Basudeb Adhikary

A New Type of Higher Associate Cyclical Association Schemes

1 .. Introduction. In earlier publications (Nandi and Adhikary, 1965; Adhikary, 1966) it has been shown how the concept of two associate cyclical association scheme of Bose and Shimamoto (1952) can be extended to yield such higher associate schemes. The group of treatments was there divided into subgroups and some of the subgroups were again divided into two subsets satisfying conditions similar to those of Bose and Shimamoto (1952). The purpose of this note is to show that, more generally, if the group of treatments can be divided into more than two subsets (not necessarily subgroups) satisfying certain conditions, it is possible to obtain higher associate cyclical association schemes. This generalisation includes the association schemes given earlier as particular cases when some of the subsets are subgroups. It will further be shown that the cubic designs of Raghavarao and Chandrasekhararao ( 1964) belong to this general class.

Higher Associate Cyclical Association Scheme and Orthogonal Factorial Structure

Cotter, John and Smith (1973) proposed a class of inter-effect orthogonal designs which we shall refer to as Ω-structure. We have established that a necessary and sufficient condition for a block design to have Ω-structure is that the levels of first (n-1) factors give a block design having higher associate cyclic association scheme. while the block design given by the levels of the n-th factor satisfy K-structure [ Mukherjee (1979)] for single-factor experiment. Structure of the design when all the n factors give a block design having higher associate cyclical association scheme has also been discussed. We have further identified higher associate cyclic partially balanced incomplete block designs as a class of block designs having orthogonal factorial strueture.

On Properties and Construction of GEGD ⁄3n -1 PBIB Designs

A definition of subfactor balance is given and it is proved that a necessary and sufficient condition for a design to have subfactor balance is that it is a GEGD⁄3 n -1 PBIB design. Some properties and methods of construction are also studied.

On Two Types of Cyclic Generalisations of EGD/2n- 1 PBIB Designs and Intereffect Orthogonality

The purpose of this paper is to suggest two classes of block designs having orthogonal factorial structure (O.F.S.)

On Two Products of Fractional Factorial Designs

Analysis of direct product (DP) of a number of fractional factorial designs (unblocked) has been presented in brief to show that the estimates of the parameters of the new design can be expressed as a suitable (asterisk) product of the estimates of the parameters of the component designs. It has also been proved that the analysis of symbolic product (SP) of the fractional designs (unblocked) can be translated into that of their direct product using pseuofactors. Also using the above results the Raos lower bound for the number of level combinations has been improved for a class of orthogonal arrays.

Construction of Grouped Arrays

Grouped arrays were introduced in Adhikary and Das (1986 a). Some constructional methods have been considered here.

On Some Mixed Order Response Surface Designs Useful in Sequential Experiments

In this paper we have introduced some mixed order response surface designs like FORDSORD, FORDTORD, SORDTORD and discussed their analysis and construction. We have also explained how we can improve upon a FORDSORD where p factors are first order and (v-p) factors are second order to get one in which p 1 factors are first order, p 2 factors are seeond order and other (v - p) factors are second order so that the design becomes a FORDGDSORD. We can then improve it further whenp, factors are first order and (v-p+p2) are second order.