ed 28 May 1980; revi manuscript received 1 July 1981 Recommended by J.N. Srivastava

Abtmct: In this arti&, we develop and examine a group-theoretic approach to the notion of simplicity in the analysis of block designs. This calls for invariance considerations (in a specialized sense). The defining conditions (for simplicity) in this approach, as they stand, appear to be quite rigid. We have, therefore, imposed some weaker conditions and exmained forther aspects of the problem. The results seem to be quite interest@.

Key wo&: Simplicity of Analysis of Block Designs; BIB and PBIB Designs; inter- and Intra- Group Balanced Block Designs; Latin Square Property of C-Matrix of a Rlock Design and its Moore-Penrose inverse.

AiUS C&s.@cution: 62K 10.

1. Introduction

Throughout, our discussion will be confined to the analysis aspect of block designs admitting of a fixed-effects additive model without interaction. Further, we will restrict aurselves only to connected designs. We refer to Chakrabarti ( the usual notations and terminologies adopted here.

Since the ANQVA rests on a solution to the system of reduced normal equations

cf = Q!, (1.1)

the simplicity aspect of the analysis has rightly been concentrated around (1.1). Here C is the usual C-matrix of a block design. We want to achieve :;im licity (in some sense) in solving the system (1.1). The earlier works along this line a?e those of Calinski (1971) and Saha (1976). (See :aiso Saha and Sinha (i97Y) for other related results.)

In this article, we intend to develop t. examine a group-theoretic approach to the notion af simplicity in ck designs. In Section 2, we pro new simplicity criterion a lications. In Sect “.un 3: we study an tive (less stringent) criterion. While doing all this, we co up w-ith a col~~e~ture 0

f th l& through the structures of their C-matrices.

~378-3758/82/oooo-oooo/$Q2.75 Q 19 2 North-Holland

B. K. Sinhu / Simplicity in block &igm

2. A new simpkiay criterion1

2. jr. Preliminaries

Denote by &, the symmetric group of permutations of the numblers (1,2, ..* s 0).

Denote by G&I x U) the permutation matrix based on ge SO. Let C(r) x U) stand for the usual Grkatrix of a certain ctDnnecred design in a given context. Note that C has he usual properties: Cl =O (the null vector), C is p.s.d. and Rank(C) = v- 1. Here I=(1 1 l me 1)’ of appropriate order. Set P= {g Ig E $, G&X3~ = C}. It is easy to verify that P is a subgroup of S,. Further, let C+ stand for the Moore--Penrose g-inverse of C (see Rao and Miara (1971)) and set P+ ={~~~~ES,,G~C+C+C+}. It then follows (Sinha (1979)) that P+ (which is also a subgroup of Sb) is identical to P. In fact, starting with any arbitrary g-inverse C- of C and defining C= as

C” = c GgF-G&&P, REP

one observes, as in Sinha (1979), that PE P= E So, where

P” ={g~gEs,,G;c=G,=c=).

2.2. The simplicity criterion

We shall say that the analysis of a design is srinrpire if the group P corresponding to its C-matrix is (singly) transitive. The reason for this is tw-folId:

First, if P is transitive, in finding a g-inverse of C, it is enough to find the first row of a g-inverse C= of the type considered above. For, as G,$Xi$ = C= for all g E P, taking in turn a member of P which takes the first element into the 2nd, 3rd, . . ,, 0th as g,, we get the other rows of C=.

Secondly, when P is transitive, in estimating the treatment differences q - rj, the number of accuracy ?evels is limited, Let C= = ((ct)) be a g-inverse of C of the type considered above. For the treatment difference ri - tj9 we have

where! Q is a u-element column vector whose ith element is + 1, jth elelment is -II and the rest is zero.

Now E; C= eij = E~(G~C”G~)&~=(GREO.)‘C=(~G~&~) for any gtr P, Thus, if for two treatment differences ri - rj and rie - rj*, G$u = (EiIi’, then the two will1 be estimated with the same accuracy. Hence, if P is transitive, the number of accuracy levels in estimalting ri - rj, 1 s i#j= o is the same as the number of accuracy levels: in estimating q - r2, q - ~~~ . . . , r1 - q, (for, here;” for every j+ j’, (QIl= +Q, for some

’ The author rempins grateful to Professor SK Chatter&e of the Department of Statistics, Calwlstta

Universlity for kindly posing this notion of simplicity tq him in 1970 whiic he (the author) was developing invariance consider tions in the stu (1st of the results of this section were derived at that time.

B, K. Sinha / Simplicity in block desigtis 167

g E P and some I# 1). In fact, accuracy levels in estimating q - r2, q - z3, . . . , q - rU may also be less than (u- 1). If Pi is the subgroup of P consisting of just those permutations which take the first element into first, and the number of orbits (sets of transitivity) of PI is rn, then the number of accuracy levels in estimating

r1 - r2, q - f3, l ** ) q - zu will be m (for if g~pr takes I to I’, then Giq,= e,,).

2.3. Investigatims

(a) We want to investigate under what conditions a connected C-matrix would have its p-group transitive. Towards this end, we present the following results:

Tbeorm 1. A necewry condition &f~r the P-group to be transitjve is that qp =cn= l ** =:qu.

Proof. Given that the p-group Is transitive, taking in turn, for gE P, g=2+ 1, 3-4 , l ,U+, we get immediatebr from G&‘G,=C that ql =cz2= g**=q,o.

Note 5. IF the design considered is binary and proper, then it must necessarily be e++iq mate.

The setting for the next result (as also for the results of Section 3) would be as follows:

A square matrix A(n x n) is said to have the Latin Square Property (UP) if(i) it iuvolves only n elemento, say, (al ,a2, . . . , a,) and (ii) every element occurs exacily once in each row and in each column of the matrix.

If (in the above definition) some of .he ai’s are equal, A is said to have extended LSP (&UP). Suppose ahe elements in every row of a matrix A(n XI?) can be generated <on permutation) from the elements in one of its rows. Then, whenever A is symmetric, it has the LSP (or the ELSP). The verification is left to the reader. Further, it must be observed that a symmetric matrix A possessing (E)LSP can be represented as A = C aipi where Pi’s are distinct (0,l) symmetric matrices adding up to llf = ((111) and pi1 0~ 1 for every i, Were ((I)) is a matrix (of appropriate order) with all element unity.

Our next result goes like this:

Theorem 2, Whenever la is transitive, C has the LSP (or thtp ELSP).

Then

g=(l-+i,g-l(j)-*j, j(#i)= 1,2, . . ..v)EP.

i.e., '_"ij =2= Cl;<-l(j), I rjr v where, of course, Cii = ~11. This it tr ue for all i. conchtsion (since C is a symmetric matrix).

163 B.K. Sinhcr / Simplicity in block designs

Note 2. .An interesting situation arises when the elements in the first row of C(U x u) are all distinct a& one calls for transitivity of the underlying P-group. We irnmzdiately derive that w must be an even integer and that every g,gE P, g+e (th: identity), must necess&iy be of order 2. Further, P must be a regular subgroul~ (Abelian) with a specified structure. The only possibility, on account of symmetrqr of C, then corresponds to o = 4 and one might describe such a C as (rows displayed one by one) @I a2 9 ad, (at ~1 a4 a3), (a3 at al 02) and (a4 ~3 ~2 at Ia

(b) It would be interesting to verify whether the C-matrices of the commonlv *

employed designs have transitive P-groups. (i) IHM%. The verification is immediate in the positive. In fact, F=:: &.

(ii) PBIBD’S. We have considered the following subtypes of PBIBD”s with two associate classes and in each case the P-group has been verified to be transitive. Group divisible type, triangular (T), kxtipa square with m (22) constraints (LS,).

(iii) Inter- and Intro-group Buhced Block Designs ~IKBBD’S). EIere also the verification is immediate in the negative except for the BIBD*s and the GDD’s which form special cases and fall under (i) and (ii) above.

Wcussion. For quite some time, we had been holding the idea that the P-group corresponding to any PBIBD would prove to be transitive. That this is not so has only recently been observed. We get the counter example by referring to a recent result on strongly regular graphs (which, in graph theoretic terminology, exhibit the association relation among the symbols (treatments) in two-associate PBIBD’s). It was :Seidel (1974) who first established the striking result that the strongly regular graph (eq, 2.1 of Paulus (1973)) has a trivial automorphism group. Later, Mendelson (1978) also observed that every (finite) group can be regarded as the grotip of automorphism of some (finite) strongly regular graph? Observe that, in our setting, P refers to the group of automorphism of the strongly regular graph underlying the association scheme of any PBIBD with two associate classes. Thus, according to Seidel and Mendelson, we might come across examples of PBIBD’s where the P-group could as well be just the trivial subgroup consisting of the idenlity alone CII it could be any group - not necessarily a transitive one.

2.4. Remarks

The criterion of simplicity as discussed in this section seems to be a stringent one. Suppose g=(l--G, . . . ). Then GiCGg = C would imply

(c2 1 ? c22 P l ..rC2v)=(Clg-l(l),Clas’(2), l ‘*B Cl*-l(v))

2 The author tban~~~~Ily acknswle es the help received fr vision, ISI (Calculatta), Dr. Rae brought to the author’s

nd eison. works of Seidel and

169

as also

(Cj 19 CjZ* l ** CjJ = (Cig”l(I)* C&-l(2), l ** 9 Cii-‘(v)l

for f:very pair (i, j) such that in g,i*j. This seems to be a stronger demand on the nature of t? (i.e., of C+), One could perhaps replace this b:f the weaker lcoxldition that C-, a g-inverse of CI should possess the ELSP so that all its rows can be generated from the first row. We want to investigate this aspect of sim.plicity in details in the next section.

3. An alternative criterion

3.1. Preliminaries

In this subsection we would like to develop the auxiliarlrr mathematicr; that we shall need afterwards. Suppose C(o x U) is the C-matrix of a connected desi,i;n, i.e., a :symmetric p.s.d. matrix of Rank@ - 1) and Cl = 0. Suppose, further, that C has the (E)LSP (see Subsection 2.3) so that C can be represented as C= zg-’ CiPi with

c ‘-‘Ci=O, C*>O, CisO* IlisU - 1, Ci’S may or may not be all distinct; PO= I,, I$ ’ Pi:= W-I,; Pi’s being (0, I) matrices. -.

First we want to investigate under what conditions C+ (the Moore-Penrose g- inverse of C) also retains the (E)LSP. CTbserve that C has the (E)LSP iff C + (P~‘)/v has this property. Let C*= C+(ft’)Ju so that Cs is p.d. trirrite C* as l,;t;-’ cF?‘i. Under the condition that C has the (E)LSP, C* has the reprlzsentation

= [H&*(H;c*) 0.. 1 H;_ lc*], (3.1)

where c*=(c~c~*~~c,*,J and (Go(-:I,,),Gt,...,G,_,) and (Ho(=I,),HI,...,Hu_l) are permutation matrices satisfying XX-’ Gi= Ci-’ Hi= 1

Suppose now that C +A has the representation

(3.2)

‘Then a set of n.!,s. conditions for this to hoi

170 B.K. $idta / Simpticity in block ddgnj

The subsett of conditions in (I) determines da completely and this d+ satisfies (II) and (III; iff for every pair (i, .j) with 0~ i; j 11c u - 1, GiGj = Gk for some I(r. In other words, a set of n.s. con&ions can be stated as follows:

Gaatition A. The set of permutations correspondin to (GgJ%*..*CiuJ fwM a subgroup of order I), involving the u symbols (1,2, l a. 1, u), i.e., they form a regular subgroup of &.

Whenever condition A obtains, the &Ioore-Penrose g-inverse C’ of 6 becomes one to possess the (E)LSP as it is easy to check that

c+ = c+lll _-~~*-‘---I ( > -I 11’ ” =Cc,‘P, v u V

1 with ci’ =@--, BS~SV- 1.

u

Under the same condition, it is a possible to characterize ail (E)LSP-type g- inverses of C. This can be done as lows:

Gbserve that any g-inverse of C can be written as (see Rae and Mitra (1971)) C-. =: c+ -t_ (&- c+cucc1*+ f or an arbitrary (v x v) matrix U. However, for a connected C-matrix, CC+ =C+C=I,-(ll’)/u. Hence, C’=C++(crl’+lb’)/u- (1 P’)./o for some u-element with column vectors o and b. And this will have the @)LSP iff a, &a 1. Thus, C’- = C + + 811’ represents the most general form* possess- ing the (E)LSP, of any g-inverse of C which itself has the @)LSP. In~terestingly all such g-inverses are symmetric. [Ngte that a matrix possessing the (E)LSP need not bg: symmetric.) Further, s%ze (@ +)+ = C, the same condition A, a so guarantees the (EQLSP of C whenever Cf’ has the (E)LSP.

Remlark. In the above, we have rightly assumed the representation, as in (X2), for p I. One might start with a general form sf representation of C*-’ as

and the equations as C*‘GiQj z 6, z C*‘Ni Pj * (=I if i==j; f=: equations are consistent iff Gi i aud Ui c Pi for every i.

.

!i s SOF “f si hat at least one g-inverse of

waittei; down by knowing just one row, say, the rst row3. We set er condition that this particular g-inverse should be symmetric. Hence

tion problem WI be stated as foilows: ~~~~~c~et~~~ ult C-ma&es which provide symmetric g-inverse,s possessing the

C* could be demanded to have the (E)LSP. Iln view of the auxiliary ted in the previous subsection, we observe that condition A is to be

by the relevant C-matrices whenever they possess the (E)LSP. For certain the verification is done below. sitivity of P-group e (3” possess the (E&S

(b) PlBIjllED%. The C-matrices do, of course, possess the (E)LSP. Further, the es also possess the (E)LSP as will be evident from a piece of interesting se and Mesner (1959) on Li ear associative algebras corresponding to the

association shcemcs of PB1BD’s. For PBIBD with m-associate classes, one has for the C-matrix

C=r(l- I/&)&-&/k)B, ---(Am/k)

where the symbols have their significance as in BOSY and Mesner (1959). We have, for our C*, the form

C*= $ aiBi with ao=r(l -l/k)+ l/v, Qi=-Ai/k+ I/V, 1 =iSrn.

Now in view of the results in the appendix, it follows that C*- 1 is of the fo:m Cr biBi for some bi ‘se Hence c+ possesses the (E)LSP.

(c) IIGBBD’S. Here the verification is made in the negative (except for the partgL\u!x cases of the BIBD’s and the GDD’s).

3.3. Concluding remurks

At this stage, we set forth the following conjecture: enever 4 t?=marf& has the (E SP, it necessarily cwrespond~ I 3 the C-matrix

of some PBIBD ( WM Y~~-CISSOC in& classes, for some m > 1; m = 1 kin,;? the MM of Q BlB.L?). We b&eve the conjeclure to hold ,. pod in which case this a‘ltelnative sim- plicity criterion will speak only for the FNLD’s.

3 is, of course, excludes the possibility that a g-inverse could be just a cliaganal matrix (with elements not all equal) as in the case of e$jkiemy4afa~c~d (EB) designs (see Sinta and Saha (1979)).

in. the context c*f a PBIBD, Usin, this fact, one derives the following set at’ equations for determination of the &s:

i.e,,

(using p@~ = Q$ ; p& = tfik ; 6 being the Kronecker delta). The matrix of the coeffi- cients can be displayed very easily. To determine its rank, one may observe that adding all the columns, one: gets the vector (1 1 l 1)’ and this may replace the first column. Further, one may transform this vector to (1 0 0 l a* 0)’ by subtracting the first row from each of the other rows. The resulting wf xm lower submatrix coincides with the :matrix .A = (QQ)) occurring in the analysis of PBIBD’s (see Chakrabarti (1962)). It is known hat A is non-singular anti hence the result folilow?.

Acknowledgement

The author would like to thank Dr. Subir Ghosh of the University of California (Rive;_side) and Dr. G.M. Saha ;tnd Dr. Anish C. Mukhopadhyay of the Indian Statistical Institute, Calcutta for some helpful discussions.

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