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f UNIVERSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. MULTIDIMENSIONAL PARTIALLY BALANCED DESIGNS AND THEIR ANALYSIS, WITH APPLICATIONS TO PARTIALLY BALANCED FACTORIAL FRACTIONS by ,; R. C. Bose and J. N. Srivastava October 196:3 •J. The purpose of this p~er is three fold~ The first one is to introduce a class of .... tidimensional designs (under the model of addivity of factori•. effects) which involve partial balance • The next purpose is to" xamine closely the pattern in the matrices which one has to invert in order to carry out the analysis. The third aim is to apply the theory so developed to the problem of analysing irregular fractions of certain kinds. This research was supported by the Mathematics Division of the Air Force Office of Scientific Research. Institute of Statistics Mimeo Series No. :376 1 MULTIDIMENSIONAL PARTIALLY BALANCED DESIGNS AND THElIR ANALYSIS, WITH APPLICATIONS TO PilRTIALLY BALANCED FACTORIAL FRACTIONS 1 by R. C. Bose and J. N. Srivastava = = = = = = = = = = = = = = = == = = = = = == = = = = = 1. == = == = = = == = Introduction and Summary. The use of multidimensional designs like !atin Square, Graeco-Latin Square, Youden Square is now well established. 12, 14, l5J. A good discussion is available in [5, 6, Consider a Latin Square: We have three factors (i) row blocks, (ii) column blocks, and (iii) treatments. This is a special kind of 3-dimensional (i.e. 3-factor) design. Notice that if the number of rows, columns or treatments is t, we take only t 2 observations and not t 3 , as one would get by trying all combinations of rows and columns and treatments, if that were possible. A struc- turally similar design would be obtained if the factors were (i) blocks (ii) varieties of corn (iii) insecticides, each having t prOVided one takes the same t 2 levels (i.e. being t in number), combinations of factors as in the above design. The ordinary designs involving blocks and treatments may be called 2-dimensional. Designs of three and higher dimensions have been discussed in detail in [7, 8, 9, 10J, both when the effects of the various factors are assumed to be additive, and when interaction is present between various pairs of factors. The purpose of this paper is three fold. The first one is to introduce a class of multidimensional designs (under the model of addivity of factorial effects) which involve partial balance. Such designs would be useful for economising on ~his research was supported by the Mathematics Division of the Air Force ,~ Office of Scientific Research. t 2 the number of observations to be taken1 and also for many kinds of special experimental situations. Hm-lever "i'le shall not be able to consider here the impor- tant problem of constructing such designs l and shall confine ourselves to their analysis. The next purpose is to examine closely the pattern in the matrices which one has to invert in order to carry out the analysis. This leads us to some powerful methods for inverting such patterned matrices. The third aim is to apply the theory so developed to the problem of analysing irregular fractions of certain kinds. These include (i) balanced fractions which are essentially partially balanced arrays (of strength 41 if one is interested in all effects involving two or a lesser number of fractions)1 as defined by Chakravarti balanced. [4J; and (ii) certain more general fractions, termed here as partially In factI now that a quick method of analysis becomes available1 the problem of the examination of the properties of newly constructed (partially balanced) fractions becomes much easier. This in turn would greatly help in the construction of good economic fractions. Much of the work in this paper was first discussed in [ l}]. 2. Partially balanced association schemes. Corresponding to an ordinary PBIB design, one has an association schemel the definition of which can be found in Bose and Mesner [}]. Given a set of v f' r obj.ects l a relation satisfying the following conditions is said to be an association scheme with m classes:C(a) Any two distinct objects are either the first l second, or ••• m-th associates l the relation of association being symmetrical1 i.e. if the object a is the i-th e associate of the object C(b) of a. ~, Each object a has then n i ~ is the i.th associate of a. i-th associates, the number n i being independent 3 C(c) If any two objects and ~ a are i-th associates, then the number of ob- jects which are the j-th associates of and is independent of the pair a, are ~ ~ and k-th associates of a is i Pjk with which we start so long as a and ~ i-thassociates. We consider a generalisation of the above scheme. Definition 2.1. Suppose there are m sets of objects Sl' S2' ••• 'Sm' the objects in the i-th set Si being denoted by xil ' xi2 , ••• ,xin .• Then the sets J. (Sl' S2'··. ,Sm) will be said to have a multidimensional partially balanced association if the following conditions are satisfied:C (i) m With respect to any X. J.a E S., the objects of J. S. J can be divided into n .. J.J (n > 0), where each element of the a-th class is the a-th assoij in Sj. The number of objects in the a-th associate class is disjoint classes ciate of xia a nij (i and j may take any value between 1 and m, and may in particular be identical) • The number n .. is independent of the particular object in Si' and depends only on i. J.J Cm(ii) The relation of association is symmetrical, i.e. if x jb X. J.a chosen is the k-th associate of xia in Sj' then xia is the k-th associate of xjb in Si. Notice that this implies n = n for all i and j. ji ij Cm(iii) Let S., S. and Sk be any three sets, where i, j, k are not J. J necessarily distinct. associate of be the a.th ia € Si and x jb € Sj. Let x jb in S., so that x.W is the a-th associate of xjb in Si. J . Consider the set Q Let (Xia ' x ~, k) of the ~-th associates of xia in Sk' and the 4 The scheme defined above will be referred to as the multidimensional partially balanced (MDPB) association scheme. :3. Multidimensional partially balanced des igns. F " F " ••• " F • Suppose we have m factors 2 m Suppose the i-th factor has Here the word level does not imply that the l levels F11" F i2 " ... " F • iSi levels of any factor are necessarily ordered according to some quantitative s1 criterion~ This if Fl denotes varieties of wheat" then Fll" F12" ••• "Fl sl may stand for sl different varieties of wheat. There are sl x s2 x ••• x sm = No" say, combinations of levels. (j." j2' ••• , jm) e Let denote a ty:pical treatment combination in which the r-th factor occurs at level F. (1 < j < s ; r = 1" 2" ••• , m). The observed response to rJ r - r - r this treatment combination will be denoted by y(jl' j2' ••• , jm). Also we shall v7rite E[y(jl" j2" •• ·,jm)] = Y*(jl' j2' ••• , jm)' where E denotes ex,pected value. The main aspect in which multidimensional designs differ from the factorial designs is that in the former, we do not define main effects, interactions etc. as in the latter. In fact in the latter, the factorial effects are generally assumed to be additive, which we in particualr shall assume. Thus we shall take as model: (1) Y*(jl' j2' •• ·,jm) = T(l, Var. [y(jl" j2' ••• " jm)J where T(r, j ) (1 < j < s ; r r - r - r level Frj of the factor jl) + T(2, j2) + ... + T(m, jm) = ~2 " = 1, 2, ••• , m) denotes the 'true effect' of the Also vIe shall suppose all the observations to be r independent. e An example of a 4-dimensional situation will be given here. Suppose one has 20 different varieties of Wheat, 4 methods of cultivation, and 10 different insectisides. In addition to these :3 factors, one usually has another nuisance factor . 5 viz. a set of blocks. A system of blocks may be called for because of soil heterogeneity, for example. As appears likely, the interaction between these factors may be negligible, and the model (1) may be valid. The problem may then be to estimate the true effect of the various factor levels. Since the total number of level-combinations is large and the number of effects to be estimated is relatively small, a design is required which may satisfactorily cope with our needs in relatively fewer observations. This is exactly the purpose served by MOPB designs to be defined now. l be the total number of times the combination Let h ,2, ••• ,m jl,j2,···,jm .u, jm) is tried in our experiment. Let (2) E k~r,t , = i.e. appears at level F .~ in the r r'd r t r set level-combinations selected for experimentation, and h.'. the numJr,J t ber of times the levels F . and Ft j occur together in O. r,J r , t Def. 3.1. Let the set of the sk levels of F (k = 1,2, ••• ,m) be denoted by k 8 • Then the set 0 will be said to be a multidimensional partially balanced k design if M(i) is the number of times the factor the sets 8 , 8 , ••• ,8 1 2 m F have a multidimensional partially balanced asso- ciation scheme defined over them, M(ii) h: Jr = ~r (r = 1,2, ••• ,m), is independent of the level F where ., r,l.r ~ r depends only on the r-th factor, and 6 hr,t. M( ~i~) • • j ,J . r t = dar,t' a cons t ant dependi ng on the pa ir f 0 fac t ors Fr' Ft' an d also upon a, where F . € Sand Ft. € St are ~th associates of each r r,J r ,J t a other under the association scheme in M(i). Obviously we must have d =0 } r,r for all permissible r. Consider now the analysis of the above designs. Only the linear estimation part of the analysis will however be discussed here; for once the best linear unbiased esti.ms.tes of the parameters have been obtained, the sum. of squares etc. and the analysis of variance table can be easily computed using standard methods discussed, for example, in [1, 6]. Consider the model (1). The number of parameters T to be estimated is m s = E s. Let y denote a (fixed) column vector of the observations r=l ;r y{jl,j2,···,jm) where (jl,j2, ••• ,jm) € O. Let e (3) it = (T(l,l), ••• ,T(l,sl); T(2,l), T(2,2), ••• ,T(2,s2); •••• ; T{m,l), T(m,2), ••• ,T(m,sm» be the sxl vector of parameters. Then we can write (4) E{X) = - , At P where A is a certain matrix with elements equations (1). i Then it is well-known [lJ 0 and 1, and is obtained by using that the normal equations for obtaining can be written A At i In general = (M I) is singular. A 1. • However we can overcome this difficulty by pro- ceeding as in the case of 2-dimensional designs. respect to factor Call a design connected with F., if after eliminating from (5) the parameters corresponding 1. of to all the factors except F., we get a se/ (s.-l) independent .J. involving the T(i,r) alone. be called well-connected. J. 7 equations A design connected with respect to each factor may For a well connected design, suppose we make the (usual) assumptions si ~ T(i,r) = i = l,2, ••• ,m 0, r=l Then equations (5) are changed to (6) (M') + rJ.E = AX 1 where Q's being real numbers, and Q being repeated si times. By properly choosing i r , the matrix (M' + r) becomes nonsingular. Then solving (6) we have a solution for .E. r, Due to the special diagonal nature of have the same pattern. the matrices (M.') and [(M I) + r] Thus the problem now is to invert a nonsingular matrix with the same pattern as of (M'). In the remaining part of this section, we shall examine this pattern more closely. The element in the cell (k, f) of M' is obtained by taking the sum of the products of the corresponding elements in the k-th column and the ~-th column of the matrix At. Let the elements in the k-th row and the f-th row of 12 be respectively T(x, jx) and T(YI j ). y (q, k) and cell (q, f) of AI the vector X contains both Then the product of the elements in the cell will be unity, if and only if, the q-th element in jx and jy. Hence the element (It, equal to the number of level-combinations in ~ n f) in which the symbols of AA' is x and j y both occur together (respectively at the position of the x-th and the y-th factors). j 8 hX'Y. by definition. Again this in turn equals lJ, if jx,J y x,y j x E Sand j y E Sy are the a-th a~soc~taG of~ch other. x a Let nml the matrices D be defined as follows. x,y a between the s levels of F First we define association matrices B This however equals x ~ x and sy levels of F, i.e. between Sand Sy (where x and y mayor may not y x a is of size s x s , and has unity in the cell (i,j) be equal). The matrix B x,y x y if the i-th level of F and the j-th level of F a.re a-th associates under the x y a multidimensional association scheme, and zero otherwise. Next we define D~, of size s x s each, such that each such matrix contains m2·submatrices. Let M ij denote the si x Sj sub-matrix in the i-th row block and j-th column bloek. This corresponds to the factors a and F ·• Then D j i ~ t is a zero matrix if x ~ x or yt ~ Y and Mxy t • F a is such that M = B ~ or both. Thus ~ a , is an D ~ s x s matrix in which the rows and columns correspond in order to the elements in -p , and in which the cell (j ,j ) corresponding to the elements T(x, j ) x T(y, j ) in x y and and j are a-th associates, and contains x y a which do not correspond to zero otherwise. Also all other elements of D y p - contains unity if j ~ factors F x and Fy are zero. From the developments in the last two paragraphs, it therefore follows that Mt = I: a,x,y The next two sections will be devoted to the development of certain algebraic properties of our association scheme, which "l'ill be used later to obtain an algoritbm for inverting a matrix of the type (7). 4. Linear associative algebra of the MDPB association scheme. We first establish certain properties of the matrices It element will be denoted by b':lu whose (t,u) 9 Lemma 4.1. (i) nij (ii) ni n .. 0: bat'll , for all permissible t, i, j, a ij u 0: = n j n ji for all pemmissible i, j, 0:. = r: 0: ~J = (iv) r: Co: B~. a J ninj = 0nin ~J j (the n x n j i matrix of all unities) (the n x n . zero matrix) implies i J c~ =0 , for all \oN permissible The linear functi ons of Bij , B~j' same nij matrices as a basis. form a vector space with these (v) Proof: (i) 0:. For fixed t, we have r: u b atu ij = number of elements in Sj which are the o:-th associates of object t (e Si) • = naij , by definition. (ii) For each object t e Si' we have n~j elements in Sj a-th associates of t. which are the Thus from (i), we get r: u a n .. J~ waich givea the required result. (iii) This relation holds, since for all t e Si' u e Sj' the pair (t,u) are 'C¥.. th ,associates fa.+' one (iv) and only one value of a. This is obvious. (v) This holds by virtue of (iv). Lemma. 4 • 2 • r: p(i, k, y; j, a, ~) B~k Y 10 Proof: The matrices on the l.h.s. are of dimensions n x n j and nj x i The element in the cell (t, u) respectively so that the product exists. ~ of the product matrix is n. = 'i/ '1=1 Suppose now that t e Si and u e Sj bo:tq b13qu ij jk and t, u are y-th associates. the last expression above equals the number of elements in S. which are common J to the set of ex-th associates of t and the set of 13-th associates of u and is, by definition, equal to p(i,k,y; j,o:,13). On the other hand, the element in the (t, u) cell of the matrix on the r.h.s. of (9) is Z p(i, k, q; k, Since e (t,u) Then 0:, f3)bi~U • '1 are y-th associates, only one member in this last sum is nonzero, and the sum reduces to p(i, k, y; j, 0:, 13). This ccmpletes the proof. Lemma 4.3;. = Z p(j, f, y; k, j, f, 0:, 13) D}r' if k = k' , Y for all permissible values of k, ex and 13, and where 0ss' denotes the zero matrix of order s x s'. Proof: Consider the product D~k D~'f obtained by mUltiplying the two matrices blockwise (blocks formed by a row or coltunn of submatrices). The element in the ql-th row block and %-th column block of the product will be a zero either the matrill~ if 'll-th row block of D~k consists entirely of zero submatrices, or the ~-th column block of De'f has zero submatrices only, or both. Since all the row ex blocks of Djk consist of zero submatrices except the j-th row block, and all the column blocks (except f-th) of D~'f contain zero submatrices only, the only possi- 11 ble nonzero submatrix of the product is and f-th column blocl.... HOY7ever, if M (' which stands in the j-th rm'1' block j k! k' the two nonzero submatrices will get mUltiplied with zero submatrices, and Mjf i'7ill also be zero. If k = k t, then obviously by the last lemma. This completes the proof. a The above lennna shows that the product of any two matrices Dij pressed as a linear function of these same matrices. a of any D ij are 1 and 0 Also since the elements only, and since = (11) can be ex- J ss it is clear that D~,j (for all permissible a, i, j) are all linearly independent. Hence the vector space algebra. algebra L formed by the linear functi ons of D~j is a linear Further since matrix multiplication follov7s the associative law, the L is associative too. Hence we have proved The set of all linear tunctions of the D~j Theorem 4.1. eiative a.lgebra L with the . n •• .(;: E n... ) matrices ~J By using the properties of the algebra. L a D J. i form a linear asso" as a basis. established above, one could prove a number Of useful necessary relations uhich the parameters etc must satisfy, if the multidimensional association ~cheme are summarized below. .Theorem 4.2 • n (12) ik (i) E p{k,i,a; f,f3 I Y) p(j,1,o; k,~,a) CX=1 njf = E a=l p(j, f,a; k,~,(3) p (j,1,o; f,a,y) , p(i,j,a; k,f3,y), n exists. ij Many Of these 12 4It for all permissible values of f, ~, i , j, k, y, 0 and ~. n ij E p(i, k, a; j, ~, y) (ii) = ~=l for all permissible (iii) , i, j, k, a and y. p(j, j, a; k, y, y) if each object in set S. J Proof: ~j is the ~-th associate of itself. The first set of relations can be established by considering the equality a and equating coefficients of D atter expansion. The other results can be ji obtained by using the same techniques on B~k together with (11). ~ Consider the matrices D~j , a = 1,2, ••• ,n jj , Note that because of the re- lation of association being symmetrical (see condition C (ii) in sec, 2,), m these matrices are symmetric, Hence D~j D~j = (D~j D~j)' Again each product function of the D}j' D~j D~j could obviously be written down as a linear Thus we have Hence the set of all linear functions of the algebra (L n.. JJ matrices a D . jJ is a sub- say) of L, and further, L is commutative. The algebra's L. j j J correspond to the algebra of ordinary PB association scheme, and have been studied by Bose and Mesner [3]. 13 4It The algebra D~k D~( (I j. L however is not commutative, since for example the product D~( D~k is in general nonzero, the product Since commutativity of L. J a will always be so, if is partly a conseq~enCe of the synmletry of the Djj , one might wonder whether one could generate a commutative algebra by con- a sidering instead, the linear functions of A defined by ij , (14) which are symmetric. The answer however is negative, since it can be shown that a: the Aij do not in general generate an algebra. 5. Inversion of matrices belonging to linear algebras. Consider the algebra L. Clearly the identity matrix Is e: L. Let D e: L, and suppose , and that D is nonsingular. Since every matrix satisfies ita own characteristic polynomial, we must have ... + for some real numbers Q's. Q r r D = 0 ss , Since D is nonsingular, i'le then get (16) (A similar equation could be written dovIn if Q I 0, and i j = 0, 1, ••• , i-l.). Since D e: L, each power of D also belongs J to L, and hence so does the expression on the r.h.s. of (16). Hence D- l e: L. provided Q o Q. = 0 for I O. (This result is well known but we gave a proof for reader's convenience). -1 D = W, and write Let 14 = where vi'S are real numbers, to be obtained if D- l is needed. To solve this latter problem, let (18) Since W D = Is' we get Z = e {f ~,k.f o:,i,j 0: Z = d Z o:,~,k,i,j {i d ij y Hence equating coefficients of D kj = Z rv R · ~k. u,~,~ ~ ~f D~j 0: ij ~j p(k, j, y; i, ~, Z DY kj 0:) y=l we get d~. p(k,j,y; i,~,o:), for all permissible y,k,j. ~J The number of unknowns and the number of equations in the set (19) are both Z n~ . ). Further since liJ € L , these equations O:,i,j J must all be independent and consistent and must offer a unique solution for the obviously equal to w's. ~. Now, in m = .Zl J= ~j (19), n •• (= keeping k fixed, vary y ~. equations involving only the fixed, and x, y take all permissible values. broken up into sets, the k-th set containing set of equations. For fixed y and Z d~. p(k,j,y; i,~,o:). Varying y, j 0: ~J values, we get an ~. and j. This gives '{X unknmms Thus the ~. n.. unknowns. j, the coefficient of and ~, i over the inwhich k is equations can be Consider the k-th v~i is ~. permissible Hence the e 15 problem of inversion of D is thrO'WD. on that of inverting 01' 1:"2 2 ", .,Om • The above method of inversion is valid (up to equations (19» for any linear algebra. are small. For matrices in L, it is further useful if the in case ~. However are large, another more powerful method to be presented below is much ~. more useful. Consider a commutative linear algebra, say L.. Let Lj denote the equivaJ 0 n jj lent algebra generated by Bjj, ••• ,B • Let the n j characteristic roots of jj a Bjj , arranged in a certain order to be explained below, be the elements of the row vector (20) 2ja = (0 ja,l' 5 ja,2' ... , ° ja,n} The order of the roots in (20) is I as required by the following well known Theorem (Frobeniu~). +. 2 n .. B , Bjj , ••• , Bj~J , say, which are pairwise jj a f3 commutative, i.e. for which Bjj B = Bf3jj Bajj for any permissible a and f3. jj Suppose there exist matrices Then there exists an ordering (say, as in (20» of the roots of B~j' a = 1,2, ••• ,n jj such that the elements in the vector (Oja,l + °jf31' °ja,2 + °jf32' ••• , °ja,n + Oj f3n j B~j represent in some order the roots of (Oja,l 0j~,l' 0ja,2 0jf3i 2 ' ••• , 0ja,n ( Bajj B13) j + ~j' 0jf3 n ) j j ) Similarly those of represent in some order the roots of • jj A direct application of Frobenius Theorem shows that if Bj = (21) Z b a 0; a jj Bjj , and iB LjO ' 1.e. denotes the row vector of the roots of B., then Bj E J j = n .. JJ 0; Z bjj a:l 2ja if 16 Suppose now that B, -1 of Bj • Let Bj is nonsingular" and consider the problem of inversion J = Cj " and let the vector of roots of C j be denoted by ~ej • Further let (22) ~B = (OB l j = .2C, J =I n Then since BjC j ' °B j 2" j (OC ,1' °c 2' j J ..., OBjnj ) ... , • °c n ) j j Frobenius l Theorem gives j' write n e (24) .2C = jj r. cx=l j cx C jj E.jCX Further, write (25) so that tl 6.1 B,,, and c J vector - ~j j j = [.2..11' (b~j' :2.1 = ~j = (c~jl °'2 -J , ••• , .2.jn n, , b2 , .,., bj~J) jj n jj 2 C c jjl • • e_ J jj ) ] jj 1 n x n.1j matrix, Ej is the known vector of coefficients for .1 l • (The is the unknown vector, to be calculated for evaluating is a. B:J will be spoken as being the inverse of b). -.1 be rewritten as (26) .2B j .9.c = j ~. c. J -J • Thus (24) and (21) can 17 Thus the solution for .£j is given by _ (, )-1, !J,j~' c - j ~. J J 5 -c. J Consider the element in the cell (a,13) of (6 ' !J,.). which by Frobenius' Theorem must equal to (B~j I j .). Now I e J J nj j J O!N 5.~ , It clearly equals -J~ -J~ L., and if every J object is defined a certain (say y-th) associate of itself and of none alse, then for some value of a' (here a a' elements of each B.. we have BYj' J = I n. ,and all the diagonal a' J = are zero for a' ~ y, which implies that to B. . JJ JJ for a' ~y. = y), 0, (9) Henceby tr(a~ ~.)=p(j,j,Yjj,a,13) tr BY j. J . 'Jj. jJ =tljP(jlj,Yjjla,13). The last term is clearly zero if a ~ 13 1 since distinct associate classes of any object do not have any common elements. (a,a) is cell 6J 6j Thus is a diagonal matrix and the element in the n.), i.e. J (28) which can be easily calculated. For the algebra Lj made very easy as soon as therefore, the problem of inversion of any B j € L. is JO from is known. For then one just calculates ~B j j (26) (which just involves the multiplication of a vector by a matriX), and ~ 6 j from .2-B. of by taking reciprocals, and finally multiplying J tl j ~C. by the transpose J • A few remarks regarding the computation of Corresponding to the algebra L. J will not be out of place here. a generated by the B.. , there is a standard 6.. J JJ representation of L., say L. generated by certain (n .. x n. ) matrices which J JO JJ Jj may be denoted by Pja. The interested reader is referred to Bose and Mesner [3J. It is shown there that the nonzero roots of B~. JJ are the same. and of the corresponding P. Ja a Further the multiplicities of these nonzero roots as roots of B jj 18 can be easily obtained by noting that ex tr(B .. ) JJ = O. The ordering of the roots doS required by Frobenius is usually easily done by trial and error. The above holds only due to commutativity. algebra.s of However since the Ljl,s are sub- L, we shall in the next section use the above property to invert matrices in L. If D is a matrix belonging to an algebra multidimensiona.l (orMnary) L, then D will be said to be a partially balanced matrix, if L is the algebra generated by a multidimensional (ordinary) partially balanced association scheme. 6. An algorithm for inverting multidimensional partially balanced matrices We shall first indicate the method for the cases generalization for higher m. verted. e (29) Let m=2, and let D(eL) m=2 and 3, and later the be the matrix to be in~ Suppose D = ex dij I: ex, i,j and write D in the partitioned form B ll D = (30) ex Dij 1 B 12 1 where (30) Let M = D-1 , and suppose a partitioned form for M, similar to ( 30) is MIl (31) M = \. i M2 M 12 M22 ! ...l 19 It can be easily checked tillat the M ij relations are connected with the B ij M22 = -1 B r1 [B22 - B2l Bll 12 M 12 = Mll = -B-1 B M , M = M• 12 ll 12 22 2l B-1 + B-1 B M B B-1 ll ll 12 22 2l ll (32) where without , loss of generality, we assume the matrices under the inversion sign to be nonsingular. This formula holds whatever the pattern in D may be. now that = Then we have where (35) Q(a',y') by the = ~ ~,~' ~2 ~~ E Y p(2,1,y;1,~,ar) p(2,2,yJ;lJYJ~t) E p(i,k,y; j,~,ar) p(1,f,yt;k'Yl~t), y Suppose 20 n. j ••• , di~ ) , for all permissible i" j" k and f · Hence we get (37) -1 M 22 = yl (d 22 yl M 22 = E - a l Q( a' ,y I )c~~) Byl22 E E yl YI d22 ,1 B22 , yl = E say . Thus (38) yl B 22 ~f m 22 yf , say could be easily obtained by the method of the last section. Fer MIl' we go backwards. Thus, E crf312 = - f3 , f3 I the vectors ~ll' !22 = (E = E which shows that a a I (c lla2 I 11 tt •• <~ w f3 11 !lltj2 ]BlIl~ . = i'tl... oller being defined analogously to C~l B~l) + E L l f3"a • - , f3 • t E f3 I I m12 _B t ] a Bll M 12 and I B12 (36) • Similarly, I 1121 crf312 (~ll ~a !12 E a a mIl Bll ' say MIl OPe .·g~tting ,say " 21 Next, using an obvious generalization of the above notation, consider (for the case m:::3) the problem of obtaining M where ji M ; (41) M M M 12 ll M 13 22 21 = M M 23 ~3 ~2 ~l 1- ,.. --f \ I i M ::: D-1 , B12 B13 1 B21 B22 B LB31 B 32 B 33 I and D I1 T Bll =1 II 2, i ;. ! .... -: Since, Bij = we can denote D (42) E c/, bc/' ij c/, Bij in short by a such that the element in e , for all permissible and (i,j) cell is • The suffix (ij) at the end of the bracket indicate the two sets volved. j , 3x3 matrix D* whose elements are row vectors, nij 2 (bi j , bij , ... , b ij \ j (43) i Si and Sj in- We shall consider the expression (43) to be identical with (42), except for an abbreviation. D* will be called the coefficient matrix of D. The computation proceeds as indicated in the partitioning of D. first stage, we obtain M , the inverse of Bil' l m=2, one computes M , where 2 -1 B B ll 12 i 1 r- I 21 B ~ Then proceeding as for the case M2 • 11 M2 •12 = ! B 22 At the M2 • 21 , say • M2 • 22 The computations can all be neatly and briefly expressed by ua;lng the vector (4,}. This ,nll be seen in an example (sec. 8) where we illustrate the working of the algorithm. 22 From the formula$ (32) one sees that we must next obtain (44) Using computations similar to those at (34), one finds that each term in the last bracket is a linear function Of the matrices B~j' so that (44) belongs to , and hence can be easily inverted. Let this inverse be 3 L ~3. Then to get M, we further use (32) to obtain f~ ) ;::: M~; - M 2 B13 \ ~3 B \ ,- 23 -; -' = (M2 • 11 H 13 + M2 • 12 B23 ) M --1 33 (M2 '.21 B 13 + M2 • 22 B ) M 23 33 • each term of which could be computed using product formulae Of· the form where (46) The sub-matrices of M can be similarly computed using (32). The method for inverting D remains the same for general values of m. 2 are m submatrices B ij of D. There At the r-th stage, we invert the top-left large e 23 submatrix of D, viz B 12 B ll D r = • ! L B 12 ·_·i • •• • Brl • •• B r2 • •• B rr II 1 .J Formulae of the form (45) suffice for all computations. The algoritm is carried is obtained. at the end of which D-1 v The,·above algorithm. offers a speedy method of inverting any multidimensional on for r = l,2,~ •• ,v, partially balanced matrix, in particular one of the form (7), which occurs in the analysis of a MDPB design. The algorithm has the further advantage that at every stage, the relation D D- l = I provides a check on the calculations. r r r The method of linear associative algebras appears to be a very powerful tool e for inverting several large classes of JlS.tterned matrices. Such matrices arise in various areas of research, particularly design of experiments. In the next section we consider an important application to the latter case. 7. Applications to the an~y.sis of partially balanced factorial fractions. The theory that "Te have developed so far has an important application to the analysis of factorial fractions haVing some balance or symmetry in them. Due to lack of space, we shall not be able to demonstrate this in complete generality. m l m 2 shall also restrict ourselves to fractions from 2 x 3 We factorials, firstly in order to make ideas clearer and the presentation elaborate, and secondly to avoid certain complications that arise when some factors have more than 3 levels. How- ever the generalization for the latter case is not too difficult, and we shall give some hints in that direction. e The factors at two levels each may be denoted by a , a , ••• , a and those 2 l ml at 3 levels by b , b2 , ••• , b~ • The corresponding symmetrical case is obtainable l 24 by taking or m as zero. We shall suppose that interest lies in estimating 2 the general mean, all the main effects and all the two factor interactions. For ~ an elaborate definition of the various effects, see [2J, to which we shall constantly refer. As usual, let the general mean be den cted by = 1,2, ••• ,ml ), Ai (i AiAif (i« if) 1,2, ... ,m -l) 2 AiB~ and B~ (j Bj = 1,2, ... ,ml -l), and pure interactions by B?~f(j«jf) BjBjf(j(<jf) = l,2, .....m2-l) .. 2 = 1,2, ... ,m2 ) and BjB (j ~ k k (i = 1,2, ••• ,m , j l = 1,2, ••• ,m2 ), Il, the main effects by = and the mixed ones by AiB j and = l,2, ••• ,m2 ). The total number of effects is and these can be grouped in an obvious manner into 10 sets containing 8 4 = 2 (B j } Il alone; (ii) each with m 2 8 2 = {Ail, elements; (iii) with m l 8 5 viz elements, = {AiAj }, 8 6 (i) 8 3 8 1 = {Il}, = (B j }, = (BiB j }, 7 = (B~ B~} With ~ ml{ml - I), ~ m2 (m2 - 1) and ~ m2 (m2 - 1) elements respectively; (iv) 8 = (BiB~J with m (m -l) elements; and (v) 8 = (AiB } j 2 2 8 9 8 and 8 10 = Now, in 2 (A B } 1 j with m m l 2 elements each. r2J , "le have shown that if we are interested in the above effects, the main problem in the analysis of a factorial fraction (T to the inversion of a certain row and each column of EE f effects. (Q, e Thus if ~ and ~ \)X\) matriX, denoted there by EEf. \) say) reduces Further, each corresponds to one and only one among these \) are any two such effects, then there is a unique cell 9) corresponding to them, and the element in this cell being denoted by Now take any set, say 8 • 2 Corresponding to this there is a EEf, say M , whose diagonal elements are of the form 22 mlXln €(Ai,A:t.) l €(Q,~). submatrix of and off-diagonal 25 ones of the form £(Ai,A j ). Similarly if we take the rows corresponding to S2 and the columns corresponding to S4 we get a m xm submatrix M24 of EEl ~ l 2 whose elements are of the form £(Ai,B~). Thus EEt is subdivided into 100 sub- = 1,2, ••• ,10), matrices Mij(i,j, where Mij corresponds to the sets Si and Sj. We wish to apply the properties of the linear algebroas developed in the earlier sections for the inversion of EEt. For this purpose we define an association scheme between and within the different sets. The following table gives, for a typical element of each set, its various associates in the same set. TABLE 1. Set S1 e Associates 1st i 2nd 1 IJ. 2 A. Ai Aj 3 B i 2 B i B.J. B~J. B j 2 B j 4 3rd 4th 5th 2 2 BiBk' BkB j l\B~ I..l. J. 5 AiAj AiAj Ai~ ~A~ 6 BiB j BiBj BiBk 7 B~B~ B?~ B~~ BkB( 2 2 BB k f 8 2 BiB j 2 BiB j B?j 9 AiB j 2 AiBj A.B. J. J 2 AiB j AiBk 2 AiBk 10 e Typical Element In the above table, the subscripts B?k' i,j,k and they occur in interactions belonging to the same set. ( BjB~ ~Bj ~Bf ~B~ ~B( are all unequal, whenever Thus in S5 the second 26 associates of A A are Al~' A A etc, i.e. those interactions between A-factors 1 2 1 4 which involve Ai but not Aj • Similarly the third associates are those which do not involve either Ai or A.• J ft can be shown, from considerations of the symmetry of the factors involved that, for each set Sj (j = 1,2, ••• ,10), the association relation defined above satisfies the conditions of an ordinary partially balanced soheme. (EE I) • Suppose that (EE I) is a balanced of the diagonal sub-matrices M.. JJ Consider now matrix, i.e. is symmetrical with respect to all the factors involved. Such a matrix arises in particular, when T is a partially balanced array of strength 4, as defined by Chakravarti [4]. Then it can be shown (using the theory developed in [2]) that if 9, iJ 1 and e CP2 are any three elements of Sj (j rf 8), such that (e, (?l) are the same associates l ) = 6(e, q(2)' This im(j ~ 8) belongs to the commutative linear algebra Lj generated (under Table 1) as (e, CP2)' then in EEl we have E(e, plies that M jj by the association scheme for Sj. CfJ On the other hand even for a balanced matrix (EEl), it can be shown that in general, we do not have , where i, j, k are all unequal. long to L • In order that M 8 cient. 88 This implies that in general M 88 does not be- E L , the condition (49) is necessary and suffi- 8 This condition can be expressed more directly in terms of the fraction T itself, when it is equivalent to Arst denotes the number of treatment ijk combinations in T in each of which the factors B , B , B occur respectively j k i for all permissible i, j and k, where e 27 at levels r, sand t. A ba.lanced fraction for which this easily verifiable condition is satisfied vdll be called a commutative balanced fraction. In order that we may be able to use the :properties of the algebra inverting the matrices for L. J M E Ljl we must know the roots of the association jj matrices B~j of Lj , or equivalently the matrices sec. 5). These matrices are now :presented below for /).j (defined for an L , L , L and 2 5 8 Lj in L , since 9 the rest are covered under these cases: (51) /)., (i) 2 = -j r 1 l~-l J'm ..1 1 (-l)J' ~- 1 -'j /)., (ii) 5 = I 2(ml -2) mil (iii) /).8 I J' mi I (-2)J', Jt m1 -1 (m -4)J' 1 m -1 1 1 I' where mi = ~ml(ml-3), -(ml -3 )J~ -1 1 1 J' 1 J' m -1 2 ~-l J,.~I J' m'2 . J' , m 2 ml m~ i J I II m 2 (-l)J'm" 2 = ~(~-2)(ml-3) -r I J' m -l 2 (-l)J' 1 1)12- , llI. l 2(m -2) 2 (m -4)J' 2 m -1 2 2 (m2-4) 2("2- ) (-2)J', m 2 2J t" m 2 (-2)J' , m 2 ( -2)J~" (m2-2)J~ -1 (O)J t It m2 (O)J~ -1 m2 ","2) (m 7"3) ( ... 2)(m2-3)J~ -1 2J' 2 m'2 2 -em -2)J' 2 m -l 2 2 2 2 , I -.¥ 28 e (iv) b,' 9 = J ~ml-l)(m2-l) (m -1) 2 (-l)J(m -l)(m -1) 1 2 {-l)J~ -1 2 (m2-l)J~_1 (ml-l) (-l)J(m -l)(m -1) 1 2 (m -l)J' 1 m -1 2 (-l)J' 1 ml - (m -l)(m -l) l 2 where Jt r JI m -1 1 1 J'(m -l)(m -1) 2 l denotes a row vector of length above matrices and the results of sec. r J'm -1 2 - (lIJ.-l)J'm -1 2 , -(m2-1)J~-1 with every element unity. Using the 5., it is a very simple matter to invert the If the fraction is balanced but not commutative, then also (by using the properties of a linear algebra, or otherwise), the inversion of M 88 e (see [l'l). to the inversion of matrices of very low order. can be reduced The formulae, however, are more complicated and less elegant and explicit than in the commutative case, and will not be presented here due to lack of space. The algorithm developed in sec. 6. can be gainfully employed for the inversion. of (EE') from any balanced fraction may be effected in the following way. T. If T is commutative, a simplification He rearrange the row blocks and column blocks of EE', so that now they correspond in order to and 8 10 • M 88 by inverting M 88 , 8 9 instead of M • The algorithm can now be started (for (EEt)*) ll first, and then continuing with the rest. In fact, writing it can be shown that get 8 1 2 1 This will mean that in the nei'T EE' (say (EEI)*) the top left hand submatrix is e 8 , 8 , 8 , ••• , 8 p = (M - Q,' M8~ Q,) ,which one has to invert (see (32» {EE' )*-1 ,belongs to an algebra L which has as subalgebras Ll ,·· .'L.r' to L 9 29 e and LIO.The multidimensional PB association scheme to which L corresponds is detailed below. Since the association scheme within each Si exhibited in Table 1, we consider the association between the elements of Si and Sj for i ~ j. The motivation behind the scheme is this. Sj. Then (a, is already <f'l) Let e Si and <f'l and CP2 £ are defined to be the same associates of each other as £ (e,CP2)' only if To be more specific, suppose that between CPl and PI A-factors and ql B-factors common (0 ~ PI + ql ~ 2). <f'2' let these numbers be respectively P2 of the A.factors common between <f' denoted by Xl and in CPl common, and the indices ~. Let If PI there are exactly Similarly for ~ 1, let the index of then we get the ordered pair e and CPI (~ll' yil). = 1, ••• ,Pl; P2 + ~ pairs Similarly if a j (X , x 2i ), (Y2j' Y2j). 2i Between <f' and be (XII' XiI) Yl and He. shall thus get PI + ql = l,···,ql· e b-factor (say B) are respectively e A be one A in by xf-.· This gives us one ordered pair of B in (Xli' Xii)' (Ylj' yij)' i and and '!rl. corresponding to the common factor A. similarly get e, is Yi' ordered pairs and *2 we shall Then it has been found that (53) holds if (i) (ii) (iii) the set of ordered pairs (Xli' xii)' i = 1, •••Pl' is the same as the set (X , x ), i = 1, ••• 'PI , 2i 21 .. the set (Ylj' Yij) and (Y , y ), j 2j 2j = 1, ••• ,ql Hence in accordance With the above, rTe shall say that <f'l and <f'2 associate class in Sj generated by e£ Si' if and only if (54) are the same. are in the same is true. This e ;0 means for example that ~(e Sl); (ii) (i) all elements of S5 are the same associates of the different associates of Bj (e S;) in the set (say 1st associates) and ~B( k, ( ~ j, (2nd associates), (iii) in SlO' AiBj , etc. Ai~' ~Bj and ~B( Bj~ S6 are AiBj(e S9) has as its four different kind of associates; It can be shown that, leaving S8 aside, the association scheme defined by Table 1, and (54), (to be called the factorial association scheme) is of the MDPB type. The proof, which follows from considerations of symmetry be~ween the factors, will be omitted here. The case when some factors, say C , C , ... , Cm' l 2 can be dealt with in a similar manner. [C~}, x = l, ••• ,s-l, and s-l There are s-l are at s(>;) levels each, sets of main effects sets of interactions of the form [C~ C~}, x = 1,2, .... ,s-l which give rise to submatrices in the Qj.agonal of EEt, which are such or L • Also the 5 process of inversion of submatrices which correspond to sets of the form. fC~C~}, that they can be inverted by using one of the algebras (x L 2 ~ y) is exactly the same as for M88 (which arises out of [B?j})' except that for commutativity, other conditions are needed. The above theory permits many further applications. As mentioned earlier, the main problem in analyzing a fraction and examining its properties, is to invert its (EEt). In many cases, although the fraction may not be balanced, it may be such that EEt. breaks up into two or more matrices of smaller size, which in turn are invertible by using the algorithm of sec. 6., or just the algebras Lj such fractions will be called partially balanced. found in [2J. • All Some examples of these will be Construction of good partially balanced fractions will be discussed in separate papers, wherein the above theory which inspired them Will be greatly exemplified. e 31 8. An example. We shall now illustrate the preceding theory by considering the inversion of 27 factorial fraction, a (22 x 22) matrix 0, which arises in the analysis of a discussed in Example 2 [2]~ of 4 , and then correspond, in order, to the effects ~,A2'~' A In this fraction the factors are divided into two groups BI , B21 B}. The rows and columns of (1 (~,~,~,~~,~~,~;Bh,~,~;AA'YI'~~'~~' AI B2 , A2B2, ~B2' A4B2 , A B , A B , A B , A4B ; J.L). The four sets of effects, I 3 2 3 3 3 3 separated from each other by semicolons, may be denoted by 8 , 8 , 8 and 8 • 4 2 1 3 An association scheme within each set may be defined as in Table 1, sec. 7. If for distinction, the sets in the table are denoted by clearly the sets types 81 and 8 1 8~ and 8 2 are of the type 8~, 8~ ~ instead of 8., then ~ and similarly 8 4 and 8 3 of respectively. In addition to the above, a factorial association scheme (defined in the last section) also exists between the different sets. Thus the total scheme between and within 8 , 8 , 8 and 8 is of the MDPB type. This can be checked directly 1 2 3 4 0: as well from the corresponding association matrices B • ij To save space, we shall indicate only the matrices B~. (i ~ j), since the ~J others can be immediately written down using table 1: B1 12 1 B 13 = J 63 = [Q. Q. , 1-- Q.], where 1 1 1 Q. = 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 e I L. B2 13 = 1 J 6,12 - B13 , 1 i j i , "-, B123 = J J 14 J 14 °141 14 °14 J 14 J J 14 °14 In the above J ~n or J 14 ~ B223 = J ,12 3 , I 1 I - B123 , J is an mxn matrix with unity everywhere, and ° ~ is of the same size with zero everywl':e reo As at (42), (43), we shall v~ite (for all i andj) where n is the number of associate classes between 8 and 8 • The number ij j i of associate classes within 8 , 8 , 8 and 8 respectively is 3, 2, 4 and 1. 1 2 3 4 These include the zero-th or the self associate class. For convenience we shall make a departure from (55) for 8 , am write 2 8ince B~2 terms of =I3 (I , J 3 ' and B~2 = (J33 - 1 ), this essentially implies 'Working in 3 33 ) in place of the usual basis (B~2' B~2). With the above notation at hand, 0 is given by (ll,l'-l)l~_J (-1)21 (10,1)22 I ____ .__._.J (0,2)32 __. ---------- . (11,1,1,-1)33 (0)43 l 33 In order to apply the algorithm of sec, 6, we use the partitioning indicated above~ The successive square matrices indicated by this partitioning will be denoted by 01' 02' ~ and 04 (= 0) respectively. L , particularly To invert 01 we must obviously use algebra ml = 4) of (51) (2), together with (27). = bt b t bo' = (14,12.1 13 , 6 '.1 .2.'°1-1 8 J 12 ) bo = (8 bo 5 (with Here (ll, I, -1) =.2,'°1 , = (lk), (l~ 5 say (in the notation as at (27», )J 13 ,{-§-)J12 -3 1 ) W"']Ji:' W'" ' = diag (6, 24, 6) • bot b. °1 .., Hence 8/6 1 W" \ -3/24 [ L 1/6_ , and Proceeding with the algorithm we next invert 02 using (32). (58) it can be easily checked that The inverse of the last matrix is easily obtainable, and is Since e 34 r Supposing Xl 0- 1 2 1 Y ] = y' 1 we have using (32) with (58) Xl Next, to invert 1 1 B32 B23 1 2 B32B23 0, = 01-1 1 Zl J () -1 Y-1 21 ~ 1 = 48 x41 we need the following results: 2 2 = ( 2,1,2,1)33' B32 B23 = ( 1,0,1,0)33 2 1 = B32 B23 = (0,1,0 1 1)33 1 1 2 1 B13B33 = (2,0)13 ' B13B33 = (0 1 2)13 2 = (1,2)13' B2 B2 = ( 21 1)13 B113B33 13 33 Bi3B~3 = (2,4\3 ' Bi3B~3 = (4,2)13 • , e 35 Then for the first formula in (32), lIe compute = (11,1,1,-1)33 - "4}- [ ~ (60, 60, 19, 19)+ 6(1,1,1,1)33 + = 41 1 x To find 30 -1 (11682, -126, 202, -1766)33 = Z2 Z2' we use algebra L 9 Using notation as at b f 1.:1' ~~ = 1.:1 2 (l.:1 fb,) 1.:1 9(vlith (156,-8,156,-8)33 J ' say m1 = 4, m2 = 3) of (52) (iv). (27), we have here = [(48/41), 8 J 16, (144/10)J12, 12 J 13 J = ~f_1 ' .. Z2 2 ~z and io [(41/48), (1/8)J16 , (10/144)J12 , (1/12)J13 J = lU4 [287, 190, 285, 714J , = diag Renee = 1 12 x 144 so that - vlriting now II X Y2 I Y'2 Z2 2 0;1 = L ·lJ (12, 24, 36, 72) • I -' 287 -1 95 I 95 ; 119 , say "tit we have (0,2)13 i • J _ (0,2)23 r Using (59) J this can be quickly reduced to 12-~ 1~~ Y2 = (123, 147)13 • L(13 0 , 154)23 Similarly X 2 = 0;1 - Y2 [(0,2)31 {O,2)32 J 0;1 (327, 135, 159)11 1 = 12 x 144 -1 , the above process is repeated once more. n which border ~ are of the type 1 J (192, 140)22 L To obtain 0 (162)12 k. J 1r' "There k ,calculation is easy and will not be reproduced. Since the matrices in is a constant, the REFERENCES 37 38 [12] Shrfkhande, S. experiments. s. Some combinatorial problems in the design of (1950). Unpublished doctoral dissertation, University of North Carolina, Chapel Hill, N. ·C. [13] Srivastava, J. N. (1961). designs. Contributions to the construction and analysis of Institute of Statistics, mimeo series no. 301, Univ. of North Carolina, Chapel Hill, N. C. [14J Yates, Frank (1937). Impe~.al [15J The design and analysis of factorial experiments. Bureau o:f Soil Science, Technical Conununication no. 35. Youde:'J., H. J. (1937). tobacco mosaic virus. Use of incomplete block replications in estimating Contributions from Boyce Thompson Institute .2 317-326.