On the construction of group divisible incomplete block designs

In this paper a method of constructing group-divisible incomplete block designs has been suggested. A series of balanced incomplete block designs has also been obtained.     

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.     

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.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

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.     

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.     

A generating algorithm for linear trend-free and nearly linear trend-free block designs

The theory and properties of trend-free (TF) and nearly trend-free (NTF) block designs are wel1 developed. Applications have been hampered because a methodology for design construction has not been available.

This article begins with a short review of concepts and properties of TF and NTF block designs. The major contribution is provision of an algorithm for the construction of linear and nearly linear TF block designs. The algorithm is incorporated in a computer program in FORTRAN 77 provided in an appendix for the IBM PC or compatible microcomputer, a program adaptable also to other computers. Three sets of block designs generated by the program are given as examples.

A numerical example of analysis of a linear trend-free balanced incomplete block design is provided.     

On a class of variance balanced incomplete block designs

In this paper variance balanced incomplete block designs have been constructed for situations when suitable BIB designs do not exist for a given number of treatments, because of the contraints bk=vr, λ(v-1) = r(k-l). These variance balanced designs are in unequal block sizes and unequal replications.     

Incomplete block designs for complete diallel crosses and their analysis

A large number of incomplete block designs for Griffing's complete diallel cross-systems I, II and III, involving from five to 12 lines, are suggested, using two-associate triangular partially balanced incomplete block designs. Analysis of incomplete block designs for complete diallel cross-systems has been carried out assuming the most appropriate model for genetic yield, as advocated by Hinklemnann. This includes estimation of the general combining ability, specific combining ability and reciprocal cross- effects. An illustration of the design for each system is presented.     

ON DERIVATION OF INITIAL BLOCKS OF B.I.B. DESIGNS WITH MORE THAN ONE INITIAL BLOCK

A good amount of work has been done on the construction of balanced incomplete block (B.I.B.) designs by Bose (1939, 1942), Sprott (1954, 1956), Rao (1961), Takeuchi (1962) and others. Sprott (1954, 1956) obtained several series of B.I.B. designs through difference sets. The main purpose of the present investigation is to provide two methods of construction of B.I.B. designs obtainable through more than one initial block. The first method derives initial blocks of a series of designs from some specified blocks of a B.I.B. design obtainable by developing one or more initial blocks. The second method attempts to obtain one of the initial blocks (the basic initial block) through the different powers of an element of a finite field; then an appropriate method for generating the other initial blocks from it is discussed. A table showing the basic initial block for different designs has been presented. By these methods several solutions of some B.I.B. designs could be obtained from different initial blocks. An examination was therefore made to see if these designs were all isomorphic.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

D-1-Partially efficiency-balance designs with at most two efficiency classes

A wide class of block designs admitting a simple analysis has been considered. The statistical properties of such designs have been indicated and the problems relating to their characterization and construction have been investigated.     

On the optimalizes of the mbgdd's under mixed effects model

In the present paper, under the assumption of a mixed effects model with random block effects, the type 1 optimality of the most balanced group divisible designs of type 1 has been established within the general class of all proper and connected block designs with k<v.     

SOME SERIES OF EFFICIENCY BALANCED DESIGNS

Two series of efficiency balanced designs with v*+ 1 treatments have been constructed using balanced incomplete block designs having v* treatments.     

Single replicate orthogonal block designs for circulant partial diallel crosses

Some incomplete block designs for partial diallel crosses have been given in the literature. These designs are obtained by regarding the number of crosses as treatments, and consequently require several replications of each cross. The need for resorting to a partial diallel cross itself implies that it is desired to have fewer crosses. A method for constructing single replicate incomplete block designs for circulant partial diallel crosses is provided in this paper. The designs are orthogonal, and thus they retain full efficiency for estimation of the contrasts of interest.     

Optimal block designs with non additive mixed effects interference

This paper deals with optimality aspects of block designs balanced for interference effects from neighboring units on both sides under a general non additive model along with random block effects. Here, a class of complete, circular block designs strongly balanced for interference effects has been shown to be universally optimal for the estimation of direct effects and interference effects (left and right) of treatments under a non additive mixed effects model.     

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.     

On totally balanced block designs for competition effects

Competition between neighbouring units in field experiments is a serious source of bias. The study of a competing situation needs construction of an environment in which it can happen and the competing units have to appear in a predetermined pattern. This paper describes methods of constructing incomplete block designs balanced for neighbouring competition effects. The designs obtained are totally balanced in the sense that all the effects, direct and neighbours, are estimated with the same variance. The efficiency of these designs has been computed as compared to a complete block design balanced for neighbours and a catalogue has also been prepared.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

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.