Minimum Aberration Criterion Research Articles

Construction of mixed-level fractional factorial split-plot designs with combined minimum aberration

Running agricultural experiments in split-plot configurations is an efficient way to estimate main-effect contrasts with higher precision. In a fractional factorial split-plot design, the hard-to-vary factors are called whole-plot (WP) treatment factors and the easy-to-vary factors are called subplot treatment factors. Yan and Zhao (2022) proposed the combined minimum aberration criterion of WP (WP-MAc) for mixed-level split-plot designs. This article studies the construction methods of the WP-MAc 2 ( n 1 + n 2 ) − ( k 1 + k 2 ) 4 1 designs with four-level factor in subplot. Some examples are given to illustrate the theories. Finally, the results of 8, 16, 32, 64 runs WP-MAc 2 ( n 1 + n 2 ) − ( k 1 + k 2 ) 4 s 1 designs are tabulated for possible employ by practitioners.

Theory of [formula omitted]-characteristics of four-level designs under quaternary codes

The J-characteristics play an instrumental role in the development of generalized resolution and minimum aberration criteria for two-level designs. In this paper, the concept of J-characteristics is naturally generalized to four-level designs via quaternary codes, which maps the four-level designs to two-level designs. Based on the relationship between the minimum G2-aberration criterion of two-level design and the generalized minimum aberration criterion of its projection designs, the properties of J-characteristics of four-level designs are explored. The relationship between J-characteristics of four-level design and J-characteristics of corresponding effective two-level sub-designs is built. The generalized resolution, confounding frequency vector and B-vector of four-level design are respectively defined based on the J-characteristics and their upper bounds, and minimum G-aberration and minimum G2-aberration criteria of four-level design are proposed, which are useful to assess the goodness of four-level designs.

Optimal fractions of three-level factorials under a baseline parameterization

The paper proposes the minimum aberration (MA) criterion for three-level designs under a baseline parameterization. An algorithm is put forward to search the MA designs. A catalogue of MA 18-run and 27-run three-level baseline designs is tabulated.

A study on average Lee discrepancy measure

In this paper, we proposed the average Lee discrepancy to measure the uniformity of designs and studied its connection with the standard optimality measures used in the context of design of experiments. Lower bounds to such measure are provided as a benchmark to obtain uniform designs. We also defined the minimum average Lee-moment aberration criterion used to compare and screen the optimal design, and some lower bounds to such criterion are also provided.

Minimum Aberration Split-Plot Designs When the Whole Plot and Subplot Factors Do Not Have the Same Importance

In practical factorial experiments, we sometimes find that complete randomization of the order of the runs is infeasible because it is more difficult to change the levels of some factors than the others, especially in some engineering experiments. Then, fractional factorial split-plot (FFSP) designs represent a practical option in such situations. The difficult-to-change factors are called whole plots (WP) factors, and the other factors are called subplot (SP) factors. The WP and SP factors do not have the same importance in many experiments. Then, the popular minimum aberration criterion is not suitable any more for choosing FFSP designs. This paper proposes two criteria for selecting FFSP designs. Algorithms for constructing the optimal FFSP designs under the two criteria are proposed. Some optimal designs under the two criteria are tabulated.

Selecting baseline designs using a minimum aberration criterion when some two-factor interactions are important

ABSTRACT This article considers the problem of selecting two-level designs under the baseline parameterisation when some two-factor interactions are important. We propose a minimum aberration criterion, which minimises the bias caused by the non-negligible effects. Using this criterion, a class of optimal designs can be further distinguished from one another, and we present an algorithm to find the minimum aberration designs among the D-optimal designs. Sixteen-run and twenty-run designs are summarised for practical use.

Wordlength enumerator for fractional factorial designs

While the minimum aberration criterion is popular for selecting good designs with qualitative factors under an ANOVA model, the minimum $\beta$-aberration criterion is more suitable for selecting designs with quantitative factors under a polynomial model. In this paper, we propose the concept of wordlength enumerator to unify these two criteria. The wordlength enumerator is defined as an average similarity of contrasts among all possible pairs of runs. The wordlength enumerator is easy and fast to compute, and can be used to compare and rank designs efficiently. Based on the wordlength enumerator, we develop simple and fast methods for calculating both the generalized wordlength pattern and the $\beta$-wordlength pattern. We further obtain a lower bound of the wordlength enumerator for three-level designs and characterize the combinatorial structure of designs achieving the lower bound. Finally, we propose two methods for constructing supersaturated designs that have both generalized minimum aberration and minimum $\beta$-aberration.

An addendum on “Three‐level designs: Evaluation and comparison for screening purposes”

AbstractIn our recent paper “Three‐Level Designs: Evaluation and Comparison for Screening Purposes”, by the same authors, we investigated and compared the original definitive screening designs and the designs that were constructed from weighing matrices. For this purpose, we used the generalized resolution and minimum aberration criteria as well as the projection estimation capacity criterion from the literature to perform the comparison. After our paper was accepted, it came to our attention that there were four significant papers, published by a number of authors, having related content and these were not mentioned in our paper. With this short addendum, we would like to kindly acknowledge the important contributions from those authors in those newly discovered papers in recent literature and also to include them in the related reference list.

Construction of Blocked Factorial Designs to Estimate Main Effects and Selected Two-Factor Interactions

AbstractTwo-level factorial designs are widely used in industry. For experiments involving n factors, the construction of designs comprising 2n and 2n-p factorials, arranged in blocks of size 2q is investigated. The aim is to estimate all main effects and a selected subset of two-factor interactions. Designs constructed according to minimum aberration criteria are shown to not necessarily be the most appropriate designs in this situation. A design construction approach is proposed which exploits known results on proper vertex colourings in graph theory. Examples are provided to illustrate the results and construction strategies. Particular consideration is given to the special case of designs with blocks of size four and tables of designs are given for this block size.

Three‐level designs: Evaluation and comparison for screening purposes

AbstractSince their introduction by Box and Hunter, resolution criteria have been widely used when comparing regular fractional factorials designs. In this article, we investigate how a generalized resolution criterion can be used to assess some recently developed three‐level screening designs, such as definitive screening designs (DSDs) and screening designs from weighing matrices. The aim of this paper is to capture the projection properties of those three‐level screening designs, complementing the work of Deng and Tang, who used generalized resolution and minimum aberration criteria for ranking different two‐level designs, particularly Plackett‐Burman and other nonregular factorial designs. An advantage of generalized resolution, extended here to work on three‐level designs, is that it offers a useful criterion for ranking three‐level screening designs, whereas the Deng and Tang resolution is used mainly for the assessment of two‐level designs. In addition, we applied a projection estimation capacity (PEC) criterion to select three‐level screening designs with desirable properties. Practical examples and the best projections of the designs are presented in tables.

Quadrupling: construction of uniform designs with large run sizes

Fractional factorial designs are widely used because of their various merits. Foldover or level permutation are usually used to construct optimal fractional factorial designs. In this paper, a novel method via foldover and level permutation, called quadrupling, is proposed to construct uniform four-level designs with large run sizes. The relationship of uniformity between the initial design and the design obtained by quadrupling is investigated, and new lower bounds of wrap-around $$L_2$$-discrepancy for such designs are obtained. These results provide a theoretical basis for constructing uniform four-level designs with large run sizes by quadrupling successively. Furthermore, the analytic connection between the initial design and the design obtained by quadrupling is presented under generalized minimum aberration criterion.

On the aberrations of mixed level orthogonal arrays with removed runs

Given an orthogonal array we analyze the aberrations of the sub-fractions which are obtained by the deletion of some of its points. We provide formulae to compute the Generalized Word-Length Pattern of any sub-fraction. In the case of the deletion of one single point, we provide a simple methodology to find which the best sub-fractions are according to the Generalized Minimum Aberration criterion. We also study the effect of the deletion of 1, 2 or 3 points on some examples. The methodology does not put any restriction on the number of levels of each factor. It follows that any mixed level orthogonal array can be considered.

Minimum aberration criterion for screening experiments at two levels from an entropy-based perspective

ABSTRACTThis paper presents a new criterion for selecting a two-level fractional factorial design. The theoretical underpinning for the criterion is the Shannon entropy. The criterion, which is referred to as the entropy-based minimum aberration criterion, has several advantages. The advantage of the entropy-based criterion over the classical minimum aberration criterion is that it utilizes a measure of uncertainty on the skewness of the distribution of word length patterns in the selection of the “best” design in a family of two-level fractional factorial plans. The criterion evades the trauma associated with the lack of prior knowledge on the important effects.

Minimum aberration designs for discrete choice experiments

A discrete choice experiment (DCE) is a survey method that gives insight into individual preferences for particular attributes. Traditionally, methods for constructing DCEs focus on identifying the individual effect of each attribute (a main effect). However, an interaction effect between two attributes (a two-factor interaction) better represents real-life trade-offs, and provides us a better understanding of subjects’ competing preferences. In practice it is often unknown which two-factor interactions are significant. To address the uncertainty, we propose the use of minimum aberration blocked designs to construct DCEs. Such designs maximize the number of models with estimable two-factor interactions in a DCE with two-level attributes. We further extend the minimum aberration criteria to DCEs with mixed-level attributes and develop some general theoretical results.

A Note on Foldover of 2k − p Designs With Column Permutations

Foldover is a commonly used follow-up strategy in experimental designs. All existing foldover designs were constructed by reversing the sign of columns of the initial design. We propose a new methodology by allowing the permutation of columns in foldover. Focusing on resolution IV designs, we show that almost all designs are better than existing results with respect to the minimum aberration criterion. While augmenting a design by a foldover with column permutations may result in a nonregular combined design, the proposed designs all have a resolution of 4.5 or higher, for which no two-factor interaction is fully aliased with any other two-factor interactions.

Generalized minimum aberration mixed-level orthogonal arrays: A general approach based on sequential integer quadratically constrained quadratic programming

ABSTRACTOrthogonal fractional factorial designs and in particular orthogonal arrays (OAs) are frequently used in many fields of application, including medicine, engineering, and agriculture. In this article, we present a methodology and an algorithm to find an OA, of given size and strength, which satisfies the generalized minimum aberration criterion. The methodology is based on the joint use of polynomial counting functions, complex coding of levels, and algorithms for quadratic optimization and puts no restriction on the number of levels of each factor.

Minimum Lee-moment aberration and its applications

Various criteria have been proposed to compare and assess fractional factorial designs. Among them the minimum moment aberration criterion is conceptually simple and computationally cheap. However, the power moments in the minimum moment aberration criterion depend on the Hamming distances between two runs, which leads to minimum moment aberration criterion having a few defects. In this paper, we use Lee distance instead of Hamming distance to de ne power moments, and propose the minimum Lee-moment aberration criterion to compare and assess multi-level fractional factorial designs. Some relationships between minimum Lee-moment aberration criterion and other criteria are also studied, which provides a signi cant statistical justi cation of minimum Lee-moment aberration criterion.

Minimum Aberration Regular Two-Level Designs in the Presence of Dispersion Factors

In practice, the variance of the response variable may change as some specific factors change from one setting to another in a factorial experiment. These factors affecting the variation of the response are called dispersion factors, which can violate the usual assumption of variance homogeneity. In this study, we modify the conventional minimum aberration criterion to take the impact of dispersion factors into account. The situations of one or two dispersion factors are investigated. As a result, we present regular 2n − p designs with run sizes equal to 16 and 32 using the modified minimum aberration criterion.

Some properties of β-wordlength pattern for four-level designs

Fractional factorial designs have played a prominent role in the theory and practice of experimental design. For designs with qualitative factors under an ANOVA model, the minimum aberration criterion has been frequently used; however, for designs with quantitative factors, a polynomial regression model is often established, thus the β-wordlength pattern can be employed to compare different fractional factorial designs. Although the β-wordlength pattern was introduced in 2004, its properties have not been investigated extensively. In this paper, we will present some properties of β-wordlength pattern for four-level designs. These properties can help find better designs with quantitative factors.

Optimality of pairwise blocked definitive screening designs

Definitive screening designs are a new class of three-level designs which are shown superior to the classical central composite designs in response surface methodology. They can be constructed by inserting conference matrices into their fold-over structures. How to block definitive screening designs in an optimal way is of practical importance, and lacks systematic theoretical research up to now. Pairwise blocking schemes are usually adopted which assign each pair of fold-over runs into the same block. In this paper, the optimality of such pairwise blocking schemes is thoroughly studied in theory. It is shown that under the linear model consisting of main effects, quadratic effects and block effects, pairwise blocked definitive screening designs are universally optimal for the main effects among all balanced blocking schemes. Moreover, such blocked designs are proved to have the same generalized wordlength pattern and are also shown optimal under the generalized minimum aberration criterion.