Construction of Balanced Incomplete Block Designs Using Cyclic Shifts

Balanced incomplete block designs (BIBDs) play important role in design of experiments, especially in field experiments. These designs ensure that treatments are compared with equal precision. Several methods are available in the literature to construct BIBDs but in this article some infinite series of these designs are presented by method of cyclic shifts. This method expresses some standard properties of a design just through examining the sets of shifts rather than studying the whole design.     

Incomplete Split-Plot Designs Supplemented by a Single Control

The article deals with the constructing methods for experiments carried out in an incomplete split-plot design supplemented by an additional treatment, called a single control. The control treatment has been treated usually as one specific factor level while not necessarily. The control cannot be connected with treatment combinations in an experiment. This distinguishes this article from others in the area considered. The proposed supplementation of whole incomplete split-plot designs leads to the designs with generally accepted methodological requirements, especially randomization. Moreover, we propose a few methods for constructing considered types of the designs with desirable statistical properties such as general balance and efficiency balance of the design with respect to treatment contrasts.     

Linear Integer Programming Approach to Construction of Balanced Incomplete Block Designs

This article proposes a linear integer programming-based algorithm to construct balanced incomplete block designs. Working of the algorithm is illustrated with the help of an example. The algorithm is able to generate balanced incomplete block designs very fast in most of the cases. The performance of the proposed algorithm is compared with other algorithms proposed in the literature. It is demonstrated that the proposed algorithm is competitive with the existing algorithms.     

Example of Block Designs for Plant and Tree Breeding Trials

Experimental designs which use extensive blocking and which are particularly useful in plant and tree breeding trials are discussed. They can be constructed either to accommodate field restrictions or take advantage of favourable plot layouts. Computer software is available to generate these design types for use in practice. Examples cover latinized row-column designs, t -latinized and partially-latinized designs and designs with unequal block sizes.     

Some Classes of Binary Block Neighbor Designs

Neighbor designs are useful to neutralize the neighbor effects. In this paper two classes of these designs are constructed in circular binary blocks of size 4, 8, …, 24. First class consists of six infinite series of nearest neighbor designs in which each pair of distinct treatments appears once as neighbors. Second class also deals with six infinite series of these designs in which each pair of distinct treatments appears twice as neighbors. A catalog of nearest neighbor designs is also compiled in circular binary blocks for odd number of treatments from 23 to 99.     

The Use of Balanced Incomplete Block Designs in Designing Randomized Response Surveys

This paper investigates the block total response method proposed by Raghavarao and Federer for providing accurate estimates of the base rates of sensitive characteristics during surveys. It determines the best balanced incomplete block design to use to estimate the base rates for three, four, five and six sensitive attributes respectively, given a maximum total number of 13 questions. The estimates obtained from this method have smaller variance than estimates obtained using the similar, but more popular, unmatched count technique.     

Designs to guard against outliers in the présence or absence of model bias

Optimality of experimental design is considered in the situation in which individual observations may be subject to a shift in mean. The criterion of minimum average integrated mean square error is examined in general and the consequences for first and second degree models in the présence or absence of model bias are discussed.     

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.     

Remark on Construction of Efficiency-Balanced Designs

In this note, all the efficiency-balanced block designs constructed by utilizing two methods of Dey & Singh (1980) are completely presented within a practical range of parameters.     

Estimation of Missing Cells in Randomized Block and Latin Square Designs

Classical missing cell formulas for the randomized block design and Latin square design are derived using the modified cell means model. The role of the assumptions of no interaction for these designs is made clear in the development. Examples are given for the randomized block and Latin square.     

Comparison of Three Sampling Designs

Three sampling designs are considered for estimating the sum of k population means by the sum of the corresponding sample means. These are (a) the optimal design; (b) equal sample sizes from all populations; and (c) sample sizes that render equal variances to all sample means. Designs (b) and (c) are equally inefficient, and may yield a variance up to k times as large as that of (a). Similar results are true when the cost of sampling is introduced, and they depend on the population sampled.     

Optimal Sample Designs With Preliminary Tests of Significance

A common problem faced in social experiments is that of designing a sampling strategy before it is known whether various control groups can be pooled. The classical sample choice is between a large supposedly unpoolable sample or a smaller supposedly poolable sample. This article suggests a compromise strategy between these two extremes based on preliminary tests of significance that allow one to embed judgments about the likelihood of pooling into a classical sample design problem.     

An Algorithmic Construction of Four-Level Response Surface Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing four-level design matrices in order to explore and optimize response surfaces where the predictor variables are each at four equally spaced levels, by utilizing a genetic algorithm. The produced designs achieve both properties of near-rotatability and estimation efficiency.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

Modified Wynn's Sequential Algorithm for Constructing D-Optimal Designs: Adding Two Points at a Time

Partial least squares (PLS) is a class of methods for modeling relations between sets of observed variables by using the latent components where the predictors are highly collinear. SIMPLS is a commonly used PLS algorithm that calculates the latent components directly as linear combinations of the original variables. However, SIMPLS is known to be very sensible to outliers since it is based on the empirical cross-covariance matrix. RoPLS is a recently proposed iterative method for robust SIMPLS. In this article, the influence function for the RoPLS coefficient estimator is derived. It is demonstrated that under certain conditions, the RoPLS estimator has infinitesimal robustness.     

The Construction of Optimal Balanced Incomplete Block Designs When Adjacent Observations are Correlated

An algorithm is described for the optimal rearrangement of the treatments within each block of a Balanced Incomplete Block Design when a specified “nearest neighbour” correlation structure exists among observations from plots in the same block. The procedure uses results obtained by Kiefer & Wynn (1981). Designs obtained using the algorithm are found to compare favourably with those produced by combinatorial methods given in Cheng (1983). The algorithm produces optimal designs for all BIBD parameter sets, including those not covered by the results of Kiefer & Wynn or Cheng.     

Block Designs Robust Against the Presence of an Aberration in a Treatment-Control Setup

The author considers the problem of finding designs insensitive to the presence of an outlier in a treatment-control block design setup for estimating the set of elementary contrasts between the effects of each test treatment and a control treatment. The criterion of robustness suggested by Mandal (1989 Mandal , N. K. ( 1989 ). On robust designs . Cal. Stat. Assoc. Bull. 38 : 115 – 119 . [Google Scholar]) in block design setup for estimating a full set of orthonormal treatment contrasts is adapted. A new class viz. partially balanced treatment incomplete block designs (PBTIBD) is introduced and it is shown that balanced treatment incomplete block designs (BTIBD) and PBTIB designs, under certain conditions, are robust in the previous sense. Such designs are important in the sense that the inference on the treatment contrasts under consideration remain unaffected by the presence of an outlier.     

Robustness of a Class of Partial Diallel Cross Designs to the Unavailability of a Complete Block of Observations

Complete and partial diallel cross designs are examined as to their construction and robustness against the loss of a block of observations. A simple generalized inverse is found for the information matrix of the line effects, which allows evaluation of expressions for the variances of the line-effect differences with and without the missing block. A-efficiencies, based on average variances of the elementary contrasts of the line-effects, suggest that these designs are fairly robust. The loss of efficiency is generally less than 10%, but it is shown that specific comparisons might suffer a loss of efficiency of as much as 40%.     

On the Optimality of Circular Block Designs Under a Mixed Interference Model

Filipiak and Markiewicz (2012 Filipiak, K., Markiewicz, A. (2012). On universal optimality of circular weakly neighbor balanced designs under an interference model. Comm. Stat. Theor Methods 41: 2356–2366.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proved the universal optimality of circular weakly neighbor balanced designs (CWNBDs) under the interference model with fixed neighbor effects among the class of complete block designs. In two special cases where a CWNBD cannot exist, Filipiak et al. (2012 Filipiak, K., Markiewicz, A., Ró?ański, R. (2012). Maximal determinant over a certain class of matrices and its application to D-optimality of designs. Linear Algebra Appl. 436(4): 874–887.[Crossref], [Web of Science ®] , [Google Scholar]) characterized D-optimal designs. The aim of this paper is to show the universal optimality of CWNBDs and to characterize D-optimal designs under the interference model with random neighbor effects.     

Cyclic quasi-symmetric designs and self-orthogonal codes of length 63

The enumeration of binary cyclic self-orthogonal codes of length 63 is used to prove that any cyclic quasi-symmetric 2-(63,15,35)$2\text{-}(63,15,35)$ design with block intersection numbers x=3$x=3$ and y=7$y=7$ is isomorphic to the geometric design having as blocks the three-dimensional subspaces in PG(5,2)$\text{<italic>PG</italic>}(5,2)$.