On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also presented.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.     

Robustness of complete diallel crossing plans against exchange of one cross

The robustness aspects of block designs for complete diallel crossing plans against the exchange of one cross using connectedness and efficiency criteria have been investigated. The exchanged cross may have either no line in common or one line in common with the original cross. It has been found that randomized complete block (RCB) designs for complete diallel crosses and binary balanced block designs for complete diallel crosses are robust against the exchange of one cross in one observation. The RCB designs for diallel crosses have been shown to be robust against the exchange of one cross with another cross in all the blocks. The non-binary balanced block designs obtainable from Family 5 of Das et al. (1998) have also been found to be robust against the exchange of one cross.     

Designs for cropping systems research

The present article establishes equivalence between extended group divisible (EGD) designs and designs for crop sequence experiments. This equivalence has encouraged the agricultural experimenters to use EGD designs for their experimentation. Some real life applications of EGD designs have been given. It has also been shown that several existing association schemes are special cases of EGD association scheme. Some methods of construction of EGD designs are also given. A catalogue of EGD designs obtainable through methods of construction along with efficiency factors of various factorial effects is also presented. In some crop sequence experiments that are conducted to develop suitable integrated nutrient supply system of a crop sequence, the treatments do not comprise of a complete factorial structure. The experimenter is interested in estimating the residual and direct effect of the treatments along with their cumulative effects. For such experimental settings block designs with two sets of treatments applied in succession are the appropriate designs. The correspondence established between row–column designs and block designs for two stage experiments by Parsad et al. [2003. Structurally incomplete row–column designs. Comm. Statist. Theory Methods 32(1), 239–261] has been exploited in obtaining designs for such experimental situations. Some open problems related to designing of crop sequence experiments are also given.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

A SEQUENTIAL METHOD OF CONSTRUCTING OPTIMAL BLOCK DESIGNS

An algorithm for the construction of a wide class of block designs including Balanced Incomplete Blocks (BIB) is described. The algorithm which allows the experimenter to give weights for a set of treatment contrasts uses an initial starting design to generate an optimal block design sequentially. The performance of the algorithm is illustrated by examples, and designs constructed by the algorithm compare favourably with designs generated by other methods.     

Non-binary neighbor balance circular designs for v=2n and λ=2

Neighbor balance designs were first introduced by Rees (1967) in circular blocks for the use in serological research. Subsequently several researchers have defined the neighbor designs in different ways. In this paper, neighbor balance circular designs for (k≤v) block size are constructed for even number of treatments i.e. v=2n. No such series of designs is known in literature. Two theorems are developed for circular designs. Theorem 1 gives the non-binary circular blocks, whereas Theorem 2 generates binary circular blocks when n≤4 and non-binary blocks for n>4. In suggested designs no treatment is ever a neighbor of itself. Blocks are constructed in such a way that each treatment is a right and left neighbor of every other treatment for a fixed number of times say λ. Sizes of initial circular blocks are not same. One main guiding principle for such designs is to ensure economy in material use.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

On the recovery of intra- and interblock information in n-ary designs

This note presents an extension of Q-method of analysis for binary designs given by Rao (1956) to n-ary balanced and partially balanced block designs. Here a linked n-ary block (LNB) design is defined as the dual of balanced n-ary (BN) design. Having a note on Yates’ (1939, 1940) method of P-analysis, we further extend the expressions for binary linked block (LB) designs given by Rao (1956) to linked n-ary block (LNB) designs which admit easy estimation of parameters for these type of all n-ary designs.     

Generalized neighbor designs with block size 3

A generalized neighbor design relaxes the equality condition on the number of times two treatments as neighbors in the design. In this article, we have considered the construction of some classes of generalized neighbor designs with block size k=3 by using the method of cyclic shifts. The distinguishing feature of this construction method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. A catalog of generalized neighbor designs with block size k=3 is compiled for v∈{5,6,…,18} treatments and for different replications. We provide the reader with a simpler method of construction, and in general the catalog that gives an open choice to the experimenter for selecting any class of neighbor designs.     

Optimal orthogonal block designs for four-mixture components in two blocks based on F-squares for Becker's models and K-model

Orthogonal block designs in mixture experiments have been extensively studied by various authors. Aggarwal et al. [M.L. Aggarwal, P. Singh, V. Sarin, and B. Husain, Mixture designs in orthogonal blocks using F-squares, METRON – Int. J. Statist. LXVII(2) (2009), pp. 105–128] considered the case of components assuming the same volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we have used the class of designs presented by Aggarwal et al. and have obtained D-, A- and E-optimal orthogonal block designs for four components in two blocks for Becker's mixture models and K-model, respectively. Orthogonality conditions for the considered models are also given.     

Construction of balanced incomplete block designs with partial cycles of blocks

When considering systems statisfying balanced incomplete block design structure, one is interested only in distinct configurations in the sense of isomorphism. We have determined additional non-isomorphic solutions to designs with previously established solutions through consideration of automorphism groups. This effort utilizes a classic technique of direct construction, namely, symmetrically repeated differences, modified slightly when constucting designs with partial cycles of blocks. To determine te base blocks, we utilize the computer to expedite some formidable enumeration     

The efficiency and optimality characterization of certain six-Treatment incomplete-Block designs emanating from some quasi-Semi-Latin squares

Incomplete-block designs were constructed from three types of three-factor block-structured combinatorial designs which are not necessarily semi-Latin squares but whose treatments are ab initio arranged as in the semi-Latin square. The basic consideration for the construction involves the exploitation of the different block structures of the experimental units of the Quasi-semi-Latin Squares (QSLS) for the same treatments. All the emanating designs are found to be connected, only the designs in nine blocks are both variance- and efficiency-balanced and offer gain in efficiency but lead to about 46 percent loss of information.     

Adaptively combining independent jonckheere statistics in a randomized block design with unequal scales

Jonckheere (1954) proposed a test statistic which is commonly used in testing for ordered alternatives in block designs.- We consider the application of Jonckheere's test statistic in block designs which have unequal scale parameters for the blocks. Estimates of the unknown scale parameters ar-fcrmed and are used to construct a modification of Jonckheere's test statistic using adaptive ideas. A Monte Carlo study shows that the modified Jonckheere is significantly more powerful than the original Jonckheere in many unequal scale situations.     

Use of n-ary block designs in diallel crosses evaluation

To fill the gap between theory and practice of modern statistical designs, this paper presents the use of n-ary block designs in the evaluation of balanced incomplete block (BIB) designs for all the four Griffing's (1956) complete dialled crosses (CDC) systems, the construction of which is proposed here by adopting a suitable BIB design and associating its treatments with the crosses under a CDC system.     

Optimum designs for estimation of regression parameters in a balanced treatment incomplete block design set-up

The use of covariates in block designs is necessary when the experimental errors cannot be controlled using only the qualitative factors. The choice of values of the covariates for a given set-up attaining minimum variance for estimation of the regression parameters has attracted attention in recent times. In this paper, optimum covariate designs (OCD) have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs. It is seen that the OCDs depend much on the methods of construction of the basic BTIB designs. The series of BTIB designs considered in this paper are mainly those as described by Bechhofer and Tamhane (1981) and Das et al. (2005). Different combinatorial arrangements and tools such as Hadamard matrices and different kinds of products of matrices viz Khatri-Rao product and Kronecker product have been conveniently used to construct OCDs with as many covariates as possible.     

REDUCED RESIDUE CLASSES CYCLIC PBIB DESIGNS

A new general class of m-class cyclic association scheme is defined for v treatments, where v is a composite number. A simple method of construction of PBIB designs having this association scheme using more than one initial block and some methods using only one initial block are proposed. A complete analysis of this type of PBIB designs is given. Also given is a list of 39 useful PBIB designs of this type having v≤15 and r≤10 and having only three associate classes together with their efficiency factors for all types of comparisons and over all efficiency factors.     

The analysis of mixed and random effects models for equireplicate variance-balanced block designs

Equirephcate variance-balanced block designs possess many desirable properties and include several important subclasses of designs, such as balanced incomplete block designs. However, while attention has been paid to the derivation of properties and the construction of these designs, methods of analyzing experiments using these designs have not been developed completely. Thus, we consider models appropriate to these designs, construct test statistics for basic hypotheses and derive the distributions for these statistics. Four additive models that include all of the possible combinations of fixed and random effects are examined. The analyses are generalizations of the analyses for balanced incomplete block designs, and include the intra-block, intra-treatment and inter-block analyses, Estimators of fixed treatment effects and their distributions also are given.     

A note on generalized binary proper efficiency-balanced block designs

Some methods of construction of generalized binary proper efficiency-balanced block (GB-EB) designs with two different replications are derived and the solution of the unknown designs given by Das and Kageyama (1991) are obtained.