8 - Balanced 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.

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.

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

In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.

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).

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.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

: 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.*

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.

A balanced n-ary design is a design on V elements arranged in B blocks of size K such that each element can occur 0,1,2,..., or n-1 times in each block (so that the blocks are collections of elements rather than subsets) and such that Open image in new window , constant, where nij is the number of times the ith element occurs in the jth block, i=1, ...,V, j=1,...,B. So a balanced binary design is merely a balanced incomplete block design (BIBD).

In this article, we present a gauge R & R study that is different from the standard setup in several ways. A balanced incomplete block design (BIBD) was used to collect the data because the measurement is destructive. Besides addressing reproducibility, differences in repeatability (i.e., tester variances) were also of interest. Here we use Bayesian methods to analyze the study data to address whether the three tester means and variances are similar. We also consider what happens if the BIBD structure of the data and possibly different tester variances are ignored in analyzing the study data.