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On some new search designs for 2m factorial experiments
Journral of Statist
North-H
Planning Infmnce 3 (1981) 381-389 Publishing Company
381
ON E NEW SEARCH DESIGNS FOR 2mFACTORIAL ~EXPERIMENTS
Received26 April 197%;revisedmu:usctipt received20 July 1981 Retxmmcndcdby J.N. Srivastava 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.
Abstmct;
AMS I970 Subject C/ks@ution:
Secondary 62K 15.
Key WIOFQS,’ LinearModels; SearchDesigns; Factorial Experiments; Fractional Factorials.
1.
Introduction
The search designs of various kinds for 2Mfactorial 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 experiment, 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),
(1 0 1 .‘* i),
(0 0 0 “’
O),
(1 1 0
J), .
l
*a
(0 ..)
11
l
‘* I),
(1 1 1
“’
1 O),
which permits the estimation of the general mean and main effects under the assumption that all o&hereffects 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,mtreatments 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/$02.75 0 !981 North-Holland
2, Search linear models Consider the linear model &Y)=4b V(y)
4-42629 = 62JN,
IO !a
where y(Nx 1) is a vector of observations and for i= 1,2, A&@x wd) aft? known matrices, (vi X 1) are vet ors of fixed parameters, and c6 is a constant which may or may not be known. Furthermore, St is completely unknown, and we have partial information about &. We know that atmost k elements of & are non-zero and the rests are negligible, where k is a non-negative integer which may or may not be known. In this paper, we assume k is known. However, we do not know exactly which elements of & are nonzero. The problem is to search the nonzero elements of & and make inferences about them along with the elements of Ct. Such models are called search linear models with fixed effects and were introduced by J.N. Srivastava (1975). We want y (and hence At and AZ) to be such that the above problem can be resolved; the underlying design corresponding to such y is called a ‘searchdesign’. The case when cr2=0 is called the cnoiseless case’. In practice, we always have a2 ~0. The noiseless case is, however, of great importance from the design point of view. The reason is that any design which is not good in the noiseless case, can not be expected to be good in case Q?>O either. The designs described in this paper will find the nonnegligible parameters with probability one in the noiseless case. In case u 2>O the probability will depend on the *size’ or ‘amount’ of the errors. The following result is fundamental in search theory.
I’Reorem I. Consider the model ( 1,2) and let o2 = 0. A necessary and syf$icient condition (AH. C.) that the search and inference can be camp rely solved in the h!oiselesscase, 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$s, which is indeed a lar is small compared ‘to v2,
3. Conditions on existence of se Consider an FE 1, I’=
383
era1 mean, F;:the main effect of ith factor, E”,izthe two factor h the factors it and fz9and so on; ai= t or - 1 according as Xi= 1 rved response correspondin to (xr ‘4x,,J. Then our l
(5) ith N treatments by nN x m). Let y(N x 1) be the correor. We now define two vectors of parameters &(vr x 1)
(6)
consider the model (1,2) with vr , v2, &, tJ2 as in (6), (7), 4 factor and higher order interactions are zero and Al )I A2 are determined by T and (4), and k = 1. Let Ci (i=O, 1, . . . . m) be the cobJmns of A 1 corresponding to the elements in & ; C’ and CD are the columns of AZcorresponding to any two members, say F” and Ffi in 52. We consider the equation t!&c()+ e,&; + ***+
emcm + eucu + 6+.ca= 0,
(8)
where 8’s are real constants not all zero. Let T&VI xm), with Al, = m + 2, be the design as described in section 1. It is well known that Tr allows the estimation of & under the assumption that t2 =O. Suppose T2(N2x m) is another design with A$ 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 **a0) in Tl is
The row
rres
80+
ei+t&+ i=l
.a*
S,=O.
is
304
Consider the m rows in (8) corresponding to m treatments in r1 other than [S 0 l), arrd then adding all these rows we have and (1 P
l
0
l
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 1) and (0 0 0) in TI. l
l
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,ilUndF+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 independentof than, IItcan now be checked that one more treatment is * rqurred satisfying xi1*xi2 and xia=O. This comg!etes. the proof. l
Theorem4.Let the cohtmtts OfAm correspond CO the factors Fi,iiand &.li2i, Then 011 N.S.C. that Rank(A, : Aa) = m + 3 is that there exists a treatment (xl 0.l - x,,) in Tz satisfring l
Xj? =
W(Xj,,X,,) . = 1,
0,
Thtwan Theorem3. Let the columns ofAZ0correspond to the factor&T Fi ,iz andF;,ij14. N.S.C. that Rank(Al : A& = m + 3 is thot there exists a tretTtment (xl x,,) rn Tl l
satisfying one of the folio wing conditions: (0 Nq,, xi,) = 1 w
(ii) (iii)
-21, =Q,
wcq,
xi,, -
= 1) =o,
=2.
Proof. In (8), considerin the rows corresponding to a11treatments in Tl except (1 1 ‘. I)and(OO **0). we get Bil= Bipand l$ = 0 fbr u 4 {i,, i2,i3,i,}. ?‘hus we save l
l
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) Xi1t=Xi4*Xi, s Xi28(iii) Xi3!i= Xi4= 0, Xj,
20~~~r~s~~~dto t.%qfactorsfii,izand 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)
(ii) (iii) (iv) 0V (vi)
&Qp -Q = f
9
21, =O, =2,
=O, =2, = 1,
=O,
=I,
=l,
=2,
x’,=o, =l, ==: 0.
Th~mm 7. Eef the C~U~W of AZ0cormpond to thefactors Fiii3ie andFiyg4.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 A20correspond to the factors Fi,i2i5 and F&. Then an N.S.C. that Rank(Al : AzO)= m + 3 is thatthere existsa treatment(xl, . ...&) in Tzsatisfyingone of thefollowing conditions:
ii) o(xi, r xi21 = (ii) (iii) (iv) (v) (vi)
2,
=Q, =l, = 1, = 2, =O,
&i39
Xi41 = 0,
=2, =O, =2, = 1, =l,
Xis" 1,
=o, =o, =l.
Theorem 9. Let the columns of A20w-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 satisfyingone of the following conditions: (i) dxi, 9xi,9 xi,) 5 1 s d-Qt xi7 9XiJ 2 2 9 (ii) 51. 12, Proof. In (8), considering the rows corresponding all treatment in ‘r, except l)and(OO ‘0. 0), Etcan be checked that #iI= 0i2= 0i,, @id = @i,= l)ii6,and 8,,= 0 (1 1 . for u B {i,, iz,i3,id1is,i6}. Thus v*‘ehave l
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 <:asesto 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:!;srchdesign 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+1xm), with A!~,,,,.*=2&,, is
..*....*........ I=..7im I:01,
TL??l+
l
T2m i j
where j’==(f I 0-w1) and O’= (0 0 **a0). 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 19
stinct the integer m + 1 integers i (k= 1, . . . . 5) belonging to the set {1,2, . . . . m), we need to verify whether
s. G~~./scor~~*~fordmf~~~
383 Tzm+1
satisfies
the
c0IuiitionsIn TImrem
lo,; FOr the above ik’(kgl,-p~j)
distiiact integers & belonging to the set ( I, 2, .; ;,
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
m},
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 observethat our designsare
m
*1
Y?
can also be
N1
N2
2 4 5 9
N
'I 10 12 13
3
4
4
5
4 5
§ 6
6
7
10 20 35
7
8
56
6 1 8 9
10
19
8 --
9
84
10
19
29
Table 2 T2 for 3sms8
-8 1 I 1 1 ..._l_----____-__-_~___--~ ___-.“.‘..w..YW 1 1 1 1 1 _______.___ _._L_. -___._-.-_-----.-“_.~_.-.---I^I1C*II~-I1 0 0 1 1 a 0 -_______-_-I_ .__-__._ _____-_--__.--*___I___ clm"Wvlq-M"-Ml 1 ___c1I"u____nl_cP 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 interd 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 (Srivaso consider them in subseque t communications.
%NR, R.C. (lW7). Mathematical theory of symmetrical factorial designs. Sank!ry~ 8, 107-166. arid muMstage factor screening proccdur es. J. Combinatorics, Inform&ion Ghosh, S. (H79). On sin 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.
North-H
Planning Infmnce 3 (1981) 381-389 Publishing Company
381
ON E NEW SEARCH DESIGNS FOR 2mFACTORIAL ~EXPERIMENTS
Received26 April 197%;revisedmu:usctipt received20 July 1981 Retxmmcndcdby J.N. Srivastava 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.
Abstmct;
AMS I970 Subject C/ks@ution:
Secondary 62K 15.
Key WIOFQS,’ LinearModels; SearchDesigns; Factorial Experiments; Fractional Factorials.
1.
Introduction
The search designs of various kinds for 2Mfactorial 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 experiment, 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),
(1 0 1 .‘* i),
(0 0 0 “’
O),
(1 1 0
J), .
l
*a
(0 ..)
11
l
‘* I),
(1 1 1
“’
1 O),
which permits the estimation of the general mean and main effects under the assumption that all o&hereffects 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,mtreatments 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/$02.75 0 !981 North-Holland
2, Search linear models Consider the linear model &Y)=4b V(y)
4-42629 = 62JN,
IO !a
where y(Nx 1) is a vector of observations and for i= 1,2, A&@x wd) aft? known matrices, (vi X 1) are vet ors of fixed parameters, and c6 is a constant which may or may not be known. Furthermore, St is completely unknown, and we have partial information about &. We know that atmost k elements of & are non-zero and the rests are negligible, where k is a non-negative integer which may or may not be known. In this paper, we assume k is known. However, we do not know exactly which elements of & are nonzero. The problem is to search the nonzero elements of & and make inferences about them along with the elements of Ct. Such models are called search linear models with fixed effects and were introduced by J.N. Srivastava (1975). We want y (and hence At and AZ) to be such that the above problem can be resolved; the underlying design corresponding to such y is called a ‘searchdesign’. The case when cr2=0 is called the cnoiseless case’. In practice, we always have a2 ~0. The noiseless case is, however, of great importance from the design point of view. The reason is that any design which is not good in the noiseless case, can not be expected to be good in case Q?>O either. The designs described in this paper will find the nonnegligible parameters with probability one in the noiseless case. In case u 2>O the probability will depend on the *size’ or ‘amount’ of the errors. The following result is fundamental in search theory.
I’Reorem I. Consider the model ( 1,2) and let o2 = 0. A necessary and syf$icient condition (AH. C.) that the search and inference can be camp rely solved in the h!oiselesscase, 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$s, which is indeed a lar is small compared ‘to v2,
3. Conditions on existence of se Consider an FE 1, I’=
383
era1 mean, F;:the main effect of ith factor, E”,izthe two factor h the factors it and fz9and so on; ai= t or - 1 according as Xi= 1 rved response correspondin to (xr ‘4x,,J. Then our l
(5) ith N treatments by nN x m). Let y(N x 1) be the correor. We now define two vectors of parameters &(vr x 1)
(6)
consider the model (1,2) with vr , v2, &, tJ2 as in (6), (7), 4 factor and higher order interactions are zero and Al )I A2 are determined by T and (4), and k = 1. Let Ci (i=O, 1, . . . . m) be the cobJmns of A 1 corresponding to the elements in & ; C’ and CD are the columns of AZcorresponding to any two members, say F” and Ffi in 52. We consider the equation t!&c()+ e,&; + ***+
emcm + eucu + 6+.ca= 0,
(8)
where 8’s are real constants not all zero. Let T&VI xm), with Al, = m + 2, be the design as described in section 1. It is well known that Tr allows the estimation of & under the assumption that t2 =O. Suppose T2(N2x m) is another design with A$ 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 **a0) in Tl is
The row
rres
80+
ei+t&+ i=l
.a*
S,=O.
is
304
Consider the m rows in (8) corresponding to m treatments in r1 other than [S 0 l), arrd then adding all these rows we have and (1 P
l
0
l
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 1) and (0 0 0) in TI. l
l
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,ilUndF+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 independentof than, IItcan now be checked that one more treatment is * rqurred satisfying xi1*xi2 and xia=O. This comg!etes. the proof. l
Theorem4.Let the cohtmtts OfAm correspond CO the factors Fi,iiand &.li2i, Then 011 N.S.C. that Rank(A, : Aa) = m + 3 is that there exists a treatment (xl 0.l - x,,) in Tz satisfring l
Xj? =
W(Xj,,X,,) . = 1,
0,
Thtwan Theorem3. Let the columns ofAZ0correspond to the factor&T Fi ,iz andF;,ij14. N.S.C. that Rank(Al : A& = m + 3 is thot there exists a tretTtment (xl x,,) rn Tl l
satisfying one of the folio wing conditions: (0 Nq,, xi,) = 1 w
(ii) (iii)
-21, =Q,
wcq,
xi,, -
= 1) =o,
=2.
Proof. In (8), considerin the rows corresponding to a11treatments in Tl except (1 1 ‘. I)and(OO **0). we get Bil= Bipand l$ = 0 fbr u 4 {i,, i2,i3,i,}. ?‘hus we save l
l
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) Xi1t=Xi4*Xi, s Xi28(iii) Xi3!i= Xi4= 0, Xj,
20~~~r~s~~~dto t.%qfactorsfii,izand 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)
(ii) (iii) (iv) 0V (vi)
&Qp -Q = f
9
21, =O, =2,
=O, =2, = 1,
=O,
=I,
=l,
=2,
x’,=o, =l, ==: 0.
Th~mm 7. Eef the C~U~W of AZ0cormpond to thefactors Fiii3ie andFiyg4.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 A20correspond to the factors Fi,i2i5 and F&. Then an N.S.C. that Rank(Al : AzO)= m + 3 is thatthere existsa treatment(xl, . ...&) in Tzsatisfyingone of thefollowing conditions:
ii) o(xi, r xi21 = (ii) (iii) (iv) (v) (vi)
2,
=Q, =l, = 1, = 2, =O,
&i39
Xi41 = 0,
=2, =O, =2, = 1, =l,
Xis" 1,
=o, =o, =l.
Theorem 9. Let the columns of A20w-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 satisfyingone of the following conditions: (i) dxi, 9xi,9 xi,) 5 1 s d-Qt xi7 9XiJ 2 2 9 (ii) 51. 12, Proof. In (8), considering the rows corresponding all treatment in ‘r, except l)and(OO ‘0. 0), Etcan be checked that #iI= 0i2= 0i,, @id = @i,= l)ii6,and 8,,= 0 (1 1 . for u B {i,, iz,i3,id1is,i6}. Thus v*‘ehave l
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 <:asesto 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:!;srchdesign 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+1xm), with A!~,,,,.*=2&,, is
..*....*........ I=..7im I:01,
TL??l+
l
T2m i j
where j’==(f I 0-w1) and O’= (0 0 **a0). 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 19
stinct the integer m + 1 integers i (k= 1, . . . . 5) belonging to the set {1,2, . . . . m), we need to verify whether
s. G~~./scor~~*~fordmf~~~
383 Tzm+1
satisfies
the
c0IuiitionsIn TImrem
lo,; FOr the above ik’(kgl,-p~j)
distiiact integers & belonging to the set ( I, 2, .; ;,
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
m},
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 observethat our designsare
m
*1
Y?
can also be
N1
N2
2 4 5 9
N
'I 10 12 13
3
4
4
5
4 5
§ 6
6
7
10 20 35
7
8
56
6 1 8 9
10
19
8 --
9
84
10
19
29
Table 2 T2 for 3sms8
-8 1 I 1 1 ..._l_----____-__-_~___--~ ___-.“.‘..w..YW 1 1 1 1 1 _______.___ _._L_. -___._-.-_-----.-“_.~_.-.---I^I1C*II~-I1 0 0 1 1 a 0 -_______-_-I_ .__-__._ _____-_--__.--*___I___ clm"Wvlq-M"-Ml 1 ___c1I"u____nl_cP 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 interd 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 (Srivaso consider them in subseque t communications.
%NR, R.C. (lW7). Mathematical theory of symmetrical factorial designs. Sank!ry~ 8, 107-166. arid muMstage factor screening proccdur es. J. Combinatorics, Inform&ion Ghosh, S. (H79). On sin 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.