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     

A class of non binary unequally replicatedE-optimal designs

In a situation where the given set of parameters (b, k andv) precludes the existence of any known optimal block designs, but an optimal block design is known to exist with parametersb, k andv*>v, a new design is shown to be useful. This (b, k, v) design is obtained from the (b, k, v*) optimal design by collapsing the classes of a suitable paritition of the treatment set (of the latter design) to treatments (of the former). We call the new design a quotient of the original design. Although the quotient is non binary and unequally replicated, it turns out to beE-optimal within the class of all proper and connected designs withb, k andv, provided the replication number of the optimal design we start with is not too large.     

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.     

Algorithms for optimal design with application to multiple polynomial regression

The approximate theory of optimal linear regression design leads to specific convex extremum problems for numerical solution. A conceptual algorithm is stated, whose concrete versions lead us from steepest descent type algorithms to improved gradient methods, and finally to second order methods with excellent convergence behaviour. Applications are given to symmetric multiple polynomial models of degree three or less, where invariance structures are utilized. A final section is devoted to the construction of efficientexact designs of sizeN from the optimal approximate designs. For the multifactor cubic model and some of the most popular optimality criteria (D-, A-, andI-criteria) fairly efficient exact designs are obtained, even for small sample sizeN. AMS Subject Classification: 62K05.Abbreviated Title: Algorithms for Optimal Design.Invited paper presented at the International Conference on Mathematical Statistics,ProbaStat '94, Smolenice, Slovakia.     

Optimal block designs for diallel crosses

Dey and Midha (Biometrika 83(2):484–489, 1996) constructed optimal block designs for complete diallel cross experiment using triangular partially balanced incomplete block designs with two associate classes. They listed optimal block designs for the lines in the range from 5 ≤ v ≤ 10. In this paper, we are also proposing additional optimal block designs for complete diallel cross experiment using two associate class partially balanced block designs for some additional values of v. Our method yields designs for proper and non-proper settings for complete diallel cross experiments. The proper and non proper designs are optimal in the sense of Kempthorne (Genetics 41:451–459, 1956) and non-proper designs are universally optimal in the sense of Kiefer (A survey of statistical design and linear models, North Holland, Amsterdam, 1975). The list of practically important designs is also given.     

A note on the determination of A-optimal type block designs under fixed and mixed effects models

In this paper, we consider the problem of estimating a selected set of contrasts between v treatments using a block design consisting of b blocks of size k. Traditionally, the construction of A-optimal block designs for such situations has been carried out assuming a fixed effects model. In this paper, we show that A-optimal designs constructed under a fixed effects model are robust in the sense that these designs have maximal minimal efficiency when considered among all available designs and under all possible mixed effects models. AMS 1991 subject classifications: Primary 62K05; Secondary 62K10     

Some orthogonal arrays with 32 runs and their projection properties

Screening designs are useful for situations where a large number of factors are examined but only a few, k, of them are expected to be important. Traditionally orthogonal arrays such as Hadamard matrices and Plackett Burman designs have been studied for this purpose. It is therefore of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties. In this paper we present 15 inequivalent Hadamard matrices of order n=32 constructed from circulant cores. We study their projection properties using several well-known statistical criteria and we provide minimum generalized aberration 2 level designs with 32 runs and up to seven factors that are embedded into these Hadamard matrices. A concept of generalized projectivity and design selection of such designs is also discussed.AMS Subject Classification: Primary 62K15, Secondary 05B20     

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.     

sk−p Fractional factorial designs in sb blocks

Draper and Guttman (1997) shows that for basic 2^k−p designs, p≥0, k − p replicates of blocks designs of size two are needed to estimate all the usual (estimable) effects. In this work, we provide an algebraic formal proof for the two-level blocks designs results and present results applicable to the general case; that is, for the case of s ^k factorial (p=0) or s ^k−p fractional factorial (p >0) designs in s ^b blocks, where 0<b<k− p, at least replicates are needed to clear up all possible effects. Through the theoretical development presented in this work, it can provide a clearer view on why those results would hold. We will also discuss the estimation equations given in Draper and Guttman (1997).  Research supported in part by the National Science Council of Taiwan, R.O.C., Grant No. NSC 89-2118-M110-010. Acknowledgement. The authors would like to thank the referee for very helpful comments.     

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     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

On the E-optimality of truncated block designs

Sufficient conditions are found for designs, derived from completely symmetric designs by deleting binary blocks, to be E-optimal. Sufficient conditions are also found for E-optimality of designs obtained from other E-optimal designs by deleting all blocks forming a balanced incomplete block design on a subset of the treatments. The results include many binary and non-binary designs for which E-optimality was previously unknown.     

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.     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Optimal and efficient repeated measurements designs for comparing test treatments with a control

Rearranging the sequence of test treatments and control treatment within each block of an A-optimal balanced treatment block design such that certain conditions are satisfied, the resulting design is an A-optimal repeated measurements designs when blocks are regarded as units or periods. The efficiencies of designs which are obtained from universally optimal repeated measurements designs with test treatments only by changing some treatment labels into control treatment are given.     

Universally optimal designs under mixed interference models with and without block effects

The literature on neighbor designs as introduced by Rees (Biometrics 23:779–791, 1967) is mainly devoted to construction methods, providing few results on their statistical properties, such as efficiency and optimality. A review of the available literature, with special emphasis on the optimality of neighbor designs under various fixed effects interference models, is given in Filipiak and Markiewicz (Commun Stat Theory Methods 46:1127–1143, 2017). The aim of this paper is to verify whether the designs presented by Filipiak and Markiewicz (2017) as universally optimal under fixed interference models are still universally optimal under models with random interference effects. Moreover, it is shown that for a specified covariance matrix of random interference effects, a universally optimal design under mixed interference models with block effects is universally optimal over a wider class of designs. In this paper the method presented by Filipiak and Markiewicz (Metrika 65:369–386, 2007) is extended and then applied to mixed interference models without or with block effects.     

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.     

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.     

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     

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