1. Introduction

Generalized Group Divisible Designs and their optimality have been studied by Jacroux (1980) [1], Srivastav and Morgan (1998) [2], Thannippara et al. (2009) [3], Ghosh et al. (2012) [4] and others. In this article we present a new method for constructing generalized group divisible designs with two groups from balanced incomplete block designs. We prove that the constructed generalized group divisible designs are also E-optimal.

Balanced Incomplete Block Designs (BIBD): An incomplete block design with treatments allocated over blocks, each of size (), such that each treatment appears in blocks, no treatment appears more than once in a block and each pair of treatments appears in exactly blocks, is called a balanced incomplete block design (BIBD). The numbers and are called the parameters of the design.

Generalized Group Divisible Designs with two groups (GGDD(2)): Let be any design having treatments allocated in blocks each of size . Then the design is a generalized group divisible design with two groups if the treatments can be divided into two mutually disjoint sets and , each with size and respectively , such that

(i) for and for all ,

(ii) for all and , (say),

(iii) for all and , , (say), and

(iv) for all and , (say)

where denotes the th entry of the concurrence matrix .

E-Optimality: Let be a set of designs each having treatments allocated in blocks of size each. A design is said to be E-Optimal if it maximizes where are the non-zero eigenvalues of the C-matrix of the design.

2. Method of Construction

Theorem 2.1. The design obtained by merging two BIBDs with same block sizes is a generalized group divisible design with two groups.

Proof. Let and be two balanced incomplete block designs with treatments labelled and , respectively. Suppose that we merge these two designs to obtain a design, say, . Obviously, the new design has treatments and blocks. The treatments in the design can be grouped into two, viz., and such that the treatments in i.e., will be replicated times and those in , i.e., will be replicated times. In addition,

for all and , (say),

for all and , , (say), and

for all and , , (say).

Thus, the design is a GGDD(2).

3. Example

Consider the design

and the design

Note that is a BIBD(7, 7, 3, 3, 1) and is a BIBD (9, 12, 3, 4, 1). Combing these two balanced incomplete block designs, we get a new design

Obviously, the design is a GGDD(2) with and The parameters of are , , and The number of replications for the treatments in is and that for the treatments in is

4. E-Optimality

Theorem 4.1. The class of GGDD’s constructed in section 2 is E-optimal.

Proof. We will prove the theorem using the Lemma 4.1 given below. It can be verified that for the class of GGDD’s constructed as per Theorem 2.1, the smallest off-diagonal entry is and the smallest value of replication is Keeping this in mind, we shall proceed as follows:

Let denote the -matrix of the design . Consider the matrix

where is a real number, is the identity matrix, and is the matrix of ones. The matrix has the following form where

,

The eigenvalues of are

(1)

For a given value of let denote the th entry of Note that for all

(2)

for all and

(3)

for all

(4)

for all and

(5)

and for all and

(6)

First let us find an upper bound for Letting in (4), we get, for all Similarly, for all and from (6) we get

and for all and from (3),

.

Hence for all Thus must possess a negative eigenvalue or at least two zero eigenvalues. Now from (1), we get i.e., Next we obtain a lower bound on Note that Consider the matrix with . From (1), we have Now substituting and rearranging terms we have Combining the upper and lower bound obtained above, we have

Further if for all , then so that

This gives Hence, by the Lemma 4.1 given below, the class of GGDD’s constructed using Theorem 2.1 is E-optimal.

We state the Lemma 4.1 which is given in Jacroux (1980) [1] and used in the proof of the above theorem.

Lemma 4.1. Suppose has -matrix and is the smallest off-diagonal element occurring in the matrix Then Further, if for all then and hence is E-optimal in

5. Illustrations

Consider the example of GGDD(2) constructed in Section 3. The incidence matrix of this design is given by

The concurrence matrix of this design is given by

The information matrix is given by Since , therefore . In this example and finally we get the information matrix as

Here On substituting the values of and other parameters, we obtain the matrix as

The nonzero eigenvalues of are 5.33 and 3. The minimum nonzero eigenvalue of is and satisfies the conditions of Lemma 4.1. Hence the design constructed in Section 3 is E-optimal.

ACKNOWLEDGEMENTS

The authors would like to thank the referee for his/her constructive suggestions on the earlier version of the paper.