On the construction of balanced and orthogonal arrays

Two series of three symbol balanced arrays of strength two are constructed. Using special classes of BIB designs, two classes of two symbol orthogonal arrays of strength three are constructed.     

Minimal point second order designs

This paper deals with the problem of finding nearly D-optimal designs for multivariate quadratic regression on a cube which take as few observations as possible and still allow estimation of all parameters. It is shown that among the class of all such designs taking as many observations as possible on the corners of the cube there is one which is asymptotically efficient as the dimension of the cube increases. Methods for constructing designs in this class, using balanced arrays, are given. It is shown that the designs so constructed for dimensions ≤6 compare well with existing computer generated designs, and in dimensions 5 and 6 are better than those in literature prior to 1978.     

Orthogonal arrays of strength three from regular 3-wise balanced designs

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.     

Efficient designs based on orthogonal arrays of type I and type II for experiments using units ordered over time or space

For a wide variety of applications, experiments are based on units ordered over time or space. Models for these experiments generally may include one or more of: correlations, systematic trends, carryover effects and interference effects. Since the standard optimal block designs may not be efficient in these situations, orthogonal arrays of type I and type II, which were introduced in 1961 by C.R. Rao [Combinatorial arrangements analogous to orthogonal arrays, Sankhya A 23 (1961) 283–286], have been recently used to construct optimal and efficient designs for many of these experiments. Results in this area are unified and the salient features are outlined.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

Crossover designs based on type I orthogonal arrays for a self and simple mixed carryover effects model with correlated errors

We investigate the performance of crossover designs based on type I orthogonal arrays for a self and simple mixed carryover effects model in the presence of correlated errors. Assuming that between-subject errors are independent while within-subject errors behave according to the stationary first-order autoregressive and moving average processes, analytical optimality results for 3-period designs are established and, as an illustration, numerical details for a number of 4-period cases are tabulated.     

Mixture designs with orthogonal blocks

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Construction of more Flexible and Efficient P–rep Designs

Early generation variety trials are very important in plant and tree breeding programs. Typically many entries are tested, often with very few resources available. Unreplicated trials using control plots are popular and it is common to repeat the trials at a number of locations. An alternative is to use partially replicated (p–rep) designs, where a proportion of the test entries are replicated at each location. We extend a method for the generation of p–rep designs based on α–arrays to allow for a much broader class of designs to be constructed. Updating procedures for the average efficiency factor and its upper bound are developed for application to the computer generation of efficient p–rep designs.     

A Comparison of Three-level Orthogonal Arrays in the Presence of a Possible Correlation in Observations

When we want to compare two designs we usually assume the standard linear model with uncorrelated observations. In this paper we use the comparison method proposed by Ghosh & Shen (2006) to compare three level orthogonal arrays with 18, 27 and 36 runs under a possible presence of correlation in observations.     

The Hamming distances of saturated asymmetrical orthogonal arrays with strength 2

Abstract

Orthogonal arrays have many connections to other combinatorial designs and are applied in coding theory, the statistical design of experiments, cryptography, various types of software testing and quality control. In this paper, we present some general methods to find the Hamming distances for saturated asymmetrical orthogonal arrays (SAOAs) with strength 2. As applications of our methods, the Hamming distances of SAOA parents of size less than or equal to 100 are obtained. We also provide the Hamming distances of the SAOAs constructed from difference schemes or by the expansive replacement method. The feasibility of Hamming distances is discussed.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

2 m 4 n designs with resolution III or IV containing clear two-factor interaction components

The orthogonal arrays with mixed levels have become widely used in fractional factorial designs. It is highly desirable to know when such designs with resolution III or IV have clear two-factor interaction components (2fic’s). In this paper, we give a complete classification of the existence of clear 2fic’s in regular 2 ^m 4 ⁿ designs with resolution III or IV. The necessary and sufficient conditions for a 2 ^m 4 ⁿ design to have clear 2fic’s are given. Also, 2 ^m 4 ⁿ designs of 32 runs with the most clear 2fic’s are given for n = 1,2.      

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

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 article, we have identified all the geometric non isomorphic 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 when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices

In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004 Basso , D. , Evangelaras , H. , Koukouvinos , C. , Salmaso , L. ( 2004 ). Nonparametric testing for main effects on inequivalent designs. Proc. 7th Int. Workshop Model-Oriented Design Anal. Heeze, Netherlands, June 14–18 . [Google Scholar]) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified.     

Space-filling orthogonal arrays of strength two

This paper considers the use of orthogonal arrays of strength two as experimental designs for fitting a surrogate model. Contrary to standard space-filling designs or Latin hypercube designs, the points of an orthogonal array of strength two are well distributed when they are projected on the two-dimensional faces of the unit cube. The aim is to determine if this property allows one to fit an accurate surrogate model when the computer response is governed by second-order interactions of some input variables. The first part of the paper is devoted to the construction of orthogonal arrays with space-filling properties. In the second part, orthogonal arrays are compared with standard designs for fitting a Gaussian process model.     

Some new constructions of strength 3 mixed orthogonal arrays

Orthogonal arrays of strength 3 permit estimation of all the main effects of the experimental factors free from confounding or contamination with 2-factor interactions. We introduce methods of using arithmetic formulations and Latin squares to construct mixed orthogonal arrays of strength 3. Although the methods could be well extended to computing larger arrays, we confine computing to at most 100 run orthogonal arrays for practical uses. We find new arrays with run sizes 80 and 96, each has many distinct factor levels.     

Efficient arrangements of two-level orthogonal arrays in two and four blocks

When orthogonal arrays are used in practical applications, it is often difficult to perform all the designed runs of the experiment under homogeneous conditions. The arrangement of factorial runs into blocks is usually an action taken to overcome such obstacles. However, an arbitrary configuration might lead to spurious analysis results. In this work, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed. This method is based on the so-called J-characteristics of the corresponding array. General theoretical results are given for studying up to four experimental factors in two blocks, as well as for studying up to three experimental factors in four blocks. Finally, we provide best blocking arrangements when the number of the factors of interest is larger, by exploiting the known lists of non-isomorphic orthogonal arrays with two levels and various run sizes.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

Constructions of group divisible designs and related combinatorial structures

Several methods of constructing group divisible (GD) designs are given by use of rectangular designs and nested balanced incomplete block designs. The GD designs obtained here are rather large, but as series they appear to be new. In the process, some series of rectangular designs, balanced arrays, and orthogonal arrays are also provided.