Design-based Causal Inference for Incomplete 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.

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.

BCH codes and Designs

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.

An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\\lambda$ blocks. These parameters obviously satisfy the equations \\begin{equation*}\\tag{1} bk = vr\\end{equation*}\\begin{equation*}\\tag{2} \\lambda(v - 1) = r(k - 1)\\end{equation*} Fisher [1] has also proved that the inequality \\begin{equation*}\\tag{3} b \\geq v, \\quad r \\geq k\\end{equation*} must hold. If $v, b, r, k$ and $\\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \\leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \\lambda = 2$ and $v = b = 29, r = k = 8, \\lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \\lambda = 2,$ is definitely impossible. Symmetrical designs with $\\lambda \\leq 5, r, k \\leq 20$, which are impossible combinatorially, are also listed.

8 - Balanced Incomplete Block Designs

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.

Youden [8] introduced a new class of designs called Linked Block (LB) designs, where a LB design is defined as that in which any two blocks of the design have the same number of treatments in common. Later, the same author together with Connor [9] presented a set of designs called `chain block designs' useful in the field of Physical Sciences. These designs can be considered as `not fully linked block designs'. In this paper we shall extend the idea of LB designs in yet another way by defining Partially Linked Block (PLB) designs. The basic motive in introducing PLB designs is to generalise the notion of LB designs and also to see how far the existing designs yield new PBIB designs. Consider an arrangement of $v$ treatments in $b$ blocks of $k$ plots each $(k < v)$, such that each treatment occurs at most once in any block and altogether in $r$ blocks. This is called an incomplete block design and is denoted by $D(v, b, k, r)$. We have $bk = vr$. When $b = v$, the design becomes symmetric. The incidence matrix $N = (n_{ij}) (i = 1, 2, \cdots, v; j = 1, 2, \cdots, b)$, where $n_{ij} = 1$ or 0 according as the $i$th treatment occurs or does not occur in the $j$th block characterises such a design completely.

: 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)

With the balanced incomplete block designs [1] $r$ replicates of each of $v$ treatments are arranged in $b$ `blocks', and the number of `plots', $k$, in each block is smaller than $v$. In order to eliminate systematic block differences from the comparison of treatment means and to obtain treatment comparisons of equal precision the well known conditions of balance specify (a) that no treatment should occur more than once in any block, (b) that the number of blocks in which any two particular treatments are both applied should be a constant number of blocks ($\lambda$ blocks) for all possible treatment pairs. When these designs are used in this form, the position of treatments within each block is not specified and can normally be regarded as unimportant. Accordingly the treatments in each block are randomized. Situations, however, arise in which the `plots' occupy certain characteristic positions in each block. Thus, if in a field experiment the blocks are vertical columns the plots would fall into $k$ horizontal rows which may also have systematic effects on the yields. In this case it will often be advantageous to balance the design with regard to rows (as well as with regard to columns) in a manner similar to the Latin square. Such an arrangement was first developed by Youden [2] who used the particular incomplete block designs with $b = v$ and in these rearranged the treatments in each block in such a way that every treatment occurred precisely once in each row. More recently one of us (S.S.S.) has carried out similar rearrangements for the other incomplete block designs with $b > v, v \leqq 10, r \leqq 10$ (ie., for those tabulated in standard tables and book), and has used a definition of balance resulting in at most two different precisions for treatment comparisons. In this note we show that (i) balancing with regard to rows resulting in an equal precision of all treatment comparisons is possible if $b = mv$ ($m$ integral), (ii) in all incomplete block designs with $r = mk \pm 1$ a row balance is possible resulting in treatment comparisons of two different precisions. One of us (W. B. T.) has prepared complete tables of double balanced designs suitable for practical use which it is hoped to publish together with the analysis of variance procedure with recovery of interblock information.*

Kim and Ko proposed second type second order slope rotatable designs (SOSRD) utilizing central composite designs (CCD) wherein the two digits   1 2 a , a denotes the position of star points. In this study, we propose SOSRD of second type utilizing balanced incomplete block designs (BIBD) with unequal block sizes. In specific cases, the recommended procedure results in fewer design points than SOSRD of second type acquired through pairwise balanced designs (PBD), symmetrical unequal block arrangements (SUBA) with two unequal block sizes and balanced incomplete block designs (BIBD).

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.

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.

Many factors, such as the number of vertices and the resolution of texture, can affect the display quality of 3D objects. When the resources of a graphics system are not sufficient to render the ideal image, degradation is inevitable. It is therefore important to study how individual factors affect the overall quality, and how the degradation can be controlled given limited resources. The essential factors determining the display quality are reviewed and a 3D perceptual estimation method is described. One of the major concerns in designing perceptual experiments is the large number of evaluations to be performed by judges, which results in fatigue and errors in judgement. To reduce judging fatigue and increase reliability of evaluations we propose using a statistical approach, following the design of experiments technique of balanced incomplete block design (BIBD). We develop models following BIBD for perceptual experiments and validate our model with experimental results. Even though the BIBD framework for perceptual evaluations is described in the context of 3D quality estimation, the approach can be used in other perceptual evaluation scenarios as well.