Regular Fractional Factorial Designs Research Articles

Fast construction of efficient two-level parallel flats designs

Two-level parallel flats designs (PFDs) are nonregular designs that retain some of the simplicity of regular fractional factorial designs. PFDs have many desirable properties, such as flexible run sizes and block diagonal information matrices. In this article, we develop a fast algorithm for constructing efficient two-level PFDs, optimizing the parallel flats structure via a novel application of coordinate exchange. The resulting designs fill gaps among existing nonregular designs and outperform many comparably-sized designs in terms of G- and G2-aberration. An efficient implementation of the proposed algorithm in Matlab and GNU Octave can be found in the online Supplementary Material. Several new designs with desirable properties are generated and tabulated.

Optimal two-level regular fractional factorial split-plot designs when the effects of subplot factors are more important

Optimal two-level regular fractional factorial split-plot designs when the effects of subplot factors are more important

Enumeration of regular fractional factorial designs with four-level and two-level factors

Abstract Designs for screening experiments usually include factors with two levels only. Adding a few four-level factors allows for the inclusion of multi-level categorical factors or quantitative factors with possible quadratic or third-order effects. Three examples motivated us to generate a large catalogue of designs with two-level factors as well as four-level factors. To create the catalogue, we considered three methods. In the first method, we select designs using a search table, and in the second method, we use a procedure that selects candidate designs based on the properties of their projections into fewer factors. The third method is actually a benchmark method, in which we use a general orthogonal array enumeration algorithm. We compare the efficiencies of the new methods for generating complete sets of nonisomorphic designs. Finally, we use the most efficient method to generate a catalogue of designs with up to three four-level factors and up to 20 two-level factors for run sizes 16, 32, 64, and 128. In some cases, a complete enumeration was infeasible. For these cases, we used a bounded enumeration strategy instead. We demonstrate the usefulness of the catalogue by revisiting the motivating examples.

A class of space‐filling designs with low‐dimensional stratification and column orthogonality

AbstractStrong orthogonal arrays are suitable designs for computer experiments because of stratification in low‐dimensional projections. However, strong orthogonal arrays may be very expensive for a moderate number of factors. In this article, we develop a method for constructing space‐filling designs with more economical run sizes. These designs are not only column‐orthogonal but also enjoy a large proportion of low‐dimensional stratification properties that strong orthogonal arrays ought to have. Moreover, a class of proposed designs can be 3‐orthogonal. In addition, some theoretical results on regular fractional factorial designs are obtained as a by‐product.

Construction of Optimal Split-Plot Designs for Various Design Scenarios

When performing fractional factorial experiments in a completely random order is impractical, fractional factorial split-plot designs are suitable options as an alternative. It is well recognized that the more there are lower order effects of interest at lower order confounding, the better the designs. From this viewpoint, this paper considers the construction of optimal regular two-level fractional factorial split-plot designs. The optimality criteria for two different design scenarios are proposed. Under the newly proposed optimality criteria, the theoretical construction methods of optimal regular two-level fractional factorial split-plot designs are then proposed. In addition, we also explore the theoretical construction methods of some optimal regular two-level fractional factorial split-plot designs under the widely adopted general minimum lower order confounding criterion.

Utilizing individual clear effects for intelligent factor allocations and design selections

Extensive studies have been conducted on how to select efficient designs with respect to a criterion. Most design criteria aim to capture the overall efficiency of the design across all columns. When prior information indicated that a small number of factors and their two-factor interactions (2fi’s) are likely to be more significant than other effects, commonly used minimum aberration designs may no longer be the best choice. Motivated by a real-life experiment, we propose a new class of regular fractional factorial designs that focus on estimating a subset of columns and their corresponding 2fi’s clear of other important effects. After introducing the concept of individual clear effects (iCE) to describe clear 2fi’s involving a specific factor, we define the clear effect pattern criterion to characterize the distribution of iCE’s over all columns. We then obtain a new class of designs that sequentially maximize the clear effect pattern. These newly constructed designs are often different from existing optimal designs. We develop a series of theoretical results that can be particularly useful for constructing designs with large run sizes, for which algorithmic construction becomes computationally challenging. We also provide some practical guidelines on how to choose appropriate designs with respect to different run size, the number of factors, and the number of 2fi’s that need to be clear.

Partially orthogonal blocked three-level response surface designs

When fitting second-order response surface models in a hypercuboidal region of experimentation, the variance matrices of D-optimal continuous designs have a particularly attractive structure, as do many regular unblocked exact designs. Methods for constructing blocked exact designs which preserve this structure and are orthogonal, or nearly orthogonal, are developed. Partially orthogonal designs are built using a small irregular fraction of a two- or three-level design and a regular fractional factorial design as building blocks. Results are derived which relate the properties of the blocked design to these components. Moreover, it is shown how the designs can be augmented to ensure that the model can be fitted and a method for constructing designs with small blocks is presented. Examples illustrate that partially orthogonal designs can compete with more traditional designs in terms of both efficiency and overall size of the experiment.

Constructing 2-level foldover designs with minimal aliasing

This paper introduces an algorithmic approach to constructing 2-level foldover designs using the minimum G2-aberration criterion (Tang & Deng, 1999). Our algorithms can find designs without fully aliased 2-factor interactions (2FIs). As a result, follow-up runs which are used to disentangle the 2FI from another might become unnecessary. The constructed designs are compared with the efficient 2-level foldover designs of Errore et al. (2017), the 2-level designs of strength 3 for 32, 40 and 48 runs of Schoen & Mee (2012), and some regular fractional factorial designs for up to 64 runs. A catalogue of minimum G2-aberration 2-level foldover designs for up to 32 factors is given.

An algorithm for blocking regular fractional factorial 2-level designs with clear two-factor interactions

Regular fractional factorial designs with 2-level factors are among the most frequently used experimental plans. In many cases, designs should be blocked for dealing with inhomogeneity of experimental units. At the same time, the research question at hand may imply a focus on specified sets of two-factor interactions, while it is not justified to assume negligibility of other low order effects. An algorithm is provided for blocking a regular fraction into – possibly small – blocks while keeping specified two-factor interactions clear from confounding with main effects or other two-factor interactions. The proposed algorithm is implemented in the R package FrF2 and combines an estimability algorithm by the author with an automated implementation of a recent proposal for blocking fractions by hand.

Direct construction of globally D‐optimal designs for factors at two levels and main effects models

AbstractIn developing screening experiments for two‐level factors, practitioners typically are familiar with regular fractional factorial designs, which are orthogonal, globally D‐optimal (ie, 100% D‐efficient), and exist if is a power of two. In addition, nonregular D‐optimal orthogonal designs can be generated for almost any a multiple of four, the most notable being the family of Plackett and Burman1 designs. If resource constraints dictate that is not a multiple of four, while an orthogonal design for two‐level factors does not exist, one can still consider a D‐optimal design. Exchange algorithms are available in commercial computer software for creating highly D‐efficient designs. However, as the number of factors increases, computer searches eventually fail to find the globally optimal design for any or require impractical search times. In this article, we compile state‐of‐the‐art direct construction methods from the literature for producing globally D‐optimal designs for virtually any number of two‐level factors as well as any greater than the number of factors. We summarize the known methods as well as areas for continued research, with the intention of catalyzing research in extending these construction methods.

Integer programming approaches to find row–column arrangements of two-level orthogonal experimental designs

Design of experiments is an effective, generic methodology for problem solving as well as for improving or optimizing product design and manufacturing processes. The most commonly used experimental designs are two-level fractional factorial designs. In recent years, nonregular fractional factorial two-level experimental designs have gained much popularity compared to the traditional regular fractional factorial designs, because they offer more flexibility in terms of run size as well as the possibility to estimate partially aliased effects. For this reason, there is much interest in finding good nonregular designs, and in orthogonal blocking arrangements of these designs. In this contribution, we address the problem of finding orthogonal blocking arrangements of high-quality nonregular two-level designs in scenarios with two crossed blocking factors. We call these blocking arrangements orthogonal row-column arrangements. We propose two strategies to find row-column arrangements of given two-level orthogonal treatment designs such that the treatment factors’ main effects are orthogonal to both blocking factors. The first strategy involves a sequential approach which is especially useful when one blocking factor is more important than the other. The second strategy involves a simultaneous approach for situations where both blocking factors are equally important. For the latter approach, we propose three different optimization models, so that, in total, we consider four different methods to obtain row-column arrangements. We compare the performance of the four methods by looking for good row-column arrangements of the best two-level 24-run orthogonal designs in terms of the G-aberration criterion, and apply the best of these methods to 64- and 72-run orthogonal designs.

An Evaluation of Estimation Capacity Under the Conditional Main Effect Parameterization

The conditional main effect (CME) parameterization system can enable more informative analyses of regular two-level fractional factorial designs compared to the traditional orthogonal components system. However, formal evaluations of estimation capacities for designs under CME models have yet to be performed. We establish a necessary and sufficient condition for a model consisting of all of the main effects and a selection of CMEs to be estimable in a regular two-level design of resolution at least III. Our condition illuminates the implications of the maximum estimation capacity criterion for analyses of such traditional and conditional effects. A novel aspect of our evaluations is the direct derivation of D-efficiencies for regular designs of resolution at least III with respect to CME models in which the selected CMEs are not siblings or family members, and their corresponding two-factor interactions are not completely aliased with a main effect.

Row-column arrangements of regular and nonregular fractional factorial two-level designs

Row–column designs involve two crossed blocking factors. We propose three combinatorial criteria and a criterion based on D-efficiency to identify optimal row–column arrangements of fractional factorial two-level designs. We search for such arrangements in complete catalogs of 16-run and 24-run orthogonal arrays of strength 2 with two-level treatment factors and two multi-level blocking factors. One of the combinatorial criteria is a good surrogate for the efficiency-based criterion. For searching similar catalogs of strength-3 orthogonal arrays, we adapt the criteria and conduct a complete search for optimal row–column arrangements in catalogs with 64 and 72 runs.

Robust two-level regular fractional factorial designs

In this paper, we introduce the concept of model quality for two-level regular fractional factorial designs. Under the effect hierarchy principle, this paper raises the definition of model quality and introduces robust model-number pattern (RP) to choose the optimal robust design. Some theoretical results on this optimality and comparisons with GMC and MEC criterion are given.

Signal aliasing in Gaussian random fields for experiments with qualitative factors

Signal aliasing is an inevitable consequence of using fractional factorial designs. Unlike linear models with fixed factorial effects, for Gaussian random field models advocated in some Bayesian design and computer experiment literature, the issue of signal aliasing has not received comparable attention. In the present article, this issue is tackled for experiments with qualitative factors. The signals in a Gaussian random field can be characterized by the random effects identified from the covariance function. The aliasing severity of the signals is determined by two key elements: (i) the aliasing pattern, which depends only on the chosen design, and (ii) the effect priority, which is related to the variances of the random effects and depends on the model parameters. We first apply this framework to study the signal-aliasing problem for regular fractional factorial designs. For general factorial designs including nonregular ones, we propose an aliasing severity index to quantify the severity of signal aliasing. We also observe that the aliasing severity index is highly correlated with the prediction variance.

De-aliasing in two-level factorial designs: A Bayesian approach

Given limited resources for conducting follow-up trials, the inability to separate aliased factorial effects hinders the ubiquitous practicality of regular fractional factorial designs in the analysis of experiments. Wu (2015) proposed a frequentist remedy for “de-aliasing” aliased effects by using conditional main effects. Although Su and Wu (2017) systematized the remedy to make it easy to implement, it might miss truly active effects. Missing active effects can be a severe drawback if the purpose of experimentation is to determine the mechanism of a process rather than to make predictions. In this paper, we propose a Bayesian remedy for de-aliasing in two-level regular factorial designs. Through numerical studies, we show that our method can yield desirable model fittings and reliable de-aliasing results.

A Bayesian approach to the selection of two-level multi-stratum factorial designs

In a multi-stratum factorial experiment, there are multiple error terms (strata) with different variances that arise from complicated structures of the experimental units. For unstructured experimental units, minimum aberration is a popular criterion for choosing regular fractional factorial designs. One difficulty in extending this criterion to multi-stratum factorial designs is that the formulation of a word length pattern based on which minimum aberration is defined requires an order of desirability among the relevant words, but a natural order is often lacking. Furthermore, a criterion based only on word length patterns does not account for the different stratum variances. Mitchell, Morris and Ylvisaker [Statist. Sinica 5 (1995) 559–573] proposed a framework for Bayesian factorial designs. A Gaussian process is used as the prior for the treatment effects, from which a prior distribution of the factorial effects is induced. This approach is applied to study optimal and efficient multi-stratum factorial designs. Good surrogates for the Bayesian criteria that can be related to word length and generalized word length patterns for regular and nonregular designs, respectively, are derived. A tool is developed for eliminating inferior designs and reducing the designs that need to be considered without requiring any knowledge of stratum variances. Numerical examples are used to illustrate the theory in several settings.

A note on extending the alias length pattern

The alias length pattern (ALP) was proposed to characterize the aliasing among two-factor interactions for regular two-level fractional factorial designs of resolution IV. Experimenters often pay more attention to joint estimation of main effects and some two-factor interactions. Hence, resolution III and IV designs are commonly used in practice. In this article, we first extend the ALP for regular two-level fractional factorial designs of resolution III and IV by modeling justification. Then we study the properties of the ALP and provide the relationships between the ALP and other popular criteria.

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.

A note on constructing clear compromise plans

The regular two-level fractional factorial designs of n factors and N runs, having resolution IV and allowing experimenters to clearly estimate all main effects and a set of required two-factor interactions (2fi’s), are called clear compromise plans. Four classes of clear compromise plans have been discussed in the literature. The general minimum lower order confounding (GMC) is an elaborate criterion, which was proposed to select optimal fractional factorial designs. This paper gives a theory on constructing a set of class three clear compromise plans with 5N∕32+3≤n≤N∕4+1, called partially general minimum lower order confounding (P-GMC) designs. We first prove that each P-GMC design is constructed by a GMC design and two specified columns. Then we study the properties of these designs. For N=32,64 and 128, we illustrate that the P-GMC designs are admissible designs. Furthermore, they all have GMC, except for the P-GMC 213−7 and 223−16 designs.