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.     

A note on the robustness of Box-Behnken designs to the unavailability of data

Box-Behnken designs and central composite designs are efficient designs for fitting second order polynomials to response surfaces, because they use relatively small numbers of observations to estimate the parameters. In this paper we investigate the robustness of Box-Behnken designs to the unavailability of observations, in the sense of finding t _max , the maximum number of arbitrary rows in the design matrix that can be removed and still leave all of the parameters of interest estimable. The results are compared to the known results for the central composite designs found in MacEachern, Notz, Whittinghill & Zhu (1995). The blocked Box-Behnken designs are equally as robust as those that are not blocked. Received December 1997     

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.     

On the construction of trend free row-column 2-level factorial experiments

In this paper we consider experimental situations where a complete or fractional factorial experiment having all factors at 2 levels is to be conducted using a 2 ^m × 2 ⁿ row-column design and where there may be an unknown trend effect that can be expressed as a polynomial function of the position in which observations are obtained in the row-column design. Methods are given for allocating the treatments from a complete or fractional 2-level factorial experiment to rows and columns so that the resulting design yields estimates for main effects that have a high level of resistance against trend effects. Research supported by NSF Grant No. DMS-8700945.     

Optimal nested row-column designs for blocks of size 2×2 under dependence

This paper examines nested 2 ×2 row-column designs when within-block observations are assumed to be dependent. The model considered has fixed block effects, which may also include row and/or column effects. Optimal binary and non-binary designs, constructed from semi-balanced arrays, are given under both generalised and ordinary least squares estimation. It is shown that binary designs are optimal when dependence is low. In general, however, the optimal designs are highly specific to the correlation values. Received: October 1999     

Optimality results on orthogonal arrays plus <Emphasis Type="Italic">p</Emphasis> runs for <Emphasis Type="Italic">s</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> factorial experiments

The optimality of designs obtained by adding p runs to an orthogonal array is studied for experiments involving m factors each at s levels. The optimality criterion used here, is the Type 1 criterion due to Cheng (1978) which is an extension of Kiefer (1975) universal optimality criterion. Unlike what happens with orthogonal array plus one run designs, the behavior of designs obtained via augmentation of an orthogonal array by p runs depends on the particular runs added.     

Robustness of Williams Square change-over designs

In literature, Williams Square change-over designs balanced for first residual effects of treatments have been proved to be universally optimal. Here, an attempt has been made to show that these designs are robust against missing of last α [≤v−1; v being the number of periods in the design for v treatments] observations from an experimental unit. Received May 2000     

Connections between uniformity and aberration in general multi-level factorials

Discrepancy is a kind of important measure used in experimental designs. Among various existing discrepancies, the discrete discrepancy, centered L ₂-(CD ₂) and wrap-around L ₂-discrepancy (WD ₂) have been well justified and widely used. In this paper, using the second-order polynomials of indicator functions for these three discrepancies, we investigate the close relationships between them and the generalized wordlength pattern, and provide some conditions under which a design having one of these minimum discrepancies is equivalent to having generalized minimum aberration (GMA). These results provide further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrepancies in the quadratic forms of the indicator functions are useful for us to find optimal designs under any of them.     

Projection properties of certain three level orthogonal arrays

Two orthogonal arrays based on 3 symbols are said to be isomorphic or combinatorially equivalent if one can be obtained from the other by a sequence of row permutations, column permutations and permutations of symbols in each column. Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a ``projection design'. In this paper we have identified all the inequivalent projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k=3,4 and 5 factors. It is shown that the generalized wordlength pattern criterion proposed by Ma and Fang [23] can distinguish between most, but not all, inequivalent classes. We propose an extension of the Es² criterion (which is commonly used for measuring efficiency of 2-level designs) to distinguish further between the non-isomorphic classes and to measure the efficiency of the designs in these classes. Some concepts on generalized resolution are also discussed.     

Asymmetrical Factorial Designs Containing Clear Effects

The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s ²)s ⁿ design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s ²)s ⁿ design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s ²)s ⁿ designs according to the clear 2fi components, main effects and 2fis they have.     

Nonregular designs from Hadamard matrices and their estimation capacity

Deng and Tang (1999) proposed the generalized minimum aberration (GMA) criterion to assess fractional factorial designs, and a design with GMA is often regarded as the best. However, there exist situations where some other designs may suit practical needs better. In this article, we propose an algorithm to sequentially examine designs obtained from Hadamard matrices under estimation capacity (EC) and provide designs with maximum or high EC for various combinations of run-size and number-of-factors. The usefulness of maximum or high EC designs is discussed.2000 Mathematics Subject Classification: 62K15, 05B20.Acknowledgements The research of Yingfu Li is supported by a Faculty Research Support Fund through the School of Science and Computer Engineering, University of Houston - Clear Lake. The authors are very grateful to the editor and two referees for their helpful comments that have led to the improvement of the paper.     

Optimal mixed-level supersaturated design

A supersaturated design is essentially a fractional factorial in which the number of potential effects is greater than the number of runs. In this paper, E(f _NOD ) criterion is employed for comparing supersaturated designs from the viewpoint of orthogonality and uniformity, and a lower bound of E(f _NOD ) which can serve as a benchmark of design optimality is obtained. It is shown that the existing E(s ²) and ave ² criteria (for two- and three-level supersaturated designs respectively) are in fact special cases of this criterion. Furthermore, a construction method for mixed-level supersaturated designs is proposed and some properties of the resulting designs are investigated. Key words:Discrepancy; Hamming distance; Orthogonal array; Supersaturated design; Uniformity; U-type design. 2000 Mathematics Subject Classifications62K15, 62K05, 62K99. Corresponding author.     

Orthogonal main effect plans for two and three factors in small blocks

Orthogonal main-effect plans for two and three factors in small blocks are obtained from the dual of adjusted orthogonal row-column designs. The method for constructing efficient plans is presented, and a relationship between the average efficiency factors of the row-column design and the corresponding main effects is given for the two-factor case. Orthogonal Main Effect.     

DS-optimal designs in one way ANOVA

Characterization and construction of optimal designs using the familiar optimality criteria, for example A-, D- and E-optimality are well studied in the literature. However the study of the Distance Optimality (DS-) criterion introduced by Sinha (1970) has very recently drawn attention of researchers. In the present article, we consider the singularly estimable full rank problem of estimating the full set of elementary treatment contrasts using the DS optimality criterion in the set up of a one way ANOVA model. Using a limit argument it turns out that a CRD in which difference between any two allocation numbers is at the most unity is uniquely DS-optimal. Acknowledgement. We are thankful to Prof. B. K. Sinha for suggesting the problem to us and many helpful discussions with him. We are also thankful to the referees for drawing our attention to the reference of Bischoff (1995) and many helpful comments.     

Lower bounds of various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this paper, some lower bounds of various discrepancies of combined designs, such as centered L ₂-discrepancy, symmetric L ₂-discrepancy and wrap-around L ₂-discrepancy, under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

A note on connections among criteria for asymmetrical factorials

Various optimal criteria have been proposed to rank asymmetrical fractional factorial designs. Among them, the generalized minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention. Recently, Liu et al. (Stat Sin 16:1285–1297, 2006) proposed the minimum χ ² criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum χ ² criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an analytic relationship between generalized minimum aberration and minimum moment aberration is obtained for asymmetrical fractional factorial designs.     

A Lower Bound for the Centered <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-Discrepancy on Asymmetric Factorials and its Application

The role of uniformity measured by the centered L ₂-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L ₂-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L ₂-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.     

On the information criterion for selecting regressors

Summary It has been asserted in the past that any Bayesian treatment of the model selection problem in regression using some form of continuous loss structure would always lead to using the largest possible model (Leamer 1979; Chow 1981). We show in this paper that, provided the distinction between the choice of a model and the estimation of its parameters is maintained, the Kullback-Leibler information measure can be used in a Bayesian context to derive a criterion which may lead to parsimony of parameters in regression analysis. The regression models are taken as restrictions of a general class of distributions which includes the truen-variate distribution of the variabley. Separate criteria for the cases of known and unknown variance ofy are obtained. In the limiting situation when prior opinions about the parameters are weak, these criteria reduce to special cases of the generalizedC _p and AIC criteria (Atkinson 1981). Relationship with other criteria is discussed.     

Bounds on the efficiency of the residual design of extended BIB designs

This gives a complete proof for bounds of the efficiency, conjectured by Das and Kageyama (1992), on robustness of extended balanced incomplete block designs against the unavailability of any number of observations in a block.     

Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs

Recently, Xu and Wu (2001) presented generalized minimum aberration criterion for comparing and selecting general fractional factorial designs. This criterion is defined using a set of _u(D) values, called J-characteristics by us. In this paper, we find a set of linear equations that relate the set of design points to that of J-characteristics, which implies that a factorial design is uniquely determined by its J-characteristics once the orthonormal contrasts are designated. Thereto, a projection justification of generalized minimum aberration is established. All of these conclusions generalize the results for two-level symmetrical factorial designs in Tang (2001).Acknowledgements The authors are grateful to the editor, the associate editor and the referees for their valuable comments. This paper is supported by NNSF of P.R.China grant No. 10171051. and RFDP grant No. 1999005512.