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The existence of additive balanced incomplete block (BIB) designs has been discussed with direct and recursive constructions in Sawa et al. (2006). In this article, pairwise additive BIB designs are proposed and then new recursive methods of constructing such designs are provided. It is finally shown that for any integer v, two pairwise additive B( ) can be constructed. As a by-product we present a recursive construction of multiply nested BIB designs of block sizes for some .

Some methods of constructing affine resolvable semiregular group divisible (SRGD) designs are newly presented. By these methods, new series of affine resolvable SRGD designs can be obtained. Furthermore, new individual examples in a practical range are illustrated.

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 concept of affine α -resolvability has been discussed for block designs in literature since 1942 for α = 1 and in particular since 1963 for α ⩾ 2 . Among group divisible (GD) designs, affine α -resolvable designs are known for both classes of singular GD and semi-regular GD designs. However, no example has been found for an affine α -resolvable regular GD design in literature. In this paper, the validity of such concept will be disproved for regular GD designs in general. Thus the regular GD design does not possess any property of the affine α -resolvability.

The existence on affine αresolvability with some properties has been discussed for block designs in literature since 1942 for a =1 and in particular since 1963 for a ≥ 2. Recently, Kageyama (2007) disproved the validity of such concept for regular group divisible designs. A similar problem for 2-associate triangular designs will be tackled here. No example has been found for an affine αresolvable triangular design for any a in literature. In this paper, when a = 1,2, the non-existence of an affine αresolvable triangular design will be shown completely, whereas, when a ≥ 3, partial non-existence results are provided. AMS Subject Classification: Primary 62K10, Secondary 05B05.

Several methods of constructing group divisible (GD) designs are given by use of rectangular designs and nested balanced incomplete block designs. The GD designs obtained here are rather large, but as series they appear to be new. In the process, some series of rectangular designs, balanced arrays, and orthogonal arrays are also provided.

A-optimality of block designs for control versus test comparisons in diallel crosses is investigated. A sufficient condition for designs to be A-optimal is derived. Type S 0 designs are defined and A-optimal type S 0 designs are characterized. A lower bound to the A-efficiency of type S 0 designs is also given. Using the lower bound to A-efficiency, type S 0 designs are shown to yield efficient designs for test versus control comparisons.

Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.

It is shown that a group divisible design with r = λ 1 +1 is regular and symmetric. Some block structure of the design is also determined. This characterization is given by a constructive approach.

Some rectangular designs are presented along with a table of new designs in the range of parameters r,k⩽10. Some of these are E-optimal designs. A subclass of designs produced by Theorem 2.6 are E-optimal. A series of E-optimal designs, two series of group divisible (GD) designs and a new cyclic solution of a GD design are also obtained.

A sufficient condition for an incomplete split-block design to be universally optimal is given. Optimal properties are examined under two linear models, i.e., with interaction effects and without these effects. Furthermore, some methods of constructing universally optimal incomplete split-block designs are presented.

A (k,λ)-frame of type hu is a (k,λ)-GDD ( X , G , A ) of type hu in which the collection of blocks A can be partitioned into partial parallel classes each of which partitions X −G i for some G i ∈ G . In this paper, we investigate the existence of (4,3)-frames of type hu and show that the necessary conditions are also sufficient with a small number of possible exceptions.

In this paper, a balanced incomplete split-block design (BISBD) is shown to be universally optimum for the estimation of the interaction effects with respect to each of the intra-plot stratum estimation and the GLSE. Furthermore, the efficiency factor of an optimal design is investigated.

It is shown that the complement of a universally optimal design derivable from a triangular design is again universally optimal in a class of connected designs. Furthermore, some series of universally optimal designs for diallel cross experiments are provided. These observations strengthen the result on universally optimal designs by Das, Dey and Dean (1998).

A special type of balanced bipartite block designs is defined and a lower bound of the number of superblocks of the design is given. Furthermore, some constructions are provided for the balanced bipartite block design attaining the lower bound.

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.

Nested balanced incomplete block designs and balanced incomplete block designs with nested rows and columns are considered. It is shown that the necessary conditions for the existence of such designs with superblock size 4 are also sufficient.

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.

Some methods of constructing a class of block designs which are pairwise-balanced as well as variance-balanced are presented. Several series of such designs are given. A table of pairwise-balanced as well as variance-balanced designs with replications no larger than 30 is provided.

Some regular and semi-regular group divisible designs with block size three are constructed through triple systems to provide new non-isomorphic solutions, most of which are α-resolvable for α ϵ {2, 3, 4}.

A general randomization model for experiments in block designs and conditions for obtaining the best linear unbiased estimators of treatment parametric functions under the model are re-examined. It is shown how the best linear unbiased estimators under the overall randomization model can be obtained when the variance components are known. Furthermore, a general method of estimating these, usually unknown, variance components is described and properties of the resulting estimators of treatment parametric functions are considered. In connection with this the recovery of inter-block information in a general block design is discussed.

Some new variance-balanced designs with unequal block sizes and unequal replications are presented by use of group divisible and triangular partially balanced incomplete block designs. Tables of such variance-balanced designs with replications no larger than 30 are provided.

Under the assumptions that a linear model corresponding to two-way layouts is the mixed model without interaction, necessary and sufficient conditions are derived under which the minimum norm invariant quadratic unbiased estimators have uniformly minimum variance among all unbiased invariant quadratic estimators for data with arbitrary kurtosis. The conditions are expressed in terms of the incidence matrix N and C -matrix of the design. Under the model considered in this paper, the results obtained here strengthen those by Anderson et al. (1984), where data are required to satisfy a special balance condition.

Some incomplete block designs for partial diallel crosses have been given in the literature. These designs are obtained by regarding the number of crosses as treatments, and consequently require several replications of each cross. The need for resorting to a partial diallel cross itself implies that it is desired to have fewer crosses. A method for constructing single replicate incomplete block designs for circulant partial diallel crosses is provided in this paper. The designs are orthogonal, and thus they retain full efficiency for estimation of the contrasts of interest.

SUMMARY Optimality of complete diallel crosses in incomplete blocks is investigated. Optimal complete diallel crosses are characterized, and such binary plans are derived using nested balanced incomplete block designs. A table of optimal complete diallel crosses for up to 15 lines is also provided.

This gives a complete proof for bounds of the efficiency, conjectured by Das and Kageyama (1992), on robustness of extended balanced incomplete block designs against the unavailability of any number of observations in a block.

Several patterned methods of construction of nested group divisible (NGD) designs, starting from self-complementary group divisible (GD) designs and Hadamard matrices or GD designs, are given with some series of NGD designs. As a by-product, a new regular GD design is found.

Some new methods of construction of rectangular designs which are 3-associate partially balanced incomplete block designs with a rectangular association scheme are presented. Such rectangular designs which may be new are tabulated under the range of r, k ≤ 10.

The paper provides methods of construction of group divisible designs with block sizes three and four through balanced incomplete block and partially balanced incomplete block designs of block sizes 3 and 4. Furthermore, four non-isomorphic solutions are given.

Robustness of block designs is investigated, when any observations in a block are lost, in terms of efficiency of the residual design. The designs are balanced incomplete block (BIB) and extended BIB designs. The investigation shows that the covered designs are fairly robust against the unavailability of any number of observations in a block. Furthermore, the robustness of any Youden design and Latin square design against the loss of a whole column is shown.

In this paper the problem of finding the design efficiency is considered when a single observation is unavailable in a connected binary block design. The explicit expression of efficiency is found for the resulting design when the original design is a balanced incomplete block design or a group divisible, singular or semiregular or regular with λ1>0, design. The efficiency does not depend on the position of the unavailable observation. For a regular group divisible design with λ1>0, the efficiency depends on the position of the unavailable observation. The bounds, both lower and upper, on the efficiency are given in this situation. The efficiencies of designs resulting from a balanced incomplete block design and a group divisible design are in fact high when a single observation is unavailable.

Supplemented balance in block designs with nested rows and columns is considered, under the name of typeS designs with nested rows and columns. Analysis and construction of such designs are also discussed.

Methods of construction of proper efficiency-balanced designs from group divisible designs and balanced incomplete block designs are presented together with the tables of these designs in the range of ri≤ 20 or 30 and k ≥ 10.

SUMMARY Proper efficiency-balanced designs are considered. Upper bounds are given for the efficiency factor of proper efficiency-balanced designs. A class of E-optimal efficiency-balanced designs is observed as designs which attain such an upper bound.

Robustness of group divisible (GD) designs is investigated, when one block is lost, in terms of efficiency of the residual design. The exact evaluation of the efficiency can be made for singular GD and semi-regular GD designs as ell as regular GD designs with λ1 = 0. In a regular GD design with λ1 > 0, the efficiency may depend upon the lost block and sharp upper and lower bounds on the efficiency are presented. The investigation shows that GD designs are fairly robust in terms of efficiency. As a special case, we can also show the robustness of balanced incomplete block design when one block is lost.

Hedayat and John (1974) defined the concept of a resistant BIB design as one that remains variance-balanced upon loss of all the observations on a treatment. In this paper we consider the designs with supplemented balance (Pearce (1960)), or BTIB designs (Bechhofer and Tamhane (1981)). We investigate a subclass of these designs to determine which of these retain their ‘balance’ upon loss of all observations on a test treatment. We study the properties of such designs, give various methods of construction and examine their efficiencies with respect to the A-criterion.

ItisourgreatpleasuretopresenttostatisticalcommunitythememorialvolumeofProfessorJunjiroOgawa,ProfessorEmeritusofUniversityofCalgary,oftheJournalofStatisticalPlanningandInference.Hehadpassedawayatage84,onMarch8,2000.Hewasinternationally well known for his important works in mathematical statistics, particularly in the areas of design of experiments,multivariate analysis and order statistics. Many eminent authors submitted excellent papers and they were grouped two parts.The first includes all design of experiments and inference papers and the second includes papers in the area of multivariateanalysis, order statistics and related topics.The papers were gone through the refereeing process. We would like to thank the many referees who gave of their time andeffort to make this issue possible. Without the unselfish contributions of the following referees, we would never have been ableto bring this issue to being.: Amari, S., Adler, R., Bailer, R.A., Bentler, P.M., Burke, M.D., Chen, J., Copas, J., Fuji-Hara, R., Fujikoshi,Y., Ghosh, S., Glaz, J., Gupta, S., Gupta, V.G., Hayter, A., Hedayat, A.S., Helland, I.S., Horvath, L., Iwashita, T., Jammalamadaka,S.R.,Jimbo,M.,Jupp,E.P.,Kamakura,T.,Kano,Y.,Koutras,M.V.,Kuriki,S.,Letac,G.,Levina,E.,Lou,W.,Mejza,S.,Miao,Y.,Mishima,M., Morgan, J.P., Mukerjee, R., Naus, J.I., Preda, C., Provost, S.B., Rosenbaum, P.R., Savalei, V., Shechtman, E., Shinozaki, S., Sinha,B.K., Stufken, J., Tanaka, K., Taniguchi, M., Tarpey, T., Taylor, J.E., Vos, P., Watanabe, M., Wilson, S.R., Wu, Y., Yajima, Y., Yan, B. andYin, J.We are greatly indebted to Professor J.N.Srivastava, Editor-in-Chief, for his continuous encouragement and unfailing supportwhile this issue was being prepared.

In a proper non-binary variance-balanced design with parameters ν, b,r1,…, rν, k (

A new series of partially balanced incomplete block designs with nested rows and columns is presented. This has a triangular association scheme.

This forms a sequel to Chacko and Dey (1979) who characterized chemical balance weighing designs optimum for the estimation of total weight. In this note, such optimum designs are completely characterized by showing the general construction.

Rao (1947) provided two inequalities on parameters of an orthogonal array OA(N,m,s,t). An orthogonal array attaining these Rao bounds is said to be complete. Noda (1979) characterized complete orthogonal arrays of t=4 (strength). We here investigate complete orthogonal arrays with s=2 (levels) and general t; and with t=2, 3 and general s.

The family of t -designs is, without any doubt, the most important family of statistical designs. Their importance is due to their statistical optimalities, desirable symmetries for analyses and interpretations, and uses for constructing other important designs and structures such as Youden designs, generalized Youden designs, optimal fractional factorial designs, error defecting and correcting binary codes, balanced arrays, combinatorial filing systems, Hadamard matrices, finite projective and affine planes, strongly regular graphs, and so on. Research in the area of t -designs has been steadily and rapidly growing, especially during the last three decades. The number of publications in this area is in the several hundreds. Since papers on t -designs are published in a variety of journals, and because of the extensive role of these designs in design of experiments and other areas we believe it is imperative to gather these results and present them in varied form to suit diverse interests. This paper is an instance of such an attempt.

The reduction of the number of associate classes of some hypercubic association schemes by clubbing certain associate classes has been studied in the paper. It has been found that the reduction of anm-class hypercubic association scheme forv=2m treatments into a 2-class association scheme is always possible. Further it is proved herein that them-class hypercubic association scheme forv=sm treatments is reducible (i) to a 3-class association scheme, whens=3 and (ii) to a 2-class association scheme, whens=4, which really hasp111=p112 and hence leads to a series of balanced incomplete block designs.

Sufficient conditions for the existence of certain optimum chemical balance weighing designs are investigated.