Department of Statistics Chapel Hill, N. C.

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1 UNIV ltsity OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. Mathematical Sciences Directorate Air Force Office of Scientific Research Washington 25, D. C. AFOSR Report TN ON SOME METHODS OF CONSTRUCTION OF PARTIALLY BALANCED ARltAYS by I. M. Chakravarti May, 1960 Contract No. 49-(638)-213 Methods of construction of partially balanced arrays are considered in this report. Two methods of construction are given. One of them derives partially balanced arrays from (A - ~ - v) configurations and the other is an extension of Bose-Shrikhande method of construction of orthogonal arrays. Qualified requestors may obtain copies of this report from the ASTIA Document Service Center, Arlington Hall Station, Arlington 12, Virginia. Department of Defense contractors must be established for ASTIA services, or have their "need-to-know" certified by the cognizant military agency of their project or contract. Institute of Statistics Mimeograph Series No. 260

2 ON" SOME jyiethods OF CONSTRUCTION OF PAB.'l'IAj~LY BALANCED ARRAYS* by I. M. Chakravarti University of North Carolina O. Introduction Suppose A = (~ij» is a matrix i =1, 2,, m, j =1, 2, N and the elements a.. of the matrix are symbols 0, 1, 2, "" s-l. J.J Consider the st matrices xl = (xl' x 2 ' X t ) that can be formed by giving different values to x., 6, X. = 0, 1, 2,, s-l; i = 1, 2, t. J.. J. Suppose associated with each matrix X there is a positive integer ~ (xl' x 2 '..., x t ) which is invariant under permutations of (xl' x s ' "', x t ). If for every t-rowed submatrix of A, the st matrices X occur as columns A(x l,x 2,.e., x t ) times, then the matrix A is called a partially balanced array of strength t in N assemblies~ m constraints (or factors), s symbols (or levels) and the specified A (Xl' x 2 ', x t ) parameters. When A(x l,x 2, x t ) = A for all (Xl' x 2,, x t ), the array is called an Orthogonal array. Orthogonal arrays were definec1in <[,4_7, ['5_7) and construction of orthogonal arrays were considered in (1:1_7, f:2j, 1:3_7, L4J, ["5_7). Partially balanced arrays were defined in ['6J where their use as multifactorial designs is also discussed.., r " *This research was supported in part by the United states Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF49(638)-2l3. Reproduction in whole or in part is permitted for any purpose of the United states Government.

3 2 In this paper, some methods of construction of partially balanced arrays are considered. One of the methods is applicable when s ::t 2 and derives partially balanced arrays from the well-known (A - lj. - v) configurations. The other method is an extension of Bose-Shrikhande ['2_7 method of construction of orthogonal arrays. 1. An example: An example of a partially balanced array of strength 2, s = 3, m = 6 constraints in N = 15 assemblies is given below Partially balanced array (15, 6, 3, 2) assemblies constraints G ~ This array has the A(x, x 1 2 ) parameters A(Xl'x2) = 2 if Xl and x 2 are unlike =1 otherwise This array was constructed by cutting out 3 assemblies and omitting one row from the orthogonal array A (18, 7, 3, 2) ( 2. Construction of partially balanced arrays for s = 2 from A - lj. - v configurations.

4 3 Definition: A A - IJ. - \I configuration, of m elements is defined ["7_7 as the configuration of m elements taken v at a time so that each set of IJ. elements shall occur together in just A of the sets. Suppose there are N sets of v e:}.ements each'in the configuration. o Let Nt denote the number of sets each containing a fixed subset of t elements. Then it is easily seen that (2.1) Nt.. A f m-t) / r- t ) 'IJ.-t \IJ.-t t.. 0, 1, 2, $" IJ. Consider the matrix A = «a..» of form (mx No') derived from a J.J A - IJ. - v configuration of m elements in N ' sets in the following o manner: Let al' a2'.'" am denote the m elements and sl' s2'... s~o denote the No sets of the configuration. Then let a ij = 1 if a i occurs in the set Sj.. 0 otherwise. Consider a IJ.-rowed submatrix of A with elements a.. as defined above. J.J Amongst the No columns of the submatrix, a column matrix X 1 where IJ., its transpose Xl'.. (Xl' x 2 ', X ), x. = 0 or 1 i = 1, 2,, IJ.,IJ. IJ. J. occurs A. (xl' x 2,..., xlj.) times. Specifically, let Xi = 1 for i = 1, 2, '0', r; Xi = 0 for i = r+l,, IJ. in X. Then it is easy to show that for such an X (2.2)

5 where n stands for the symbol of finite difference, viz., 4 Value of A(x l, x 2 ', x~) depends only on the count r of unities in its argument and hence it is invariant under permutation of its arguments. Now provided A(xl'~', x~) > 0 for all s~ sets of X, we have Theorem 2:1, The existence of a (A - ~ - v) of m elements With A (xl' x 2,, x~) all positive, implies the existence of a partially balanced array of strength ~ with parameters s = 2 and A(xl,x2' '~) as defined in (2.2) Well-known examples of A - ~ - v configurations are the triple systems, quadruple systems, etc _, which are defined in :7J. 3. An extension of Bose-Shrikhande method of construction of orthogonal arrays and its use in the construction of partially balanced arrays. Definition: A pairwise partially balanced design with parameters (v, k l, k 2,, k m ; b l, b 2,, b m ; A l, A 2,, At; n l, n2', ~) is defined as an arrangement of v varieties in blocks of m different sizes k l, k 2, _, k m, there being b i blocks of size k i, m Zbi = b, satisfying the following conditions: i=l (i) No block contains a single variety more than once. (ii) With respect to any variety, the remro.n.ing (v-l) varieties fall

6 into t categories, there being 5 th n i varieties in the i category, called the i th associates of the variety; (iii) Two varieties which are i th associates, occur together in A; blocks, i = 1, 2,, t. Then the following relations ~ong the parameters hold, m l:b.k.(k.-l) J. J. J. i=l t = l:niva i =- i=l t 'ij l: i=l n.a. J. J. Suppose there exist the orthogonal arrays i = 1, 2, "" m of strength two and index A and in k i symbols. Consider the pairwise partially balanced design defined earlier. There are b. blocks J. each of size k i These b i blocks provide b i sets of k i symbols each. Using each set of k. symbols once in the orthogonal array A., one J. J. gets b i such orthogonal arrays. If all such orthogonal arrays are arranged side by side, then one gets a matrix A with number of columns I. m 2 N = A l:bik i and number of rows q = min (ql' q2' "" qm)' In any i=l t two-rowed submatrix of matrix A, every ordered pair \ t u) of two v distinct symbols of varieties which are ith associates will occur AA i

7 6 times snd every ordered pair (:~ \ of two like symbols occur Arj times, if the variety t j occurs in r j blocks of the pairwise partially balanced design. Hence the Theorem 3.1: The existence of a pairwise partially balanced design with parameters ( V; kl' k 2,...k m ; b l, b 2, b m, AI' A2' A t, n t ) and of the orthogonal arrays ~, n 2,..., Ai(Aki' q., 2 J. k i, 2) i = 1, 2,..., m, imply the existence of the partially balanced array of strength two in V symbols and q = min (ql' Q2', qm) constraints and A (xl' x 2 ) = AA i where Xl' x 2 stand for two varieties which are ith associates and A(X,X) = Ar j where the variety X occurs r j times in the pairwise partially balanced design. As an illustration, a partially balanced array which has been constructed using the method described above, is given below. 111is is a partially balanced array in v = 6 symbols, N = 48 assemblies m = 5 constraints and = 1 if (x I,x 2 ) are second associates = 2 if Xl and x 2 are like where Xi' i = 1, 2,, 6 are the variety symbols. I

8 ''\ f 7 In constructing this array, the partially balanced design (v = 6, r = 2, b = 3, k = 4, n l = 1, n 2 = 4, A l = 2, A 2 = 1) in three blocks Xl' x4' x2' x'!2 x2' xs' X y x 6 ~y x6' xl' ~ and the orthogonal array A (16, 5, 4, 2) pave been used. Orthogonal Array A (16, 5, 4, 2) e, R t t t t t 2 t 2 t 2 t 2 C 0 1 t t t t t t t t 2 1: t t t 2 t t t t 2 t 1 0 L t t 2 t t t 2 t t 2 t t t 2 t 2 t t 2 t t t Making successively the identifications (1) (2) (3) 1 = Xl 1= x 2 1 = Xl t :: x 2 t=x t = x 3 3 t = x4 t = x s t = x 4. j and 0 =- x s o = x6 o = x6 using them on the above array in place 2 of (0, 1, t, t ) one gets

9 8 is the desired partially balanced array in 6 symbols and 48 assemblies. Let A* denote the array derived from A by truncating the first row and the first four columns (as indicated by the horizontal and vertical lines) Then the arrays AI' * A* 2 and A* are obtamed 3. from A* using the three identifications of variety -symbols given above. E denote the array Let Xl x 2 x 6 xl x 2 x6 E xl x 2 x 6 xl x 2 '.. x6 xl x 2 x 6 xl x 2 x6 Then the array A~ =- LE A~ A* is a partially balanced array in ~ =- 6 symbols, N = 42 assemblies, m = 4 constraints and... A(X,X) = 1, A(x.,x.) =2 1. J and x. J are first associates and A(Xi,x j ) = 1 if they are second associates. 4. AcknOWledgement: Thanks are due to Professor R. C. Bose for kindly going through the manuscript and for his helpful comments.

10 ~ J. 9 /-1 7 Bose, R. C. and Bush, K. A. (1952): Orthogonal arrays of strength - - two and three. Ann. Math. Stat. 23, 508. ["2_7 Bose, R. C. and Shrikhande, S. S. (1960): On the com1=c~ition o.f balanced incomplete block designs. Canad. J. Math. Vol 12 (2), 177. ~3_7 Bush, K. A. (1952) stat., 23, 426. Orthogonal arrays of index unity. Ann. Hath. ~4-7 Rao, C. R. (1946): On hypercubes of strength d and a system of confeundiilg in factorial experiments. Bull. Cal. Math. Soc. 38, /:5_7 Rao, C. R. (1947): Factorial experiments d~rivable from combinatorial arrangements of arrays. J. Roy. Stat. Soc. (Suppl) 9, 128. L6J Chakravarti, 1. M. (1956): Fractional replication in asymmetrical factorial designs and partially balanced arrays. Sankhya 17, 143. L7~ Carmichael, R. D. (1937) Introduction to the Theory of Groups of Finite Order, Dover Bublications, Inc.,I -(... ;._h