A catalogue of three-level regular fractional factorial designs

A common problem that experimenters face is the choice of fractional factorial designs. Minimum aberration designs are commonly used in practice. There are situations in which other designs meet practical needs better. A catalogue of designs would help experimenters choose the best design. Based on coding theory, new methods are proposed to classify and rank fractional factorial designs efficiently. We have completely enumerated all 27 and 81-run designs, 243-run designs of resolution IV or higher, and 729-run designs of resolution V or higher. A collection of useful fractional factorial designs with 27, 81, 243 and 729 runs is given. This extends the work of Ch93, who gave a collection of fractional factorial designs with 16, 27, 32 and 64 runs.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments

This article provides an overview of the recent literature on the design of blocked and split-plot experiments with quantitative experimental variables. A detailed literature study introduces the ongoing debate between an optimal design approach to constructing blocked and split-plot designs and approaches where the equivalence of ordinary least squares and generalized least squares estimates are envisaged. Examples where the competing design strategies lead to totally different designs are given, as well as examples in which the optimal experimental designs are orthogonally blocked or equivalent-estimation split-plot designs.     

Optimality and construction of some rectangular designs

We obtain a sufficient condition forE-optimality of equireplicate designs. As an application, we proveE-optimality of certain types of three-class PBIBDs based on rectangular association scheme — in short — rectangular designs. These designs turn out to be highly efficient with respect to theA-criterion as well. We also observe that these designs, though themeselves not regular graph designs (RGD's) are yet strictlyE-better than every competing RGD, wheneverv≥26 andv=2 (mod 4). This provides an infinite series of counter examples to the conjecture of John and Mitchell (1977). We also present two methods of construction of the rectangular designs. Apart from providing infinitely many examples of the designs provedE-optimal in this paper and in Cheng and Constantine (1986), this construction also provides — as a special case — the first known infinite series of most balanced group divisible designs, which were proved optimal with respect to all type 1 criteria by Cheng (1978).     

Optimality and efficiency of incomplete split-block designs

In this paper, a balanced incomplete split-block design (BISBD) is shown to be universally optimum for the estimation of the interaction effects with respect to each of the intra-plot stratum estimation and the GLSE. Furthermore, the efficiency factor of an optimal design is investigated.     

Economic designs of single and double screening procedures

Summary Economic designs of single and double screening procedures for improving outgoing product quality based on two screening variables are presented for the case of one-sided specification limit. Two screening variables are observed simultaneously in single screening procedure and are observed sequentially in double screening procedure. It is assumed that the performance variable and the two screening variables are jointly normally distributed. Three quality cost functions — constant, linear, and quadratic — are considered. Cost models are constructed which involve screening inspection cost, and costs of accepted and rejected item. Methods of finding the optimal cutoff values are presented and a numerical example is given.     

Construction of equidistant and weak equidistant supersaturated designs

Supersaturated designs (SSDs) have been highly valued in recent years for their ability of screening out important factors in the early stages of experiments. Recently, Liu and Lin (in Statist Sinica 19:197–211, 2009) proposed a method to construct optimal mixed-level SSDs from smaller multi-level SSDs and transposed orthogonal arrays (OAs). This paper extends their method to construct more equidistant optimal SSDs by replacing the multi-level SSDs and transposed OAs with mixed-level SSDs and general transposed difference matrices, respectively, and then proposes two practical methods for constructing weak equidistant SSDs based on this extended method. A large number of new optimal SSDs can be constructed from these three methods. Some examples are provided and more new designs are listed in “Appendix” for practical use.     

Economic designs of two-sided single and double screening procedures

Economic designs of single and double screening procedures for improving outgoing product quality based on two screening variables are presented for the case of two-sided specification limits. Two screening variables are observed simultaneously in the single screening procedure. In the double screening procedure, one variable is used first to make one of three decisions — accept, reject, or undecided — and after the first screening, the second variable is employed to screen the undecided items. It is assumed that the performance and the two screening variables are jointly normally distributed, and the deviation of the performance variable from the ‘ideal’ value causes dissatisfication to the consumers. Two quality cost functions — constant and quadratic — are considered. Cost models are constructed which involve screening inspection cost, and costs of accepted and rejected item. Methods of finding the optimal cutoff values are presented and a numerical example is given.     

浅谈我国中小企业合理避税问题

随着社会主义市场经济机制的不断完善,中小企业要想在经济大浪潮中站稳脚跟,就必须适当调整经营策略,改革企业管理体制。而在实践的众多方法中,合理避税被认为是最有效的方法之一。在当今社会,税收对于每个企业来说都是一项较大的开支,所以,必须将合理避税纳入企业组织经营管理活动中以增强企业的竞争能力。现阶段我国许多中小企业已经采取...     

The Engel curve for alcohol and the rank of demand systems

This paper shows that the quadratic shape of the Engel curve for alcohol is induced by preference heterogeneity between drinkers and abstainers in a Japanese data set. With controlling the heterogeneity, it is shown that the Engel curve for alcohol slopes monotonically downwards for drinkers, and that the probability of being a drinker is an increasing function of total expenditure. These two relationships generate a quadratic shape for the Engel curve for alcohol. Other goods in this data set appear to have nearly linear Engel curves, so if the alcohol Engel curve for drinkers is also linear, then after controlling for this preference heterogeneity the rank of this demand system would be two. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimality and efficiency of small chessboard designs for correlated errors

Optimal p × q row–column designs are obtained via complete enumeration of all possible designs for two treatments in some fixed effects models with errors specified by a doubly geometric covariance structure. This is done, in part, by a computer search, for a finite set of sizes of the correlation coefficients and in cases where p and q are small enough to make such a search feasible.     

Optimal variance- and efficiency-balanced designs for one- and two-way elimination of heterogeneity

Summary In this paper, a series ofE-optimal non-binary variance balanced (block or row-column) designs and a series ofE-optimal non-binary efficiency balanced (block or row-column) designs are provided in certain broad classes of competing designs. Furthermore, their high efficiencies by the usualA- andD-optimality criteria are shown.     

Optimality and efficiency of circular neighbor balanced designs for correlated observations

The purpose of this paper is to apply the Kunert and Martins (2000b) method for finding optimal designs into the case of dependence. Using this method we study optimality of circular neighbor balanced designs at distances 1 and 2 under the one-dimensional interference model with errors correlated according to a circular autoregressive process. We determine the efficiency of binary designs for specified values of correlation coefficient, for which these designs are not optimal.Research partially supported by the KBN Grant Number 5 P03A 041 21.     

Barycentric algorithm for computing D-optimal size- and cost-constrained designs of experiments

In this paper, we study the problem of D-optimal experimental design under two linear constraints, which can be interpreted as simultaneous restrictions on the size and on the cost of the experiment. For computing a size- and cost-constrained approximate D-optimal design, we propose a specification of the “barycentric” multiplicative algorithm with sequential removal of redundant design points. We analytically prove convergence results for the proposed algorithm and numerically demonstrate its favorable properties compared to competing methods.     

A relationship between efficiencies of marginal designs and the product design

We derive theD- andG-efficiencies of product designs in a multifactor experiment in terms of theD- andG-efficiencies of the designs in the marginal models. Work supported by grants Ku 719/2 and 477/645/96 of the Deutsche Forschungsgemeinschaft     

SomeE andMV-optimal row-column designs having equal numbers of rows and columns

In this paper we consider experimental situations in whichv treatments are to be tested using a row-column design consisting ofb columns andb rows and wherev does not divideb ². Some sufficient conditions are obtained for a design to beE orMV-optimal in such an experimental setting and methods for constructing row-column designs satisfying the sufficient conditions obtained are also given. This research was supported by NSF Grant No. DMS-8401943.