On Partially Linked Block Designs

In agricultural, animal, fisheries and industrial experimentation under block design setup, systematic trend may affect the response under consideration. Although remote, these effects may still have high influence on response and hence should be incorporated in the model for proper model specification. Completely trend resistant block designs, also known as trend free block designs, are an important class of designs where the analysis can be done in usual manner as in case of normal block designs since trend effects and treatment effects are orthogonal. Literature is available on different aspects of trend resistant incomplete block designs mainly focusing on balanced incomplete block designs. This article deals with the construction of completely trend resistant Partially Balanced Incomplete Block (TR-PBIB) designs. Two methods of constructing such designs have been discussed. The information matrices for the class of designs developed have been derived and SAS code for the generation of the developed designs has also been provided.

BCH codes and Designs

Partially Balanced Incomplete Block (PBIB) designs are the primary subject of this investigation of independent dominance in block designs. In the combinatorial structure of PBIB-designs, where treatments and blocks are depicted as vertices and edges, respectively, the graph-theoretic parameter known as independent dominance is applied. Within the framework of PBIB-designs defined by different association schemes, the study delves into the theoretical foundations, computation, and consequences of independent domination. Finding autonomous dominating sets that effectively represent all treatments while minimizing redundancy is the goal of the study, which offers a paradigm for doing so by examining the links between blocks and treatments. Experiment design optimization using independent dominating parameters is demonstrated in case studies of 2-associate and 3-associate PBIB-designs. This study sheds light on the structural features of PBIB-designs by demonstrating the interaction between graph theory and combinatorial design theory. In situations where resources are limited, it highlights the value of independent dominance as a method for assessing and improving block designs’ efficiency. Our current understanding of block designs and how to optimize them for use in experimental planning, data analysis, and resource allocation is enhanced by these findings.

An Erratum has been published for this article in Journal of Combinatorial Designs 14: 83–83, 2006.We enumerate a list of 594 inequivalent binary (33,16) doubly‐even self‐orthogonal codes that have no all‐zero coordinates along with their automorphism groups. It is proven that if a (22,8,4) Balanced Incomplete Block Design were to exist then the 22 rows of its incident matrix will be contained in at least one of the 594 codes. Without using computers, we eliminate this possibility for 116 of these codes. © 2005 Wiley Periodicals, Inc. J Combin Designs.

This paper is concerned primarily with a Monte Carlo investigation into the small sample robustness and power of an aligned rank transformation statistic. The aligned rank transformation statistic (ART) is compared to the classical F-test and to Durbin's (1951) rank test for its ability to detect treatment effects in balanced incomplete block (BIB) designs.

Bose and Shimamoto [1] have shown that partially Balanced Incomplete Block (PBIB) designs are characterized by association schemes. But in the PBIB designs discussed so far by the various authors the relation of association is symmetrical or reflexive. The object of this paper is to study a class of useful designs in which the relation of association is not necessarily symmetrical. First the definition of this new design is given. The relation between the parameters and the number of independent secondary parameters are discussed next. Two types of association schemes are given defined as type I and type II, the latter being based on Galois Fields. For the general method of analysis, the model assumed is that of a design for two-way elimination of heterogeneity. The final equations arrived at are the same as those for PBIB designs. A numerical example illustrates the method still further. Two methods of construction, one each for type I and type II are given. The method for type II is based on the property of Galois Fields.

In the problem of recovery of inter-block information in incomplete block designs, the ratio ρ of the inter-block variance to the intra-block variance which plays a key role is usually unknown. This article studies the loss of information on treatment estimation and treatment contrast estimation due to using some value ρ* in place of ρ. It is seen that for the group divisible two-associate partially balanced incomplete block (PBIB) designs, this loss is generally small.

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

Tables of Parameters of BIBDs with r⩽41 including Existence, Enumeration, and Resolvability Results

A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \subset V$ with $|S|=n$ and every $T \subseteq S$, there exists a vertex $x \in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$. Given a combinatorial design ${\cal D}$ with block set ${\cal B}$, its block-intersection graph $G_{{\cal D}}$ is the graph having vertex set ${\cal B}$ such that two vertices $b_1$ and $b_2$ are adjacent if and only if $b_1$ and $b_2$ have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed. We characterise the BIBDs with block size $k \geq 3$ and index $\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\lambda=1$ that are $n$-e.c. where $n \geq 3$. For $\lambda \geq 2$ and $n \geq 2$, we prove that only simple $\lambda$-fold designs can have $n$-e.c. block-intersection graphs. In the case of $\lambda$-fold triple systems we show that $n \geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\lambda=2$) have 2-e.c. block-intersection graphs.

When a block design is needed for controlling the effects of nuisance factors, but the blocks cannot be made sufficiently large to accommodate all the treatments, incomplete block designs can be used instead. Basic design issues of block size, connectedness, and randomization are discussed in this chapter. Balanced incomplete block designs, group divisible designs, and cyclic designs, which are three efficient types of incomplete block designs, are described in greater detail. Formulas for the analysis of incomplete block designs are given, including simplifications for balanced incomplete block designs and group divisible designs. A specific experiment designed as a cyclic group divisible design is described and analyzed. A discussion of sample size calculation and factorial experiments in incomplete block designs is also included. The concepts introduced in this chapter are illustrated through several real experiments. The use of SAS and R software is illustrated for both the design and analysis of incomplete block designs.

It is shown that, under certain hypotheses, the dual of a symmetric partially balanced incomplete block design with $m$ associate classes is also a PBIB design with $m$ associate classes and all parameters the same as before. In the case $m = 1$, and in the case that the design is group divisible, these hypotheses coincide with assumptions previously known to be sufficient to ensure duality. To show that some hypotheses are needed for duality, an example is given of a group divisible design whose dual is not a group divisible design.

The purpose of this paper is to give some methods of construction of balanced bipartite block designs. These designs are useful for comparing a set of test treatments to a set of control treatments. The methods of construction of non-proper balanced bipartite block designs are based on the UBe of incidence matrices of block designs with known patterns, such as balanced incomplete block (BiB) designs, group divisible (GD) designs and variance balanced (VB) designs. Using resolvable BIB designs and certain GD designs, two methods of construction of proper balanced bipartite block designs have also been given.

: A class of nearly balanced incomplete block designs is defined. This extends the concept of regular graph designs of John and Mitchell (1977) to the unequally replicated case. Some necessary conditions on the existence of such designs are derived. Methods of construction are given for some special cases. For five or six varieties, the 'best' nearly balanced incomplete block designs and their efficiencies are tabulated. (Author)

8 - Balanced Incomplete Block Designs