Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

Cubic lattices

In some experiments, the problem is to compare many unstructured treatments in small blocks, the classical example being the study of new plant varieties on variable land. A common method is to use lattice designs, i.e. block designs based upon rows and columns of a square format, with ficrther replicates being formed, required, from orthogonal squares applied to the format. It has been known for some time that cubes can be used instead; this paper sets out to explore the possibilities. There are two cases. In one case, the blocks are formed from the planes of the cube and, in the other case, from its lines. The cubic lattice basically has three replicates-one from each dimension-but, two or four replicates are required, a design can be found by omitting or duplicating one of the dimensions. Where standard treatments need to be introduced, -a useful device is to reinforce, i.e. supplement each block with additional plots of standards, with each block of a replicate being supplemented in the same way. These possibilities are examined. It emerges that cubic lattices with two or three replicates usefully extend the range of available designs, but that those with four replicates are disappointing. However, there is the alternative of using designs based upon Latin cubes. This matter is not taken far but it is shown that, where the Latin cube exists, it gives a better design. A quick way of calculating an approximate analysis of variance is given, which is applicable in a wide range of cases.     

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.     

Orthogonal sets of balanced incomplete block designs with nested rows and column

Known series of balanced incomplete block designs with nested rows and columns are used to find orthogonal sets of these designs, producing main effects plans in nested rows and columns. Two infinite series are so constructed and shown to be universally optimum for the analysis with recovery of row and column information, a benefit produced by the additional higher strata orthogonality they enjoy. One of these series achieves orthogonality with just v − 1 replicates of v treatments, fewer than required by Latin squares.     

Block-circulant matrices for constructing optimal Latin hypercube designs

Computer simulations are usually needed to study a complex physical process. In this paper, we propose new procedures for constructing orthogonal or low-correlation block-circulant Latin hypercube designs. The basic concept of these methods is to use vectors with a constant periodic autocorrelation function to obtain suitable block-circulant Latin hypercube designs. A general procedure for constructing orthogonal Latin hypercube designs with favorable properties and allowing run sizes being different from a power of 2 (or a power of 2 plus 1), is presented here for the first time. In addition, an expansion of the method is given for constructing Latin hypercube designs with low correlation. This expansion is useful when orthogonal Latin hypercube designs do not exist. The properties of the generated designs are further investigated. Some examples of the new designs, as generated by the proposed procedures, are tabulated. In addition, a brief comparison with the designs that appear in the literature is given.     

Projection array based designs for computer experiments

In this paper we define a new class of designs for computer experiments. A projection array based design defines sets of simulation runs with properties that extend the conceptual properties of orthogonal array based Latin hypercube sampling, particularly to underlying design structures other than orthogonal arrays. Additionally, we illustrate how these designs can be sequentially augmented to improve the overall projection properties of the initial design or focus on interesting regions of the design space that need further exploration to improve the overall fit of the underlying response surface. We also illustrate how an initial Latin hypercube sample can be expressed as a projection array based design and show how one can augment these designs to improve higher dimensional space filling properties.     

Mixture designs for five components in orthogonal blocks

The methods developed by John and Draper et al. of partitioning the blends (runs) of four mixture components into two or more orthogonal blocks when fitting quadratic models are extended to mixtures of five components. The characteristics of Latin squares of side five are used to derive rules for reliably and quickly obtaining designs with specific properties. The designs also produce orthogonal blocks when higher order models are fitted.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

EQUINEIGHBOURED XPERIMENTAL DESIGNS

This paper introduces a new class of designs called equi-neighboured designs. An equineighboured design has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at every level. These designs are defined in Section 4 and shown to be balanced when neighbouring observations are correlated. Some equineighboured designs are constructed using a complete set of orthogonal Latin squares. Cyclic equineighboured designs are also defined.     

Crossover designs in the presence of carry-over effects from two factors

Experiments, used in the telecommunications industry and elsewhere, are considered that involve the simultaneous application of levels of two unrelated factors, treatments and stimuli, to each of several subjects in a succession of time periods. The existence is suspected of carry-over effects of treatments and stimuli, in the period immediately following the period of their application. Methods are given for the construction of separate sequences of treatments and of stimuli; these methods are based on the Latin squares of Williams and of Russell. In the resulting designs, the treatments and stimuli are either orthogonal or nearly orthogonal, and the coincidence of the direct and carry-over effects of each factor is either balanced or nearly balanced. The efficiencies of the designs are assessed by comparing the average variances of elementary contrasts in the levels of each factor with appropriate lower bounds.     

ON RESOLVABLE INCOMPLETE BLOCK DESIGNS1

A method of constructing resolvable incomplete block designs for v(=ks, 2 ≤ k ≤ s - 1) treatments in blocks of size k using mutually orthogonal Latin squares is proposed. It has been seen in particular that when the number of replications is s — 1 (or s), which is feasible if s is a prime or a prime power, the method gives PBIB (3) (or semi-regular GD) designs. The analysis of such designs has also been discussed.     

Cutting Experimental Designs into Blocks

Most experimental material in agriculture and industry is heterogeneous in nature and therefore its statistical analysis benefits from blocking. Many experiments are restricted in time or space, and again blocking is useful. This paper adopts the idea of orthogonal blocking of Box & Hunter (1957) and applies it to optimal blocking designs. This approach is then compared with the determinant-based approach described in the literature for constructing block designs.     

Symmetry in mixture experiments

Two types of symmetry can arise when the proportions of mixture components are constrained by upper and lower bounds. These two types of symmetry are shown to be useful for blocking first-order designs, as well as for finding the centroid of the experimental region. Orthogonal blocking of first-order mixture designs provides a method of including process variables in the mixture experiment, with the mixture terms orthogonal to the process factors. Symmetric regions are used to develop spherical and rotatable response surface designs for mixtures. The central composite design and designs based on the icosahedron and the dodecahedron are given for four-component mixtures. The uniform shell designs are three-level designs when applied to mixture experiments.     

A class of single replicate orthogonal designs

A simple method is given to calculate the number of degrees of freedom confounded with blocks of a specific factorial effect in a single replicate orthogonal design. Two classes of designs having partial orthogonality are also discussed     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

Some new construction of circular weakly balanced repeated measurements designs in periods of two different sizes

Abstract

Balanced repeated measurements designs (RMDs) balance out the residual effects. Williams Latin square designs work as minimal combinatorial balanced as well as variance balanced for RMDs for p (period sizes) = v (number of treatments). If minimal balanced RMDs cannot be constructed for the situations where p must be less than v then weakly balanced RMDs should be preferred. In this article, some generators are developed to generate circular weakly balanced RMDs in periods of two different sizes. To obtain the proposed designs, some construction procedures are also described for some of the cases where we could not develop generators.     

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.     

Change-over designs with complete balance for first and second residual effects

This paper presents a unified method of constructing change-over designs that permit the estimation of direct effects orthogonal to all other effects when the residual effects of treatments last for two consecutive periods. Explicit methods of analysis of these designs have been obtained for the situations where the first period observations are omitted from the analysis and where the first period observations are included.     

Orthogonal Latin hypercube designs for Fourier-polynomial models

Latin hypercube designs (LHDs) are widely used in computer experiments because of their one-dimensional uniformity and other properties. Recently, a number of methods have been proposed to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear interactions. This paper focuses on the construction of LHDs with the above desirable properties under the Fourier-polynomial model. A convenient and flexible algorithm for constructing such orthogonal LHDs is provided. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new and very suitable for factor screening and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).