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.     

非重复析因试验数据分析的两个方法

本文介绍了非重复析因试验数据分析的两个比较实用的方法 :Lenth方法和Dong方法 ,并对这两个方法进行了比较全面的比较     

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.     

Theory of optimal blocking for fractional factorial split-plot designs

The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) are developed for discriminating among rival nonisomorphic blcoked FFSP designs. A general rule for identifying MSA or MSEC blocked FFSP designs through their blocked consulting designs is established.     

Bounds on the maximum numbers of clear two-factor interactions for 2~((n1+n2)-(k1+k2)) fractional factorial split-plot designs

Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures. There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors, which can form three types of two-factor interactions: WP2fi, WS2fi and SP2fi. This paper considers FFSP designs with resolutionⅢorⅣunder the clear effects criterion. It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs, and gives some methods for constructing the desired FFSP designs. It further examines the performance of the construction methods.     

具有特定纯净效应折中设计的若干理论及构造

许多因析试验中,试验者只关心指定的一部分因子效应的估计效果.针对此类问题,Addelman(1962)首次提出了折中设计的方法,并定义纯净折中设计以保证指定的因子效应被有效地估计出来,但此类纯净折中设计的分辨度限定为Ⅳ.本文研究了四类全新的折中设计,指定因子效应的集合分别记为{G1,G1×G1}、{G1,G1×G1,G...     

Adding more observations to saturated D‐optimal resolution III two‐level factorial designs

We consider the class of saturated main effect plans for the 2^k factorial. With these saturated designs, the overall mean and all main effects can be unbiasedly estimated provided that there are no interactions. However, there is no way to estimate the error variance with such designs. Because of this and other reasons, we like to add some additional runs to the set of (k+1) runs in the D‐optimal design in this class. Our goals here are: (1) to search for s additional runs so that the resulting design based on (k+s+1) runs yields a D‐optimal design in the class of augmented designs; (2) to classify all the runs into equivalent classes so that the runs in the same equivalent class give us the same value of the determinant of the information matrix. This allows us to trade runs for runs if this becomes necessary; (3) to obtain upper bounds for determinant of the information matrices of augmented designs. In this article we shall address these approaches and present some new results. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 51–77, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10026     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .     

无重复因析试验中一种估计散度效应的方法

自从Box和Meyer首次提出无重复因析试验中散度效应的识别和估计问题, 各种散度效应的估计方法(包括迭代和非迭代)被提出. 特别地, Brenneman 和Nair 给出了这些方法的一个综述, 并且他们验证了改进的Harvey方法优于其它的方法.本文中对于对数线性模型, 一个基于多个位置模型残差平均的非迭代的散度效应估计方法在模型选择阶段被提出. 在大多数的模拟实验模型中, 本文方法具有比MH方法更小的均方误差, 且它可以应用于MH方法不适用的0或小的绝对残差情形. 我们也考虑了这个估计的理论性质, 并进行了实例分析.     

两水平无重复因析试验散度效应BH估计的性质

本文研究了两水平无重复因析试验散度效应BH估计的性质,给出了BH估计无偏性的充分必要条件,求得了它的近似方差.并在多个模型下对BH与MH估计进行了模拟比较.     

Zero‐sum flows in designs

Let D be a t ‐ ( v, k , λ) design and let N _i (D) , for 1 ≤ i ≤ t , be the higher incidence matrix of D , a ( 0 , 1 )‐matrix of size , where b is the number of blocks of D . A zero‐sum flow of D is a nowhere‐zero real vector in the null space of N ₁ ( D ). A zero‐sum k‐flow of D is a zero‐sum flow with values in { 1 , …, ±( k ? 1 )}. In this article, we show that every non‐symmetric design admits an integral zero‐sum flow, and consequently we conjecture that every non‐symmetric design admits a zero‐sum 5‐flow. Similarly, the definition of zero‐sum flow can be extended to N _i ( D ), 1 ≤ i ≤ t . Let be the complete design. We conjecture that N _t ( D ) admits a zero‐sum 3‐flow and prove this conjecture for t = 2 . © 2011 Wiley Periodicals, Inc. J Combin Designs 19:355‐364, 2011     

Two‐strata rotatability in split‐plot central composite designs

A rotatable design (Ann. Math. Stat. 1957; 28 :195–241) for k factors is one such that the prediction variance is purely a function of distance from the design center. Of special interest in this paper is the rotatable central composite design (CCD), which most software packages use as the typical default choice for a second‐order design. In many cases some factors are hard to change while others are easy to change, which creates a split‐plot experiment. This paper establishes that the split‐plot structure precludes the possibility of any second‐order design being rotatable in the traditional sense. As an alternative this paper proposes the two‐strata rotatable split‐plot CCD, where the resulting prediction variance is a function of the whole plot (WP) distance and the subplot (SP) distance separately instead of the sum of them. The resulting design is rotatable in the WP space when the SP factors are held fixed, and vice versa. In the special case where the WP variance component is zero, the two‐strata rotatable split‐plot CCD becomes the standard rotatable CCD. Copyright © 2009 John Wiley & Sons, Ltd.     

无重复因析试验中多个散度效应检验的一种改进方法

在无重复因析试验的多个散度效应分析中,常常出现错误识别的现象,即两个显著的散度效应可能在它们的交互列上产生一个错误的散度效应,并且现有的许多方法都存在这样的问题.为了解决这种模棱两可性,McGrath和Lin(2001)提出了一种基于残差样本方差几何平均的检验方法(ML方法),但是这个方法不能应用于零残差样本方差的情形.鉴于此,提出了一种基于修改残差的改进方法,适用于零残差样本方差的情形,并且通过实例验证了方法的合理性.最后,通过模拟和ML方法做了比较.     

Optimum covariate designs in a binary proper equi-replicate block design set-up

The choice of covariates values for a given block design attaining minimum variance for estimation of each of the regression parameters of the model has attracted attention in recent times. In this article, we consider the problem of finding the optimum covariate design (OCD) for the estimation of covariate parameters in a binary proper equi-replicate block (BPEB) design model with covariates, which cover a large class of designs in common use. The construction of optimum designs is based mainly on Hadamard matrices.     

On the robustness of balanced fractional 2m factorial designs of resolution 2l+1 in the presence of outliers

Summary By use of the algebraic structure, we obtain a simplified expression for the outlier-insensitivity factor for balanced fractional 2^m factorial (2^m-BFF) designs of resolution 2l+1 derived from simple arrays (S-arrays), whose measure has been introduced by Ghosh and Kipnegeno (1985,J. Statist. Plann. Inference,11, 119–129). It is defined by use of the measure suggested by Box and Draper (1975,Biometrika, 62 (2), 347–352). As examples, we study the sensitivity ofA-optimal 2^m-BFF designs of resolution VII (i.e.,l=3) given by Shirakura (1976,Ann. Statist.,4, 515–531; 1977,Hiroshima Math. J.,7, 217–285). We observe that these designs are robust in the sense that they have low sensitivities. Research supported in part by Grant 59530012 (C) and 60530014 (C), Japan.     

Restricted b‐factors in bipartite graphs and t‐designs

We present a new equivalence result between restricted b‐factors in bipartite graphs and combinatorial t‐designs. This result is useful in the construction of t‐designs by polyhedral methods. We propose a novel linear integer programming formulation, which we call GDP, for the problem of finding t‐designs that has a noteworthy advantage compared to the traditional set‐covering formulation. We analyze some polyhedral properties of GPD, implement a branch‐and‐cut algorithm using it and solve several instances of small designs to compare with another point‐block formulation found in the literature. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 169–182, 2006     

Efficiency measure,modelling and estimation in combined array designs

In off‐line quality control, the settings that minimize the variance of a quality characteristic are unknown and must be determined based on an estimated dual response model of mean and variance. The present paper proposes a direct measure of the efficiency of any given design‐estimation procedure for variance minimization. This not only facilitates the comparison of different design‐estimation procedures, but may also provide a guideline for choosing a better solution when the estimated dual response model suggests multiple solutions. Motivated by the analysis of an industrial experiment on spray painting, the present paper also applies a class of link functions to model process variances in off‐line quality control. For model fitting, a parametric distribution is employed in updating the variance estimates used in an iteratively weighted least squares procedure for mean estimation. In analysing combined array experiments, Engel and Huele (Technometrics, 1996; 39:365) used log‐link to model process variances and considered an iteratively weighted least squares leading to the pseudo‐likelihood estimates of variances as discussed in Carroll and Ruppert (Transformation and Weighting in Regression, Chapman & Hall: New York). Their method is a special case of the approach considered in this paper. It is seen for the spray paint data that the log‐link may not be satisfactory and the class of link functions considered here improves substantially the fit to process variances. This conclusion is reached with a suggested method of comparing ‘empirical variances’ with the ‘theoretical variances’ based on the assumed model. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of means approach in advanced designs

Analysis of means (ANOM), similar to Shewhart control chart that exhibits individual mean effects on a graphical display, is an attractive alternative mean testing procedure for the analysis of variance (ANOVA). The procedure is primarily used to analyze experimental data from designs with only fixed effects. Recently introduced, the ANOM procedure based on the q‐distribution (ANOMQ procedure) generalizes the ANOM approach to random effects models. This article reveals that the application of ANOM and ANOMQ procedures in advanced designs such as hierarchically nested and split‐plot designs with fixed, random, and mixed effects enhances the data visualization aspect in graphical testing. Data from two real‐world experiments are used to illustrate the proposed procedure; furthermore, these experiments exhibit the ANOM procedures' visualization ability compared with ANOVA from the point of view of the practitioner.     

Run orders and quantitative factors in asymmetrical designs

Results on run orders leading to trend-free symmetrical factorial designs are extended to the asymmetrical case, using the character theory of abelian groups. The tools developed apply equally to the construction of designs for quantitative treatment factors with eight or more regularly spaced levels. Abelian group theory can also be used to find minimum-cost run orders for asymmetrical designs, with a cost based on the number of changes of levels between successive runs.     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013