Semi-regular Group Divisible Designs Research Articles

A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized to<i>α</i>-resolvability and affine <i>α</i>-resolvability in 1963. When <i>α</i> = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and <i>α</i>-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine <i>α</i>-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L<SUB>2</SUB>), cyclic. First, it could be proved that an affine <i>α</i>-resolvable cyclic 2-associate PBIB design does not exist for any <i>α</i> ≥ 1. Also, Kageyama proved the non-existence of an affine <i>α</i>-resolvable triangular design for 1 ≤ <i>α</i> ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L<SUB>2</SUB> designs with parameters <i>υ</i> ≤ 100 and <i>r</i>, <i>k</i> ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L<SUB>2</SUB> design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L<SUB>2</SUB> design which is left as only one unknown among L<SUB>2</SUB> designs.

Factorial designs robust against the presence of an aberration

In this paper, designs, insensitive to the presence of an outlier for estimating a complete set of orthonormal treatment contrasts representing the factorial effects, are obtained for both complete and fractional factorial designs. Fractional factorials incorporated in blocks are also considered in this study. In case of blocked complete factorials, semi-regular group divisible designs and for fractional factorials, universally optimal plans are found to be robust.

New Construction Methods of Affine Resolvable SRGD Designs

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.

Non-validity of affine [formula omitted]-resolvability in regular group divisible designs

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.

Optimality of partial geometric designs

We find a sufficient condition on the spectrum of a partial geometric design d* such that, when d* satisfies this condition, it is better (with respect to all convex decreasing optimality criteria) than all unequally replicated designs (binary or not) with the same parameters b, v, k as d*. Combining this with existing results, we obtain the following results: (i) For any q \ge 3, a linked block design with parameters b = q2, v = q2 + q, k = q2 -1 is optimal with respect to all convex decreasing optimality criteria in the unrestricted class of all connected designs with the same parameters. (ii) A large class of strongly regular graph designs are optimal w.r.t. all type 1 optimality criteria in the class of all binary designs (with the given parameters). For instance, all connected singular group divisible (GD) designs with \lambda_1 = \lambda_2 + 1 (with one possible exception) and many semiregular GD designs satisfy this optimality property. Specializing these general ideas to the A­criterion, we find a large class of linked block designs which are A­optimal in the un­restricted class. We find an even larger class of regular partial geometric designs (including, for instance, the complements of a large number of partial geometries) which are A­optimal among all binary designs.

Non-isomorphic solutions of group divisible designs with k = 3 and λ1 = 0

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}.

Semiregular group divisible designs with dual properties

A construction method for group divisible designs is employed to construct (i) infinitely many non-symmetric semiregular group divisible designs whose duals are semiregular group divisible designs, and (ii) infinitely many transversal designs whose duals are group divisible 3-associate designs. A construction method for affine α−resolvable balanced incomplete block designs is also given and illustrated.

Maximal arcs and group divisible designs

The existence of maximal arcs of a certain type in symmetric designs is shown to yield semiregular group divisible designs whose duals are also semiregular group divisible. Two infinite families of such group divisible designs are constructed. The group divisible designs in these families are, in general, not symmetric.

Robustness group divisible designs

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.

Methods of construction of group divisible designs

Some new series of regular and semi-regular group divisible designs are obtained from BIB designs with v = 2 k and with v ≠ 2 k, generalizing earlier results of the first author. Use in this construction is also made of normalized Hadamard matrices.

Incomplete block designs for parallel line assays

Abstract A systematic method of constructing incomplete block designs for parallel line assays from semi-regular group divisible designs is proposed. Many new series of designs are obtained through this procedure. These designs estimate all important bio-assay contrasts with maximum efficiency in addition to having simple analysis. They need a smaller number of experimental units as compared to known designs in literature.

On cyclic semiregular group divisible designs

On cyclic semiregular group divisible designs

A structural classification of SR group divisible designs

A simple method of construction of a semi-regular (SR) group divisible design from another SR group divisible design is given. Using this method, 111 available SR designs from Clatworthy (1973) and John and Turner (1977) are systematically classified into 20 classes. This procedure may produce new nonisomorphic solutions for known designs.

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with λ1=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.

Some families of group divisible designs

Abstract Generalizing methods of constructions of Hadamard group divisible designs due to Bush (1979), some new families of semi-regular or regular group divisible designs are produced. Furthermore, new nonisomorphic solutions for some known group divisible designs are given, together with useful group divisible designs not listed in Clatworthy (1973).

Series of semiregular group divisible designs

Two series of semiregular group divisible designs are given. The first has v=sN + sN−1 , m=s + 1, k = v/s. The second has v = 6t + 6, m = 3, k = 2t + 2. Both series have λ1 =λ2−1. Designs in these series exist for v = 12, 18, 20, 24, 30, etc.

Inequalities for Semiregular Group Divisible Designs

Inequalities for Semiregular Group Divisible Designs

On some constructions of modular hadamard matrices

Some new constructions for modular Hadamard and Hadamard matrices are given. Incidentally, it is shown that the Williamson series of H-matrices can also be constructed by using modular Hadamard matrices and resolvable semi-regular group divisible designs.

Some New Families of Partially Balanced Designs of the Latin Square Type and Related Designs

Four very general families of partially balanced incomplete block designs with two associate classes and having the Latin square type of association scheme are presented. These families have simple, yet general, constructions and the treatment extends particular solutions presented by Chang and Liu (1964). The general treatment brings out relations between the class studied and certain other closely related designs, including balanced incomplete block designs, regular and semi-regular group divisible designs, and various square lattice designs. A by-product is the discovery of one new L 3(5) type and four new regular group divisible designs.

Some New Families of Partially Balanced Designs of the Latin Square Type and Related Designs

Four very general families of partially balanced incomplete block designs with two associate classes and having the Latin square type of association scheme are presented. These families have simple, yet general, constructions and the treatment extends particular solutions presented by Chang and Liu (1964). The general treatment brings out relations between the class studied and certain other closely related designs, including balanced incomplete block designs, regular and semi-regular group divisible designs, and various square lattice designs. A by-product is the discovery of one new L 3(5) type and four new regular group divisible designs.