AN ALGORITHM FOR CONSTRUCTING OPTIMAL RESOLVABLE ROW-COLUMN DESIGNS

This paper describes an effective algorithm for constructing optimal or near-optimal resolvable row-column designs (RCDs) with up to 100 treatments. The performance of this algorithm is assessed against 20 2-replicate resolvable RCDs of Patterson & Robinson (1989) and 17 resolvable RCDs based on generalized cyclic designs (GCDs) of Ipinyomi & John (1985). The use of the algorithm to construct RCDs with contiguous replicates is discussed.     

CONSTRUCTION OF RESOLVABLE ROW-COLUMN DESIGNS USING SIMULATED ANNEALING

The paper describes an algorithm which constructs optimal or near optimal resolvable row-column designs. Its performance is assessed against available tables of two-replicate designs.     

THE USE OF A CLASS OF FOLDOVER DESIGNS AS SEARCH DESIGNS

It is shown that members of a class of two-level nonorthogonal resolution IV designs with n factors are strongly resolvable search designs when k, the maximum number of two-factor interactions thought possible, equals one; weakly resolvable when k = 2 except when the number of factors is 6; and may not be weakly resolvable when k≥ 3.     

t-Latinized Designs

A variety trial sometimes requires a resolvable block design in which the replicates are set out next to each other. The long blocks running through the replicates are then of interest. A t -latinized design is one in which groups of these t long blocks are binary. In this paper examples of such designs are given. It is shown that the algorithm described by John & Whitaker (1993) can be used to construct designs with high average efficiency factors. Upper bounds on these efficiency factors are also derived.     

UPPER BOUNDS FOR LATINIZED DESIGNS

Upper bounds axe derived for the efficiency factor of a class of resolvable incomplete block designs known as latinized designs. These designs are particularly useful in glasshouse and field trials, and can be readily extended to two-dimensional blocking structures. Existing bounds for resolvable designs axe also reviewed and a comparison is made between the third moment bounds discussed by Jarrett (1989) and the second moment bounds of Tjur (1990).     

Resolvable Designs With Unequal Block Sizes

Resolvable block designs for v varieties in blocks of size k require v to be a multiple of k so that all blocks are of the same size. If a factorization of v is not possible then a resolvable design with blocks of unequal size is necessary. Patterson & Williams (1976) suggested the use of designs derived from α -designs and conjectured that such designs are likely to be very efficient in the class of resolvable designs with block sizes k and k – 1. This paper examines these derived designs and compares them with designs generated directly using an interchange algorithm. It concludes that the derived designs should be used when v is large, but that for small v they can be relatively inefficient.     

αn–Designs

This paper defines a broad class of resolvable incomplete block designs called α_n–designs, of which the original α–designs are a special case with n = 1. The statistical and mathematical properties of α–designs extend naturally to these n –dimensional designs. They are a flexible class of resolvable designs appropriate for use in factorial experiments, in constructing efficient t –latinized resolvable block designs, and for enhancing the existing class of α–designs for a single treatment factor.     

A REVIEW OF BOUNDS FOR THE EFFICIENCY FACTOR OF BLOCK DESIGNS

This paper draws together bounds for the efficiency factor of block designs, starting with the papers of Conniffe & Stone (1974) and Williams & Patterson (1977). By extending the methods of Jarrett (1983), firstly to cover supercomplete block designs and then to cover resolvable designs, a set of bounds is obtained which provides the best current bounds for any block design with equal replication and equal block size, including resolvable designs and two-replicate resolvable designs as special cases. The bounds given for non-resolvable designs apply strictly only to designs which are either regular-graph (John & Mitchell, 1977) or whose duals are regular-graph. It is conjectured (John & Williams, 1982) that they are in fact global bounds. Similar qualifications apply to the bounds for resolvable designs.     

Multilevel Augmented Pairs Second-Order Response Surface Designs and Their Robustness to Missing Data

Missing observations can occur even in a well-planned experiment. The effect of missing observations can be much more serious when the design is saturated or near saturated. The levels of factor settings that make a design more robust to missing observations are of great importance in the sense that the loss for missing observations becomes minimum. In this study, new augmented pairs minimax loss designs are constructed, which are more robust to one missing design point than the augmented pairs designs presented by Morris (2000 Morris , M. D. ( 2000 ). A class of three-level experimental designs for response surface modeling . Technometrics 42 : 111 – 121 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). New designs are compared with augmented pairs designs, central composite designs, and small composite designs under generalized scaled standard deviations. The model used is also studied for the regression coefficient estimates.     

On the equivalence of definitions for regular fractions of mixed-level factorial designs

The notion of regularity for fractional factorial designs was originally defined only for two-level factorial designs. Recently, rather different definitions for regular fractions of mixed-level factorial designs have been proposed by Collombier [1996. Plans d’Expérience Factoriels. Springer, Berlin], Wu and Hamada [2000. Experiments. Wiley, New York] and Pistone and Rogantin [2008. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138, 787–802]. In this paper we prove that, surprisingly, these definitions are equivalent. The proof of equivalence relies heavily on the character theory of finite Abelian groups. The group-theoretic framework provides a unified approach to deal with mixed-level factorial designs and treat symmetric factorial designs as a special case. We show how within this framework each regular fraction is uniquely characterized by a defining relation as for two-level factorial designs. The framework also allows us to extend the result that every regular fraction is an orthogonal array of a strength that is related to its resolution, as stated in Dey and Mukerjee [1999. Fractional Factorial Plans. Wiley, New York] to mixed-level factorial designs.     

Constructing 1-rotational NRDFs through an optimization approach: new (46,9,8), (51,10,9) and (55,9,8)-NRBDs

In this paper we formulate the problem of constructing 1-rotational near resolvable difference families as a combinatorial optimization problem where a global optimum corresponds to a desired difference family. Then, we develop an algorithm based on scatter search in conjunction with a tabu search to construct many of these difference families. In particular, we construct three new near resolvable difference families which lead to an equal number of new 1-rotational near resolvable block designs with parameters: (46,9,8), (51,10,9) and (55,9,8). Our results indicate that this conjunction outperforms both scatter search and tabu search.     

Efficient Block Designs for Dependent Observations—A Computer-Aided Search

In this article, the exchange and interchange algorithm of Zergaw (1989 Zergaw , G. ( 1989 ). A sequential method of constructing optimal block designs . Austral. J. Statist. 31 : 333 – 342 .[Crossref] , [Google Scholar]) and Martin and Eccleston (1992 Martin , R. J. , Eccleston , J. A. ( 1992 ). Recursive formulae for constructing block designs with dependent errors . Biometrika 79 : 426 – 430 .[Crossref], [Web of Science ®] , [Google Scholar]) have been modified and used for searching efficient block designs for making all possible pairwise treatment comparisons when observations are dependent. The lower bounds to the A- and D-efficiencies of the designs in a given class of the designs have been obtained for correlated observation structure and the procedure of computing lower bounds to A- and D-efficiencies has been incorporated in the algorithm. The algorithm has been translated into a computer program using Microsoft Visual C++. Using this program, a search for efficient designs for making all possible pairwise treatment comparisons has been made for v ≤ 10, b ≤ 33, k ≤ 10 such that bk ≤ 100 and v > k. The block designs considered are usual block designs (rectangular block designs) and circular block designs. Nearest neighbor (NN), autoregressive of order 1 (AR(1)) correlation structures are studied. The ranges of correlation coefficients for different correlation structures investigated are |ρ|≤0.50 for NN correlation structure in rectangular blocks, |ρ|≤0.45 for NN correlation structure in circular blocks, and |ρ|≤0.95 for AR(1) correlation structure. For these ranges, the matrix of correlation coefficients among observations within a block is positive definite. Robustness aspects of designs that are efficient for a given value of correlation have been investigated against other values of correlation coefficients. Robustness aspects of designs that are efficient for independent observations have also been studied for experimental situations with dependent observations.     

Construction of circular partially balanced-Repeated measurement designs using cyclic shifts

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. These designs balance out the residual effects. The situations where balanced repeated measurements designs require a large number of the subjects, partially-balanced repeated measurements designs should be used. In this paper some infinite series are developed which provide circular partially-balanced repeated measurement designs for p (periods) even. Catalogues of circular partially-balanced repeated measurement designs are also presented for v (treatments) ≤ 100 with p = 5, 7 & 9.     

CONSTRUCTION OF RESOLVABLE ROW-COLUMN DESIGNS BASED ON CONTRACTIONS

This paper defines the contraction of a resolvable row‐column design for more than two replicates. It shows that the (M,S)‐optimality criterion for the row‐column designs can be expressed simply in terms of the elements of the row and column incidence matrices of the contraction. This allows the development of a very fast algorithm to construct optimal or near‐optimal resolvable row‐column designs. The performance of such an algorithm is compared with an existing algorithm.     

Rotatability is a Sufficient Condition for A- and D-Rotatability

Designs for estimating axial slopes of a response surface are considered. A design is said to be A- (D-) rotatable if the trace (determinant) of variance–covariance matrix of the estimated slopes at a point in the factor space is a function of the distance of the point from the center of the design. It is shown that for any order d and any dimension k, a design which is rotatable in the Box and Hunter (1957 Box , G. E. P. , Hunter , J. S. ( 1957 ). Multifactor experimental designs for exploring response surfaces . Annals of Mathematical Statistics 28 : 195 – 241 .[Crossref] , [Google Scholar]) sense is automatically A- and D-rotatable. Some specific examples illustrating the validity of this general result are provided.     

Supersaturated designs with high searching probability

A supersaturated design is essentially a fractional factorial design whose number of experimental variables is greater than or equal to its number of experimental runs. Under the effect sparsity assumption, a supersaturated design can be very cost-effective. In this paper, our prime objective is to compare the existing two-level supersaturated designs for the noisy case through the probability of correct searching—a powerful criterion proposed by Shirakura et al. [1996. Searching probabilities for nonzeroeffects in search designs for the noisy case. Ann. Statist. 24, 2560–2568]. An algorithm is proposed to construct supersaturated designs with high probability of correct searching. Examples are given for illustration.     

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2^n-p$2^{n-p}$ fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.     

Designs Robust against Violation of Normality Assumption in the Standard F-test

In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962 Atiquallah, M. (1962). The estimation of residual variance in quadratically balanced least-squares problems and the robustness of the F-test. Biometrika 49: 83–91.[Crossref], [Web of Science ®] , [Google Scholar]) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012 Biswas, A. (2012). Block designs robust against the presence of an aberration in a treatment-control setup. Commun Statist.Theor Methods 41: 920–933.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.     

Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978 Hader , R. J. , Park , S. H. ( 1978 ). Slope-rotatable central composite designs . Technometrics 20 : 413 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.     

Families of proper generalized neighbor designs

A generalized neighbor design relaxes the equality condition on the number of times two treatments occur as neighbors in the design. In this article we have constructed a new series of generalized neighbor designs with equal block sizes, a series of neighbor designs of Rees [1967. Some designs of use in serology. Biometrics 23, 779–791] and a series of neighbor designs with two distinct block sizes. Two more new series of GN₂ designs are also constructed for even number of treatments. It has been shown that quasi neighbor designs introduced by Preece [1994. Balanced Ouchterlony neighbor designs. J. Combin. Math. Combin. Comput. 15, 197–219] are special cases of generalized neighbor designs with t=2$t=2$. All the designs given here are binary. A new definition—partially balanced circuit design is introduced which is a special case of generalized neighbor designs with binary blocks.