APPENDIX Some matrix results
The main matrix results required for this book are given in this Appendix. Most of them are stated without proof as they are relatively straightforward and can be found in most books on matrix algebra; see for instance Searle (1966), Basilevsky (1983) and Graybill (1983). The properties of circulant matrices are considered in more detail as they have particular relevance in the book and are perhaps not widely known. A detailed account of circulant matrices can be found in Davis (1979).
A.1 Notation
An n x m matrix A = ((aij)) has n rows and m columns with aij being the element in the ith row andjth column. A' will denote the transpose of A. A is symmetrie if A = A'. If m = 1 then the matrix will be called a eolumn veetor and will usually be denoted by a lower ca se letter, a say. The transpose of a will be a row veetor a'. A diagonal matrix whose diagonal elements are those ofthe vector a, and whose ofT-diagonal elements are zero, will be denoted by a6• The inverse of a6 will be denoted by a- 6• More generally, if each element in a is raised to power p then the corresponding diagonal matrix will be denoted by a P6• Some special matrices and vectors are:
In' the n x 1 vector with every element unity In = 1~, the n x n identity matrix Jnxm = I nl;", the n x m matrix with every element unity
J n = J nxn
Kn = (l/n)Jn the n x n matrix with every element n -1. Suffices can be omitted if, in doing so, no ambiguity results. 206 SOME MATRIX RESULTS A.2 Trace and rank
If A = ((au)) is a square matrix of order n, i.e. an n x n matrix, then
n trace(A) = L ajj ;= 1 The number of linearly independent rows or columns of the n x n matrix A is given by r = rank(A). If r = n then A is non-singular, if r < n A is singular. Provided the matrices are conformable, and not necessarily square, then
trace(AB) = trace(BA) (A.l)
rank (AB) :s.;; min[rank(A), rank(B)] (A.2)
rank(AA') = rank(A' A) = rank(A) = rank(A') (A.3) It follows from (A.l) that the trace of the product of matrices is invariant under any cyclic permutation ofthe matrices. For example, trace(ABC) = trace(BCA) = trace(CAB)
A.3 Eigenvalues and eigenvectors Let A be a square symmetrie matrix of order n. An eigenvalue of A is a seal ar A such that Ax = AX for some vector x # o. The vector x is called an eigenvector of A. If Al' A2' ... , An are the eigen val ues of Athen
n trace(A) = L A; (A.4) ;= 1 rank(A) = number of non-zero eigenvalues (A.5)
n lAI = n A; (A.6) ;= 1 where lAI is the determinant of A. If Al ,A2 , ••• ,Am are the distinct eigenvalues of A (m:s.;; n) then the determinant can be written as
lAI = A~I Ai' ... A~'" where n; is the multiplicity of Ai> and where Ln; = n. Repeated application ofAx = AX shows that, for some positive integer h, x is also an eigenvector of Ah with corresponding eigtmvalue IDEMPOTENT MATRICES 207 A\ i.e. (h = 1,2, ... ) (A.7) The eigenvalues of a symmetrie matrix are real. For eaeh eigenvalue of a symmetrie matrix there exists areal eigenveetor. If B is a non-square matrix then B'B and BB' have the same non-zero eigenvalues. For every symmetrie matrix A there exists a non-singular matrix X sueh that
(A.8) where the elements of Aare the eigenvalues of A. For the symmetrie matrix A there exists a set of orthogonal and normalized eigenveetors XI' X2, •.. , Xn satisfying , {1, i=j xix j = 0, i#j The canonical or spectral deeomposition of A is then given by
n A = LAiXiX; (A.9) i= I where Ai is the eigenvalue eorresponding to Xi' If A is non-singular then
n -1 "1-1 , A = L... Ai XiXi (A.1O) i= I
It also follows from (A.9) that the eigenveetors Xi (i = 1,2, ... , n) satisfy
n L xix;=I (A.ll) i= I
A.4 Idempotent matrices
A square matrix A is idempotent if A 2 = A. If A is idempotent then so is I - A. land Kare examples of idempotent matriees. The eigenvalues of an idempotent matrix A of order n are 1 with multiplieity r, and 0 with multiplieity n - r, where r = rank(A). It then follows from (A.4) and (A.5) that for an idempotent matrix
rank(A) = traee(A) (A.12) 208 SOME MATRIX RESULTS If A and Bare idempotent then AB is idempotent if and only if AB=BA.
A.5 Generalized inverses Let A be a square matrix of order n with rank(A) = r. If r = n then A has an unique inverse A - 1 which satisfies (A.13) Further, an unique solution to the equations Ax = y is then given by x=A-ly (A.14)
If r < n, A - 1 does not exist. There will, however, be an infinite number of solutions of the consistent equations Ax = y. As an example, consider the equations
The second equation is twice the first so that rank(A) = 1. Replacing 6 by 5, say, in the vector y would lead to an inconsistent set of equations for which no solution will exist. One solution to the above equations is given by Xl = 3, X2 = 0 and can be written as x = By where B=(~ ~}
Although A - 1 does not ex ist, the matrix B plays the role of an inverse in that its use leads to a solution to the equations. Bis called a generalized inverse or g-inverse of A, and is denoted by A -; it is sometimes called a pseudo-inverse or conditional inverse of A. It is not unique since
A-=G _~) also leads to the solution Xl = 3, X2 = 0, whereas
A-= C~3 1~6) leads to a different solution, namely Xl = x2 = 1. GENERALIZED INVERSES 209 A - is ag-inverse of A if and only if it satisfies (A.15) A solution to the eonsistent equations Ax = y is then given by (AI6) It ean be verified that the three g-inverses given in the above example satisfy (A.15). The matrix A + whieh satisfies the following four eonditions: AA+A=A A+AA+ =A+ (A.l7) (AA +l' = AA + (A + Al' = A + A is ealled the Moore-Penrose g-inverse of A; it ean be shown to be an unique matrix. If A is given in eanonieal form as n A= L AjXjX; j= I where Al, A , .•• , A, (r< n) are the non-zero eigenvalues, then: 2 , + "1-1 I (a) A = L Aj XjXj (AI8) j= 1 is the Moore-Penrose g-inverse of A;
, n (b) A - = L Aj- 1 XjX; + L (XjXjX; (A.19) i=l i=r+l is ag-inverse of A for a given set of eonstants C(, + l' C(, + 2, ... , C(n. If these eonstants are zero then A - '7 A + .
If A is asymmetrie idempotent matrix then two useful g-inverses of Aare the Moore-Penrose inverse A + = A and the inverse A - = I. The following two results are frequently used in the book: (a) AA - is an idempotent matrix with rank(AA -) = traee(AA -) = rank(A) (A.20) (b) If (A/A)- is ag-inverse of A/A then A(A/A)- A' A = A (A21) 210 SOME MATRIX RESULTS A.6 Kronecker products
Let A = ((aij)) be a p x q matrix and B be an r x s matrix. The Kronecker product of A and B, denoted by A ® B, is the pr x qs matrix defined as
allB al2B ... alqB) A®B= ( f21 B f22 B i·· f2qB aplB ap2B ... apqB
Two special results are that In ® Im = Inm and Kn ® Km = Knm . If a is a scalar and matrices are conformable, then
(aA)®B = A®(aB) = a(A®B) (A®B)®C = A®(B®C) (A®B)' = A'®B' (A.22) (A®B)(C®D) = AC®BD (A + B)®C = A®C + B®C Let A and B be square matrices of order n and m respectively; then (A®B)- =A-®B (A.23) trace(A ® B) = trace(A)·trace(B) (A.24)
IfxI isan eigenvector of A with eigenvalue Al' and x2 is an eigenvector of B with eigenvalue A2' then
(A ® B)(x I ® X2) = Al A2(X I ® X2) (A.25) i.e. Xl ®X2 is an eigenvector of A®B with eigenvalue AI A2.
A.7 Circulant matrices If A = ((ai)) is an n x n matrix with aij = alm where j - i + 1, r;~ i { m= n-(j-i+ 1), jcirculant matrix completely determines the matrix, the n elements in the first row of A will be denoted by ao, al , ... , an-I. For example, the CIRCULANT MATRICES 211 circulant matrix for n = 4 is
a 1 az a3) ao a 1 a2 a3 ao a 1 a2 a3 ao
Let the n x n matrix r h be a basic circulant matrix whose first row has 1 in the (h + 1)th column and zero elsewhere. For instance, for n=4,
0 0 1 0) o 0 0 1 r 2 = ( 1 0 0 0 o 1 0 0 The general circulant matrix A can then be written as
(A.26)
The eigenvectors and eigenvalues of the circulant matrix A can be determined from those of r 1, in view of (A.26) and the fact that (h=0,1, ... ,n-1) (A.27)
Eigenvectors of r 1 are given by
(A.28)
with corresponding eigenvalues
Au = wj for j = 0, 1, .. . ,n - 1, where w is given by w = exp(2nijn) = cos(2n/n) + i sin(2n/n) (A.29) and where i = J=1. This result follows from the fact that w" = 1. Using (A.7) and (A.27), eigenvectors ofrh are then given by (A.28) 212 SOME MATRIX RESULTS with eigenvalues for h,j = 0, 1, ... , n - 1. Henee, eigenveetors of the general eireulant matrix A are also given by (A.28) with eigenvalues n-1 Aj = L ahw jh (A.30) h=O for j = 0, 1, ... , n - 1. Note that the eigenveetors rj are independent of the elements of the matrix A. If A is symmetrie then a j = an _j (j = 1,2, ... , m) so that
m Aj=ao + L ah[wjh+W(n-j)h] (A.31) h=l where n/2, n even m- { (n - 1)/2, n odd Now sinee
sin [2n(n - j)h/n] = - sin(2njh/n) it follows that
wjh + w(n- j)h = eos(2njh/n) + eos [2n(n - j)h/n] so that the eigenvalues of asymmetrie eireulant matrix Aare given by n-1 Aj = L aheos(2njh/n) (A.32) h=O A speetral deeomposition of A is given by n-1 A= L Ajr/ij (A.33) j=O where Yj is the conjugate of rj given by replacing w in rj by its eomplex eonjugate w= exp( - 2ni/n) = eos(2n/n) - i sin(2n/n) Equation (A.33) ean be verified by first showing that CIRCULANT MATRICES 213 The (k,l)th element of r/yj is (1In)w(k-1li, and since for k> I
W(k-Ilj = w-[n-(k-Il]j it follows that rj rj is a circulant matrix, which can be written as
(A.34) Hence since h=l L W-(h- 1lj = {n, j 0, h#l Then
which establishes (A.33). Note that j=k I _ {I, rjrk = 0, j#k
The Moore-Penrose inverse A + of A is also a circulant matrix since it can be written as
A+ = L.A. j- 1 rdj where the summation is over all r non-zero eigenvalues of A, and where r = rank(A). If r = n then A + = A -1. Now using (A.34),
where
hj eh = (I/n) L.A. j- 1 w- j
If A is a symmetrie matrix then, from (A.3I), .A. j = .A.n - j so that
eh = (Ijn) L ).j- 1 cos(2njhln) (A.35) j
Let r hk be abasie circulant of order nk and let the n x n matrix r h be defined by 214 SOME MATRIX RESULTS where n = n1 n2 ••• nm • Then following (A.26) a general block circulant matrix A is defined by
(A.36) where "h,h2 ... hm is an element in the first row of A. It folio ws from (A.25) that eigenveetors of Aare given by rit ® rh® ... ®rjm (A.37) where rj is given by (A.28), with eorresponding eigenvalues
A··1112".)m . = L..'" '"L., ... '" L., ah1 h2··. hrn WS h, h2 hm where m S = n I jkhk/nk k= 1 for jk = 0, 1, ... ,nk-1. If A is symmetrie then Aith ... jm=I···Iah,h2 ... hmeos[ f (2nj khk/nd ] (A.38) h, hm k=l References
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Combinatorial Mathematics, 8, 304~313. Lecture Notes in Mathematics 884. Ed. K.L. McAvaney. Springer-Verlag, Berlin. Voss, D.T. and Dean, A.M. (1987) A comparison of dasses ofsingle replicate factorial designs. Ann. Statist., 15, in press. Wheeler, N.C. and Kshirsagar, A.M. (1981) Uniformly better estimators with applications in two-way designs. Commun. Statist. Theor. Meth., AIO, 699-711. White, D. and H ultquist, R.A. (1965) Construction of confounding plans for mixed factorial designs. Ann. Math.Statist., 36, 1256~1271. Williams, E.R. (1975a) A new dass ofresolvable block designs. Ph.D. Thesis, Univ. of Edinburgh. Williams, E.R. (1975b) Efficiency-balanced designs. Biometrika, 62, 686~689. Williams, E.R. (1976) Resolvable paired-comparison designs. J. Roy. Statist. Soc., B, 38,171-174. Williams, E.R. and Patterson, H.D. (1977) Upper bounds for efficiency factors in block designs. Austral. J. Statist., 19, 194~201. Williams, E.R., Patterson, HD. and John, J.A. (1976) Resolvable designs with two replications. J. Roy. Statist. Soc., B, 38, 296~301. REFERENCES 221 Williams, E.R., Patterson, H.D. and John, J.A. (1977) Efficient two-replicate resolvable designs. Biometries, 33, 713-717. Worthley, R. and Banerjee, K.S. (1974) A general approach to confounding plans in mixed factorial experiments when the number of levels of a factor is any positive integer. Ann. Statist., 2, 579-585. Yates, F. (1936a) Incomplete randomized designs. Ann. Eugenics, 7,121-140. Yates, F. (1936b) A new method of arranging variety trials involving a large number of varieties. J. Agric. Sei., 26, 424-455. Yates, F. (1937) A further note on the arrangement of variety trials: Quasi Latin squares. Ann. Eugenics, 7,319-331. Yates, F. (1939) The recovery of inter-block information in variety trials arranged in three-dimensional lattices. Ann. Eugenics, 9, 136-156. Yates, F. (1940a) Lattice squares. J. Agric. Sei., 30, 672-687. Yates, F. (1940b) The recovery of inter-block information in balanced incomplete block designs. Ann. Eugenics, 10, 317-325. Author index
Agrawal, H.L., 107, 112-3 Graybill, F.A., 205 Anderson, V.L., 150 Greenfield, A.A., 150 Gupta, S.c. 178 Bailey, R.A., 127-8, 150, 179-80 Banerjee, K.S., 149 Hall, W.B., 31, 73, 77, 79-80, 83, Basilevsky, A., 205 104, 110, 113 Bose, R.C., 41, 57, 62, 126, 129, 149 Harshbarger, B., 54 Box, G.E.P., 150 Hedayat, A., 28 Butz, L., 100 Hultquist, R.A., 149 Hunter, E.A., 86-7 Calinski, T., 26 Hunter, J.S., 150 Ceranka, B., 36 Hunter, W.G., 150 Clatworthy, W.H., 41-2,54,58,62, 81, 104, 167, 194 Ipinyomi, R.A., 112-3, 115 ConnitTe, D., 33 Izumi, K., 169 Connor, W.S., 57 Cotter, S.c., 149 James, A.T., 26 Cox, D.R., 4 Jarrett, R.G., 33-4, 55, 77, 79-80, 83, 110, 113 David, H.A., 54, 62, 64, 67-8, 73, John, J.A., 4, 16, 19, 28, 30, 33, 58, 81,92, 104, 194 62,64,67-9, 71-3, 91-2, 104, Davis, P.J., 205 107, 112-3, 115, 142, 150, 168, Dean, A.M., 138-9, 150, 165 174,180-1,194,203 Dey, A., 112-3 John, P.W.M., 42, 48, 58, 75 Draper, N.R., 4 Jones, B., 29, 35 Jones, R.G., 17 Eccleston, J.A., 28, 35, 97-8, 107, 109,203 Kempthorne, 0., 126, 129 Kiefer, 1., 98 Federer, W.T., 100 Kishen, K., 149 Fisher, R.A., 1, 4, 46, 48, 73, 75, Kshirsagar, A.M., 106, 161, 203 166,174 Kurkjian, B., 160-1 Franklin, M.F., 150 Freeman, G.H., 29, 58, 75, 107 Lamacraft, R.R., 31, 73, 104 Fries, A., 150 Lewis, S.M., 150, 165, 180-1
Gilchrist, F.H.L., 127-8 MarshalI, T.F. de c., 26 Giovagnoli, A., 139 McLean, R.A., 150 224 AUTHORINDEX Mejza, S., 36 Shimamoto, T., 41 Mukerjee, R., 158, 178 Shrikhande, S.S., 41, 107 Sihota, S.S., 149 Nair, K.R., 41, 62 Silvey, V., 50 National Bureau of Standards, 150 Singh, M., 112-3 NeIder, JA, 183, 186, 189, 191,203 Smith, H., 4 Smith, T.M.F., 16 Paterson, L.J., 31-2, 87 Stone, 1., 33 Patterson, H.D., 33, 35, 50, 54-5, Street, D.J., 112 80-2, 86-9, 91, 106, 127-8, 150,179,191 Thompson, R., 191 Pearce, S.c., 26, 29, 99, 100, 106 Turner, G., 58 Prasad,1., 112-3 Preece, DA, 112 Voss, D.T., 138-9, 150
Quenouille, M.H., 4, 142, 174 Wolock, F.W., 62, 64, 67-8, 73, 92, 104, 194 Raghavarao, D., 42, 48, 58, 100, Wilkinson, G.N., 26 107 Worthley, R., 149 Raktoe, B.L., 149 White, D., 149 Rao, C.R., 179 Williams, E.R., 28-9, 30, 33, 35, Rice, c.G., 169 54-5,80-3,85-9,91-2, 106, Roy, 1., 203 108, 168 Russen, K.G., 97, 99-100, 107, 109 Wheeler, N.C., 203
Se arie, S.R., 11, 16, 205 Yates, F., 40, 46, 48, 73, 75, 111, Shah, B.V., 19,41 166,174,179,191,194 Shah, K.R., 28, 107, 203 Zelen, M., 160-1 Subject index
A-optimal designs, 30-2, 34, 39, 60, Analysis of variance, 12-14, 16, 68, 72, 90, 92, 99, 102, 166, 101,103,173,175-6,190,197 168 null, 184-6 A-optimality, 28 APL,196 Additive etTects, 119 Assumptions underlying analysis, Additive model, 4 4, 12, 199 Adjusted block totals, 15 Asymmetrical factorial designs, 117 Adjusted orthogonality, 97-8, 100, 106, 107-9, 180-1,203-4 Balance canonical efficiency factors, 98 factorial, 160, 166-9, 175, 177 condition for, 98 general, 96, 100, 191, 203 importance of, 204 in factorial experiments, 160-1, information matrix, 98 166-7 properties of, 98 see also Efficiency-balance; Adjusted treatment totals, 8 Variance-balance Aliased components, 145 Balanced incomplete block designs, Aliasing system, 145 40-2,45-9,50-1,56-62,73, 1X-designs, 80, 82-8, 90-2, 105, 75,90,96, 103-4, 107, 109, 113 106-10,198 basic contrasts, 49, 153 1X(0, 1)-designs, 85, 198 canonical efficiency factors, 48, 1X(0, 1, 2)-designs, 84-5 153 1X-optimality, 85, 90, 92 complement, 47, 49 basic arrays, 86-7, 106, 108, 110 construction of, 45 choice of designs, 83-8 decomposition of sums of computer construction, 87 squares, 49 concurrences, 83 dual designs, 58 construction of, 82-3, 88 efficiency-balance, 48, 155 dual,83 estimating stratum variances, equivalence to generalized cyc1ic 194 designs, 82 existence of, 45 existence of 1X(0, 1)-designs, 87 factorial balance, 160-1 red uced array, 84-5 factorial structure, 160 row-column, 105, 106-10 information matrix, 48 tables of, 85 in factorial experiments, 153-6, two-replicate, 90-1 159, 166, 168 unequal block sizes, 88 optimality, 48 Analysis of experiments by recovery of inter-block computer, 169 information, 191 226 SUBJECT INDEX Balanced incomplete block in n-cyclic designs, 77, 162-5, designs (contd) 181 relationship among parameters, in nested row-column designs, 45 114 tables of, 48 in rectangular lattice designs, unreduced designs, 45-6, 48,57, 54-5 109 in resolvable designs, 55 use of orthogonal squares, 46-9 in row-column ex-designs, 107-9 Balanced lattices, 51-2, 58, 111-2 in row-column designs, 96, 99- Basic contrasts 100, 181 definition of, 26-7, 36 in row-orthogonal designs, 102 factorial balance, 160 in square-lattice designs, 51-2 factorial structure, 159 largest,27 in balanced incomplete blocks, optimality criteria, 28, 30-1 49, 153 properties of, 26-7 in complete block designs, 43 smallest, 27-8 in group divisible designs, 59-60 unequal replication, 36 unequal replication, 36 upper bounds, 33 Binary designs, 16 Combined analysis, 3 Block concurrence graph, 89 Combined estimates, 192, 194, 197, Block concurrence matrix, 51, 54 199 Block effects, 4, 187 in block designs, 192-201 Block stratum, 94, 184 in row-column designs, 201-4 Block structure variance-covariance matrix, 194, crossed, 186 203 nested, 186 Column-orthogonal designs, 103 orthogonal, 182-6 Comparisons simple, 183, 186, 188-9 between blocks, 1-3,27, 182 between columns, 182 Canonical efficiency factors between rows, 182 adjusted orthogonality, 98 of interest, 2, 12, 27, 29, 44, 60, computer program, 169 155, 157 definition of, 25-6, 36 within blocks, 1-2,22-3,27,43, dual designs, 37-8 182 factorial structure, 157-61 Complete set of binary matrices, for binary designs, 42 123, 183 harmonie mean, 26-8 Complete block designs, 22, in balanced incomplete block 24-8,30,36,40,42-4,90,96, designs, 48, 153 98,101-2,117,167-8 in cyclic designs, 64-6, 114 basic contrasts, 43 in factorial experiments, 152-3 efficiency balance, 43 in generalized cyclic designs, 79 information matrix, 43 in group divisible designs, 59, treatment sum of squares, 43-4 156 see also Randomized block in Kronecker product designs, designs 178 Complimentary designs, 47, 73 in Latin squares, 101 Component designs, 95 in lattice squares, 112 SUBJECT INDEX 227 column component, 95 E-optimality, 68, 72 having common sets of efficient designs, 64, 66-73 eigenvectors, 96, 98, 100 efficiency factors, 64-6 row component, 95 estimating stratum variances, Components of interaction, 120, 194 126-9, 133, 151 information matrix, 64-5 complete set of, 128 initial block, 61 exhaustive, 128 pairwise efficiency factors, 65 for symmetrical p" designs, 126, permutation theory, 64, 66-7 129 resolvable, 80-2 general definition of, 127-9 row-column designs, 63-4, 104, homogeneous, 126 113-5 identifying, 126-9 tables of, 67 orthogonal, 126 Cyclic sets, 62 Computer analysis of experiments, full, 62, 91 169, 178, 196-7 partial, 62-3, 67-8, 75, 81, 114-5 Computer construction of designs, 29, 87, 139-40, 149-50, 169 D-optimal designs, 39, 60, 99 Computer packages, 169 D-optimality, 28, 30 Concurrence matrix Defining contrasts, 143 block, 51, 54 Degrees of freedom confounded, treatment, 17 135 Confounding Design graph, 89 degrees of freedom, 135 Design matrix in 2" experiments, 142-3 block, 6 partial, 117 treatments, 6 principle of, 118 Determining the confounding scheme, 133-40 scheme, 133-40 total, 117, 120 Dicyc1ic designs, 73-4, 166 Connected designs, 16-19,24-5, Disconnected designs, 17-19,25, 31, 35-7,196 63, 99, 109, 119, 132, 150 Connectedness, 16-19,99-100 DSIGN algorithm, 150, 179-80 Contraction of resolvable designs, Dual designs, 35, 37-40, 51, 54-5, 88-9 58, 68, 80, 83 Cubic partially balanced designs, Duality, 37 75 Cyclic designs, 42, 61-73, 79-82, E-optimal designs, 30, 39, 60, 68, 90-2,96,109-10,113-5,163- 72, 99 8 E-optimality, 28 A-optimality, 68, 72 EfTectiveness of blocking, 182, 200 advantages of, 63 Efficiency-balance, 28-9, 33-4, 38, average efficiency factor, 65-6 40, 43, 48, 96, 99, 101, 155, canonical efficiency factors, 64-6, 160 114 condition for, 29 complement, 73 Efficiency factors, 23-4, 27, 35-6 construction of, 61-3 see also Canonical efficiency dual, 68 factors; Pairwise efficiency 228 SUBJECT INDEX
Efficiency factors (contd) computer algorithm for, 149 factors; Upper bounds for confounding in, 147 efficiency factors; Weighted construction, 144 efficiency factors DSIGN algorithm, 150 Efficiency of a design, 23, 200 in 2" experiments, 148-9 Error covariance structure, 186, reasons for, 144 189 in row-column designs, 150, 180 for block designs, 186 tables, 150 for row-column designs, 186 usefulness of, 145 spectral form, 187-8 Estimability, 10 GCjn designs, see n-CycIic designs condition for, 11, 159 GCIB designs, see Generalized in hypothesis testing, 13 cycIic designs Estimable function, 11-12, 16, 19- General balance, 96, 100, 191,203 21, 24-5, 95, 99, 119-20, 125, definition of, 191 132, 135 Generalized cycIic designs, 73, 77- variance of, 11, 95, 125 83, 90, 104-5, 110, 113-5 Estimating error from high-order canonical efficiency factors, 79 interactions, 117, 124 choice of efficient designs, 80 Estimation of stratum variances, concurrence matrix, 78-9 190-2, 194-5, 203-4 construction of, 77-8 Even-odd rule, 143 dual designs, 80 equivalence to IX-designs, 82 Factor, 117 information matrix, 79 pseudo-factor, 129, 138-9 resolvable, 80-3,90-1 Factorial balance, 160, 166-9, 175, row-column, 104-5, 110, 113-5 177 tables of, 80 Factorial experiments, 16, 27, 42, unequal block sizes, 79 60, 75, 116-81 Generalized interaction, 123, 154 asymmetrical, 117 Generalized least squares analysis, model,121 189-90,192-4,202-3 symmetrical, 117 GENSTAT, 169, 196 see also Fractional replication; Graphs Multi-replicate factorial block concurrence, 89 experiments; Single replicate design, 89 designs treatment concurrence, 18,31-2, Factorial structure 125, 154, 156- 34 61, 178-9 Group-divisible designs, 41-2, 51, canonical efficiency factors, 158 55-62, 75, 81, 104, 109 condition for, 158 association scheme, 56, 58 definition of, 157 average efficiency factor, 59 designs having, 160 basic contrasts, 59-60 FORTRAN, 196 canonical efficiency factors, 59, 156 Fractional replication, 118, 143-50, concurrence matrix, 59 180 construction of, 57-8 aliasing, 145 existence of, 56 analysis, 149 factorial balance, 160-1, 167 SUBJECT INDEX 229 factorial structure, 160 orthogonality, 101 in factorial experiments, 155-7, pairwise orthogonal, 46 159, 167-8 Latin square type PBIB/2 designs, information matrix, 59 41, 75, 95, lO6, 112, 152, pairwise efficiency factors, 59-60 160-1 properties of, 58-60 association scheme, 152 regular, 57-8 factorial balance, 161 resolvability,81 factorial structure, 160 sem i-regular, 57-9 Lattice designs, 40, 42, 49-55, 62, singular, 57, 59, lO9 80, 82 canonical efficiency factors, 51-2 Incidence matrix, 6 estimating stratum variances, Incomplete block designs, 40, 94, 194 102,108,117,152-3 recovery of inter-block Information matrix, 8, 190 information, 191 canonical form of, 9, 11, 25, 194 resolvability, 49 Interaction see also Rectangular lattice components of, 120 designs; Square-Iattice designs definition of, 118-9 Lattice squares, 94-6, 111-2, 186 generalized, 123 analysis, 112 orthogonal decomposition of, canonical efficiency factors, 112 121 variance-balance, 111-2 three-factor, 123 Least squares analysis, 7 two-factor, 119 estimators from, 24 Inter-block generalized, 189-90, 192-4 analysis, 3, 182 Levels of a factor, 117 information, 15 Intra-block (M, S)-optimal designs, 30, 34, 60, analysis, 3-4, 8, 14, 23, 52, 182, 107, lO9, 168 192-3, 197 (M, S)-optimality, 28, 30 analysis of variance, 14, 16, 173, Main efTect, 117-8 190, 198 Models, orthogonal, 21 model, 4, 14 Multi-replicate factorial sums of squares, 12-16 experiments, 151-81 treatment estimates, 9, 192 n-Cyclic designs, 73-7, 79, 96, lO4, Kronecker product designs, 178-9 132, 134, 150, 152, 160-9 analysis, 75 Latin squares, 46, 93, 97, 101-2, canonical efficiency factors, 77, 111,179 162-4 anal ysis, 10 1 computer program to construct, analysis of variance, 101 169 canonical efficiency factors, lOl concurrence matrix, 75-7 complete sets of orthogonal construction of, 73-5, 165-7 squares, 46, 111 degree of balance, 165 efficiency-balance, lOl example of experiment using, nested row-column designs, 111 169-78 230 SUBJECT INDEX n-Cyclic designs (contd) Latin squares, 46 factorial structure, 160 162 models, 21 in factorial experiments, 132, partition of sums of squares, 134, 150, 161-9, 179 157, 159-60, 174 information matrix, 75 polynomials, 174 partial sets, 75, 132, 168 row-column designs, 104, 113, Paired -com parison designs, 91-2 180-1 Partial confounding, 117, 151 single replicate designs, 132, 134, Pairwise efficiency factors 150 average, 24, 26-7 Nested row-column designs, 100- definition of, 24 1,111-5,186 harmonic mean of, 24, 65 average efficiency factor, 112 in cyclic designs, 65 balanced, 112-3 in group-divisible designs, 59-60 canonical efficiency factors, 114 in partially balanced designs, 41 cyclic, 113-5 Partially balanced incomplete generalized cyclic, 113-5 block designs, 41, 62, 194 group-divisible, 113 association scheme, 41 n-cyclic, 113 canonical efficiency factors, 41 partial sets, 114 definition of, 41 rectangular, 113 pairwise efficiency factors, 41 tables of, 113 Pascal, 196 Non-binary designs, 19, 30, 33, 42, PBIB/2 designs, 41-2,54,62,81, 90, 106, 168 95, 104, 107, 152, 168 Non-orthogonal designs, 23 classification of, 41 Normal equations, 7-8, 10, 12, Property A designs, 160-1 20-1,189-90,192-4,202-3 Pseudo-factors, 129, 138-9, 166 reduced, 8-9, 20 Null Quadratic identity, 189 analysis of variance, 184-6 Quasi-Latin squares, 179-80 block design model, 187 randomization distribution, 186 Randomization, 3-4, 170, 186, 198 row-column design model, 188 distribution, 186 theory, 186 Optimality criteria, 3, 8, 27-8, 30, Randomized block designs, 22, 43, 36, 48, 66, 96-104 95, 97, 101, 112, 192 use of circuits, 31-2, 34, 87 factorial balance, 160-1 use of simple counting rules, 29- factorial structure, 160 30, 66, 79, 85, 90 in factorial experiments, 124-6, see also A-optimality; D 132, 153-5, 159 optimality; E-optimality; (M, see also Complete block designs S)-optimality Recovery of inter-block Orthogonality, 15,21-2,40,97 information, 15, 23, 178, 191- condition for, 21, 97 2, 196-7 see also Adjusted orthogonality gain from, 194 Orthogonal see also Combined estimates block structure, 182-6 Rectangular lattice designs, 53-4, designs, 22-3, 35,97-9, 102, 111 80,87 SUBJECT INDEX 231 block concurrence matrix, 54 general balance, 203 canonical efficiency factors, 54-5 generalized cyclic, 104-5 construction of, 53 in factorial experiments, 170, dual design, 54 179-81 properties of, 54 information matrix, 94-5, 97, resolvable, 53 100 simple, 53, 90 (M, S)-optimality, 107-109 tripie, 53 model,94 Reduced normal equations, 8-9, n-cyclic, 104 20,48,95 null model, 188 Reduced model, 12-13 orthogonality, 97-9, 102 Replication, 6 partition of degrees of freedom, average, 35 94 fixed,35-6 recovering information in, 182 Residual maximum likelihood, 191 reduced normal equations, 95 Residuals, 4, 12, 197, 199 row-orthogonality, 102-6 Resolvable designs, 40, 50, 53-4, single replicate, 180 80-92, 110, 195-6, 198 treatment information in, 94 canonical efficiency factors, 55 variance-balance, 106-7 contraction of, 88-9 Youden squares, 103-4 paired comparison, 91-2 see also Nested row-column t-resolvable, 110 designs two-replicate, 88-91 Row-orthogonal designs, 102-6, 17 upper bounds for average efficiency factor, 54-5, 85 SAS, 196 use in variety trials, 50 Screening experiments, 146 Resolvability, 49 Simple block structure, 183, 186, practical importance of, 50 188-9 Row-column ex-designs, 105-10 restrictions for, 183 canonical efficiency factors, 107- Simple lattices, 50-2, 90 9 Simple rectangular lattices, 53, 90 construction of, 107 Single replicate designs, 118-53 efficient designs, 109 analysis of, 132-3 efficiency factors, 108 choice of generators, 131, 140-2 Row-column designs computer algorithm, 139-40 ex-designs, 105-10 construction of, 129-32 adjusted treatment totals, 95, 97 degrees of freedom confounded, blocking structure, 94, 184 135 canonical efficiency factors, 96, DSIGN algorithm, 150 99, 100, 181 generator of, 130 column-orthogonality, 103 information matrix, 133 combined analysis, 201-4 n-cyclic sets, 132, 134, 150 connectedness, 99-100 randomized blocks, 124-6 covariance structure, 186 row-column, 150, 180 definition of, 93 tables of, 150 disconnectedness, 99 Single set of treatments, 2-3, 116 estimating stratum variances, 203 Square-lattice designs, 50-3, 58, fractional replication in, 180 80,85,87 232 SUBJECT INDEX
Square-Iattice designs (contd) Symmetrie incomplete block balanced lattice, 51-2, 58 design, 57 block concurrence matrix, 51 canonical efficiency factors, 52 Total confounding, 117, 120 construction of, 50 Treatment combination, 117 dual of, 51, 58 Treatment concurrence graph, 18, harmonie mean efficiency factor, 31-2,34 52 adjacency matrix of, 32 information matrix, 52 Treatment concurrence matrix, 17 intra-block analysis, 52 Treatment structure, 121-4, 153 properties of, 51 TripIe lattices, 50-2, 58 simple lattice, 50-2, 90 TripIe rectangular lattices, 53 tripIe lattice, 50-2, 58 Two-replicate designs, 88-91 Standard errors of differences, 176- average efficiency factor, 90 7, 196-7 optimality, 88 average, 197, 199-200 Statistical computer packages, 169 Upper bounds for efficiency Strata, 184-5 factors, 32-5, 39, 54-5, 85 Stratum variances, 187-9, 190-2, Unequal 194, 202 block sizes, 19, 79, 88 Sums of squares replication, 19,23,29, 35-6,96 adjusted block, 14, 16, 194-5, Unreduced designs, 45-6, 48, 57, 197,204 109 adjusted column, 204 adjusted row, 204 Variance-balance 29, 96, 104, 106, adjusted treatment, 13-15, 22, 111-3 26,43,95, 125, 157 Variance components, 203-4 fitting intra-block model, 12 see also Stratum variances for estimable hypo thesis, 16 Variety trials, 50, 198 partitioning of, 14, 26, 44, 49, 125,157,159-60,174,184 Weighted efficiency factors, 29 residual, 12 unadjusted block, 13-15, 43 Yield identity, 183, 186, 189 unadjusted treatment, 15-16,22 Youden squares, 103-4, 107, 179 Symmetrical factorial design, 117