Mixed two- and four-level fractional factorial split-plot designs with clear effects

Mixed-level designs have become widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. It is highly to know when a mixed-level FFSP design with resolution III or IV has clear effects. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and four-level factors to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.     

Some Results on Fractional Factorial Split-Plot Designs with Multi-Level Factors

Fractional factorial split-plot (FFSP) designs have received much attention in recent years. In this article, the matrix representation for FFSP designs with multi-level factors is first developed, which is an extension of the one proposed by Bingham and Sitter (1999b Bingham , D. , Sitter , R. R. ( 1999b ). Some theoretical results for fractional factorial split-plot designs . Ann. Statist. 27 : 1240 – 1255 . [Google Scholar]) for the two-level case. Based on this representation, periodicity results of maximum resolution and minimum aberration for such designs are derived. Differences between FFSP designs with multi-level factors and those with two-level factors are highlighted.     

Experimentation order with good properties for 2 k factorial designs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders achieve the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a large number of changes in factor levels and thus make experimentation expensive and difficult. De Leon et al. [Experimentation order in factorial designs with 8 or 16 runs, J. Appl. Stat. 32 (2005), pp. 297–313] proposed experimentation orders for designs with eight or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels. This article presents a new methodology to obtain experimentation orders with the desired properties for designs with any number of runs.     

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.      

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.     

Selection of cost-optimal 2k−p fractional factorials

Selecting an optimal 2^k?pfractional factorial is structured as a mathematical programming problem. An algorithm is defined for the solution, and the case of additive costs is shown to have a known solution for resolution III designs.     

Optimal design results for 2n factorial paired comparison experiments

Four general classes of partially balanced designs for 2ⁿ factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ² of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H₀ is derived under defined local alternatives. Optimal designs d₁ maximizing λ² in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.     

Fractional Factorial Designs for the Detection of Interactions between Design and Noise Factors

In industrial experiments on both design (control) factors and noise factors aimed at improving the quality of manufactured products, designs are needed which afford independent estimation of all design×noise interactions in as few runs as possible, while allowing aliasing between those factorial effects of less interest. An algorithm for generating orthogonal fractional factorial designs of this type is described for factors at two levels. The generated designs are appropriate for experimenting on individual factors or for experimentation involving group screening of factors.     

(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.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.     

A series of single array 2m factorial search designs for even m

By means of a search design one is able to search for and estimate a small set of non‐zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non‐negligible effect in the set of 2‐ and 3‐factor interactions, assuming 4‐ and higher‐order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2^m factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number m of factors. These designs are generally determined by two arrays, one specifying a main effect plan and the other specifying a follow‐up. In this paper we develop the construction of search designs for an even number of factors m, m≠6. The new series of MEP.1 plans is a set of single array designs with a well structured form. Such a structure allows for flexibility in arriving at an appropriate design with optimum properties for search and estimation.     

Analysis of variance of balanced fractional 2n factorial designs of resolution 2l+1

By use of the algebraic structure of the triangular multidimensional partially balanced association scheme, we present the analysis of variance and the hypotheses testing of a balanced fractional 2ⁿfactorial design of resolution 2l+1, which is derived from a balanced array of strength 2l.     

Robust 2k factorial design with Weibull error distributions

It is well known that the least squares method is optimal only if the error distributions are normally distributed. However, in practice, non-normal distributions are more prevalent. If the error terms have a non-normal distribution, then the efficiency of least squares estimates and tests is very low. In this paper, we consider the 2^k factorial design when the distribution of error terms are Weibull W(p,σ). From the methodology of modified likelihood, we develop robust and efficient estimators for the parameters in 2^k factorial design. F statistics based on modified maximum likelihood estimators (MMLE) for testing the main effects and interaction are defined. They are shown to have high powers and better robustness properties as compared to the normal theory solutions. A real data set is analysed.     

Bounds on the maximum number of clear two‐factor interactions for 2m‐p designs of resolution III and IV

The authors derive upper and lower bounds on the maximum number of clear two‐factor interactions in 2^m?p fractional factorial designs of resolution III and IV. A two‐factor interaction is said to be clear if it is not aliased with any main effect or with any other two‐factor interaction. The lower bounds are obtained by exhibiting specific designs. By comparing the bounds with the values of the maximum number of clear two‐factor interactions in cases where it is known, one concludes that the construction methods perform quite well.     

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     

Switching-one-column follow-up experiments for Plackett-Burman designs

Industrial experiments are frequently performed sequentially using two-level fractional factorial designs. In this context, a common strategy for the design of follow-up experiments is to switch the signs in one column. It is well known that this strategy, when applied to two-level fractional factorial resolution III designs, will clear the main effect, for which the switch was performed, from any confounding with any other two-factor interactions and will also clear all the two-factor interactions between that factor and the other main effects from any confounding with other two-factor interactions. In this article, we extend this result and show that this strategy applies to any orthogonal two-level resolution III design and therefore specifically to any two-level Plackett- Burman design .     

On some new search designs for 2m factorial experiments

In this paper, we obtain search designs with reasonably small number of treatments which permit the estimation of the general mean and main effects and search of one more unknown possible nonzero effect among two and three factor interactions in 2^m factorial experiments, 3 ? m ? 8.     

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3^n-k fractional factorial designs.     

Optimal 2n-p fractional factorial designs for dispersion effects under a location-diapersion model

We consider main effects models for 2-level experiments that also include. Parameters characterizing potential dispersion effects due to specified factors. One special case is considered. In this case only a single specified factor is responsible for the dispersion effects. We determine the connection between alias relations and Optimality of a design for estimation of dispersion effects in the class of regu!ar fractional Y - P factorial designs of resolution III or higher. This rmectioil heips US identify those designs that are A-optimal for estimating dispersion effects by a suitable choice of defining contrasts. in particuiar, we show that an increase in efficiency with respect to dispersion effects is accompanied by a loss iii efficiency for estimating the location effects. In practice, one mmt thcrcfcre accept a trade& between the efficiencies associated with estirnates of location effects and dispersion effects.