Designing fractional factorial split-plot experiments with few whole-plot factors

Summary.  When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.     

Partial duplication in two-level fractional factorial designs

Most fractional factorial designs have no replicated points and thus do not provide an estimate for pure error. The construction methods for orthogonal main-effect plan in the literature usually do not produce designs with duplicate points. However, it is possible to combine four fractions to provide a set of duplicate points without sacrificing the orthogonality of main effects. This paper proposes two techniques of this idea to produce designs with replicate points in two-level fractional factorial designs.     

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.     

Minimum aberration and model robustness for two-level fractional factorial designs

The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution III or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

2 m 4 n designs with resolution III or IV containing clear two-factor interaction components

The orthogonal arrays with mixed levels have become widely used in fractional factorial designs. It is highly desirable to know when such designs with resolution III or IV have clear two-factor interaction components (2fic’s). In this paper, we give a complete classification of the existence of clear 2fic’s in regular 2 ^m 4 ⁿ designs with resolution III or IV. The necessary and sufficient conditions for a 2 ^m 4 ⁿ design to have clear 2fic’s are given. Also, 2 ^m 4 ⁿ designs of 32 runs with the most clear 2fic’s are given for n = 1,2.      

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

The minimum aberration criterion has been advocated for ranking foldovers of 2^k−p$2^{k-p}$ fractional factorial designs (Li and Lin, 2003); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.     

Construction of some mixed two- and four-level regular designs with GMC criterion

General minimum lower-order confounding (GMC) criterion is to choose optimal designs, which are based on the aliased effect-number pattern (AENP). The AENP and GMC criterion have been developed to form GMC theory. Zhang et al. (2015 Zhang, T.F., Yang, J.F., Li, Z.M., Zhang, R.C. (2015). Construction of regular 2ⁿ4¹ designs with general minimum lower-order confounding. Commun. Stat. - Theory Methods 46:2724–2735.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduced GMC 2ⁿ4^m criterion for choosing optimal designs and constructed all GMC 2ⁿ4¹ designs with N/4 + 1 ? n + 2 ? 5N/16. In this article, we analyze the properties of 2ⁿ4¹ designs and construct GMC 2ⁿ4¹ designs with 5N/16 + 1 ? n + 2 < N ? 1, where n and N are, respectively, the numbers of two-level factors and runs. Further, GMC 2ⁿ4¹ designs with 16-run, 32-run are tabulated.     

Robust two-level regular fractional factorial designs

Abstract

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.     

Optimal fractional factorial designs and their construction

Optimal symmetrical fractional factorial designs with n$n$ runs and m$m$ factors of s$s$ levels each are constructed. We consider only designs such that no two factors are aliases. The minimum moment aberration criterion proposed by Xu (2003) is used to judge the optimality of the designs. The minimum moment aberration criterion is equivalent to the popular generalized minimum aberration criterion proposed by Xu and Wu (2001), but the minimum moment criterion is simpler to formulate and employ computationally. Some optimal designs are constructed by using generalized Hadamard matrices.     

2(n1+n2)-(k1+k2) Fractional factorial split-plot designs containing clear effects

Whole-plot (WP) factors and sub-plot (SP) factors play different roles in fractional factorial split-plot (FFSP) designs. In this paper, we consider FFSP designs with resolution III or IV from the viewpoint of clear factorial effects, classify two-factor interactions (2FIs) into three types, and give sufficient and necessary conditions for the existence of FFSP designs containing various clear factorial effects, including two types of main effects and three types of 2FIs. The structures of these designs are also shown and illustrated with examples.     

On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada     

On the problem of augmented fractional factorial designs

Let D be a saturated fractional factorial design of the general K₁ x K₂ ...x K_t factorial such that it consists of m distinct treatment combinations and it is capable of providing an unbiased estimator of a subvector of m factorial parameters under the assumption that the remaining k-m,t (k = H it ) factorial parameters are negligible. Such a design will not provide an unbiased estimator of the varianceσ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatmentcombinations with the aim to estimate ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatment combinations with the aim to estimate σ² unbiasedly. The problem then is how to select the c treatment combinations such that the augmented design D retains its optimality property. This problem, in all its generality is extremely complex. The objective of this paper is to provide some insight in the problem by providing a partial answer in the case of the 2^tfactorial, using the d-optimality criterion.