Resistance of block designs

Journal of Statistical Planning and Inference 27 (1991) 263-269

North-Holland

263

Resistance of block designs

Rajeshwar Singh

Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut, India

V.K. Gupta

Indian Agricultural Statistics Research Institute, New Delhi 110012, India

Received 14 December 1988; revised manuscript received 23 February 1990

Recommended by AS. Hedayat

Abstract: Hedayat and John (1974) introduced resistant and susceptible balanced incomplete block

designs. The results were essentially on locally/globally resistant balanced incomplete block designs of

degree one. This paper extends the concept of resistance to general variance balanced block designs as

well as universally optimal block designs with unequal block sizes. Some methods of constructing locally

resistant designs of degree one have been given.

AMS Subject Classification: Primary 62KlO; secondary 05B05.

Key words and phrases: Resistant designs; variance balanced designs; universally optimal designs.

1. Introduction

Consider a block design d in b blocks, u treatments and u x b incidence matrix

Nd = (nd$, where ndi; is the replication number of the i-th treatment in the j-th

blockofd,i=l,..., ; , . . . , b. The row and column sums of Nd are respectively

the elements of Y d =‘;r”= ’ d,, . . . , rd")' and kd = (k,, 1.. , I$,)‘. Further, let

R, = diag (l;di, . . . , r&,) and & = diag (kdi, . . . , k&).

Under the usual homoscedastic, fixed effects, additive model, the coefficient matrix

of the reduced normal equations for estimating the linear functions of treatment

effects is

c,=R,-N,K,-‘h$. (1.1)

We shall deal only with connected designs throughout this paper. For definitions

and some notations reference may be made to Kageyama (1987).

Hedayat and John (1974) introduced and studied resistant balanced incomplete

block (BIB) designs. Other results concerning resistance of BIB designs known so

far are those given by Most (1975), Shah and Gujarathi (1977, 1983), Chandak

0378-3758/91/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland)

264 R. Singh, V.K. Gupta / Resistance of block designs

(1980), Kageyama (1982, 1987) and Kageyama and Saha (1987). Kageyama (1987)

proposed the idea of investigating resistant variance balanced designs. Such ex-

amples are found in Kageyama (1974, 1976). The purpose of this article is to study

locally resistant variance balanced block designs. In Section 2, we study the

characterization and construction of locally resistant variance balanced designs of

degree one.

This article also proposes to extend the concept of local resistance to universally

optimal block designs. Pal and Pal (1987) studied the universal optimality of non-

proper variance balanced designs in the class of designs D(o; b; k,, . . . , kb), where

D(o;b;k,, . ..) kb) denotes the class of all connected designs in o treatments, b

blocks and fixed block sizes k ,, . . . , k,. In Section 3, the concept of locally resistant

variance balanced designs is extended to locally resistant universally optimal

designs. Some series of resistant universally optimal designs of degree one and more

are given. These designs are also locally resistant variance balanced designs.

An important point about locally resistant universally optimal designs is that the

class of resulting designs is different from the class of original designs. This happens

with locally resistant BIB and variance balanced designs also. The original design

has u+ 1 treatments while the resulting design has only u treatments and therefore

the classes of designs in which the original and the resulting designs are optimal is

bound to be different. But what is important is that the optimality is retained in the

resulting design after losing information.

2. Resistant variance balanced block designs

This section obtains a necessary and sufficient condition for a variance balanced

block design to be resistant of degree one. Without any loss of generality, let Nd,

as given in (2.1), be the incidence matrix of a variance balanced design d,

(2.1)

It is assumed that the first treatment appears in the first m blocks, p’=@, . . . ,p,J

andp’J=r,, where J, is a tx 1 vector of ones. Let k =(k’,,k;) denote the b x 1 vec-

tor of block sizes, where k, = (k,, . . . , k,)’ is the m x 1 vector of first m block sizes

and k*=(k,+,,..., kb)’ is the (b-m) x 1 vector of the remaining b-m block sizes.

Also let k, = k, -p. Similarly r= (r,,r;)’ denotes the u x 1 vector of replication

numbers with r2 = (r2, . . . , ro)’ denoting the vector of the last o- 1 replication

numbers. We shall denote by R,, K, K,, K2 and JCs diagonal matrices with suc-

cessive elements of the vectors rz, k, k,, k2 and k3 on the diagonals respectively. Let

6 (> 0) denote the common value of the off-diagonal elements of N,ZC’Ni. Also

denote by

K4 = dM(k, -pJklA . . . , W,,, -n,&h,J.

R. Singh, V.K. Gupta / Resistance of block designs 265

Then it is easy to see that

R,-- N,K;‘N; - N,K;‘N; = &(I,_, - JJ’/v).

Let 25 be the design obtained after deleting the first treatment from d. The

parameters of a are D= u- 1, 6= 6, ~=r,, k= (k;, ki)’ and incidence matrix

NJ= [N,, N2]. The C-matrix of a is

Cd= R, - N,K;‘N; - N,KF’N;

= R, - N,K,‘N; - N,K,-‘N; - N,K,-‘N;.

It therefore follows that il is variance balanced if and only if N,K;‘N’, = al,_ I + /3JJ’, (x 2 0, /I? 0 and cr + (u - l)p= 6. We thus have the following theorem:

Theorem 2.1. A necessary and sufficient condition for a variance balanced block design to be resistant of degree one is that N,Ki’N’, =aZ,_, +/lJJ’, with cw+(u- l)P=S.

Remark 2.1. The above theorem gives a mathematical characterization of locally

resistant variance balanced designs in terms of N,. It would be interesting to get

this in terms of design parameters, but it does not appear to be possible.

We now give some methods of constructing locally resistant variance balanced

designs of degree one.

Method 2.1. Forpositive integers p, n and m =p(n + l)/@ + 1) there exists a locally resistant variance balanced design of degree one with parameters v= n + 2, b = m + n + 1, r = ((n + l),(m +p)JL + ,)‘, k = ((n + l)J&,(p + 1)JA + i)’ and incidence matrix

0; Nd= J’,

i J,Jin PZ,, 0,

(2.2)

Proof. Removing the first treatment from d gives

N, = oil P PZ, 1 0, ’

K,=P(P+ ~)Z,,+I, WG’N’,=b’t~+1)1~,+~,

and cx+p(u- 1)=6 for cr=6=p/(p+ 1) and p=O.

Method 2.2. Let Nd be the incidence matrix of a variance balanced block design d with parameters v, b, r and k and common value of the off-diagonal elements of


- Cd matrix of Nd as 6. Then for any integer a (2 0), there exists a locally resistant variance balanced design d * of degree one with incidence matrix

N$= 0; . . . 0; I, (a + l)JL Nd . . . Nd I,+aJJ’ 1

/ ” m times

and parameters v *=v+l, b*=bm+v,

if

r*=(v(a+ l), mr’+(av+ l)J’)‘, k*=(k’, . . . . k’, (au + a + 2)JI)‘,

ma= (a(v- l)+ l}{a(v+ 1)+2)-l.

Proof. Removing the first treatment from d* gives

N, = I, + aJJ’, K4 = { t/(a + 1)}1,,

N,K:‘N; = {(a+ l>/t}{Z,+ (a2v+ 2a)JJ’},

t = (au + a + 2)(av + l), a=(a+ 1)/t, j? = (a2v +- 2a)(a + 1)/t.

Example 2.1. Let Nd be a BIB design with parameters v = 6, b = 10, r= 5, k = 3, L=2. Then for a= 1,

Nd*= 06 2 J;

Nd Z,+JJ I

is the incidence matrix of a locally resistant variance balanced design of degree one

with respect to the first treatment.

Method 2.3. Consider a BIB design d, with parameters v’, b’= v’(v’- 1)/k, r’ = v’- 1, k, A’ = k - 1 and incidence matrix N. Then there exists a locally resistant variance balanced design d * of degree one with parameters v = VI+ 1, b = b’+ v’, r = ((r’+ k - l)JI,v’)‘, k = kJb and incidence matrix

Np= N (k - l)Z,,

o;, 1 J;, . (2.3)

Proof. On removing the v-th treatment from d*, we have N1 = (k- l)I,,, K4 = k(k - l)Z,,, and N,K;‘N; = [(k - l)/k]I,. Also (Y +/3(0 - 1) = 6 for a = 6 =

(k-1)/k and a=O.

We now give some series of BIB designs dl used in the method just described.


(i) If tk+ 1 is a prime or prime power and x is a primitive root of GF(tk+ l),

then the t initial sets (2, A!+‘, . . . ,2”‘k- ‘)(), i=O, 1, . . . , t - 1, form a difference set

for a BIB design with parameters u = tk+ 1, b = t(tk+ l), r= tk, k, A = k - 1.

(ii) If 4t- 1 is a prime or a prime power and x is a primitive root of GF(4t- l),

then the initial sets (0, 2, 2,. . . ,x4”- “), (03, x, 2, . . . ,x~‘-~) form a difference set

of a BIB design with the parameters u=4t, b=2(4t- l), r=4t- 1, k=2t, A=2t- 1.

3. Resistance of optimal designs

This section studies the resistance of universally optimal block designs. If a

universally optimal design d belonging to certain class D remains invariant under

loss or deletion of some (or any) t treatments then d is said to be locally (globally)

resistant of degree t. The design d is invariant if the resulting design obtained upon

deleting or losing t treatments remains universally optimal over a certain class.

Example 3.1. Consider a design with parameters u = 8, b = 14, r= 14, k = (12J;,4J;)

and incidence matrix

Nd=

21112120111010

22111210011101

12211121001110

21221110100111

12122111010011

11212211101001

11121221110100

22222220000000

This design is universally optimal over D(8; 14; 12&4J;). It is locally resistant of

degree one with respect to the last treatment because the design obtained after

deleting the last treatment is also universally optimal over D(7; 14; lOJ;,4J;).

We now have the following results:

Theorem 3.1. The existence of a BIB design d with parameters v, b, r, k, A such that b f 21= 3r and incidence matrix Nd implies the existence of a locally resistant universally optimal design d * of degree one and parameters v * = v + 1, b * = 2b, r*=2b 9

k*= [(u+k+2)J;, (u-k)Jh]‘,



where fid = J,,Jh - Nd.

Proof. The proof follows by noting that the design obtained after deleting the last

treatment is universally optimal over

D(o;b*;{(u+k)J;, (u-k)J;)‘).

For BIB designs satisfying b + 2A = 3r, reference may be made to Hedayat and

John (1974, p. 154).

Theorem 3.2. The existence of a symmetric BIB design d with parameters v = b, r = k, A and incidence matrix Nd implies the existence of a locally resistant universally optimal design d* of degree k or u - k and parameters o * = v, b * = b + t,

r*=(r+ab+st)J,,,


k * = [(k + au)Jb, suJi] ’

Nd* = [Nd + aJ,Jb sJ,J;].

Proof. It is easy to verify that the design d* is universally optimal over

D(v*; b*;k;: . . . . k,*,). Without any loss of generality, consider the first column of

N& Delete from Nd* the k treatments corresponding to the element one in the first

column of N& It is known that if we delete any k treatments which appear in the

same block of a symmetric BIB design then the resulting design is BIB with

parameters u, = u-k, b, = u - 1, r, = k, k, = k - A, A. Using this fact it follows that

the resulting design d ** is universally optimal over D(u- k, b*, k**) where

i

a(u - k) ifj=l,

k,**= k-A+a(u-k) ifj=2,...,b,

s(u - k) ifj=b+ l,...,b*.

The design d** is therefore locally resistant of degree k. SimilarIy if we delete the

u-k treatments corresponding to the element zero in the first column of Nd from

Nd*, then the resulting design is locally resistant of degree u-k.

Remark 3.1. The designs given in Theorems 3.1 and 3.2 are also locally resistant

variance balanced designs.

Acknowledgements

The authors are thankful to the referees for their valuable suggestions which led

to a considerable improvement in the presentation of the results.


References

Chandak, M.L. (1980). On the theory of resistant block designs. Calcufta Statist. Assoc. Bull. 29, 27-34.

Hedayat, A. and P.W.M. John (1974). Resistant and susceptible BIB designs. Ann. Statist. 2, 1488158.

Kageyama, S. (1974). Reduction of associate classes for block designs and related combinatorial ar-

rangements. Hiroshima Math. J. 4, 527-618.

Kageyama, S. (1976). Construction of balanced block designs. Utilitas Math. 9, 209-229.

Kageyama, S. (1982). The existence of locally resistant BIB designs of degree one. Stat-Math. Tech.

Rept. No. 18/82, ISI, Calcutta.

Kageyama, S. (1987). Some characterizations of locally resistant BIB designs of degree one. Ann. Inst.

Statist. Math. A 39, 661-669.

Kageyama, S. and G.M. Saha (1987). On resistant t-designs. Ars Combinaf. 23, 81-92.

Most, B.M. (1975). Resistance of balanced incomplete block designs. Ann. Statist. 3, 1149-l 162.

Shah, S.M. and C.C. Gujarathi (1977). On a locally resistant BIB design of degree one. Sankhya Ser.

B 39, 406-408.

Shah, S.M. and C.C. Gujarathi (1983). Resistance of balanced incomplete block designs. Sankhyo Ser.

B 45, 225-232.

Resistance of block designs

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