CONFOUNDING STRUCTURE OF TWO-LEVEL NONREGULAR FACTORIAL DESIGNS

In design theory,the alias structure of regular fractional factorial designs is elegantly described with group theory.However,this approach cannot be applied to nonregular designs directly. For an arbi...     

On optimal two-level nonregular factorial split-plot designs

This article studies two-level nonregular factorial split-plot designs. The concepts of indicator function and aliasing are introduced to study such designs. The minimum G$G$-aberration criterion proposed by Deng and Tang (1999) [4] for two-level nonregular factorial designs is extended to the split-plot case. A method to construct the whole-plot and sub-plot parts is proposed for nonregular designs. Furthermore, the optimal split-plot schemes for 12$12$-, 16$16$-, 20$20$- and 24$24$-run two-level nonregular factorial designs are searched, and many such schemes are tabulated for practical use.     

Matrix Image Method for Ranking Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. In this paper, we study matrix image theory and present a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Based on the matrix image theory, our proposed method captures orthogonality and projection properties. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria.     

Graphical Aids for the Analysis of Two-Level Nonregular Designs

The complex alias pattern between main effects and two-factor interactions for two-level nonregular designs has been considered a problem when analyzing these designs. If only a few two-factor interactions are active, however, the pattern induced into the estimated main effects contrasts from the active interactions may be very structured. This is, in particular, true for the 12-run and the 20-run Plackett–Burman (PB) designs, probably the two most important designs for physical experimentation. This article presents a graphical method for the analysis of nonregular two-level designs. The method consists of two steps. The first step is called contrast plots interpretation and is directed toward revealing the cause for the pattern observed in contrast plots. The second step is called alias reduction and aims at simplifying the interpretation of the contrast plots by reducing the aliasing caused by effects that, with a high degree of certainty, may be considered active. The method is tested on the 12-run PB design both with simulated and real data and on the 20-run PB design for one particular case with real data. Supplementary materials (MINITAB codes for performing calculations and plots) are available online.     

Connections Between the Resolutions of General Two-level Factorial Designs

Indicator functions are new tools for studying two-level fractional factorial designs. This article discusses some properties of indicator functions. Using indicator functions, we study the connection between general two-level factorial designs of generalized resolutions.     

Forms of four-word indicator functions with implications to two-level factorial designs

Indicator functions are new tools to study fractional factorial designs. In this paper, we study indicator functions with four words and provide possible forms of the indicator functions and explain their implications to two-level factorial designs.     

8×4×2~n设计纯净两因子交互作用成分最大数的界

混水平部分因析设计在各类试验中有广泛应用.纯净效应准则是用于选取最优部分因析设计的重要准则之一.本文考虑含有一个八水平因子、一个四水平因子和若干二水平因子的8×4×2~n混水平设计,给出了分辨度为Ⅲ和Ⅳ的该类混水平设计包含纯净两因子交互作用成分最大数的上界和下界.下界通过构造特定设计而得到.     

The Optimality Theories and Construction Methods on Regular Fractional Factorial Designs

The fractional factorial designs are widely used in various experiments. The optimality theories and construction methods of the fractional factorial designs are the core of the investigation on experimental designs. Many researchers have investigated this issue since 1980. This paper gives a summary on the optimality theories and construction methods of the regular fractional factorial designs.     

2(n1+n2)-(k1+k2)42混合水平裂区设计包含纯净效应的条件

混水平部分因析裂区设计在各类试验中有广泛应用. 在三因子及更高阶交互作用可以忽略这一很弱的假设下, 试验者可以得到纯净主效应或者纯净两因子交互作用成分的无偏估计. 本文给出了含有两个四水平因子和若干二水平因子的混水平裂区设计包含各类纯净主效应或者纯净两因子交互作用成分的条件以及构造相应设计的方法.     

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Optimum mechanism for breaking the confounding effects of mixed-level designs

Fractional factorial designs (FFD’s) are no doubt the most widely used designs in the experimental investigations due to their efficient use of experimental runs to study many factors simultaneously. One consequence of using FFD’s is the aliasing of factorial effects. Follow-up experiments may be needed to break the confounding. A simple strategy is to add a foldover of the initial design, the new fraction is called a foldover design. Combining a foldover design with the original design converts a design of resolution r into a combined design of resolution \(r+1\). In this paper, we take the centered \(L_2\)-discrepancy \(({\mathcal {CD}})\) as the optimality measure to construct the optimal combined design and take asymmetrical factorials with mixed two and three levels, which are most commonly used in practice, as the original designs. New and efficient analytical expressions based on the row distance of the \({\mathcal {CD}}\) for combined designs are obtained. Based on these new formulations, we present new and efficient lower bounds of the \({\mathcal {CD}}\). Using the new formulations and lower bounds as the benchmarks, we may implement a new algorithm for constructing optimal mixed-level combined designs. By this search heuristic, we may obtain mixed-level combined designs with low discrepancy.     

Classification of Three-word Indicator Functions of Two-level Factorial Designs

Indicator functions have been in the literature for several years, and yet only a few of their properties have been examined. In this paper, we study some properties of indicator functions of two-level fractional factorial designs. For example, we show that there is no indicator function with only two words, and also classify all indicator functions with only three words. The results imply that there is no valuable non-regular design with only three or less words in its indicator function.     

Detailed wordlength pattern of regular fractional factorial split-plot designs in terms of complementary sets

With reference to regular fractional factorial split-plot designs, we consider a detailed wordlength pattern taking due cognizance of the distinction between the whole-plot and sub-plot factors. A generalized version of the MacWilliams’ identity is employed to express the detailed wordlength pattern in terms of complementary sets. Several special features make this result intrinsically different from the corresponding one in classical fractional factorial designs where all factors have the same status. An application to robust parameter designs is indicated and examples given.     

Some optimal criteria of model-robustness for two-level non-regular fractional factorial designs

We present some optimal criteria to evaluate model-robustness of non-regular two-level fractional factorial designs. Our method is based on minimizing the sum of squares of all the off-diagonal elements in the information matrix, and considering expectation under appropriate distribution functions for unknown contamination of the interaction effects. By considering uniform distributions on the symmetric support, our criteria can be expressed as linear combinations of B _s (d) characteristic, which is used to characterize the generalized minimum aberration. We give some empirical studies for 12-run non-regular designs to evaluate our method.     

Two-level fractional factorial designs in blocks of size two for the orthogonal estimation of all main effects and two-factor interactions

In this paper we consider the problem of constructing two-level fractional factorial designs in blocks of size two that allow for the orthogonal estimation of all main effects and two-factor interactions (after adjusting for blocks). This problem has been considered in the literature, e.g., see Yang and Draper (2003), Wang (2004) and Kerr (2006). In this paper we give two systematic methods for the construction of such designs. The first construction method gives in many situations designs requiring fewer runs than those designs previously given whereas the second method gives a systematic method for constructing designs analogous to those illustrated in Yang and Draper (2003) by example.     

单次重复试验的三水平析因设计的分析

由于试验材料、费用和时间等条件的限制,仅有单次重复试验的三水平析因设计经常要应用在农业、工业和医学临床试验等领域。例如,在医学临床试验中,为找到影响治疗关节炎效果的重要因子和最佳治疗方案需要考虑2个三水平的因子:A(药物治疗)和B(运动治疗),由于只能找到9位病情相似的病人进行试验,故只能实施仅有单次重复试验的三水平析因设计3~2。不幸的是,交互作用A×B也可能存在,这样就没有剩余自由度用于估计误差的方差,从而通常的方差分析方法不再能用于数据分析。针对上述问题,本文提出了三个基于均方误差的检验统计量用于分析单次重复试验的三水平析因设计。通过实例表明用这些方法不仅能检验所考虑因子的主效应,而且还能同时检验交互效应。相应检验所用的一些常用临界值提供在附录中。并且,还通过大量的模拟研究对所提出的三个检验方法进行了比较。结果显示,T_~((3))检验在三个检验方法中具有最大的功效。     

Some properties of double designs in terms of lee discrepancy

Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.     

Comparing different fractions of a factorial design: a metal cutting case study

Full factorial designs of a significant size are very rarely performed in industry due to the number of trials involved and unavailable time and resources. The data in this paper were obtained from a six‐factor full factorial (2⁶) designed experiment that was conducted to determine the optimum operating conditions for a steel milling operation. Fractional‐factorial designs 2 (one‐eighth) and 2 (one‐fourth, using a fold‐over from the one‐eighth) are compared with the full 2⁶ design. Four of the 2 are de‐aliased by adding four more runs. In addition, two 12‐run Plackett–Burman experiments and their combination into a fold‐over 24‐run experiment are considered. Many of the one‐eighth fractional‐factorial designs reveal some significant effects, but the size of the estimates varies much due to aliasing. Adding four more runs improves the estimation considerably. The one‐quarter fraction designs yield satisfactory results, compared to the full factorial, if the ‘correct’ parameterization is assumed. The Plackett–Burman experiments, estimating all main effects, always perform worse than the equivalent regular designs (which have fewer runs). When considering a reduced model many of the different designs are more or less identical. The paper provides empirical evidence for managers and engineers that the choice of an experimental design is very important and highlights how designs of a minimal size may not always result in productive findings. Copyright © 2006 John Wiley & Sons, Ltd.     

Some Properties of β-wordlength Pattern for Four-level Designs

Fractional factorial designs have played a prominent role in the theory and practice of experimental design.For designs with qualitative factors under an ANOVA model,the minimum aberration criterion has been frequently used;however,for designs with quantitative factors,a polynomial regression model is often established,thus theβ-wordlength pattern can be employed to compare different fractional factorial designs.Although theβ-wordlength pattern was introduced in 2004,its properties have not been investigated extensively.In this paper,we will present some properties ofβ-wordlength pattern for four-level designs.These properties can help find better designs with quantitative factors.     

On the Isomorphism of Fractional Factorial Designs

Two fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and switching the levels of factors. To identify the isomorphism of two s-factor n-run designs is known to be an NP hard problem, when n and s increase. There is no tractable algorithm for the identification of isomorphic designs. In this paper, we propose a new algorithm based on the centered L₂-discrepancy, a measure of uniformity, for detecting the isomorphism of fractional factorial designs. It is shown that the new algorithm is highly reliable and can significantly reduce the complexity of the computation. Theoretical justification for such an algorithm is also provided. The efficiency of the new algorithm is demonstrated by using several examples that have previously been discussed by many others.