Deletion Designs

: In this paper a method of constructing a class of flexible single replicate factorial designs in blocks is given. Simple expressions for calculating loss of information on low order interactions is presented.

single replicate design is referred to as a j-th order deletion design if levels are deleted from j factors. Bose (1947) laid the foundation of factorial designs. He used finite Euclidean geometry to construct symmetrical factorial designs in blocks. Kishen and Srivastava (1959) extended the method of finite geometries to the construction of balanced confounded asymmetrical factorial designs thereby introducing the idea of deletion. John and Dean (1975) proposed a simple method of confounding based on the properties of generalised cyclic designs from a set of generating treatments or generators and showed that the confounding patterns could easily be determined from these generators. More recently Voss (1986) has constructed nearly orthogonal single replicate factorial designs in blocks. He uses the deletion technique where he deletes from the first factor, without loss of generality, to obtain first order deletion designs. The most recent contribution in this direction is that of Chauhan (1989) who generalized the work by Voss (1986), by constructing efficient single replicate designs using the generalized deletion technique. Starting from an single replicate generalized cyclic design, levels are deleted from the first factors, without loss of generality , to obtain an ( ) deletion design. given by Chauhan (1989) thus become special cases of the results obtained in this study. The method proposed by John and Dean (1975) is used to construct the preliminary single replicate factorial design, which is always symmetric. That is, factor occurs at s i F i s = levels for all 1, 2,..., i n = .
Conditions are given which guarantee the existence of either proper or improper deletion designs. Simple formulas for calculating the loss of information, due to confounding with blocks, on main effects and two factor interactions are given. A simple method of choosing a fraction for estimating main effects and low order interactions is also given.

Notations
We shall first assume the fixed effects linear model That is the i-th row corresponds to the i-th treatment in the above arrangement of the v treatment combinations. We shall denote the incidence matrix, the intrablock matrix, the diagonal matrix of block sizes and the number of blocks, respectively, by N, A, K and b . The i-th row of the incidence matrix N corresponds to the i-th lexicographically ordered treatment combination . The q a 1 × vectors of ones and of zeros will be denoted by 1 and , respectively. A generalized interaction will be denoted by a where

Some Properties of Deletion Designs
We start by giving results useful in constructing deletion designs which can be used to estimate the main effects and also the results are useful in calculating loss of information due to confounding in blocks.

Loss of Information on Main Effects
Dean (1978) showed that for a given vector c , the loss of information as given by John and Dean (1975). We give here two results on loss of information on main effects.
Theorem 2: Loss of information due to confounding in blocks on any factor whose levels were not deleted from d to obtain d , is given by From John and Dean (1975), and Chauhan (1989)  Theorem 3: Loss of information due to confounding in blocks on any factor whose levels were deleted from d to obtain d , is given by  4). Therefore using (4.4), (4.5) and (4.6) in (4.1) Theorem 3 follows.

Confounding in Deletion Designs
The following results in confounding in generalized cyclic designs are due to John and Dean (1975). The number of degrees of freedom confounded with blocks for any given interaction, α x is given by    4). Thus, using (5.5), Y 0 j = .
Theorem 5: All the r-factor interactions among any number of the first ( ) We can therefore conclude that all these factor interactions are partially confounded with blocks. Hence the theorem.
r Confounding in deletion designs has been studied by Chauhan (1989). Theorem 6 below is due to her. Let α x be a given interaction.