Theory of optimal blocking of nonregular factorial designs

The authors introduce the notion of split generalized wordlength pattern (GWP), i.e., treatment GWP and block GWP, for a blocked nonregular factorial design. They generalize the minimum aberration criterion to suit this type of design. Connections between factorial design theory and coding theory allow them to obtain combinatorial identities that govern the relationship between the split GWP of a blocked factorial design and that of its blocked consulting design. These identities work for regular and nonregular designs. Furthermore, the authors establish general rules for identifying generalized minimum aberration (GMA) blocked designs through their blocked consulting designs. Finally they tabulate and compare some GMA blocked designs from Hall's orthogonal array OA(16,2¹5,2) of type III.     

Minimum Aberration Regular Two-Level Designs in the Presence of Dispersion Factors

In practice, the variance of the response variable may change as some specific factors change from one setting to another in a factorial experiment. These factors affecting the variation of the response are called dispersion factors, which can violate the usual assumption of variance homogeneity. In this study, we modify the conventional minimum aberration criterion to take the impact of dispersion factors into account. The situations of one or two dispersion factors are investigated. As a result, we present regular 2^{n ? p} designs with run sizes equal to 16 and 32 using the modified minimum aberration criterion.     

The existence of the strong combined-optimal design

The technique of fold-over is useful for conducting follow-up experiments. Based on the minimum aberration criterion, Li and Lin (2003) developed an algorithm and used computer to search the corresponding optimal foldover designs for 16 and 32 runs in the 2 ^k-p design. In their study, they found that the 2¹⁰⁻⁶ design is the only one that is not a strong combined-optimal design among all the designs. However, they did not interpret the reason causing the phenomenon. This article will explore under what kind of conditions, that the strong combined-optimal design will exist, and the solutions of the related problems.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

Construction of minimum aberration blocked two-level regular factorial designs

ABSTRACT

In this article, we consider the construction of minimum aberration 2^{n ? k}: 2^p designs with respect to some existing combined wordlength patterns, where a 2^{n ? k}: 2^p design is a blocked two-level design with n treatment factors, 2^p blocks, and N = 2^q runs with q = n ? k. Two methods are proposed for two situations: n ? 2^{q ? p} ? 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2^{n ? k}: 2^p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.     

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

The minimum aberration criterion has been advocated for ranking foldovers of 2^k−p$2^{k-p}$ fractional factorial designs (Li and Lin, 2003); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.     

Generalized discrete discrepancy and its applications in experimental designs

In recent years, there has been increasing interest in the study of discrete discrepancy. In this paper, the popular discrete discrepancy is extended to the so-called generalized discrete discrepancy. Connections among generalized discrete discrepancy and other optimality criteria, such as orthogonality, generalized minimum aberration and minimum moment aberration, are investigated. These connections provide strong statistical justification of generalized discrete discrepancy. A lower bound of generalized discrete discrepancy is also obtained, which serves as an important benchmark of design uniformity.     

A new Markov chain approach for the economic statistical design of the VSS T2 control chart

An economic statistical design model for a T² chart which uses a variable sample size (VSS) feature is developed in this article. This study mainly differs from the others conducted in the field. In that a new approach is offered to achieve closed form of some statistical criteria. In other words, the proposed formulas can be considered as a better alternative approach in designing the VSS control charts in terms of simplicity and yet providing the users with better optimal solutions.     

Minimum aberration and model robustness for two-level fractional factorial designs

The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution III or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.     

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>4<Superscript>1</Superscript> designs with minimum aberration or weak minimum aberration

For measuring the goodness of 2 ^m 4¹ designs, Wu and Zhang (1993) proposed the minimum aberration (MA) criterion. MA 2 ^m 4¹ designs have been constructed using the idea of complementary designs when the number of two-level factors, m, exceeds n/2, where n is the total number of runs. In this paper, the structures of MA 2 ^m 4¹ designs are obtained when m>5n/16. Based on these structures, some methods are developed for constructing MA 2 ^m 4¹ designs for 5n/16<m<n/2 as well as for n/2≤m<n. When m≤5n/16, there is no general method for constructing MA 2 ^m 4¹ designs. In this case, we obtain lower bounds for A ₃₀ and A ₃₁, where A ₃₀ and A ₃₁ are the numbers of type 0 and type 1 words with length three respectively. And a method for constructing weak minimum aberration (WMA) 2 ^m 4¹ designs (A ₃₀ and A ₃₁ achieving the lower bounds) is demonstrated. Some MA or WMA 2 ^m 4¹ designs with 32 or 64 runs are tabulated for practical use, which supplement the tables in Wu and Zhang (1993), Zhang and Shao (2001) and Mukerjee and Wu (2001).     

Invariance of generalized wordlength patterns

The generalized wordlength pattern (GWLP) introduced by Xu and Wu [2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 1066–1077] for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang [2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285] defined the J$J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J$J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J$J$-characteristics are not. We briefly discuss some implications of these results.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 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.     

Creating catalogues of two-level nonregular fractional factorial designs based on the criteria of generalized aberration

We consider the problem of constructing good two-level nonregular fractional factorial designs. The criteria of minimum G and G₂ aberration are used to rank designs. A general design structure is utilized to provide a solution to this practical, yet challenging, problem. With the help of this design structure, we develop an efficient algorithm for obtaining a collection of good designs based on the aforementioned two criteria. Finally, we present some results for designs of 32 and 40 runs obtained from applying this algorithmic approach.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

An example arising from berkson's conjecture

Berkson (1980) conjectured that minimum x² was a superior procedure to that of maximum likelihood, especially with regard to mean squared error. To explore his conjecture, we analyze his (1955) bioassay problem related to logistic regression. We consider not only the criterion of mean squared error for the comparison of these estimators, but also include alternative criteria such as concentration functions and Pitman's measure of closeness. The choice of these latter criteria is motivated by Rao's (1981) considerations of the shortcomings of mean squared error. We also include several Rao-Blackwellized versions of the minimum logit x² the purpose of these comparisons.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

On Rao score and PearsonX 2 statistics in generalized linear models

The identity of the Rao score and PearsonX ² statistics is well known in the areas where the latter was first introduced: goodness-of-fit in contingency tables and binary responses. We show in this paper that the same identity holds when the two statistics are used for testing goodness-of-fit of Generalized Linear Models. We also highlight the connections that exist between the two statistics when they are used for the comparison of nested models. Finally, we discuss some merits of these unifying results. Work financially supported by cofin. MIUR grants 2000 and 2002.