A Lower Bound for the Wrap-around L2-discrepancy on Combined Designs of Mixed Two- and Three-level Factorials

The objective of this article is to study the issue of employing the uniformity criterion measured by wrap-around L₂-discrepancy to assess the optimal foldover plans. For mixed two- and three-level fractional factorials as the original designs, general foldover plan and combined design under a foldover plan are defined, and the equivalence between any foldover plan and its complementary foldover plan is investigated. A lower bound of wrap-around L₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Some new lower bounds to various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this article, some new lower bounds of various discrepancies of combined designs, such as the centered, symmetric, and wrap-around L₂-discrepancies, are obtained, which can be used as a better benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Reverse Foldovers for Blocked 2 m−k Fractional Factorial Designs

Two-level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. When the experimental material to be used in the experiment is heterogenous or the experiment must be performed over several well-defined time periods, blocking is often used as a means to improve experimental efficiency by removing the possible effects of heterogenous experimental material or possible time period effects. In a recent article, Li and Jacroux (2007 Li , F. , Jacroux , M. (2007). Optimal foldover plans for blocked 2 ^m?k fractional factorial designs. J. Statsist. Plann. Infer 137:2434–2452. [Google Scholar]) suggested a strategy for constructing optimal follow-up designs for blocked fractional factorial designs using the well-known foldover technique in conjunction with several optimality criteria. In this article, we consider the reverse foldover problem for blocked fractional factorial designs. In particular, given a 2^(m+p)?(p+k) blocked fractional factorial design D, we derive simple sufficient conditions which can be used to determine if there exists a 2^{(m+p?1)?(p?1+k+1)} initial fractional factorial design d which yields D as a foldover combined design as well how to generate all such d. Such information is useful in developing an overall experimental strategy in situations where an experimenter wants an overall blocked fractional factorial design with “desirable” properties but also wants the option of analyzing the observed data at the halfway mark to determine if the significant experimental variables are obvious (and the experiment can be terminated) or if a different path of experimentation should be taken from that initially planned.     

Optimal blocking and foldover plans for nonregular two-level designs

This paper discusses the issue of choosing optimal designs when both blocking and foldover techniques are simultaneously employed to nonregular two-level fractional factorial designs. By using the indicator function, the treatment and block generalized wordlength patterns of the combined blocked design under a general foldover plan are defined. Some general properties of combined block designs are also obtained. Our results extend the findings of Ai et al. (2010) from regular designs to nonregular designs. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of the generalized aberration criterion for nonregular initial design with 12, 16 and 20 runs is tabulated, respectively.     

Lower bounds of the wrap-around -discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L₂-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L₂-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L₂-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.     

A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs

Fractional factorial (FF) designs are no doubt the most widely used designs in experimental investigations due to their efficient use of experimental runs. One price we pay for using FF designs is, clearly, our inability to obtain estimates of some important effects (main effects or second order interactions) that are separate from estimates of other effects (usually higher order interactions). When the estimate of an effect also includes the influence of one or more other effects the effects are said to be aliased. Folding over an FF design is a method for breaking the links between aliased effects in a design. The question is, how do we define the foldover structure for asymmetric FF designs, whether regular or nonregular? How do we choose the optimal foldover plan? How do we use optimal foldover plans to construct combined designs which have better capability of estimating lower order effects? The main objective of the present paper is to provide answers to these questions. Using the new results in this paper as benchmarks, we can implement a powerful and efficient algorithm for finding optimal foldover plans which can be used to break links between aliased effects.     

Lower bound of average centered L2-discrepancy for U-type designs

Uniform designs are widely used in various scientific investigations and industrial applications. By considering all possible level permutation of the factors, a connection between average centered L₂-discrepancy and generalized wordlength pattern for asymmetrical fractional factorial designs is derived. Moreover, we present new lower bounds to the average centered L₂-discrepancy for symmetrical and asymmetrical U-type designs. For illustration of the theoretical results, the lower bounds for symmetrical and asymmetrical U-type designs are tabulated, and numerical results indicate that our lower bounds behave well and can be recommended for use in practice.     

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.     

An effective construction method for multi-level uniform designs

Uniform designs are widely used in various applications. However, it is computationally intractable to construct uniform designs, even for moderate number of runs, factors and levels. We establish a linear relationship between average squared centered L₂-discrepancy and generalized wordlength pattern, and then based on it, we propose a general method for constructing uniform designs with arbitrary number of levels. The main idea is to choose a generalized minimum aberration design and then permute its levels. We propose a novel stochastic algorithm and obtain many new uniform designs that have smaller centered L₂-discrepancies than the existing ones.     

Foldover Conference Designs for Screening Experiments

A conference matrix is a square matrix C with zeros on the diagonal and ±1s off the diagonal, such that C ^T C = CC ^T  = (n ? 1)I, where I is the identity matrix. Conference matrices are an important class of combinatorial designs due to their many applications in several fields of science, including statistical-experimental designs, telecommunications, elliptic geometry, and more. In this article, conference matrices and their full foldover design are combined together to obtain an alternative method for screening active factors in complicated problems. This method provides a model-independent estimate of the set of active factors and also gives a linearity test for the underlying model.     

Lower Bounds on the Symmetric L 2-Discrepancy and Their Application

The role of uniformity measured by the symmetric L ₂-discrepancy given in Hickernell (1998 Hickernell , F. J. (1998). A generalized discrepancy and quadrature error bound. Math. Computat. 67:299–322.[Crossref], [Web of Science ®] , [Google Scholar]) has been studied in fractional factorial designs. The issue of lower bounds on the symmetric L ₂-discrepancy is crucial in the construction of uniform designs. This article reports some new lower bounds on the symmetric L ₂-discrepancy for symmetric fractional factorials and for a set of asymmetric fractional factorials. It is valuable to use these lower bounds to measure uniformity of given designs.     

NESTING OPTIMAL MAIN EFFECTS PLANS AND OPTIMAL FOLDOVER DESIGNS

ABSTRACT

Optimal main effects plans (MEPs) and optimal foldover designs can often be performed as a series of nested optimal designs. Then, if the experiment cannot be completed due to time or budget constraints, the fraction already performed may still be an optimal design. We show that the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEP for 4t factors in 4t + 2 points nested within it. In general, the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEPs for 4t factors in 4t + 1, 4t + 2, or 4t + 3 points and the optimal MEP for 4t + 1 factors in 4t + 4 points does not contain the optimal MEPs for 4t + 1 factors in 4t + 2 or 4t + 3 points. We also show that the runs in an orthogonal design for 4t factors in 4t + 4 points, and the optimal foldover designs obtained by folding, should be performed in a certain sequence in order to avoid the possibility of a singular X'X matrix.     

A new extension strategy on three-level factorials under wrap-around L2-discrepancy

Sequential experiment is an indispensable strategy and is applied immensely to various fields of science and engineering. In such experiments, it is desirable that a given design should retain the properties as much as possible when few runs are added to it. The designs based on sequential experiment strategy are called extended designs. In this paper, we have studied theoretical properties of such experimental strategies using uniformity measure. We have also derived a lower bound of extended designs under wrap-around L₂-discrepancy measure. Moreover, we have provided an algorithm to construct uniform (or nearly uniform) extended designs. For ease of understanding, some examples are also presented and a lot of sequential strategies for a 27-run original design are tabulated for practice.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

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.     

On the Use of Semi-folding in Regular Blocked Two-level Factorial Designs

In this article, we consider experimental situations where a blocked regular two-level fractional factorial initial design is used. We investigate the use of the semi-fold technique as a follow-up strategy for de-aliasing effects that are confounded in the initial design as well as an alternative method for constructing blocked fractional factorial designs. A construction method is suggested based on the full foldover technique and sufficient conditions are obtained when the semi-fold yields as many estimable effects as the full foldover.     

Uniformity in factorial designs with mixed levels

The uniformity can be utilized as a measure for comparing factorial designs. Fang and Mukerjee (Biometrika 87 (2000) 193–198) and Fang et al. (in: K.T. Fang, F.J. Hickernell, H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002) found links among uniformity in terms of some non-uniformity measures, orthogonality and aberration for regular symmetric factorials. In this paper we extend their results to asymmetric factorials by considering a so-called wrap-around L₂-discrepancy to evaluate the uniformity of factorials. Furthermore, a lower bound of wrap-around L₂-discrepancy is obtained for asymmetric factorials and two new ways of construction of factorial designs with mixed levels are proposed.     

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.