TypeS designs with nested rows and columns

Summary Supplemented balance in block designs with nested rows and columns is considered, under the name of typeS designs with nested rows and columns. Analysis and construction of such designs are also discussed.     

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     

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.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

On Minimally-supported <Emphasis Type="Italic">D</Emphasis>-optimal Designs for Polynomial Regression with Log-concave Weight Function

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.     

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.     

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.     

On exact D-optimal designs with 2 two-level factors and n autocorrelated observations

In this paper we consider the exact D-optimal designs for estimation of the unknown parameters in the two factors, each at only two-level, main effects model with autocorrelated errors. The vector of the n random errors in the observed responses is assumed to follow a first-order autoregressive model (AR(1)). The exact D-optimal designs seek the optimal combinations of the design levels as well as the optimal run orders, so that the determinant of the information matrix of BLUEs for the unknown parameters is maximized. Bora-Senta and Moyssiadis (1999) gave some conjectures about the exact D-optimal designs based on their experience of several exhaustive searches. In this paper their conjectures are partially proved to be true.Received: January 2003 / Accepted: October 2003Partially supported by the National Science Council of Taiwan, R.O.C. under grant NSC 91-2115-M-008-013.Supported in part by the National Science Council of Taiwan, R.O.C. under grant NSC 89-2118-M-110-003.     

Eight-run two-level factorial designs under dependence

Dependent observations commonly arise in factorial experiments. Apart from main-effects two-level designs formed by the Cheng & Steinberg reverse foldover algorithm, which are known to be very efficient designs under dependence using the D-criterion, little is known about other designs, models and criteria, and the range of possible behaviour. In this paper, we investigate in detail 8-run two-level designs. Received February 1998     

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.     

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.     

$${\phi_p}$$-optimal designs for a linear log contrast model for experiments with mixtures

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S ^k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these -optimal designs behave. Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001.     

A note on the uniform distribution on the arcsin points

In his book Pukelsheim [8] pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains theD-optimal arcsin support design and theD ₁-optimal design for a polynomial regression. The results allow explicit representations ofD-, andD ₁-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.     

Robustness to the unavailability of data in block designs

We present general results for finding or boundingt _maxB , the maximum number of arbitrary whole blocks of observations which can be removed from a block design, and still leave all of the elementary treatment contrasts estimable. The block sizes may be larger than the number of treatments. The results are applied to BBDs, reinforced BIBDs and BBDs, BTIBDs, and a series of variance balanced incomplete block designs with two block sizes. Also given for most of these designs, are results fort _max, the maximum number of arbitrary, scattered observations that can become unavailable, and still leave all of the elementary treatment contrasts estimable.The work was undertaken while Dr. Whittinghill was visiting Ohio State University, and supported by a grant from the Natural Sciences Division, Colby College, Waterville, Maine.     

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L₂-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.     

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.     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

A general construction of E(s 2)-optimal large supersaturated designs

A general method for construction of E(s ²)-optimal, two-level supersaturated designs (SSDs) with the equal occurrence property, from supplementary difference sets is introduced. It is proved that SSDs constructed in this way are E(s ²)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of E(s ²)-optimal large SSDs.     

Optimal allocation of measurements in a linear calibration process

A problem of allocation of measurements for a linear calibration process is considered in this article. It is assumed that a total of N measurements are made some of which may be measurements on two distinct standards, while the remaining measurements are on m different unknown specimens. We discuss allocation of the N measurements for the two standards and m unknown specimens based on A-optimality criterion, which is applied to asymptotic variances of maximum likelihood estimators for the true values of unknown specimens. It can be shown that the optimal allocation depends on the true values of unknown specimens. Hence, the user may resort to locally or Bayesian A-optimal measurement designs. Some practical solution is presented. Furthermore, the impact of prior on the allocation is also discussed.     

Construction of block designs admitting an Abelian automorphism group

Construction of block designs admitting an action of a presumed automorphism group consists of two basic steps: 1.construction of orbit structures for the given automorphism group, 2. construction of block designs for the orbit structures obtained in 1. For an abelian group G, the second step, called indexing, often lasts too long, because there are too many possibilities to be enumerated with today's computer facilities. To make such a construction possible we use a principal series of the group G to obtain a refinement of the orbit structures.