Information Matrices of Irregular Factorial Designs

A new method is presented for calculating the elements of T′T, of an array T in two symbols. The concept of marginal index is introduced. It is proved that the elements of T′T can be written in terms of marginal indices. A theorem is presented for the case of Partially Balanced array and its generalization for PB1 and Extended PB1 arrays. This method is useful in calculating the information matrix or parts of the information matrix of irregular factorial designs with two level factors. Keywords: Design of Experiments, Factorial Design, Partially Balanced Array, PB1 Array, EPB1 Array, Information matrix. 1. Introduction Bose and Srivastava (1964) obtained T′T of a Partially Balanced array T (Chakravarti (1956)) using the multidimensional partially balanced ( MDPB ) association scheme, a generalization of the usual association scheme. An N x m ( 0, 1 ) matrix T is said to be a Partially Balanced array ( B-array ) of strength t, size N, m constraints, 2 levels and index set { μ0, μ1,..., μt }, denoted as B( N, m, 2, t ), if every subarray Ti1 i2...i t of T is such that every ( 0, 1 ) vector with weight i ( i= 0,..., t ) occurs exactly μi times as a row of Ti1 i2...i t. Necessary and sufficient conditions for the existence of a B-array of strength t, m factors, 2 symbols and index set { μi | i = 0,..., t } are given by Srivastava (1972) for m≤ t+ 2, and Shirakura (1977) for m= t+ 3. Orthogonal arrays are B-arrays with μi= μ,∀ i= 0,..., t. Srivastava (1970) expressed the elements of T′T ( information matrix ) of a B-array of strength t= 4 ( resolution V design ) in terms of its index set. Shirakura and Kuwada (1976) gave the information matrix of B-arrays that correspond to higher odd resolution designs, using the algebraic structure of the triangular multidimensional partially balanced (TMDPB) association scheme; their work includes that of Srivastava and Chopra (1971). Shirakura (1976b) gave the information matrix of B-arrays that correspond to designs of even resolution. Kuwada (1988, 1988b) obtained the information matrix of PB1 designs of resolution V and VII using the algebraic structure of extended TMDPB ( ETMDPB ) association scheme. A ( 0, 1 ) matrix [ T ; T ] of size N×( m1+ m2 ), in which T ( k= 1, 2 ) are of size N× mk, is called a PB1 array of strength ( t1+ t2 ), size N, m1+ m2 constraints, 2 levels and index set { μ( i1, i2 ) | 0 ≤ ik ≤ tk, k= 1, 2 }, written PB1( N, m1+ m2, 2, t1+ t2, { μ( i1, i2 ) } ), if for fixed values of tk ( ≤ mk ), every submatrix [ T ; T 0 ] of size N× ( t ) 1 ( 0 ) 2 1 + t2 ) is such that every ( 0, 1 ) vector with weight ik in T0 occurs exactly μ( i1, i2 ) times as a row of [ T ; T ], where T ( 0 ( k= 1, 2 ) are of size N× t ) 1 ( 0 ) 2 ( 0 ) k k and are consist of tk columns of T and the weight of a ( 0, 1) vector is the number of ones in the vector. (Kuwada and Kuriki (1986)). Necessary and sufficient conditions for the existence of PB1 arrays have been obtained by Kuwada and Kuriki (1986) using an argument similar to Srivastava (1972). A generalization of PB1 array, the Extended PB1 (EPB1) array was given by Katsaounis (1999). A ( 0, 1) matrix [ T;... ; T ] of size N× ( m1 +...+ mq ), in which T ( k= 1, 2,..., q ) are of size N× mk, is called an Extended Partially Balanced Array of strength ( t1+...+ tq ), size N, m1+...+ mq constraints (factors), 2 levels and index set { μ( i1,..., iq ) | 0 ≤ ik ≤ tk, k= 1, 2,..., q }, written EPB1( N, m1+...+mq, 2, t1+...+ tq, { μ( i1,..., iq ) } ), if for fixed values of tk ( ≤ mk ), every submatrix [ T ;...; T ] of size N× ( t ) 1 0 ) ( 0 q 1+...+ tq ) is such that every ( 0, 1 ) vector with weight ik in T occurs exactly μ( i ) 0 k 1,..., iq ) times as a row of [ T ;...; T ( 0 ] where T ( k= 1,..., q ) are of size Nx t ) 1 ( 0 ) q ) ( 0 k k and consist of tk columns of T . In this paper, the concept of marginal index is introduced ( Definition 2.1 ). Let (x ) … (x ) be the product of columns x u ,…, x of T, { u 1 u 1 r v u p r 1 v u 1 ,…, uv } { 1,…, m }, each raised to