Nonregular Designs Research Articles

A maximum estimability criterion for design classification and selection

A maximum estimability (maxest) criterion is proposed for design classification and selection. It is an extension and refinement of Webb's resolution criterion for general factorial designs. By using the estimability vector associated with the maxest criterion, projective properties of nonregular designs are studied from the estimability perspective. Comparisons with other criteria are also discussed.

Nonregular designs from Hadamard matrices and their estimation capacity

Deng and Tang (1999) proposed the generalized minimum aberration (GMA) criterion to assess fractional factorial designs, and a design with GMA is often regarded as the best. However, there exist situations where some other designs may suit practical needs better. In this article, we propose an algorithm to sequentially examine designs obtained from Hadamard matrices under estimation capacity (EC) and provide designs with maximum or high EC for various combinations of run-size and number-of-factors. The usefulness of maximum or high EC designs is discussed.

Blocked Nonregular Two-Level Factorial Designs

This article discusses the optimal blocking criteria for nonregular two-level designs. We extend the optimal blocking criteria of Cheng and Wu to nonregular designs by adapting the G- and G2-minimum aberration criteria discussed by Tang and Deng. To define word-length pattern for nonregular designs, we extend the notion of “word” to nonregular designs through a polynomial representation of factorial designs. We define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding. We propose four new criteria, which we use to search for optimal blocking schemes of 12-run, 16-run, and 20-run two-level orthogonal arrays.

Optimal Projective Three-Level Designs for Factor Screening and Interaction Detection

Orthogonal arrays (OAs) are widely used in industrial experiments for factor screening. Suppose that only a few of the factors in the experiments turn out to be important. An OA can be used not only for screening factors, but also for detecting interactions among a subset of active factors. In this article a set of optimality criteria is proposed to assess the performance of designs for factor screening, projection, and interaction detection, and a three-step approach is proposed to search for optimal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient, and practically useful nonregular designs are found and their properties are discussed.

Theory of optimal blocking of nonregular factorial designs

AbstractThe authors introduce the notion of split generalized wordlength pattern (GWP), i.e., treatment GWP and block GWP, for a blocked nonregular factorial design. They generalize the minimum aberration criterion to suit this type of design. Connections between factorial design theory and coding theory allow them to obtain combinatorial identities that govern the relationship between the split GWP of a blocked factorial design and that of its blocked consulting design. These identities work for regular and nonregular designs. Furthermore, the authors establish general rules for identifying generalized minimum aberration (GMA) blocked designs through their blocked consulting designs. Finally they tabulate and compare some GMA blocked designs from Hall's orthogonal array OA(16,215,2) of type III.

Minimum aberration construction results for nonregular two-level fractional factorial designs

Nonregular two‐level fractional factorial designs are designs which cannot be specified in terms of a set of defining contrasts. The aliasing properties of nonregular designs can be compared by using a generalisation of the minimum aberration criterion called minimum G2‐aberration. Until now, the only nontrivial designs that are known to have minimum G2‐aberration are designs for n runs and m ≥ n−5 factors. In this paper, a number of construction results are presented which allow minimum G2‐aberration designs to be found for many of the cases with n = 16, 24, 32, 48, 64 and 96 runs and m ≥ n/2−2 factors.

Optimal Foldover Plans for Two-Level Nonregular Orthogonal Designs

This article considers optimal foldover plans for nonregular designs. By using the indicator function, we define words with fractional lengths. The extended word-length pattern is then used to select among nonregular foldover designs. Some general properties of foldover designs are obtained using the indicator function. We prove that the full-foldover plan that reverses the signs of all factors is optimal for all 12-run and 20-run orthogonal designs. The optimal foldover plans for all 16-run (regular and nonregular) orthogonal designs are constructed and tabulated for practical use. Optimal foldover plans for higher-order orthogonal designs can be constructed in a similar manner.

Generalized minimum aberration for asymmetrical fractional factorial designs

By studying treatment contrasts and ANOVA models, we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs. The criterion is independent of the choice of treatment contrasts and thus model-free. It works for symmetrical and asymmetrical designs, regular and nonregular designs. In particular,it reduces to the minimum aberration criterion for regular designs and the minimum $G_2$-aberration criterion for two-level nonregular designs. In addition, by exploring the connection between factorial design theory and coding theory, we develop a complementary design theory for general symmetrical designs,which covers many existing results as special cases.

Generalized minimum aberration for asymmetrical fractional factorial designs

By studying treatment contrasts and ANOVA models, we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs. The criterion is independent of the choice of treatment contrasts and thus model­free. It works for symmetrical and asymmetrical designs, regular and nonregular designs. In particular, it reduces to the minimum aberration criterion for regular designs and the minimum G2 ­aberration criterion for two­level nonregular designs. In addition, by exploring the connection between factorial design theory and coding theory, we develop a complementary design theory for general symmetrical designs, which covers many existing results as special cases.

Experiments - Planning, Analysis and Parameter Design Optimization

Basic Principles and Experiments with a Single Factor. Experiments With More Than One Factor. Full Factorial Experiments at Two Levels. Fractional Factorial Experiments at Two Levels. Full Factorial and Fractional Factorial Experiments at Three Levels. Other Design and Analysis Techniques for Experiments at More Than Two Levels. Nonregular Designs: Construction and Properties. Experiments with Complex Aliasing. Response Surface Methodology. Introduction to Robust Parameter Design. Robust Parameter Design for Signal-Response Systems. Experiments for Improving Reliability. Experiments With Nonnormal Data. Appendices. Indexes.

Efficient Computational Algorithms for Searching for Good Designs According to the Generalized Minimum Aberration Criterion

SYNOPTIC ABSTRACTGeneralized minimum aberration was introduced by Deng and Tang (1999) as a criterion for comparing and assessing the “goodness” of nonregular fractional factorial designs. For small run sizes, generalized minimum aberration designs can be obtained by a complete search of Hadamard matrices; however, for larger run sizes a complete search is impractical. The main purpose of this paper is to study the use of efficient computational algorithms to construct generalized minimum aberration designs. Application to small run sizes shows that our algorithms perform almost as well as complete search, but with much less computing time. This strongly suggests that our algorithms should be successful in finding good designs for larger run sizes.

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.

Analysis of Censored Data from Fractionated Experiments: A Bayesian Approach

Abstract Censored data arise naturally in industrial experiments whose observations are failure times or measurements with mixed continuous-categorical outcomes. When censored data are observed from a fractionated experiment, likelihood-based estimates quite often do not exist, especially when the opportunity for improvement is great. To circumvent this problem, we propose a Bayesian analysis strategy that has a straightforward implementation using the data augmentation and Monte Carlo EM algorithms. For nonregular designs with complex aliasing patterns, a modified analysis strategy is proposed that substantially reduces the computation needed for model selection. The proposed strategy is illustrated using data from three real industrial experiments. For these data sets, the analysis results are fairly insensitive to the specification of the prior.

Analysis of Censored Data from Fractionated Experiments: A Bayesian Approach

Abstract Censored data arise naturally in industrial experiments whose observations are failure times or measurements with mixed continuous-categorical outcomes. When censored data are observed from a fractionated experiment, likelihood-based estimates quite often do not exist, especially when the opportunity for improvement is great. To circumvent this problem, we propose a Bayesian analysis strategy that has a straightforward implementation using the data augmentation and Monte Carlo EM algorithms. For nonregular designs with complex aliasing patterns, a modified analysis strategy is proposed that substantially reduces the computation needed for model selection. The proposed strategy is illustrated using data from three real industrial experiments. For these data sets, the analysis results are fairly insensitive to the specification of the prior.