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.     

An enumeration of graphical designs

Let Ψ(t,k) denote the set of pairs (v,λ) for which there exists a graphical t‐(v,k,λ) design. Most results on graphical designs have gone to show the finiteness of Ψ(t,k) when t and k satisfy certain conditions. The exact determination of Ψ(t,k) for specified t and k is a hard problem and only Ψ(2,3), Ψ(2,4), Ψ(3,4), Ψ(4,5), and Ψ(5,6) have been determined. In this article, we determine completely the sets Ψ(2,5) and Ψ(3,5). As a result, we find more than 270,000 inequivalent graphical designs, and more than 8,000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 70–85, 2008     

兼顾效率和成本的ARMA控制图多目标优化设计

ARMA控制图是一种有效的自相关过程质量监控方法,为了能够同时对ARMA控制图监控方案的效率和成本进行优化,本文分别研究了ARMA控制图的平均运行长度和质量成本的计算方法,并由此建立了ARMA控制图的多目标优化设计模型。采用NSGA-Ш智能优化算法,通过一个具体的算例对该模型的计算方法进行了说明,针对不同程度的过程偏移给出了多目标优化设计的非劣解解集。然后通过灵敏度分析的方法研究了模型中的主要设计参数对监控方案的效率和成本的影响程度。最后,通过与其它几种ARMA控制图优化设计方案的比较分析,说明了本文提出的设计方法的优势。     

Efficient crossover designs for comparing test treatments with a control treatment

Within a large family of crossover designs this paper characterizes the mathematical structures of A-optimal and A-efficient crossover designs for the purpose of statistical comparison between t experimental treatments with a control (standard) treatment. It further guides the user how to go about the construction of these designs and if needed doing the last minute modifications. To demonstrate the ideas some very interesting optimal and efficient small designs are constructed. The mathematical and statistical tools developed here could be very useful in other areas of design of experiments. Many interesting and not yet solved design problems for further research are implicitly stated throughout the paper.     

New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

复方厚朴片生产工艺实验研究的优化方法

为优化复方厚朴片的生产工艺，先通过预实验找出六个影响因素及其范围，对各因素如何搭配采用随机办法安排实验方案，按五种优良性准则从方案中选出较优且运行次数最少者，用ＳＡＳ回归过程分析试验结果     

On defining sets of full designs

A defining set of a t-(v,k,λ) design is a subcollection of its blocks which is contained in a unique t-design with the given parameters on a given v-set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M|∣M is a minimal defining set of D}. The unique simple design with parameters is said to be the full design on v elements; it comprises all possible k-tuples on a v set. We provide two new minimal defining set constructions for full designs with block size k≥3. We then provide a generalisation of the second construction which gives defining sets for all k≥3, with minimality satisfied for k=3. This provides a significant improvement of the known spectrum for designs with block size three. We hypothesise that this generalisation produces minimal defining sets for all k≥3.     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

A graphical method for assessing multivariate normality

Summary  In this paper we suggest a simple graphical device for assessing multivariate normality. The method is based on the characteristic that linear combinations of the sample mean and sample covariance matrix are independent if and only if the random variable is normally distributed. We demonstrate the usage of the suggested method and compare it to the classical Q-Q plot by using some multivariate data sets.     

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。