Journral of Statist Planning Infmnce 3 (1981) 381-389 North-H Publishing Company

381

ON E NEW SEARCH DESIGNS FOR 2m FACTORIAL ~ EXPERIMENTS

Received 26 April 197%; revised mu:usctipt received 20 July 1981 Retxmmcndcd by J.N. Srivastava

Abstmct; In this paper, we obtain search designs with reasonably small number of treatments which permit the estimation of the general mean and main effects and search of one more unknown possible nonzero effect among two and three factor interactions in Z’” factorial experiments, 3 s m s 8.

AMS I970 Subject C/ks@ution: Secondary 62K 15.

Key WIOFQS,’ Linear Models; Search Designs; Factorial Experiments; Fractional Factorials.

1. Introduction

The search designs of various kinds for 2M factorial experiments were obtained by Sriv~ava and Ghosh (1976, 19731, Srivastava and Gupta (1974) and Ghosh (1979, 1980;. This paper is concerned with the problem of fiuding search designs of a new kind which will be of use in practical situations not considered earlier. We assume: that four factor and higher order interactions are all zero for a 2” factorial experi- ment, FE(2”). We are interested in finding designs with reasonably small numbers of treatments which allow the estimation of the general mean and main effects and search of one more unknown possible nonzero effect from two and three factor interactions. We first take the design 7’,,,,, with N1,m = m + 2 treatments as

(1 1 1 **. I), (0 0 0 “’ O), (0 1 1 l ‘* I),

(1 0 1 .‘* i), (1 1 0 l *a J), . ..) (1 1 1 “’ 1 O),

which permits the estimation of the general mean and main effects under the assumption that all o&her effects are zero. We then obtain a necessary and sufficient condition on a design T2,m with N2,m treatments so that for the design

+ IQrn with Nm - - N1,,, + N2,m treatments the problem will be resolved. The suffix in in T’s and N’s will not be used unless it is needed. In Table 2, we present such n3 , for our choice of T1.

0378$758/81/oooO-0000/$02.75 0 !981 North-Holland

2, Search linear models

Consider the linear model

&Y)=4b 4-42629 IO

V(y) = 62JN, !a

where y(Nx 1) is a vector of observations and for i= 1,2, A&@x wd) aft? known

matrices, <j (vi X 1) are vet ors of fixed parameters, and c6 is a constant which may or may not be known. Furthermore, St is completely unknown, and we have partial information about &. We know that atmost k elements of & are non-zero and the rests are negligible, where k is a non-negative integer which may or may not be known. In this paper, we assume k is known. However, we do not know exactly which elements of & are nonzero. The problem is to search the nonzero elements of & and make inferences about them along with the elements of Ct. Such models are called search linear models with fixed effects and were introduced by J.N. Srivastava (1975). We want y (and hence At and AZ) to be such that the above problem can be resolved; the underlying design corresponding to such y is called a ‘search design’.

The case when cr 2 =0 is called the cnoiseless case’. In practice, we always have a2 ~0. The noiseless case is, however, of great importance from the design point of view. The reason is that any design which is not good in the noiseless case, can not be expected to be good in case Q? >O either. The designs described in this paper will find the nonnegligible parameters with probability one in the noiseless case. In case u 2 >O the probability will depend on the *size’ or ‘amount’ of the errors. The following result is fundamental in search theory.

I’Reorem I. Consider the model ( 1,2) and let o2 = 0. A necessary and syf$icient condition (AH. C.) that the search and inference can be camp rely solved in the h !oiseless case, is that for every (N x 2k) submatrix Aa of As, we have

Rank[A, : Azo] = vI +2k. (3)

By “completely solved”, we mean that we will be able to search the nonzero ele of & without any error, and furthermore obtain estimators of the nonzero elements of & and the elements of g, which have variance zero. Note that for a search desi condition (3) is to be satisfied for (,‘i) A$s, which is indeed a lar is small compared ‘to v2,

3. Conditions on existence of se

Consider an FE 1, I’=

383

era1 mean, F;: the main effect of ith factor, E”,iz the two factor h the factors it and fz9 and so on; ai = t or - 1 according as Xi = 1

rved response correspondin to (xr l ‘4 x,,J. Then our

(5)

ith N treatments by nN x m). Let y(N x 1) be the corre- or. We now define two vectors of parameters &(vr x 1)

(6)

consider the model (1,2) with vr , v2, &, tJ2 as in (6), (7), 4 factor and higher order interactions are zero and Al )I A2 are determined by T and (4), and k = 1. Let Ci (i=O, 1, . . . . m) be the cobJmns of A 1 corresponding to the elements in & ; C’ and CD are the columns of AZ corresponding to any two members, say F” and Ffi in 52. We consider the equation

t!&c()+ e,&; + *** + emcm + eucu + 6+.ca = 0, (8)

where 8’s are real constants not all zero. Let T&VI xm), with Al, = m + 2, be the design as described in section 1. It is well known that Tr allows the estimation of & under the assumption that t2 =O. Suppose T2(N2 x m) is another design with A$ treatments. We want to characterize T2 so that the design T= TI + Tz with N=Nr +Nz ta tments becomes a search design, or in other words, t e condition (3) holds for every NY. 2 submatrix A2* of AZ.

) corresponding to the treatment (0 0 **a 0) in Tl is

The row rres .a* is

80 + ei+ t&+ S,=O. i=l

304

Consider the m rows in (8) corresponding to m treatments in r1 other than [S 0 l 0 and (1 P l l), arrd then adding all these rows we have

The proof is now clear from eqs. (9-11).

Proof. Similar to the proof of Lemma 1.

Proof,, The proof follows by considering the rows in (8) corresponding to the treatments (1 1 l 1) and (0 0 l 0) in TI.

The pairs whose members are the elements in & can be any of the fdowing types:

weight of a vector (xl, . . . ,x,,,), denoted by cu(xl, , . . ,x,,J, is defined as the number of nonzero elements in it. We now have the following resuk

Theorem 2. Let tht? coftmns of A20 correspond to the factors F4,il Und F+i,. Then an N.S.C. that Ran&4 1 : A-& = m + 3 is that there ts 4 treatmertj x’ = (x1 ; x2, , . . , -U,,)

in Tz such that

roof. Let u be an integer such that u E ( ,2, .,., nr) and ~6 {i&. The row in (8) to the treatment in rI whe Mth position is 0 and the others are 1, is

Now, using Lemma 1, we have i$ = 0. Therefore we

rrdent equatbns in 4 unknowns and that is all nk condition, we neec

equatbn independent of than, IIt can now be checked that one more treatment is l * rqurred satisfying xi1 *xi2 and xia =O. This comg!etes the proof.

.

Theorem 4. Let the cohtmtts Of Am correspond CO the factors Fi,ii and &.li2i, l Then 011 N.S.C. that Rank(A, : Aa) = m + 3 is that there exists a treatment (xl 0. l - x,,) in Tz satisfring

W(Xj,, X,,) = 1, . Xj? = 0,

Theorem 3. Let the columns of AZ0 correspond to the factor&T Fi ,iz and F;,ij14. Thtw an N.S.C. that Rank(Al : A& = m + 3 is thot there exists a tretTtment (xl l x,,) rn Tl satisfying one of the folio wing conditions:

(0 Nq,, xi,) = 1 w wcq, xi,, = 1)

(ii) -21, - =o,

(iii) =Q, =2.

Proof. In (8), considerin the rows corresponding to a11 treatments in Tl except (1 1 l ‘. I)and(OO l ** 0). we get Bil = Bip and l$ = 0 fbr u 4 {i,, i2, i3, i,}. ?‘hus we save

c more equaticm in 5 unknowns i~~~~~n~~nt of the above 4 that we require a treatment (xl 4’. x& satisfying one of the

Xj4,Xjl *Xi,, (ii) Xi1 t= Xi4 *Xi, s Xi28 (iii) Xi3 !i= Xi4 = 0, Xj,

20 ~~~r~s~~~d to t.%qfactors fii,iz and Fi3i4i5 D Then un P4.S.C tkzt Rank(& : A& = m + 3 ,j: that there exists a treatment (xl ) . . . ,x,,) in Tz

following condifiow

(i) O(Xil~ X$ = 1) &Qp -Q = f 9

(ii) 21, =O, (iii) =O, =2,

(iv) =2, = 1, x’,=o,

0 V =O, =I, =l,

(vi) =l, =2, ==: 0.

Th~mm 7. Eef the C~U~W of AZ0 cormpond to the factors Fiii3ie and Fiyg4. TM an N.S.C. that Rank(Al : A&=m+ 3 & that them e&s a trevrtmerrt (xl, . . ..x~) satisfying

Theorem 8. Let the columns of A20 correspond to the factors Fi,i2i5 and F&. Then an N.S.C. that Rank(Al : AzO) = m + 3 is that there exists a treatment (xl, . ...&) in Tz satisfying one of the following conditions:

ii) o(xi, r xi21 = 2, &i39 Xi41 = 0,

(ii) =Q, =2, (iii) =l, =O, Xis" 1,

(iv) = 1, =2, =o, (v) = 2, = 1, =o,

(vi) =O, =l, =l.

Theorem 9. Let the columns of A20 w-respond to the factors Fi,izi3 and Fi4i5i6. Then

an N.S.C. that Rank(AI : A& = m + 3 is that there exists a treatment (x,, .,.,x,,J in Tz satisfying one of the following conditions:

(i) dxi, 9 xi,9 xi,) 5 1 s d-Qt xi7 9 XiJ 2 2 9

(ii) 12, 51.

Proof. In (8), considering the rows corresponding all treatment in ‘r, except (1 1 l l)and(OO ‘0. 0), Et can be checked that #iI = 0i2 = 0i,, @id = @i, = l)ii6, and 8,, = 0

. for u B {i,, iz, i3, id1 is, i6}. Thus v*‘e have

Hence we need one more equation in 4 unknowns independent of t

equations. The conditions can now be checked. This completes the proof. Notice that the proofs of the Theorems 3-4, 6-8 are not included; the proot’% of

the other theorem:; provide the guidelines for them. We now combine Theorems 2-9 into the following.

It is important to note that, fdr a design, it is easier to check t c; conditions in Theorem 10 than condition (3). However, the conditions in Theorem 10 are not very simple to check because the number of <:ases to be checked is large. In the next section, we shall develop some insight in that direction.

4. Construction of search designs

Let T, m (N,,, xm) with N1 Itz = m + 2, be the design described in Section 1.

Suppose ‘Trm (R’z,,@ xm) is a &sign such that Tm= 7’,,,+ K, with (Ibl,=N,,, + IV%,) treatments is a search design for FE(2m) under (I, 2), @I-7), and k = 1. We are now interested in constructing a se:!;srch design Tm + l of the similar kind for FE(2m+ l). Let T&.+ I(&,+ t x3), with IV*,,,+ l = m + 5, be of the same type as 7’I,m, i.e., the rows of T, 1 M+ 1 are all (0,l) vectors of weights 0, m, and (m + 1).

Suppose T&+ I (N-N8+1 xm), with A!~,,,,.* =2&,, is

TL??l+ I = T2m i j l . . ..*....*........ 7im I: 0 1 ,

where j’==(f I 0-w 1) and O’= (0 0 **a 0). Consider the design ‘I;,, + I = Tl, nl + 1 + 7’z ,,,, + I

with (NH, + I== N1.m + I + Nzm .+ ,) treatments. Clearly, A& + 1 - Ntn = Nz,n + 1.

*** x;,,) in s to two treat , in f 1 9

the integer m + 1 stinct

integers i (k= 1, . . . . 5) belonging to the set { 1,2, . . . . m), we need to verify whether

383 s. G~~./scor~~*~fordmf~~~

Tzm+1 satisfies the c0IuiitionsIn TImrem lo,; FOr the above ik’(kgl,-p~j)

distiiact integers & belonging to the set ( I, 2, .; ;, m}, consider the” treatments in T& satisf,@g the conditions in Theorem 10. For any such treatment in Tam with x,-p fl (u =O, I), say, consider the corresponding treatment in Tarn+ 1 with x,+i*kar. Clearly, these treatments in Tz, Itl + I also satisfy the conditions in Theorem 110, This completes the proof.

It iz ta be remarked that the Theorem I1 can also be directly proved from (3). The drawback of the result in Theorem 11 is that if Ntm is moderately large for FE@@), then Nm+ 1 will be large for FE(2m’1). But, the idea of moving from-FE(29 to FE(2’“‘+ I) in Theorem I I is a very potentiaI tool for constructing search designs because it reduces tremendously the task of checking the conditions in Theorem 10. In Table 2, we present Tam for 3 s m ~8. In Table: 1, we observe that our designs are

m *1 Y? N1 N2 N

3 4 4 5 2 'I 4 § 10 6 4 10 5 6 20 1 5 12 6 7 35 8 9 13 7 8 56 9 10 19

8 9 84 10 19 29 --

Table 2

T2 for 3sms8

. .._ l_----____-__-_~___--~ ___-.“.‘..w..YW

_______.___ _._L_. -___ ._-. -_-----.-“_.~_.-.---I^I1C*II~-I- 1 0 0 1 1 a 0 -_______-_-I_ .__- - __._ _____-_-- -__.- --*___I___ ___c1I"u____nl_cP clm"Wvlq-M"-Ml

-8

1 I 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

dQsiRIts for P fuctoriafs 389

in the sense that the numbers of treatments are reasonably small. It is inter- d m within the range 3 =rn ~8, all vectors in 7”m itre of 1)/2. For even m, abou the half of the treatments are of

are of weightjm/2)+ 1. In this paper, no attempt has bwn ns with some optimum properties discussed in (Srivas-

o consider them in subseque t communications.

%NR, R.C. (lW7). Mathematical theory of symmetrical factorial designs. Sank!ry~ 8, 107-166. Ghosh, S. (H79). On sin arid muMstage factor screening proccdur es. J. Combinatorics, Inform&ion

and System .Scienw 4 (4). 257-284. Ghosh, S. (1980). On main effect plus one plans for 2m factorials. /#rtn. Statist. 8 (4), 922-930.

Srivastava, J.N. (1975). Designs for searching nonnegligible effects. Ia: J.N, Srivastaya, ed. A Survey of Stddad &sign a~d Linear Models. North-Holland, Amsterdam, 507-5 19.

Srivastava, J.N. (1973a). Optimai search design, or designs optimal under bias free optimality criteria. In: S.S. Gupta and D.S. Moore, eds. Statistical De&or7 rheory and Related Topics /I. Academic Press, New York, 375-409.

Srivastava, J.N. (1977b). Statistical design of agricultural experiments. PresidwMul address al 3ist Annual Coq&wrce of lad. Sot. of Ag. Res., New Delhi, India.

Srivastava, 3.N. and S. Ghosh (1977). Balanced 2”’ factorial designs of resciution V which allow search and estimation of one extra unknown effect, 4 ir;m<8. Commun. Siltist. Theor. Meth. A6(2), M-166.

Srivastava, J.N. and S. Ghosh (1976). A series of balanced 2m factorial designs of resolution V which allow search and estimation of one extra unknown effect. Sunkhyb Ser. 8, 38 (3), 280-289.

Srivastava, 3.N. and B.C. Gupta (1974). Search designs for estimating the general mean, main effects and starching and estimating one more unknown possible nonzero effect in 2”’ factorials (abstract). MS &fIetin, 2, 81.