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On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1
Jounld of StatisthI Planning and Infwence S (1979) 33%34% @ IVwtMialkmnd 1%blishing Company
Received 20 February 1978; revised manuscript received 16 Novem!.a_r 1979 Recommended by J.N. Srivastava Abstract: The norm llAll= {br(A’A))1/2 of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for l/All will be given for a balanced fractional 2m factorial design of resolution 21f 1 which is obtained from a simple array with plarameters (m ; Aa, Al, . . . , A,,). This array is identical with a balanced array of strength m, M constraints and index set (A,, A,, . . . , A,,,}. In the class of the designs of resolution V (I = 2) obtained from S-arrays, ones which minimize l\Allwill be presented for any fixed N assemblies satisfying (i) m = 4, ‘I1 s NS 16, (ii) m = 5, 16~ N s32, and (iii) m = 6, 22sN~40. AMS subject classifications: Primary 62K15; Secondary OSB30. Key
words and phrases: Balanced Fractional Design; Alias Matrix; Information Balanced Array, Simple Array, Best Alias.
Matrix;
Consider an experiment with m factors each at two levels. An assembly or treatment combination will be represented by (jl, j,. . . . , i,) where jk, the level of the kth factor, elquals 0 or I. Let 6 be the y x 1 vector composed of the effects up to o”=Sfactor interactions (m > 21) and let 8* be the y x 1 vector of the effects of (I + &factor and higher order interactions, where vI = zfzo (;^) and v’(’= 2” - y. As usual, the vector 8 is to be estimated and 8” is not of interest fhr estimation. Le:lt yT be the NX 1 observation vector corresponding to a fraction I‘ with IV assemblies. Further consider the observations in yT as independent random variables with common variance a2. Then the expected value of yT can be expressed as a linear model (1.1) where E and E* denote the WX y and N x VP design matrices for respc:ctively, whose elements are 1 or --I. A fraction T is called Cl factorial (brkfly, 2”‘~FF) design o cst For ii1Y’-FF desisgn T of resol 337
338
is given by 4i=&4-‘E’K_, where M = E’E ii called the information matrix for Q of 7’. However under the general model ( 1.l), the expected value of 6 becomes g(6) = 6 + A0*, is called the alias matrix of T. This suggests that A@* may where A = M%‘E* not always be negligible even if 8* is itself negligible in comparison with 9. The norm llAll= (tr(A’A)}“2, which has been introduced by Hedayat, Raktoe and Federer (1974, t&erefor e, is used as a measure for comparing designs. In the case whet a few nonnegligible effects are included in 8*, Srivastava (1975) has introduced the concept of search designs in which one can search these effects and estimate them in addition to 8. Indeed, Srivastava and Gho;h (1977) have given constizactions of search designs of resolution V (2 = 2) for ea.ch m (4 < m < 8) and for practical values of N under the assumption that the number of nonnegiigibie effec:ts (saa, k) in 8* is one. However, for other values of k, good warch designs have s511 :o be constructed. Until that is done, one could use IlAll LISan extra criterion to select a design. This will, in a sense, “reduce” the (i\bsolutle) value of the elements of A6*. Hence it will help reduce the bias to sctme extent, if not ,eliminate it to a large extent. We now consider a balanced fractional 2” factorial (briefly, 2m-BFF) design of resolution 2 ! -+ 1 in which the covariance matrix Var[@ = M-lo2 is invariant under any permutatioir of na factors. 2” -BFF designs of resolution V were first discussed by Srivast:qva (1970). Yamamoto, Shirakura and Kuwada (1975, 1976) have developc>d 2” -EIFF designs of resolution 21+ 1. In those two papers, they have investigated some properties of a trianguiar type multidimensional partially bsrlanced (TMDPB) association algebra defined among the effects up to I-factor interactions. This algebra is irsi;ful in clarifying algebraic and combinatorial structures of 2” -BFF designs of resolutior; 21-i- 1. It is known from their results that a 2”-BEF design of resolution 21 t 1 is equivalent to a balanced array (B-array) of strength 21, m constraints and inde.rr set {CL,,, ~11, . . . , pzl} (or indices /Li (i = 0, 1, . . . ,21):1 provided the information matrix is nonsingular. In this paper, by use of the properties of a TMDPB association algebra, an explicit expression of the norm IlAll will be given for a 2” -BFF design of resolution 21’+ 1 ob,tainec: from a simple array (S-array) with Iparameters : A;;; ). T’hIs eupr6+ci: 1.vill be given by the inverse matrices of order int;h(),h*,... (I+ I) at most. The above &ray is identical with a B-array of strength m, m constraints and indices 1hi (i ==C),l .* , m). It is easy to see that this array is also a B-array of strength 21 with indick’s /hi (i = 0, 1, . . . ,22), where -21 -E
(l.Z!)
33;;
Nom of alias matrices in 2” factorial designs
Assume throughout this paper that (E)= 0 if and only if b > a * 0 or b c 0. As wil’.l be seen from Chopra (1975), Chopra axid Srivastava (1973), Shirakura (1976a, 1977), Srivastavimand Chopra (I971, 1974), etc., most of B-arrays of strength 4 or 61are of simple types for practical numbers of m and N In the class of 2”-BFF designs of resolution V obtained from S-arrays, a design whi@hminimizes /Ali will also be presented for each nulnber of N assemblies satisfying (i) m = 4, 11 G N < 16, (ii) m = 5, 16 s N s 32, and (iii) ?~t= 6, 22 6 N 6 40.
2. Expression of IlAll
We first prove the following lemma. Lemma 1. Let ‘T be a 2”-FF design of resolution 21+ 1 with N assemblies which has exactly the ihth assembly in the set of aPIassemblies with multiplicity rikfor each h=l,2,..., n (i.e., N=CE=, Ti,). Let eih be the y A 1 vector obtained from E corresponding +othe i,th assembEy. Then jlAl12=Z”(tr(M-‘) + 2 rih(ri,,- i)e:,M2eJ
-
(2.1)
~1.
h=l
Proof. From model Cl.l), [E : E*][E : E”]’= EE’ -t-E”E*’ = 2”diag(Gri,., G,,,: . e +3Gr,,) (=D, say),
where G, denotes the r >(:r matrix with all elements one. Hence AA’ = M-‘E’E*E*‘Eb{-’
= M-‘E’DEM-’
-- 1Y’
where I.. is the identity matrix of order p. Since M = cE= l 2” I:= 1 rieke;h, it is easy to show that
AA’= 2” ha-’ + i I
Tih(ri,, - l)M-‘e,,e:,,M-’
h=l
I
ri,,e,,eg
and E’DE =
- Iti.
Since tr(AA’) = tr(A’A), we have (2.1). From Lemma 1, it is found that the norm llA\l is independent of the *. For a general fractional s1 x s2x - - x s, factorial design, Shirakura (19’7%~)has obtained a result sin-&r to Lemma 1. l
For an S-array with parameters am; X0,hI, . .
. , A,,,), de
where pi-l) = pi in (1.2) and Pi
(9) =
m-2
( 3 q-i
for q = 0, I,
. . . , m.
Pufiher define
Consider now the
(i’-- /3 +
1) x (I - 6 + 1) symmetric matrices given by
(2.2) for q=O, l,...,
m; fi=O,l,...,
I, where
‘Then we can estabhsh the following theorem. Theorem 2. Let T be ct 2”-BFF design of resolution 21-k 1 obtained from ay1 S-array with pavamcrters(RI ; Ao, A 1, . . . , A,). Then the norm \\A\\ can be expressed as IlkI\’ = 2” :i + tr K,’ f f h,(A, - l)K,,,B)K,‘)p =o ( 6 =O
VI,
(2.3
where
Remark. IS has beern shown by Xamamoto, Shirakura and Kuwada (1976) that tile information matrix A4 of a 2V3FF design of resolution 2Z+ 1 is positive definite if a& anly if thz matrices & (0 = 0, 1, . . , I) are so. Following a usual procedure in the calculrtion of l\AII, we have to calculate the inverse of a large vI x y matrix M. Howe\.er the expression of (2.3) implies that we have c)nly to calculate the inverse of st most (I+ 1) x (I + 1) matrices K. and KS. For convenience, let R ==(\lAII”+ q):!:-‘T StqJ~OzXT’q) is an S-array with parameters for q = 0, I,. . . , m.
(m; Ao= 0,. . . , Aq._1= 0, A, = 1, at the S-array T , 1, . . * ) m). Javey
assembly with weight q occurs Aqtimes in 7’ for each q = 0,. can easily be shown that
Jl rk( -
rb -
. . , m.
Therefore, it
l)e6W2ei, = 2 h\(l(hg- l)tr(E{&&4-2) c;-,sO
holds, where E(qj is the c) x I+ design matrix fur 6 of T’4).By Lemma 1, we thus have R = tr(M+) + E A& - l)tr(E&&q,M-2).
(2.4)
For a &array T of strength 21, m constraints and indices pi’s in (KC the information matrix MTcan be written as
These y x y matrices I$-“)#, (p = 0, I, . . . , min(u, u); 0 G u, o G I), are known as basis of an (2+ 1) sets TMDPB association algebra. For details, see Yamamoto, Shirakura and Kuwada (1976). Also, .“4’ is itself a B-array of strength 21 with indices pp’= (“&:I) (i = 0,. . . , 21). Therefore the information matrix I&) = E{q,Etq, of Fq) can also be written as
This means that tr(M-l) = i & tr(K,*),
hold (see, e.g., Aplpendix of Shirakusa (1967b)). By (2.4), therefore, we have (2.5) This completes thi: proof of Theorem 2. An S-array with parameters (m ; Ao, h, = 0 or 1 for each q.
Al, . . . , A,)
is said to be (0, I) bmary if
Su~pposethere exist (0, I.) binary S-arrays which minimize tr(M-l) in the class mk&vzed earlier. Then thse are the only designs that minimize l!All in ax0
this Cl&SS.
ation matrix alad ray such that tr(
T. Shit&ma
is a minimum. Further let ‘s’ be another 2m-BFF design of resolution 21+ 1 obtained from an S-array with parameters A,.$. Then, since I&’ is positive definite, tr(M,,,M-,“, = ~I~(E&‘~Z~~E~~J > 0 holds. Hence by Theorem 2, llAT,ll= 2”‘ltr(A#$!) - y s Zrntr(MG”) - y
= IIA 4T.II The second inequality Iholds with equality if and only if &(& - 1) = 0 for all q=O,I,..., m. This completes the proof. 3. l&e best alias 2”.BFF designs of redution
V
From (2.2), more explic;;t expressions of KY0(p = 0, 1,2) for 1 =: 2 are given as follows: YO
K, =
m *‘2yj
wnr*
yo + (m - l)y2
&m - l>,‘nc2Y,+ (m - 2h3)
(3:<3) [ (sym_)
Yo+2(m-2)Y2+(“2*)Y4 (m
K,
-
W2(Yl
-
I
Yd
=
(2x2)
Yo+(fP-4)Y2-(W-~)Y4
I'
K* = y&--zy2+y4=24~2 Wh+ST
Y()=fL4't/~O+4(~3+~1)+411_*, Y2 = P4+PO-&529
Yi=CL4-PO+2(CC3-Pl)~
Y3=ti4--w-2(~3--~Ih
y4=~4+Ili)-4(~3+~1)+6~2~
For brevity, we write yi = $-‘) (E’= 0, 1,2,3,4). Those of K,,,, for each 4 = 0, l,..., m can be obtained from the above expressions by replacing “yi and pi with yiq’ and pig) =z(:I:), respectively. By using Theorem 2 and Cor
343 Table 1
m=4
A,
A,
A,
A,
A4
“11
0 I
1 1
1 1
0 0
1 1
1,4861 1.3125
3.5757 3.1623
1 1-i 1 ;
1 1 1 1
1 1 1 1
0 2 1 0 10 1 ?
1.3133 1.1875 0.8250 0.6875
3.1642 2.8284 1.4832 0 ._” WI0
“12 “13 “14 “15 “16 N
m= 5
m=6
WI
N
&I A,
A2
A3
A,
R
R
A,
*28 29 ‘30 *31 *32
101010 110101 201011 301011 011010 111010 111011 111062 211032 111014 011101 111101 111102 111103 011110 111110 1 1 1
N
A0 A, A, A,
‘b
As A6
1 1 2 3 4 0 1 1 1 1 0 I 1 2 2 0 1 2 2
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1
1 1 1 1 1 0 0 0 0 0 1 1 1 1 ! 0 0 0 0
“16 “17 “18 “19 20 “21 “22 23 “24 25 *26 “27
*22 23 *24 “25 *26 “27 28 *29 *30 ‘31 *32 “33 “34 “35 “36 “37 38 39 40
0 0 0 0 0 1 1 i 1 0 1 1 1 1 1 I 1 1 1
1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 3 1
1
0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0
1
1
0 1 1 1 1 1 1 2 3 0 0 0 1 1 2 1 1 1 2
il4
1.oooo 0.9688 0.9698 0.9707 0.9278 0.8438 0.8125 0.8160 0.8i94 0.8238 0.6875 0.6563 0.6584 0.6602 0.5830 0.5313 0.5000
4.0000 3.8730 3.8773 3.8809 3.6999 3.3166 3.1623 3.1798 3.1971 3.?187 2.4495 2.2361 2.2511 2.2641 1.6293 1.oooo 0.0000
R
Ii41
1.1517 1.1392 1.1405 1,141s 1.1422 0.9754 0.9641 0.9644 0.9647 0.7563 0.6875 d.6748 0.6634 0.6656 0.6679 0.6246 0.6118 0.6142 0.6155
7.1907 7.1351 7.1408 7.1455 7.1485 6.3582 6.3009 6.3026 6.3040 5.1381 4.6904 4.6027 4.5227 4.5385 4.5545 4.2395 4.1417 4.1603 4.1701
cc,
CL1
P2
cc,
c14
tx(M-‘)
0
I
1
1
1 1
0 0
1 1
1.4861 1.3125
1 0 1 1
1 1 1 1
1 1 1 1
0 1 1 1
2 0 0 1
1.2639 1.;875 0.8250 0.6875
CL0PI
cc2
F3
P4
tr(M”‘)
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
1 1 2 2 1 1 2 3 3 5 1 1 2 3 1 1 2
1.oooo 0.9688 0.9398 0.9296 0.9278 0.8438 0.8125 3.8021 0.788 1 0.7938 0.6875 0.6563 0.6300 0.6203 0.5830 0.5313 0.5000
P3
cl4
tx(M-‘)
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 3 3 3 3 1 1 2 3 1 2 2 3 3 4 2 2 2 3
I.1517 1.1392 1.1145 1.1057 1.1012 0.9754 0.9641 f-l.9371 0.9280 0.7563 0.6875 0.6748 0.6634 0.6581 0.6533 0.6246 0.61 18 0.6062 11.5943
1
2 2 3 1 2 2 2 3 2 1 2 2 2 1 2 2
PO PI 2 2 3 4 5 2 3 3 3 2 2 3 3 4 4 3 d 5 5
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3
P2
1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 i 1 1 1 1 1 1 1 2 2 2
Also we have &, = 1, dil = 5 and & = 9. By (2.3) and (2.5), therefore, R = tr(K;
*+
we have
6K~,5&Z~2)+ 5 tr(K, ’ + 6Kd,j,K, ‘)
+ 9(K;’ + 6Ktej2Ki2) = 0.96469, /IAll= {26R - 22]= 6.30395. In Table 1, the values of parameters Aq (q = 0, 1, . . . , no) of these S-arrays are given with those of indices pi (i = 0, 1,2,3,4) of the corresponding E-ai=rays of strength 4. In addition, the values of R, llAl\ and1t&W’) are given for reference. It is very interesting that many designs of Table 1 are optimal designs with respect to the trace criterion (which minimize tr(M-“)) olbtained by Srivastava and Chopra ( 197 1). These designs are indicated in Table 1 by asterisks *. We are aXso interested in knowing the values of R and l\All for the optimal designs of Srivastava and Chopra (1971) which do not minimize \lAll. These can be observed in Table 2 which is given in the same form iis ‘Table 1. Finally note that like the optimal designs with respect to the trace criterion, for the designs In Table 1, their complementary designs (obtained by an interchange of 0 and 1) are also the best alias with respect to the norm IlAll.
Table 2 N
nt = 5
4, A, A2 A, A, As
17 19 20 21 22 23a 23b 24 2Sa 25b 26 27 28 29a 2% 3i 32 t 23 28 38 39a 39b 4Oa -1CRl
2 0 210102 3 0 0 2 0 2 211011 1 2 1 2 311012 2 2 1 0 2 0 2 ,I) 3 i’) 9 ii 2 h, 2 C)
0 1 0 1 0
1
0
1
0
IlAII
1.oooo
2 1 3 1 3 1 .2 2 2 2 3 2 3 2 3 2 4 2 42113 1 2 2 2 2 2 3 2 3 2 2 2 2 2
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 2 3 1 2 2 2 3 3
2 2 2 2 2 2 2
1 1 1 1 1 2 2
1 1 2 2 2 Jl 2
0.9688 0.9296 0.9194 0.8438 0.8125 0.7980 0.7980 0.788 1 0.7815 0.7815 0.6875 0.6563 0.6300 0.6199 0.6199 0.5313 0.5000
CL,, Cal
EL2
EL4
tr(M-‘)
2 2 3 3 3 3 3
1 2 2 2 2 2 2
2 2 3 3 3 4 4
1.1242 0.9488 0.6088 0.5993 0.5993 0.5940 0.5940
1 0 0
0 1 1
1 0 0
2 1 2
0 0
1 1
0 0
2 3
0 2 2 2 2
1 0 0 1 61 1 01 1 01 1
.3 0 0 I 1
0.9734 0.9748 1.oooo 1.oooo 0.8164 0.9726 0.9734 0.8240 0.9733 1.OOOO 1.oooo ti.9709 0.97 17
01 2
2. 1 (1
21 0
0.6619 1.oooo 1.OOOO
4.0000 3.8922 3.8977 4.OOOO 4.OOdO 3.1819 3.8890 3.892 1 3.2202 3.8918 4.0000 4.0000 3.8817 3.8852 2.276 1 4.0000 4.0000
R
IIAII
2I 0
0I) 2
A, A,
A,
A,
A,
A,
0 1 1 1 2 3 2
(1 1 0 0 1 0 1
0 0 1 1 0 1 0
I t) 0 0 1 0 1
0 2 2 2 1 3 2
1 0 1 1 0 1 0
R
1.1597
0.9734 0.6265 0.6142 0.6847 0.6107 0.6910
7.1907 0.3582 4.2540 4.1604 4.6989 4.1795 4.7 142
tr( MI')
3
2 3 4 4 4 4
b,
B 1 2 2 2 2 2
Norm of alias matrices in 2,” factorial designs
345
The author wishes to thank the editor., J.N. Srivastava, for her valuable comments.
Chopra, D.V. (1975). Balanced optimal 2” fractional :factorial designs of resolution V, 52 G N c 59. In: J.N. Srivastava, ed., A supoey of statistical designs and linear models. North-Holland, Amsterdam, 91-100. Chopra, D.V. and J.N. Srivastava (1973). Optimal balanced 2’ fraction& factorial desigtis of resolution V, with N 642. Ann. Inst. Statist. Math. 25, 587-604. Hedayat, A., B.L. Raktoe and W.T. Federer (1974). On a measure of aliasing due to fitting an incomplete modei. Ann. Statist. 2, 650-660. Shirakura, T. (1976a). Optimal balanced fractional 2” factorial designs of resolution VII, 6 < m G 8. Ann. Statist. 4, 515-531. Shirakura, T. (1976’6). Balanced fractionat 2” factorial designs of even resolution obtained from balanced arrays of strength 21 with index &f = 0. Anni 9atist. 4, 723-735. Shirakura, T. (19%). A note on the norm of alias matrices in fractional replication. Austral. J. Statist. 18, 158-160. Shirakura, T. (1977). Contributions to balanced fractional 2” factorial designs derived from balanced arrays of strength 21 Hiroshima Math. J. 7, 217-2:35. Srivastava, J.N. (1970). Optimal balanced 2” fractional factorial designs. S.N. Roy memorial volume, Univ. of North Carolina, and Indian Statistical Institute. 689-706. Srivastava, J.N. (1975). Designs for searching non-negligible effects. In J.N. Srivastava, ed., A suruey of statistical designs and linear models. North-Holland, Amsterdam, 507-S 19. Srivastava, J.N. and D.V. Chopra (1971). Balanced optimal 2” fractional factorial designs of resolution V, m G 6. Technometrics 13, 257-269. Srivastava, J.N. and D.V. Chopra (1974). Balanced trace of*timal 2’ fractional factorial designs of resolution V, with 56 to 68 runs. Utilitas Muthenratics 5, 263-279. Srivastava, J.N. and S. Gosh (1977). Balanced 2” factorial designs of resolution V which allow search and estimation of one extra unknown effect, 4~ rn ~8. Cc;nmun. Statist. A6, 141-166. Yamamoto, S., T. Shirakura and M. Kuwada (197511.Balanced arrays of strength 21 and balanced fractional 2” factorial designs. Ann. Inst. Statist. Math. 27, 143-157. Yamamoto, S., T. Shirakura and M. Kuwada (1976). Characteristic polynomials of the information matrices of balanced fractional 2” factorial designs of higher (22+ 1) resolution. In S. Ikeda et al., eds., Essays in Probabilit)l and Statistics. Birthday volume in honor of Prof. Junjiro Ogawa. Shinko Tsusho Co. Ltd., Tokyo, 73-94.
Received 20 February 1978; revised manuscript received 16 Novem!.a_r 1979 Recommended by J.N. Srivastava Abstract: The norm llAll= {br(A’A))1/2 of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for l/All will be given for a balanced fractional 2m factorial design of resolution 21f 1 which is obtained from a simple array with plarameters (m ; Aa, Al, . . . , A,,). This array is identical with a balanced array of strength m, M constraints and index set (A,, A,, . . . , A,,,}. In the class of the designs of resolution V (I = 2) obtained from S-arrays, ones which minimize l\Allwill be presented for any fixed N assemblies satisfying (i) m = 4, ‘I1 s NS 16, (ii) m = 5, 16~ N s32, and (iii) m = 6, 22sN~40. AMS subject classifications: Primary 62K15; Secondary OSB30. Key
words and phrases: Balanced Fractional Design; Alias Matrix; Information Balanced Array, Simple Array, Best Alias.
Matrix;
Consider an experiment with m factors each at two levels. An assembly or treatment combination will be represented by (jl, j,. . . . , i,) where jk, the level of the kth factor, elquals 0 or I. Let 6 be the y x 1 vector composed of the effects up to o”=Sfactor interactions (m > 21) and let 8* be the y x 1 vector of the effects of (I + &factor and higher order interactions, where vI = zfzo (;^) and v’(’= 2” - y. As usual, the vector 8 is to be estimated and 8” is not of interest fhr estimation. Le:lt yT be the NX 1 observation vector corresponding to a fraction I‘ with IV assemblies. Further consider the observations in yT as independent random variables with common variance a2. Then the expected value of yT can be expressed as a linear model (1.1) where E and E* denote the WX y and N x VP design matrices for respc:ctively, whose elements are 1 or --I. A fraction T is called Cl factorial (brkfly, 2”‘~FF) design o cst For ii1Y’-FF desisgn T of resol 337
338
is given by 4i=&4-‘E’K_, where M = E’E ii called the information matrix for Q of 7’. However under the general model ( 1.l), the expected value of 6 becomes g(6) = 6 + A0*, is called the alias matrix of T. This suggests that A@* may where A = M%‘E* not always be negligible even if 8* is itself negligible in comparison with 9. The norm llAll= (tr(A’A)}“2, which has been introduced by Hedayat, Raktoe and Federer (1974, t&erefor e, is used as a measure for comparing designs. In the case whet a few nonnegligible effects are included in 8*, Srivastava (1975) has introduced the concept of search designs in which one can search these effects and estimate them in addition to 8. Indeed, Srivastava and Gho;h (1977) have given constizactions of search designs of resolution V (2 = 2) for ea.ch m (4 < m < 8) and for practical values of N under the assumption that the number of nonnegiigibie effec:ts (saa, k) in 8* is one. However, for other values of k, good warch designs have s511 :o be constructed. Until that is done, one could use IlAll LISan extra criterion to select a design. This will, in a sense, “reduce” the (i\bsolutle) value of the elements of A6*. Hence it will help reduce the bias to sctme extent, if not ,eliminate it to a large extent. We now consider a balanced fractional 2” factorial (briefly, 2m-BFF) design of resolution 2 ! -+ 1 in which the covariance matrix Var[@ = M-lo2 is invariant under any permutatioir of na factors. 2” -BFF designs of resolution V were first discussed by Srivast:qva (1970). Yamamoto, Shirakura and Kuwada (1975, 1976) have developc>d 2” -EIFF designs of resolution 21+ 1. In those two papers, they have investigated some properties of a trianguiar type multidimensional partially bsrlanced (TMDPB) association algebra defined among the effects up to I-factor interactions. This algebra is irsi;ful in clarifying algebraic and combinatorial structures of 2” -BFF designs of resolutior; 21-i- 1. It is known from their results that a 2”-BEF design of resolution 21 t 1 is equivalent to a balanced array (B-array) of strength 21, m constraints and inde.rr set {CL,,, ~11, . . . , pzl} (or indices /Li (i = 0, 1, . . . ,21):1 provided the information matrix is nonsingular. In this paper, by use of the properties of a TMDPB association algebra, an explicit expression of the norm IlAll will be given for a 2” -BFF design of resolution 21’+ 1 ob,tainec: from a simple array (S-array) with Iparameters : A;;; ). T’hIs eupr6+ci: 1.vill be given by the inverse matrices of order int;h(),h*,... (I+ I) at most. The above &ray is identical with a B-array of strength m, m constraints and indices 1hi (i ==C),l .* , m). It is easy to see that this array is also a B-array of strength 21 with indick’s /hi (i = 0, 1, . . . ,22), where -21 -E
(l.Z!)
33;;
Nom of alias matrices in 2” factorial designs
Assume throughout this paper that (E)= 0 if and only if b > a * 0 or b c 0. As wil’.l be seen from Chopra (1975), Chopra axid Srivastava (1973), Shirakura (1976a, 1977), Srivastavimand Chopra (I971, 1974), etc., most of B-arrays of strength 4 or 61are of simple types for practical numbers of m and N In the class of 2”-BFF designs of resolution V obtained from S-arrays, a design whi@hminimizes /Ali will also be presented for each nulnber of N assemblies satisfying (i) m = 4, 11 G N < 16, (ii) m = 5, 16 s N s 32, and (iii) ?~t= 6, 22 6 N 6 40.
2. Expression of IlAll
We first prove the following lemma. Lemma 1. Let ‘T be a 2”-FF design of resolution 21+ 1 with N assemblies which has exactly the ihth assembly in the set of aPIassemblies with multiplicity rikfor each h=l,2,..., n (i.e., N=CE=, Ti,). Let eih be the y A 1 vector obtained from E corresponding +othe i,th assembEy. Then jlAl12=Z”(tr(M-‘) + 2 rih(ri,,- i)e:,M2eJ
-
(2.1)
~1.
h=l
Proof. From model Cl.l), [E : E*][E : E”]’= EE’ -t-E”E*’ = 2”diag(Gri,., G,,,: . e +3Gr,,) (=D, say),
where G, denotes the r >(:r matrix with all elements one. Hence AA’ = M-‘E’E*E*‘Eb{-’
= M-‘E’DEM-’
-- 1Y’
where I.. is the identity matrix of order p. Since M = cE= l 2” I:= 1 rieke;h, it is easy to show that
AA’= 2” ha-’ + i I
Tih(ri,, - l)M-‘e,,e:,,M-’
h=l
I
ri,,e,,eg
and E’DE =
- Iti.
Since tr(AA’) = tr(A’A), we have (2.1). From Lemma 1, it is found that the norm llA\l is independent of the *. For a general fractional s1 x s2x - - x s, factorial design, Shirakura (19’7%~)has obtained a result sin-&r to Lemma 1. l
For an S-array with parameters am; X0,hI, . .
. , A,,,), de
where pi-l) = pi in (1.2) and Pi
(9) =
m-2
( 3 q-i
for q = 0, I,
. . . , m.
Pufiher define
Consider now the
(i’-- /3 +
1) x (I - 6 + 1) symmetric matrices given by
(2.2) for q=O, l,...,
m; fi=O,l,...,
I, where
‘Then we can estabhsh the following theorem. Theorem 2. Let T be ct 2”-BFF design of resolution 21-k 1 obtained from ay1 S-array with pavamcrters(RI ; Ao, A 1, . . . , A,). Then the norm \\A\\ can be expressed as IlkI\’ = 2” :i + tr K,’ f f h,(A, - l)K,,,B)K,‘)p =o ( 6 =O
VI,
(2.3
where
Remark. IS has beern shown by Xamamoto, Shirakura and Kuwada (1976) that tile information matrix A4 of a 2V3FF design of resolution 2Z+ 1 is positive definite if a& anly if thz matrices & (0 = 0, 1, . . , I) are so. Following a usual procedure in the calculrtion of l\AII, we have to calculate the inverse of a large vI x y matrix M. Howe\.er the expression of (2.3) implies that we have c)nly to calculate the inverse of st most (I+ 1) x (I + 1) matrices K. and KS. For convenience, let R ==(\lAII”+ q):!:-‘T StqJ~OzXT’q) is an S-array with parameters for q = 0, I,. . . , m.
(m; Ao= 0,. . . , Aq._1= 0, A, = 1, at the S-array T , 1, . . * ) m). Javey
assembly with weight q occurs Aqtimes in 7’ for each q = 0,. can easily be shown that
Jl rk( -
rb -
. . , m.
Therefore, it
l)e6W2ei, = 2 h\(l(hg- l)tr(E{&&4-2) c;-,sO
holds, where E(qj is the c) x I+ design matrix fur 6 of T’4).By Lemma 1, we thus have R = tr(M+) + E A& - l)tr(E&&q,M-2).
(2.4)
For a &array T of strength 21, m constraints and indices pi’s in (KC the information matrix MTcan be written as
These y x y matrices I$-“)#, (p = 0, I, . . . , min(u, u); 0 G u, o G I), are known as basis of an (2+ 1) sets TMDPB association algebra. For details, see Yamamoto, Shirakura and Kuwada (1976). Also, .“4’ is itself a B-array of strength 21 with indices pp’= (“&:I) (i = 0,. . . , 21). Therefore the information matrix I&) = E{q,Etq, of Fq) can also be written as
This means that tr(M-l) = i & tr(K,*),
hold (see, e.g., Aplpendix of Shirakusa (1967b)). By (2.4), therefore, we have (2.5) This completes thi: proof of Theorem 2. An S-array with parameters (m ; Ao, h, = 0 or 1 for each q.
Al, . . . , A,)
is said to be (0, I) bmary if
Su~pposethere exist (0, I.) binary S-arrays which minimize tr(M-l) in the class mk&vzed earlier. Then thse are the only designs that minimize l!All in ax0
this Cl&SS.
ation matrix alad ray such that tr(
T. Shit&ma
is a minimum. Further let ‘s’ be another 2m-BFF design of resolution 21+ 1 obtained from an S-array with parameters A,.$. Then, since I&’ is positive definite, tr(M,,,M-,“, = ~I~(E&‘~Z~~E~~J > 0 holds. Hence by Theorem 2, llAT,ll= 2”‘ltr(A#$!) - y s Zrntr(MG”) - y
= IIA 4T.II The second inequality Iholds with equality if and only if &(& - 1) = 0 for all q=O,I,..., m. This completes the proof. 3. l&e best alias 2”.BFF designs of redution
V
From (2.2), more explic;;t expressions of KY0(p = 0, 1,2) for 1 =: 2 are given as follows: YO
K, =
m *‘2yj
wnr*
yo + (m - l)y2
&m - l>,‘nc2Y,+ (m - 2h3)
(3:<3) [ (sym_)
Yo+2(m-2)Y2+(“2*)Y4 (m
K,
-
W2(Yl
-
I
Yd
=
(2x2)
Yo+(fP-4)Y2-(W-~)Y4
I'
K* = y&--zy2+y4=24~2 Wh+ST
Y()=fL4't/~O+4(~3+~1)+411_*, Y2 = P4+PO-&529
Yi=CL4-PO+2(CC3-Pl)~
Y3=ti4--w-2(~3--~Ih
y4=~4+Ili)-4(~3+~1)+6~2~
For brevity, we write yi = $-‘) (E’= 0, 1,2,3,4). Those of K,,,, for each 4 = 0, l,..., m can be obtained from the above expressions by replacing “yi and pi with yiq’ and pig) =z(:I:), respectively. By using Theorem 2 and Cor
343 Table 1
m=4
A,
A,
A,
A,
A4
“11
0 I
1 1
1 1
0 0
1 1
1,4861 1.3125
3.5757 3.1623
1 1-i 1 ;
1 1 1 1
1 1 1 1
0 2 1 0 10 1 ?
1.3133 1.1875 0.8250 0.6875
3.1642 2.8284 1.4832 0 ._” WI0
“12 “13 “14 “15 “16 N
m= 5
m=6
WI
N
&I A,
A2
A3
A,
R
R
A,
*28 29 ‘30 *31 *32
101010 110101 201011 301011 011010 111010 111011 111062 211032 111014 011101 111101 111102 111103 011110 111110 1 1 1
N
A0 A, A, A,
‘b
As A6
1 1 2 3 4 0 1 1 1 1 0 I 1 2 2 0 1 2 2
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1
1 1 1 1 1 0 0 0 0 0 1 1 1 1 ! 0 0 0 0
“16 “17 “18 “19 20 “21 “22 23 “24 25 *26 “27
*22 23 *24 “25 *26 “27 28 *29 *30 ‘31 *32 “33 “34 “35 “36 “37 38 39 40
0 0 0 0 0 1 1 i 1 0 1 1 1 1 1 I 1 1 1
1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 3 1
1
0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0
1
1
0 1 1 1 1 1 1 2 3 0 0 0 1 1 2 1 1 1 2
il4
1.oooo 0.9688 0.9698 0.9707 0.9278 0.8438 0.8125 0.8160 0.8i94 0.8238 0.6875 0.6563 0.6584 0.6602 0.5830 0.5313 0.5000
4.0000 3.8730 3.8773 3.8809 3.6999 3.3166 3.1623 3.1798 3.1971 3.?187 2.4495 2.2361 2.2511 2.2641 1.6293 1.oooo 0.0000
R
Ii41
1.1517 1.1392 1.1405 1,141s 1.1422 0.9754 0.9641 0.9644 0.9647 0.7563 0.6875 d.6748 0.6634 0.6656 0.6679 0.6246 0.6118 0.6142 0.6155
7.1907 7.1351 7.1408 7.1455 7.1485 6.3582 6.3009 6.3026 6.3040 5.1381 4.6904 4.6027 4.5227 4.5385 4.5545 4.2395 4.1417 4.1603 4.1701
cc,
CL1
P2
cc,
c14
tx(M-‘)
0
I
1
1
1 1
0 0
1 1
1.4861 1.3125
1 0 1 1
1 1 1 1
1 1 1 1
0 1 1 1
2 0 0 1
1.2639 1.;875 0.8250 0.6875
CL0PI
cc2
F3
P4
tr(M”‘)
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
1 1 2 2 1 1 2 3 3 5 1 1 2 3 1 1 2
1.oooo 0.9688 0.9398 0.9296 0.9278 0.8438 0.8125 3.8021 0.788 1 0.7938 0.6875 0.6563 0.6300 0.6203 0.5830 0.5313 0.5000
P3
cl4
tx(M-‘)
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 3 3 3 3 1 1 2 3 1 2 2 3 3 4 2 2 2 3
I.1517 1.1392 1.1145 1.1057 1.1012 0.9754 0.9641 f-l.9371 0.9280 0.7563 0.6875 0.6748 0.6634 0.6581 0.6533 0.6246 0.61 18 0.6062 11.5943
1
2 2 3 1 2 2 2 3 2 1 2 2 2 1 2 2
PO PI 2 2 3 4 5 2 3 3 3 2 2 3 3 4 4 3 d 5 5
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3
P2
1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 i 1 1 1 1 1 1 1 2 2 2
Also we have &, = 1, dil = 5 and & = 9. By (2.3) and (2.5), therefore, R = tr(K;
*+
we have
6K~,5&Z~2)+ 5 tr(K, ’ + 6Kd,j,K, ‘)
+ 9(K;’ + 6Ktej2Ki2) = 0.96469, /IAll= {26R - 22]= 6.30395. In Table 1, the values of parameters Aq (q = 0, 1, . . . , no) of these S-arrays are given with those of indices pi (i = 0, 1,2,3,4) of the corresponding E-ai=rays of strength 4. In addition, the values of R, llAl\ and1t&W’) are given for reference. It is very interesting that many designs of Table 1 are optimal designs with respect to the trace criterion (which minimize tr(M-“)) olbtained by Srivastava and Chopra ( 197 1). These designs are indicated in Table 1 by asterisks *. We are aXso interested in knowing the values of R and l\All for the optimal designs of Srivastava and Chopra (1971) which do not minimize \lAll. These can be observed in Table 2 which is given in the same form iis ‘Table 1. Finally note that like the optimal designs with respect to the trace criterion, for the designs In Table 1, their complementary designs (obtained by an interchange of 0 and 1) are also the best alias with respect to the norm IlAll.
Table 2 N
nt = 5
4, A, A2 A, A, As
17 19 20 21 22 23a 23b 24 2Sa 25b 26 27 28 29a 2% 3i 32 t 23 28 38 39a 39b 4Oa -1CRl
2 0 210102 3 0 0 2 0 2 211011 1 2 1 2 311012 2 2 1 0 2 0 2 ,I) 3 i’) 9 ii 2 h, 2 C)
0 1 0 1 0
1
0
1
0
IlAII
1.oooo
2 1 3 1 3 1 .2 2 2 2 3 2 3 2 3 2 4 2 42113 1 2 2 2 2 2 3 2 3 2 2 2 2 2
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 2 3 1 2 2 2 3 3
2 2 2 2 2 2 2
1 1 1 1 1 2 2
1 1 2 2 2 Jl 2
0.9688 0.9296 0.9194 0.8438 0.8125 0.7980 0.7980 0.788 1 0.7815 0.7815 0.6875 0.6563 0.6300 0.6199 0.6199 0.5313 0.5000
CL,, Cal
EL2
EL4
tr(M-‘)
2 2 3 3 3 3 3
1 2 2 2 2 2 2
2 2 3 3 3 4 4
1.1242 0.9488 0.6088 0.5993 0.5993 0.5940 0.5940
1 0 0
0 1 1
1 0 0
2 1 2
0 0
1 1
0 0
2 3
0 2 2 2 2
1 0 0 1 61 1 01 1 01 1
.3 0 0 I 1
0.9734 0.9748 1.oooo 1.oooo 0.8164 0.9726 0.9734 0.8240 0.9733 1.OOOO 1.oooo ti.9709 0.97 17
01 2
2. 1 (1
21 0
0.6619 1.oooo 1.OOOO
4.0000 3.8922 3.8977 4.OOOO 4.OOdO 3.1819 3.8890 3.892 1 3.2202 3.8918 4.0000 4.0000 3.8817 3.8852 2.276 1 4.0000 4.0000
R
IIAII
2I 0
0I) 2
A, A,
A,
A,
A,
A,
0 1 1 1 2 3 2
(1 1 0 0 1 0 1
0 0 1 1 0 1 0
I t) 0 0 1 0 1
0 2 2 2 1 3 2
1 0 1 1 0 1 0
R
1.1597
0.9734 0.6265 0.6142 0.6847 0.6107 0.6910
7.1907 0.3582 4.2540 4.1604 4.6989 4.1795 4.7 142
tr( MI')
3
2 3 4 4 4 4
b,
B 1 2 2 2 2 2
Norm of alias matrices in 2,” factorial designs
345
The author wishes to thank the editor., J.N. Srivastava, for her valuable comments.
Chopra, D.V. (1975). Balanced optimal 2” fractional :factorial designs of resolution V, 52 G N c 59. In: J.N. Srivastava, ed., A supoey of statistical designs and linear models. North-Holland, Amsterdam, 91-100. Chopra, D.V. and J.N. Srivastava (1973). Optimal balanced 2’ fraction& factorial desigtis of resolution V, with N 642. Ann. Inst. Statist. Math. 25, 587-604. Hedayat, A., B.L. Raktoe and W.T. Federer (1974). On a measure of aliasing due to fitting an incomplete modei. Ann. Statist. 2, 650-660. Shirakura, T. (1976a). Optimal balanced fractional 2” factorial designs of resolution VII, 6 < m G 8. Ann. Statist. 4, 515-531. Shirakura, T. (1976’6). Balanced fractionat 2” factorial designs of even resolution obtained from balanced arrays of strength 21 with index &f = 0. Anni 9atist. 4, 723-735. Shirakura, T. (19%). A note on the norm of alias matrices in fractional replication. Austral. J. Statist. 18, 158-160. Shirakura, T. (1977). Contributions to balanced fractional 2” factorial designs derived from balanced arrays of strength 21 Hiroshima Math. J. 7, 217-2:35. Srivastava, J.N. (1970). Optimal balanced 2” fractional factorial designs. S.N. Roy memorial volume, Univ. of North Carolina, and Indian Statistical Institute. 689-706. Srivastava, J.N. (1975). Designs for searching non-negligible effects. In J.N. Srivastava, ed., A suruey of statistical designs and linear models. North-Holland, Amsterdam, 507-S 19. Srivastava, J.N. and D.V. Chopra (1971). Balanced optimal 2” fractional factorial designs of resolution V, m G 6. Technometrics 13, 257-269. Srivastava, J.N. and D.V. Chopra (1974). Balanced trace of*timal 2’ fractional factorial designs of resolution V, with 56 to 68 runs. Utilitas Muthenratics 5, 263-279. Srivastava, J.N. and S. Gosh (1977). Balanced 2” factorial designs of resolution V which allow search and estimation of one extra unknown effect, 4~ rn ~8. Cc;nmun. Statist. A6, 141-166. Yamamoto, S., T. Shirakura and M. Kuwada (197511.Balanced arrays of strength 21 and balanced fractional 2” factorial designs. Ann. Inst. Statist. Math. 27, 143-157. Yamamoto, S., T. Shirakura and M. Kuwada (1976). Characteristic polynomials of the information matrices of balanced fractional 2” factorial designs of higher (22+ 1) resolution. In S. Ikeda et al., eds., Essays in Probabilit)l and Statistics. Birthday volume in honor of Prof. Junjiro Ogawa. Shinko Tsusho Co. Ltd., Tokyo, 73-94.