On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

An easy method of constructing latin square designs balanced for the immediate residual and other order effects

Bradley (1958) proposed a very simple procedure for constructing latin square designs to counterbalance the immediate sequential effect for an even number of treatments. When the number of treatments is odd, balance in a single latin square is not possible. In the present note we have developed an analogous method for the construction of such designs which may be used for an even or odd number of treatments. A proof has also been offered to assure the general validity of the procedure.     

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.     

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.     

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.     

Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

Construction of minimum aberration blocked two-level regular factorial designs

ABSTRACT

In this article, we consider the construction of minimum aberration 2^{n ? k}: 2^p designs with respect to some existing combined wordlength patterns, where a 2^{n ? k}: 2^p design is a blocked two-level design with n treatment factors, 2^p blocks, and N = 2^q runs with q = n ? k. Two methods are proposed for two situations: n ? 2^{q ? p} ? 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2^{n ? k}: 2^p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.     

Construction of repeated measurements designs strongly balanced for residual effects

Abstract

Repeated Measurements Designs have been widely used in agriculture, animal husbandry, education, biology, botany and engineering. Balanced or strongly balanced repeated measurements designs are useful to balance out the residual effects. In this article, some new generators and construction procedures are proposed to obtain circular strongly balanced repeated measurements designs in periods of (a) equal sizes, (b) two different sizes, and (c) three different sizes.     

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.     

A new variable selection method for uniform designs

As an important class of space-filling designs, uniform designs (UDs) choose a set of points over a certain domain such that these points are uniformly scattered, under a specific discrepancy measure. They have been applied successfully in many industrial and scientific experiments since they appeared in 1980. A noteworthy and practical advantage is their ability to investigate a large number of high-level factors simultaneously with a fairly economical set of experimental runs. As a result, UDs can be properly used as experimental plans that are intended to derive the significant factors from a list of many potential ones. To this end, a new screening procedure is introduced via penalized least squares. A simulation study is conducted to support the proposed method, which reveals that it can be considered quite promising and expedient, as judged in terms of Type I and Type II error rates.     

Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

Interruptible supersaturated two-level designs

Interruptible designs possess a robustness against possible premature termination of an experiment. We consider such two-level designs for a first-order model and present interruptible sequences which lead to the D-optimal saturated design for four to nine factors if not interrupted. Premature termination of the experiment at any stage results in a supersaturated design with minimum loss of information about the factors. The loss for these designs, which is measured by the pairwise orthogonality between columns, is compared with that of the worst case f o r randomly ordered sequences.     

Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

m-Associate PBIB designs using Youden-m squares

In this paper, we have proposed a type of arrangement that we call Youden-m square and is similar to the usual Youden square but generates PBIB designs instead of BIB designs when its columns are taken as blocks. We have also discussed its construction methodologies, introduced two new m-associate class association schemes, and also constructed some series of Youden-m square type PBIB designs.     

Experimental designs and their efficiencies for spatially correlated observations in two dimensions

Optimality of experimental designs for spatially correlated observations is investigated.come two dimensional correlation structures are discussed and an attempt has been made to find optimal or nearly optimal design for each sitution.The solution lend to designs similar to that used for repeated measurements.The relative efficiency of the proposed designs in comparison to randomized latin square designs is tabulated for some cases.     

On the analysis of multivariate repeated measures designs

The relationship between the mixed-model analysis and multivariate approach to a repeated measures design with multiple responses is presented. It is shown that by taking the trace of the appropriate submatrix of the hypothesis (error) sums of squares and crossproducts (SSCP) matrix obtained from the multivariate approach, one can get the hypothesis (error) SSCP matrix for the mixed-model analysis. Thus, when analyzing data from a multivariate repeated measures design, it is advantageous to use the multivariate approach because the result of the mixed-model analysis can also be obtained without additional computation.     

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.     

A class of new pbib designs

Two series of PBIB designs, one with three associate classes and theother with four associate classes ars developed. Efficiency factors for two designs are computed.     

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.     

Generalizing Clatworthy group divisible designs

In this paper a neat construction is provided for three new families of group divisible designs that generalize some designs from Clatworthy's table of the only 11 designs with two associate classes that have block size four, three groups, and replication numbers at most 10. In each case (namely, _λ1=4${\lambda }_1=4$ and _λ2=5${\lambda }_2=5$, _λ1=4${\lambda }_1=4$ and _λ2=2${\lambda }_2=2$, and _λ1=8${\lambda }_1=8$ and _λ2=4${\lambda }_2=4$), we have proved that the necessary conditions found are also sufficient for the existence of such GDD's with block size four and three groups, with one possible exception.