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.     

Two-level minimum aberration designs in N =2 (mod 4) runs

For two-level factorials, we consider designs in N=2 (mod 4) runs as obtained by adding two runs, with a certain coincidence pattern, to an orthogonal array of strength two. These designs are known to be optimal main effect plans in a very broad sense in the absence of interactions. Among them, we explore the ones having minimum aberration, with a view to ensuring maximum model robustness even when interactions are possibly present. This is done by sequentially minimizing a measure of the bias caused by interactions of successively higher orders.     

Minimum aberration blocking of regular mixed factorial designs

It is known by Zhang and Park (J. Statist. Plann. Inference 91 (2000) 107) that there are no minimum aberration (MA) designs with respect to both treatments and blocks for blocked regular mixed-level factorial designs. So it should be compromised between the block wordlength pattern and treatment wordlength pattern. Two methods are considered in this article. The first is MA blocking scheme of an MA design. The other is to combine the components of the two wordlength pattern vectors into one combined wordlength pattern according to the modified hierarchical assumptions and an appropriate ordering of the numbers of alias or confounding relations. The relationship between the two types of optimal blocked designs is investigated. A complete catalogue of optimal blocked regular mixed factorial designs of the above two types with 16 or 32 runs is given.     

(M,S)-optimality in selecting factorial designs

Use of the (M,S) criterion to select and classify factorial designs is proposed and studied. The criterion is easy to deal with computationally and it is independent of the choice of treatment contrasts. It can be applied to two-level designs as well as multi-level symmetrical and asymmetrical designs. An important connection between the (M,S) and minimum aberration criteria is derived for regular fractional factorial designs. Relations between the (M,S) criterion and generalized minimum aberration criteria on nonregular designs are also discussed. The (M,S) criterion is then applied to study the projective properties of some nonregular designs.     

Classification of three-level strength-3 arrays

Generalized aberration (GA) is one of the most frequently used criteria to quantify the suitability of an orthogonal array (OA) to be used as an experimental design. The two main motivations for GA are that it quantifies bias in a main-effects only model and that it is a good surrogate for estimation efficiencies of models with all the main effects and some two-factor interaction components. We demonstrate that these motivations are not appropriate for three-level OAs of strength 3 and we propose a direct classification with other criteria instead. To illustrate, we classified complete series of three-level strength-3 OAs with 27, 54 and 81 runs using the GA criterion, the rank of the matrix with two-factor interaction contrasts, the estimation efficiency of two-factor interactions, the projection estimation capacity, and a new model robustness criterion. For all of the series, we provide a list of admissible designs according to these criteria.     

A simple method to obtain minimum aberration split-plot designs

ABSTRACT

Split-plot designs have been utilized in factorial experiments with some factors applied to larger units and others to smaller units. Such designs with low aberration are preferred when the experimental size and the number of factors considered in both whole plot and subplot are determined. The minimum aberration split-plot designs can be obtained using either computer algorithms or the exhausted search. In this article, we propose a simple, easy-to-operate approach by using two ordered sequences of columns from two orthogonal arrays in obtaining minimum aberration split-plot designs for experiments of sizes 16 and 32.     

Creating catalogues of two-level nonregular fractional factorial designs based on the criteria of generalized aberration

We consider the problem of constructing good two-level nonregular fractional factorial designs. The criteria of minimum G and G₂ aberration are used to rank designs. A general design structure is utilized to provide a solution to this practical, yet challenging, problem. With the help of this design structure, we develop an efficient algorithm for obtaining a collection of good designs based on the aforementioned two criteria. Finally, we present some results for designs of 32 and 40 runs obtained from applying this algorithmic approach.     

Analytic connections between a double design and its original design in terms of different optimality criteria

In recent years, there has been increasing interest in the study of double designs. Various popular optimality criteria have been proposed from different principles for design construction and comparison, such as E(s²), generalized minimum aberration (GMA), minimum moment aberration (MMA), and minimum projection uniformity (MPU). In this article, these criteria are reviewed, and analytic connections between a double design and its original design in terms of these criteria are investigated. These connections are suitable for general original two-level factorial design, whether regular or non regular. In addition, these results provide strong insight into the relationship between double design and original design from different viewpoints.     

Minimum breakdown designs in blocks of size two

In scientific investigations, there are many situations where each two experimental units have to be grouped into a block of size two. For planning such experiments, the variance-based optimality criteria like A-, D- and E-criterion are typically employed to choose efficient designs, if the estimation efficiency of treatment contrasts is primarily concerned. Alternatively, if there are observations which tend to become lost during the experimental period, the robustness criteria against the unavailability of data should be strongly recommended for selecting the planning scheme. In this study, a new criterion, called minimum breakdown criterion, is proposed to quantify the robustness of designs in blocks of size two. Based on the proposed criterion, a new class of robust designs, called minimum breakdown designs, is defined. When various numbers of blocks are missing, the minimum breakdown designs provide the highest probabilities that all the treatment contrasts are estimable. An exhaustive search procedure is proposed to generate such designs. In addition, two classes of uniformly minimum breakdown designs are theoretically verified.     

Alternative Foldover Plans for Two-Level Nonregular Designs

Foldover is a classic technique used to select follow-up experimental runs when an initial experiment yields ambiguities. While foldover has been soundly investigated for regular designs, less research has been devoted to this technique for nonregular designs. Previous work focuses on the use of the generalized minimum aberration criterion to obtain optimal foldover plans. In contrast, this article utilizes the concept of minimal dependent sets (MDSs) and associated criteria to rank foldovers of nonregular designs. We propose an integer programming-based solution to aid in the location and enumeration of MDSs. MDS-optimal foldovers for selected nonregular designs are presented and discussed.     

A Simple Method for Obtaining Minimum Aberration Designs

Minimum aberration designs are preferred in practice, especially when it is desired to carry out a multi-factor experiment using less number of runs. Several authors considered constructions of minimum aberration designs. Some used computer algorithms and some listed good designs from the exhausted search. We propose a simple method to obtain minimum aberration designs for experiments of size less than or equal to thirty-two. Here, we use an ordered sequence of columns from an orthogonal array to design experiments and blocked experiments. When the method is implemented in MS Excel, minimum aberration designs can be easily achieved.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 factors.     

Generalized minimum aberration two-level split-plot designs

Split-plot experiments may arise when it is impractical to completely randomize the treatment combinations of a designed experiment. To provide more flexible design choices in the nonregular split-plot setting, we describe an approach for constructing minimum aberration orthogonal two-level split-plot designs having 12, 16, 20 and 24 runs. We consider five design scenarios that may be of importance to practitioners, and then propose an approach for assigning word lengths under these five scenarios. We then use the extended word length patterns to rank both regular and nonregular orthogonal split-plot designs. While most existing papers concerning orthogonal split-plot designs focus on regular orthogonal designs, we find that many minimum aberration split-plot designs are nonregular orthogonal designs.     

A measure of aliasing and applications to two-level fractional factorials

This paper examines some properties of a measure of aliasing proposed by Hedayat, Raktoe, and Federer (1974). It is shown that in the case of balanced orthogonal designs with no repeated treatments, minimizing the alias measure is equivalent to minimizing tr Yar(ψ). Lower bounds are found for fixed eigenvalues of the design matrix. These results are applied to two-level fractional factorials to show that in certain cases classical fractional-factorial designs yield minimal solutions for the alias measure.     

On minimax designs when there are two candidate models

This work is motivated by the need to find experimental designs which are robust under different model assumptions. We measure robustness by calculating a measure of design efficiency with respect to a design optimality criterion and say that a design is robust if it is reasonably efficient under different model scenarios. We discuss two design criteria and an algorithm which can be used to obtain robust designs. The first criterion employs a Bayesian-type approach by putting a prior or weight on each candidate model and possibly priors on the corresponding model parameters. We define the first criterion as the expected value of the design efficiency over the priors. The second design criterion we study is the minimax design which minimizes the worst value of a design criterion over all candidate models. We establish conditions when these two criteria are equivalent when there are two candidate models. We apply our findings to the area of accelerated life testing and perform sensitivity analysis of designs with respect to priors and misspecification of planning values.     

Optimal split‐plot orthogonal arrays

It is well known that many industrial experiments have split‐plot structures. Compared to completely randomised experiments, split‐plot designs are more economical and thus have received much attention among researchers. Much work has been done for two‐level split‐plot designs. In this article, we consider split‐plot designs with factors of three, more than three, or mixed levels and with both qualitative and quantitative factors. We show that if two designs with both qualitative and quantitative factors are geometrically isomorphic, then their generalised wordlength patterns are identical. Three design scenarios are considered for optimal designs. The corresponding wordlength patterns are defined and the minimum aberration mixed‐level split‐plot designs having 18 and 36 runs are tabulated.     

Robustness of designed experiments against missing data

This paper investigates the robustness of designed experiments for estimating linear functions of a subset of parameters in a general linear model against the loss of any t( ≥1) observations. Necessary and sufficient conditions for robustness of a design under a homoscedastic model are derived. It is shown that a design robust under a homoscedastic model is also robust under a general heteroscedastic model with correlated observations. As a particular case, necessary and sufficient conditions are obtained for the robustness of block designs against the loss of data. Simple sufficient conditions are also provided for the binary block designs to be robust against the loss of data. Some classes of designs, robust up to three missing observations, are identified. A-efficiency of the residual design is evaluated for certain block designs for several patterns of two missing observations. The efficiency of the residual design has also been worked out when all the observations in any two blocks, not necessarily disjoint, are lost. The lower bound to A-efficiency has also been obtained for the loss of t observations. Finally, a general expression is obtained for the efficiency of the residual design when all the observations of m ( ≥1) disjoint blocks are lost.     

The Use of Treatment Concurrences to Assess Robustness of Binary Block Designs Against the Loss of Whole Blocks

Criteria are proposed for assessing the robustness of a binary block design against the loss of whole blocks, based on summing entries of selected upper non‐principal sections of the concurrence matrix. These criteria improve on the minimal concurrence concept that has been used previously and provide new conditions for measuring the robustness status of a design. The robustness properties of two‐associate partially balanced designs are considered and it is shown that two categories of group divisible designs are maximally robust. These results expand a classic result in the literature, obtained by Ghosh, which established maximal robustness for the class of balanced block designs.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.