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Neighbor designs have their own importance in the experiments to remove the neighbor effects where the performance of a treatment is affected by the treatments applied to its adjacent plots. If each pair of distinct treatments appears exactly once as neighbors, neighbor designs are called minimal. Most of the neighbor designs require a large number of blocks of equal sizes. In this situation minimal neighbor designs in unequal block sizes are preferred to reduce the experimental material. In this article some series are presented to construct minimal neighbor designs in circular blocks of unequal sizes.

The paper gives explicit formulae for analysing an experiment carried out in an affine resolvable proper block design. They follow from a randomization model, decomposed into stratum submodels. Analyses within the four relevant strata, and then the combined analysis, are considered in details. The paper is essentially an extension of some results presented in recent books, by Calinski and Kageyama [2000. Block Designs: A Randomization Approach, Volume I: Analysis. Lecture Notes in Statistics, vol. 150. Springer, New York; 2003. Block Designs: A Randomization Approach, Volume II: Design. Lecture Notes in Statistics, vol. 170. Springer, New York].

The class of balanced treatment incomplete block designs is generalized to allow for comparison of v 1 test treatments and v 2 control treatments. The generalized class is equivalent to the class of balanced bipartite block designs considered by Jaggi, Gupta, and Parsad. Some results on design construction and A-optimality are given for small values of v 1 and v 2 . Algorithms are developed for computing simultaneous confidence bounds for all test treatment versus control contrasts.

The block designs considered here are nested in the unified sense of Preece (Biometrika 54 (1967) 479–486) and Federer (in: T.A. Baneroft (Ed.), Statistical Papers in Honor of George W. Snedecor, 1972, pp. 91–114), that is, each block of the larger balanced incomplete block design contains several distinguished families of mutually disjoint sub-blocks, the sub-blocks of the same family belonging to the same system, such that each system forms the collection of blocks of some balanced incomplete block design. In this paper, it is shown that the necessary conditions for the existence of such designs are also sufficient for block size 5 and sub-block sizes 2 and 3. This, together with known results, implies the entire existence of such designs with block size 5 in general.

A multiresponse experiment is one in which a number of responses are measured for each setting of a group of input (control) variables. Quite often, the experimental units are subdivided into groups — or blocks — in order to control an extraneous source of variation. This necessitates adding block effects to the hypothesized multiresponse surface model, which typically contains fixed polynomial effects. Khuri and Valeroso (1998, J. Statist. Plann. Inference, Vol. 73, pp. 7–20) discussed the analysis of such a model when the block effects are considered fixed. There are many situations, however, where these effects should more appropriately be treated as random. In the present article, we address the analysis of a multiresponse model in the latter situations. In particular, we discuss the estimation of the model's polynomial effects in two cases: 1. The block effects are additive in the model. 2. The blocks have interactive effects with the polynomial portion of the model. Multiresponse optimization in the presence of blocks will also be discussed. A numerical example is presented for illustrative purposes.

Experiments in blocks having two treatment factors are considered in which a particular treatment must be excluded. The work is motivated by medical trials of two drugs in which a double placebo cannot be administered on ethical grounds. The contrasts of interest compare the effects of having both treatment factor at non-zero labelled levels with the effects of having only one treatment factor at a non-zero labelled level. For n ×2 experiments, a class of designs containing highly efficient members is identified and a lower bound on the efficiencies of designs in this class is derived. Tables of efficient designs are provided.

The block designs considered here are nested in the unified sense of Preece (Biometrika 54 (1967) 479–486) and Federer (in: Statistical Papers in Honour of George W. Snedecor, 1972), i.e. each block of the larger design contains several distinguished families of mutually disjoint subblocks, the subblocks of the same family belonging to the same system, such that each system forms the collection of blocks of some design. In this paper, nested balanced incomplete block designs and nested group divisible designs are investigated. It is shown that the necessary conditions for the existence of nested balanced incomplete block designs are also sufficient for block size 5 and subblock size 2.

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b1,b2,k1,k2) of incomplete block designs for v treatments in b1 blocks of size k1 each and b2 blocks of size k2 each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b1,b2,k1,k2) can be constructed by augmenting b2 blocks, with k2 − k1 extra plots each, of a BIBD(v,b = b1 + b2,k1,λ) and GDD(v,b = b1 + b2,k1,λ1,λ2). It is also shown that equireplicate E-optimal designs in D(v,b1,b2,k1,k2) can be constructed by combining disjoint blocks of BIBD(v,b,k1,λ) and GDD(v,b,k1,λ1,λ2) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v

Subjects in clinical trials are frequently randomized within blocks to ensure balance between treatment groups during the accrual period. Often, the block sizes themselves are also randomly selected. If the trial is unblinded and the block size known, the treatment assignment of some subjects will be known before they are randomized. This knowledge could lead to a bias where the investigator selects healthier subjects for a particular treatment. An investigator's potential to bias study depends on two factors: knowledge of the block distribution and the extent to which he uses this knowledge. Both factors are investigated. Knowledge of the block distribution is reflected by the variability of the estimates of the distribution. Maximum likelihood estimates (MLE) and ad hoc estimates are considered. An investigator's willingness to bias is formalized by defining a class of schemes where the decision to bias depends on the probability of correctly guessing a treatment assignment. The amount of bias these different schemes can introduce is quantified. Fixed blocks are easier to estimate and more susceptible to bias than random blocks. In general, the block distribution with least bias depends on the investigator's biasing strategy. However, block distributions with larger block sizes and uniform or increasing weights should work well in most situations.

This paper is concerned with the problem of comparing test treatments to a control in incomplete blocks when the errors in each block follow a stationary, first order autoregressive correlation structure. General optimality theorems are presented as well as A-optimality and construction results for two new families of designs: CNBTIB design when the blocks are circular and ANBIB designs when the blocks are rectangular.

This paper presents some methods for constructing balanced incomplete block designs with nested rows and columns. As examples, four new infinite series of designs are given in which each of the row, column, and block component designs is a balanced incomplete block design.

A-optimal designs for comparing v test treatments with a control in b blocks of unequal sizes are considered. A new class of designs is defined, namely the BTIUB designs, which can be considered as an extension of the BTIB designs of Bechhofer and Tamhane in the case of blocks with unequal sizes. Some conditions for the existence and construction of BTIUB designs are given. Finally an algorithm for the construction of A-optimal BTIUB designs is developed and it is applied in the case of two block sizes. Tables of some A-optimal BTIUB designs are given.

In this paper we consider proper block designs and derive an upper bound for the number of blocks which can have a fixed number of symbols common with a given block of the design. To arrive at the desired bound, a generalization of an integer programming theorem due to Bush (1976) is first obtained. The integer programming theorem is then used to derive the main result of this paper. The bound given here is then compared with a similar bound obtained by Kageyama and Tsuji (1977).

The designs considered are nested in the sense of W.T. Federer, that is, each block of the larger design contains a smaller distinguished subblock and the set of subblocks also forms a design. One of the most useful constructions for obtaining new designs is the group divisible design. We use the concept of nested group divisible design to obtain many infinite classes of nested group divisible designs and nested designs and give a catalogue of all possible nested designs with larger replication number ≤15.

The purpose of this paper is systematically to derive the general upper bound for the number of blocks having a given number of treatments common with a given block of certain incomplete block designs. The approach adopted here is based on the spectral decomposition of N′N for the incidence matrix N of a design, where N' is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah 1964, 1966), Kapadia (1966)) of using well known relations between blocks of the designs and their association schemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further generalizations of the main theorem are discussed with some illustrations.

Simple recursive constructions for cyclic block designs are given. These yield many new infinite families of cyclic Steiner 2-designs.

The class B of balanced incomplete block designs with parameters ν, b, r, k, λ considered here includes designs with repeated blocks. Attention to date has centered on constructing the members of B and on finding the minimum number of distinct blocks d ≤ b. Under the classical model, the usual statistical characteristics do not distinguish among members of B ; however, effects related to block totals can be used as distinguishing characteristics. A competing effects model is introduced and is shown to distinguish among members of B . This model is appropriate for intercropping, marketing, and survey investigations. The model is applied to the ten nonisomorphic BIB designs with parameters 7, 21, 9, 3, 3. A computer algorithm for obtaining the pairwise treatment by block incidence matrix of this model is given.

Bounds on the latest root of the C -matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.