions: baseline, precise errors and noisy errors. It turns out that the noisy error case is less efficient than the others. When it is possible to use pilot experiments, the efficiency of the product design of orthogonal arrays for two-level and three-level fractional factorial designs is compared to that of the random selection of factorial design points. It is shown that the former is more efficient than the latter in a case study.

Indicator functions and the algebra of the linear-quadratic parameterization

Indicator functions are constructed under the linear-quadratic parameterization for contrasts, and applied to the study of partial aliasing properties for three-level fractional factorial designs. An algebraic operation is introduced for the calculation of indicator function coefficients. This operation connects design construction methods to the analysis under the linear-quadratic system, and helps establish simple conditions for the estimability of interactions.

Application of fractional factorial designs to study drug combinations

Herpes simplex virus type 1 (HSV-1) is known to cause diseases of various severities. There is increasing interest to find drug combinations to treat HSV-1 by reducing drug resistance and cytotoxicity. Drug combinations offer potentially higher efficacy and lower individual drug dosage. In this paper, we report a new application of fractional factorial designs to investigate a biological system with HSV-1 and six antiviral drugs, namely, interferon alpha, interferon beta, interferon gamma, ribavirin, acyclovir, and tumor necrosis factor alpha. We show how the sequential use of two-level and three-level fractional factorial designs can screen for important drugs and drug interactions, as well as determine potential optimal drug dosages through the use of contour plots. Our initial experiment using a two-level fractional factorial design suggests that there is model inadequacy and that drug dosages should be reduced. A follow-up experiment using a blocked three-level fractional factorial design indicates that tumor necrosis factor alpha has little effect and that HSV-1 infection can be suppressed effectively by using the right combination of the other five antiviral drugs. These observations have practical implications in the understanding of antiviral drug mechanism that can result in better design of antiviral drug therapy.

Interaction Graphs for 4<sup>r</sup>2<sup>n-p</sup> Fractional Factorial Designs

Interaction graphs have been developed for two-level and three-level fractional factorial designs under different design criteria. A catalogue is presented of all possible non-isomorphic interaction graphs for 4r2n-p (r=1; n=2,…, 10; p=1,…,8 and r=2; n=1,…, 7; p=1,…,7) fractional factorial designs, and nonisomorphic interaction graphs for asymmetric fractional factorial designs under the concept of combined array.

Minimum aberration blocking schemes for two- and three-level fractional factorial designs

The concept of minimum aberration has been extended to choose blocked fractional factorial designs (FFDs). The minimum aberration criterion ranks blocked FFDs according to their treatment and block wordlength patterns, which are often obtained by counting words in the treatment defining contrast subgroups and alias sets. When the number of factors is large, there are a huge number of words to be counted, causing some difficulties in computation. Based on coding theory, the concept of minimum moment aberration, proposed by Xu [Statist. Sinica, 13 (2003) 691–708] for unblocked FFDs, is extended to blocked FFDs. A method is then proposed for constructing minimum aberration blocked FFDs without using defining contrast subgroups and alias sets. Minimum aberration blocked FFDs for all 32 runs, 64 runs up to 32 factors, and all 81 runs are given with respect to three combined wordlength patterns.

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.

Catalogue of small runs nonregular designs from hadamard matrices with generalized minimum aberration

Fries and Hunter ( 1980 ) proposed the Minimum Aberration criterion (MA) for selecting regular designs. The regular designs with MA are msot commonly used because they are considered as the best designs. How ever, as pointed out by Chen, Sun and Wu ( 1993 ), there are situations that other designs may better meet the design need. Therefore, they catalogued some two-level and three-level fractional factorial regular designs with small (16,27,32,64) runs. For nonregular designs, such as the ones taken from Hadamard matrices, the MA criterion is not appUcable. Deng and Tang ( 1999 ) introduced Generalized Minimum Aberration Criterion (GMA) as a natural extension to the MA criterion. Similar to the case in the regular designs, other designs may better meet practical need, In this paper, we use the GMA criterion to give a catalogue of nonregular designs with smaU (16,20,24) runs.

Interaction Graphs for Three-Level Fractional Factorial Designs

Graph-aided methods for accommodating the estimation of interactions in factorial experiments have become popular among industrial users. Notable among them is the method of linear graphs due to Taguchi. Previously, some shortcomings of Taguchi's linear..

Tables of minimum cost, linear trend-free run sequences for two- and three-level fractional factorial designs

For the two- and three-level fractional factorial designs tabled in National Bureau of Standards Applied Mathematics Series publications 48 and 54, plus the resolution V plans developed by Addelman ( Technometrics 7, 1965), we present generator sequences that may be used with the Generalized Foldover Scheme of Coster and Cheng ( Ann. Statist. 16, 1988) to produce systematic run orders which minimize, or nearly minimize, a cost function equal to the number of times the factors change levels during the time sequence in which the runs are performed and which simultaneously have all factor main effects components orthogonal to a polynomial time trend.

A Catalogue of Two-Level and Three-Level Fractional Factorial Designs with Small Runs

Summary Fractional factorial (FF) designs with minimum aberration are often regarded as the best designs and are commonly used in practice. There are, however, situations in which other designs can meet practical needs better. A catalogue of designs would make it easy to search for 'best' designs according to various criteria. By exploring the algebraic structure of the FF designs, we propose an algorithm for constructing complete sets of FF designs. A collection of FF designs with 16, 27, 32 and 64 runs is given.