Resistance of block designs

Journal

of Statistical

Planning

and Inference

27 (1991) 263-269

263

North-Holland

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

Recommended

1988; revised

by AS.

manuscript

received

(1974) introduced

resistant

Abstract: Hedayat

and

designs.

were essentially

degree

The results

one. This paper

well as universally resistant

designs

23 February

1990

Hedayat

John extends

optimal of degree

on locally/globally

the concept

of resistance

block designs with unequal

and susceptible resistant

balanced

to general

variance

balanced

incomplete

block

incomplete

block designs

of

balanced

block designs

as

block sizes. Some methods

of constructing

locally

one have been given.

AMS Subject Classification: Primary Key words and phrases: Resistant

62KlO;

designs;

secondary

variance

05B05.

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$, where ndi; is the replication number of the i-th treatment in the j-th blockofd,i=l,..., ; , . . . , b. The row and column sums of Nd are respectively the elements of Yd =‘;r”= d,,..’.,rd")' and kd = (k,, 1.. , I$,)‘. 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 examples 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 nonproper 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 vector of block sizes, where k, = (k,, . . . , k,)’ is the m x 1 vector of first m block sizes kb)’ is the (b-m) x 1 vector of the remaining b-m block sizes. and k*=(k,+,,..., Also let k, = k, -p. Similarly r= (r,,r;)’ denotes the u x 1 vector of replication the vector of the last o- 1 replication numbers with r2 = (r2, . . . , ro)’ denoting numbers. We shall denote by R,, K, K,, K2 and JCs diagonal matrices with successive 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.

265

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

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 parameters of a are D= u- 1, 6= 6, ~=r,, k= (k;, ki)’ and incidence NJ= [N,, N2]. The C-matrix of a is Cd= R, - N,K;‘N;

= R, - N,K,‘N;

d. The matrix

- N,KF’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’, (x2 0, /I? 0 and cr + (u - l)p= 6. We thus have the following theorem: Theorem

2.1. A necessary and sufficient

design to be resistant cw+(u- l)P=S.

condition for a variance balanced block of degree one is that N,Ki’N’, =aZ,_, +/lJJ’, with

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 designs of degree one.

of constructing

locally

resistant

variance

balanced

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

Method

0; J’,

Nd=

(2.2)

i J,Jin Proof.

Removing N, =

1)=6

0,

the first treatment

oil PZ,

K,=P(P+ and cx+p(u-

PZ,,

from d gives

P 0, 1 ’ ~)Z,,+I,

WG’N’,=b’t~+1)1~,+~,

for cr=6=p/(p+

1) and p=O.

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

Method

266

R. Singh, V.K. Gupta / Resistance

of block designs

- 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; Nd . . . Nd ” /

I,

(a + l)JL I,+aJJ’ 1

m times b*=bm+v,

and parameters v *=v+l, r*=(v(a+

l), mr’+(av+

k*=(k’, . . . . k’, (au + a + 2)JI)‘,

l)J’)‘,

if ma= (a(vProof.

Removing

1)+2)-l.

l)+ l}{a(v+

from d* gives

the first treatment

N, = I, + aJJ’,

K4 = { t/(a + 1)}1,,

= {(a+ l>/t}{Z,+

N,K:‘N;

(a2v+ 2a)JJ’},

Example

2.1.

j? = (a2v +-2a)(a + 1)/t.

a=(a+ 1)/t,

t = (au + a + 2)(av + l), Let Nd be a BIB design

with parameters

v = 6, b = 10, r= 5, k = 3,

L=2. Then for a= 1, Nd*=

06 Nd

2 J; Z,+JJ

I

is the incidence matrix of a locally resistant with respect to the first treatment.

variance

balanced

design of degree one

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=

Proof.

On

N o;,

removing

K4 = k(k - l)Z,,, and (k-1)/k and a=O.

(k - l)Z,, J;,

1.

(2.3)

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

We now give some series of BIB designs

dl used in the method

just described.

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

designs

(i) If tk+ 1 is a prime or prime power and x is a primitive root of GF(tk+ then the t initial sets (2, A!+‘, . . . ,2”‘k- ‘)(), i=O, 1, . . . , t - 1, form a difference

261

l), 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(4tl), 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. and incidence

Consider matrix

a design with parameters

u = 8, b = 14, r= 14, k = (12J;,4J;)

21112120111010 22111210011101 12211121001110

Nd=

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:

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 Theorem

3.1.

k*= [(u+k+2)J;, and incidence matrix

(u-k)Jh]‘,

268

R. Singh,

V.K. Gupta / Resistance of block designs

where fid = J,,Jh - Nd. Proof. The proof follows by noting that the design obtained treatment is universally optimal over

D(o;b*;{(u+k)J;, For BIB designs satisfying John (1974, p. 154).

after deleting

the last

(u-k)J;)‘). b + 2A = 3r, reference

may be made to Hedayat

and

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, Theorem

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

k * = [(k + au)Jb, suJi] ’

and incidence matrix Nd* = [Nd + aJ,Jb

sJ,J;].

Proof. It is easy to verify that the design d* is universally optimal over consider the first column of D(v*; b*;k;: . . . . k,*,). Without any loss of generality, 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 optimal over D(u- k, b*, k**) where the resulting design d ** is universally

k,**= i

a(u - k) k-A+a(u-k) s(u - k)

ifj=l, ifj=2,...,b, 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 variance balanced designs.

3.1 and 3.2 are also locally

resistant

Acknowledgements The authors are thankful to the referees for their valuable suggestions to a considerable improvement in the presentation of the results.

which led

R. Singh,

V.K. Gupta

/ Resistance

of block designs

269

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