- Email: [email protected]
Singular factorial designs resulting from missing pairs of design points
Journal of Statistical lanning and Inference 19 (1988)325-3
525
Received 7 November 1986;revised manuscript received 23 AugusF 1987 R~ommend~ by D. Raghav~rao
AIZsrsac: In indust~, ex~~ments are often ~ouduct~ ~ue~ti~ly due to ~ui~~e~t limitations dictatirkgthat only one or two simultaneous runs may be made. In this situation, earIy termination of the experiment results in missing points, leading to a loss in efficiency or, worse, to a singular subd~i~ with nonestimable model parameters. VUeinvestigate the specific problem of singularity when two points are lost from a factorial design based on n two-level factors. The method is based on the inner products of the coordinate vectors of the omitted design points and leads to some results on the nonexistence of fractioual factorial designs.
Key words: Factorial design; factor level chmge; 2” - ‘* fraction; design reso~~Fion~defining contrast subgroi~~.
L.M. Moore / Missing pairs of design points
326
factors obtained from loss of a foldover design point pair is resohttion II, i.e., singular for the first order model. Andrews and erzberg (1979) consider measures of efficient or robust designs based on a probability measure on subsets of -measurements that may be lost as a result of premature termination or outlier probm a loss in efficiency t llems. This, penaL,;ac may ... range in s pararheters are nones siug~~ar subdesi~u from which indivi this paper the problem of obtai~ng a singular subdesign if two points are lost from a 2”-’ factorial is investigated. In Section 2 tools are provided for exami effect of losing a pair of desigu points. The method is based on the inner of the coordinate actors of the omitted design points. In Section 3 and 4 respectivening an .appropriat~ he first and second order model are assured for design for p?two kvel factors, The procedures developed can be used to show that sore relatively maI1 designs cannot exist for certain values of ~1, Examples are dismissed in Section 5.
s
The notation
used follows that in John (~~~1). ower case letters, and upper case let 3t=.. ) A,}, are uses to represent {~1~~~~3~ l **9 a,), y, The ~~ef~~i~nt +I indi~tes the p1two-levee favors and main e ote8 the N Xp matrix of coeff& the high 1eveIof a factor and - 1 th cients for a speci~ed ~ point design an parameter model. The row v~ors of the design matrix, denoted by xi’or xi fo e i-th row, represent design points. is an o~hogoual factorial design for ap parameter model, If the design e retaining subdesig~. r~inaut of the non(1)
;,esult follows from the prope~ies of determinants and the fact that zberg (~~~~) use a more general insult to compute a n the determinant of 0 or t if the corresp or not. ricer
ore /
N-p
ng pairs of design points
is at most p are wnsidere
of design points is at least the nu that p=N=2’-‘*<2p is selected if one exists.
number of model parameters.
factors that are ch
(0
43
b
ab
c
ac bc ubc
321
L.M. Moore / M&singpa& of designpoints
328
he following theorem formulates the value of $x2 for the comple%zfirst cx second order models in terrrns of k(& xi) = kr, 2. Let xi and xi denote two factorial de&n points for n two-level factors r the first order model, p = n + 1 and =
1
l+nGk.
) For the second order model,
=l+n++n(n-1)
and
= l+n-2k+in(n-1).2k(n-k) = 2k2-%(n+ lijk+p.
(5)
The proofs follow from straightforward counting arguments. are n + I- k values of + 1 contributing to x; and k values of -1 so e conrtribution corresponding to the first order model portion of the second he two factor interaction terms of the second order model - 1 and the remaining (+n(n- 1) -k(n - k)) values are
Theorem 2 is easily extended to a complete t-th order model by counting roof. In this case, the number of arguments similar to t parameters in the model is
he inner product between two factorial design points for a t-th order model is ([ .] denotes the gregtest ir;,>trfikpicfio@
ovide the investigative ts~k for exa factorial design when a first o
ore /
ng pairs
of design
points
329
for design point pairs results are state
k is an integer in [p - + further, ifJ for so e pair of design p-#N, the upper or lower values on from omission of is singular for the first oramermoaeL It is clear that a
with either zero or on
om for error will
n two level factors e with the fact that a
columns define am orthogona!
Table 1 Znwr* fractions for fiist order models, b0UPldsOn k fOr design point pah, n -~~
N=2n-r+
4=22
24-Q
k=2 lSkS4
5
25-Q
2SkS4
6
26-3=r$
k=3or4
7
27-44
k--4
a=23
28-4,
23-l
ased on
=4
16
and exhrtpies
Examples
lSkS8
{I = ABC), a saturated design (i) (I = ABED) , singui~ if any foldover pair is lost, such QS(I) and abed. (ii) {I = ABC), singularif any pair such that
k= 1 is lost, such as a and ad. t=CDE), singularif abcde and abe are lost shce k= 2, or if abe and cde are lost since k =4; nonsingularfor k = 3, such as ac and e lost. (I=ABCD=ABEF=CQEF=ACE=BDE= BCF = ADF), a 6 point subdesign is sing&r. (I=ABCDEFG=ABC=DEFG=ADE= BCFG = BCQE = AFG = BDG = ACEF = ACDG=BEF=ABEG=CDF=CDE= ABDF~ , satur$t~. ~I=ABCD=E~GH=ABCDEFGH=ABEF= CDEF = ABGH = CDGH = ACFH = BDFH = ADEG = BDEG = BCEH = ADEH = BCFG = ADFG~, singuIarif any fo~dov~ pair is Iost.
argo~in (~9~9) and bb (~9~$), resolution IV factored des two level factors must con a mi~mum of n foldover design point pairs clear in this case that the orthogonal. John (1979) showed that loss of a design leaves a singufar foldover pair U from a minimal 2n point resolution er modes, or a resolution sign. This coincides ince from (?), for a N= oint orthogonal fact design, k is an integer in [I, .I?]. n this case, if k(x;,~$) = 1 or n then omission of oints leaves a singular subde attention to IF+ fractional riai designs for 3 s 92zs 8, Table f lists the size of the minimal fraction for a first order model and, from (7), t and lower airs. last co~um
ore / Missing pairs of design points
L.M
41
The
331
range of values of k =
Let J”(k) denote the quadratic determined by (5). n terms off, the sAtions to inequality (2) are determined from the quadratic inequality
(8)
-(N-p)sf(k)sN.-p.
TO solve this inequality, it is only necessary to look at the two e
f(k) = N-p
or, equivalently, k2- (n +
pnd
fW= -(N-p)
or, equivalently, kt - (n + I)k + +
The following theorem states the solutions to ine equations.
332
fraction. owever, as stated previously, N=2n”* is the smallest fraction size required to model the p = It+ n + ~~(~ - 1) par~eters of the second order model, so we examine the possibility that (11) might hold for some values of II. In this situation the roots, r3 and r4, of equation (10) are real and on truerange of ~(~~,~~).The fo~~owimg lemma specifies such that (11) holds.
for
e~~~iity (11) hoi~s if sore integer t,
the integer n, the number of
factors,
issuch that
J22’-l
(12a)
n=2’“‘-f*
(12b)
or , the inequality(11) is strict whifein case (12b) equality he&&in (11). . consider PI= 2’+s where 0 s ss 2’0 1; loea92’ is the largest power of two in n and s is the residue, Also n 22, so t 2 1e The smallest possible fraction size, N, is evaiuated in the foilswing two cases: case 1: if 2’~n
is a decreasing f~~nctionof n, IA case 2 where N = 22t’ II d(2’+l- 1)=0, so d(n) > 0, that is (11)’ does not hoId, for
owever, for case (12b), n = ;Zt+r- 1, the situation is such t as stated, 2n--r’case 1 where N==
LX
Moore / Mihing pairs of design points
333
The range (12a) is narrow, of 1 es of t for which an integer value, n, falls in this range. s of 8 cOmputt%izedsearch for the firsi: few values in the sequence of integers rr that satisfy (12a) for seme integer t are listed in Table 2. For values oft <: 15 e is table, the range (12a) containe Thus, for values of n < from the table, there is no int (12a) is satisfied and, therefore, (11) does not hold as well. his indicates that case of Theorem 4 generally holds except for the few v s of n listed in the table. ere is a continuing sequence of vaiues for t > I5 for tains an integer and thus a continuing sequence of values of n > ) of Theorem 4 is true, but, as is evident even from the value t = 12 with corresponding value of n = 5792, the corresponding numbers of two level factors considered, by virtue of their magnitude, are of little practical interest. In fact, PI= 90 two level factors is of marginal interest for modeling the p =4!X%=2W-78 parameters of the second order model; however, if inequality (8) corresponding to ic “__ cnlved .n. - QO I_ *Y - _-; then the- range -----Y- of -- values of k is seen to be null. This indicates that a resolution V orthogonal 2RI-‘* fraction does not exist. Further, in Section 5 below, it will be demonstrated that a minimal resolution V orthogonal fraction does not exist for n =22. ) is an integer Theorem 4 indicates that the range of possible values for k in [Q,Qi -whereQ yIIu , finA r2 are mats of equation (9): j(k) = N - p that .k(x;,xi) is an integer in {1,2,3, . . . , n} so it is of interest whether or not provides further restriction on the range of possible values of k f (0) = f (n + 1) = p > N-p, it is apparent that the integers in [rr, rz] {1,53,...#). In fact it can be shown that the roots rl and r2 lie strictly within (1, n) except for values of n such that, for some integer t, either ii is in (J222t-+j4+22t+i-~)++)~
(I 3a)
n =2*.
(
or
Table 2 Values of n such that (10) has distinct real roots, 33and 1-4,as determinedby t for which Mis in the range (12a) t
L.M. Moore / Missing pairs of design points
ike the range (Da), the range (Ha) is narrow with length less than 3. or any integer in the ran (Isa), the set of roots p1an is known apriori that k(x;, the solution range [l,n]. for these select values of n there is little benefit in consi heorem 4, on the possible val
r n=2’
than the set of int
roots q and r2 (and likewise the possible values of in an orthogonal 2;1-‘* tionally, the number of parameters in the second order model (p), the minim Table 3 The size IV= Par* of a minimal fraction for modeling the p parameters of the second order model based on n two-level factors, n s 25, and the possible values of k(xi,xijS the number of level changes between a pair of design points in such a design, if oqe exists n
P
4 5 6 7 8 9 10 11 12 13 14 15 16 17 10
11 16 22 29 37 46 56 67 79 92 106 121 137 154 1-* LIL 191 211 232 254 277 301 *e&f 3&W
;; 20 21 22 23 25
N=2n-r+
error d.f. N-P
16,24-o l&2”-’ 32=26-l 32=27-2a 64=2*-2
5 0 10
a4=29-3 ~~2’0-4 128-2”
-4
*2*,212-5 128 = 2’3-6 i28~2’~-’ 128=2’5-*
256=216-* 25#j=p7-9 25~-2’*-
10
256~219’‘1 256 = 2*O-
I2
25fj=2”+‘3 25Q‘-222512&3-14 51&&4-‘5 515!&“-‘6
14 a
3 27 18 8 61 49 36 22 7 119 102 -&I 65 45 24 2 235 211 186
&,2
range
{L23,4} 2 or 4 (1,23,...,6) (3,4,5) {1,23,...,8} {53,4, . ...8) {3,4,5,...,8) (h33,w 11) (53,4,..., 11) {3,4,5 ,..., 11) (4,5,6+, 11) (67,89,10) (1,53,..., 16) {53,4, l=., 16) (3,4,5,..., 16) *[4,5,6,..., 16) (6,7,8,-0.9 15) {7,8,9 ,..., 15) 9 or 14 (A23,...,23)
L.M. Moore / A&sing pairs of designpoints
number of design points in the fraction (N=Y-‘*), dom (N-p) are listed.
335
and the error degrees of free-
4.2 Values of k such that we now investigate the existence of integer solutions, rl, r2, r3, and r4, to the quadratic equations (9) and (IO). These are of interest because k(lse;, ily integer valued and if a pair of design points or 4, solving (9) or (10) is almitted from the desi maining subdesign, Xr , is singular. Solutions to (9) or (10) are integer valued if the discriminant of the respective equation is 8 s~-m-wl intmmar . Thaf is! tbc soktions t-r and pi to equation w_uurru AUa+ya.. * uL .-, (9) are integer-valued if and only if B = 2N- n2 - 3 is a squared integer.
(14)
Equation (10) has integer solutions, r3 and r4, if and only if D = (n + 1)2- 2N is a squared integer.
(15)
Again, limiting consideration specifically to orthogon 2”-‘” fractional factorial designs for a complete second order model, we will first consider those values of n for which r3 and r4 exist and are integer-valued. From Lemma 1s the roots rS and pj of (10) are real if and only if, for some integer t, n is an integer in (12a), which is a subcase of case 1 in which iV=22r, or R =2’+’ so that N=22r+1. If n =21+1 then I3 = 0 and r3 = p4=&z + 1) which is an integer solution of (10). From Lemma 1, iP n is in the interval (12a), (d2 2’ - I, d2 2’- +), for some integer t, then r3 and r4 are distinct real roots of equation (10). It can be shown that distinct integer roots r3 and r4 occur only for n =2 and n = 5. ror n =2, p = 4 = a2 = N so, as expected, no less t the complete 22 factorial is required for the second order model and k(xi, xi) = or 2 for all design point pairs. For n = 5 etermines two level factors, p = 16= iv= 25-1, and as an example { saturated orthogonal fraction for modeling a mean, dn effects, and all two factor interactions. It is immediately clear that loss of a pair of design points would result in a sing&*: subdesign, but this result is so evident since equations (9) an (10) are identical with roots k = 2 or 4 have for a design point pair in a saturate model.
L.M. Moore / Missing pairs of design points
336
an orthogonal A/= 2”’ point resolution V fraction exists then any N- 2 point subdesign is still resolution ecial case 2: ro= 2’+ ’ - 1. his is also the case in which t rding to the above discu r n = 3 two level factors, the corn eters of the second order that loss of any pair of design points results in a singular design for the second order model. r other values of n inspection of the discriminant value ing theorem sum arizes the cases when omission of a pa pair from a resolution V orthogonal fraction leaves a singular subdesign. if t=l,
so that n=3.
be an orthogonal N= 2n-rQpoint fractional “factorialdesign for a second order model based on n two-!evel factors. (a) Ifn= 2, 3 or 5 then any N- 2 point subdesign is singular. it, xi and Xi, omitted from the =2c9’-1, tz2, then a leaves a singular su (c) if, for nrr6 and n?t2’+‘--1, a squared integer then a ir, x; and xi, ves a singular subdesign, ) =rl or r2 where rl and r2 are the integer roots of equation (9), f(k)=N-p. otherwise, an N- 2 point subdesign of is nonsingular for the second order model. .
The results of a direct search of those cases fo er so that the roots rl and r2 are inte those special cases described above where n = 2’+ ’ - 1 for 5r: 2 where rl and r2 are not in teger are listed for n s 1 n addition to comparable infor mation to tk: ble 4 lists the integer values of k, being rlr r2, r3
ing section.
ore /
ng pairs of design points
337
Table 4 The values of n I 100 for which a 2”-‘* orthogonal resolution V fraction, if one exists, is singular if a particular pair of design points is omitted P
n
N=p-r+
error d.f.
kt.2 range
M-P 2 3 5 6 7 10 15 22 30 31 37 43 58 63 91
4 7 16 22 29 56 121 254 466 497 704 947 1712 2017 4187
4=22 8=23 16=2s-’ 32=26-r 32=27-2a 64=2’0-4 128,2’s-8 256=222-14a 512,230-21 512=231-u 1()24=237-27 1024=243-33 2@gj=258-4’ 2048 = 263 - 52 8192=29’-‘*
If klv2= is lost then subdesign is singuiar l
0
WI
k=l
1 0 10 3 8 7 2 46 15 320 77 336 31
(1,531 2 or 4 (1,53,...,6) {3,4,5) {3,4,5,...,8} {6,7,&9,10) 9 or 14 (10~11,1~...,21) {13,14,15, . . . . 19) (6,7,&...,32) 515,16,17, . ...29) (16,17,18,...,43} (27,2&29 ,..., 37) (1,911
k= 1, 2 or 3
5
or2
k=2 or 4 k=l or6 k=4 k=3 k=$ k=9
or 8 or 14
k= 10 or 21 k=l6 k=6 or 32 k= 15 or 29
k=16 or 43 k=32 k=l or 91
a A 27-2 orthogonal fractional factorial design does not exist, and nonexistence of a 222- l4 fraction will be demonstrated in Section 5.
L.M. Moore / Miming pairs of design points
338
to the group structure. or n = 7 and n = 10, such constraint n elements of a defining contrast subgroup can be used to show that a resolution fraction cannot exist. For example, for n = 7 a resolution V quarter of a 2’ factorial must be g contrast subgroup
are interaction effects of length five or more. owever, in this case cessarily of iength less than fi that the resolution V pro ne of argument does not w r n = 22, but the following exploits the restrictions on k(lc;,xi) listed 1~4 to show that a minim tion does not exist for n =22 two level factors. or n =22 two level factors, a second order model contains p =254 6 = 222-14 design points are required for om Table 4, k=9 or 14 for every pair of ; and xi in a resolution V 222- I4 fraction, if one exists. This result can be used to show that a resolution V 222-14 fraction does not exist as follows. Assume a resolution V 222-*4 fraction exists. Then, there is a defining contrast subgroup which determines ZIfraction of treatment combinations which form a of the group of 222treatment combinat ere are 256 = 222- l4 g the treatment combination (1). other than (l), must be such that plies that the string length ’ is nine or fourteen for than (1). Say t elements have a string length o string length of fourteen. Then, the cumulative 9r c 14(255- t) = 3570 - 9. so, every letter must appear in ex half the treatment combinations in 110~ssince the set of elements o hat do not include the letter a a), is a subgroup of that is distinct from . It is necessary that so that the main effect A is estimable. The remaining elements of all contain the Thus, each of the 22 let -14-’ of the elements o 22- 14- 1 his implies that the cumulati 1.
et
fb
ss
ore /
ng pairs of design points
ges
between factorial design
er of factors, it is
is va
co
ts
a
3. am Uvi. Herzberg (1979). The robustness and optimal&y of response surface designs.
ctors at two levels
P-4 plans robust against linear an
340
L.M. Moore / Missingpairs of design points
John, P.W.M. (1966). On identity relationships for 2n-r designs having words of equal length. Ann. Math. Statist. 37, 1842-1843. John, P.W.M. (1971). Statistical Design and Analysis of Experiments. Macmillan, New York. John, P.W.M. (1979). Missing points in 2” and 2n-k factorial designs. Rchnometrics 2 = fiw*hof experimental design in engineering. Technical Report # 9, Center J&n, P.W.M. (1984).Th,p ,r,.... for Statistical Sciences, University of Texas, Austin, TX. Margolin, B.W. (1969). Orthogonal main effects plans permitting eStimi3tiOn Of all tWO-factor S for the 2n3m factorial series of designs. Technometrics , 747-762. (1985). Ordering the Points in Pactorial Experiments to Protect Against Early Termination. Ph.D. Dissertation, The University of Texas, Austin, TX. Plackett, R.L. and J.P. Burman (1946).The design of optimum mn!tifactorial experiments. Biometrika 33, 305-325. Smith, D. and D.D. Schmoyer (1982). First-order ‘interruptible’ designs. 55-58. Steinberg, .M. and WG. Hunter (!984). Experimental design: Review and metrics , 71-97. W&l?i SR !1968), Non-orthogonal designs of even resolution. Technometrics
525
Received 7 November 1986;revised manuscript received 23 AugusF 1987 R~ommend~ by D. Raghav~rao
AIZsrsac: In indust~, ex~~ments are often ~ouduct~ ~ue~ti~ly due to ~ui~~e~t limitations dictatirkgthat only one or two simultaneous runs may be made. In this situation, earIy termination of the experiment results in missing points, leading to a loss in efficiency or, worse, to a singular subd~i~ with nonestimable model parameters. VUeinvestigate the specific problem of singularity when two points are lost from a factorial design based on n two-level factors. The method is based on the inner products of the coordinate vectors of the omitted design points and leads to some results on the nonexistence of fractioual factorial designs.
Key words: Factorial design; factor level chmge; 2” - ‘* fraction; design reso~~Fion~defining contrast subgroi~~.
L.M. Moore / Missing pairs of design points
326
factors obtained from loss of a foldover design point pair is resohttion II, i.e., singular for the first order model. Andrews and erzberg (1979) consider measures of efficient or robust designs based on a probability measure on subsets of -measurements that may be lost as a result of premature termination or outlier probm a loss in efficiency t llems. This, penaL,;ac may ... range in s pararheters are nones siug~~ar subdesi~u from which indivi this paper the problem of obtai~ng a singular subdesign if two points are lost from a 2”-’ factorial is investigated. In Section 2 tools are provided for exami effect of losing a pair of desigu points. The method is based on the inner of the coordinate actors of the omitted design points. In Section 3 and 4 respectivening an .appropriat~ he first and second order model are assured for design for p?two kvel factors, The procedures developed can be used to show that sore relatively maI1 designs cannot exist for certain values of ~1, Examples are dismissed in Section 5.
s
The notation
used follows that in John (~~~1). ower case letters, and upper case let 3t=.. ) A,}, are uses to represent {~1~~~~3~ l **9 a,), y, The ~~ef~~i~nt +I indi~tes the p1two-levee favors and main e ote8 the N Xp matrix of coeff& the high 1eveIof a factor and - 1 th cients for a speci~ed ~ point design an parameter model. The row v~ors of the design matrix, denoted by xi’or xi fo e i-th row, represent design points. is an o~hogoual factorial design for ap parameter model, If the design e retaining subdesig~. r~inaut of the non(1)
;,esult follows from the prope~ies of determinants and the fact that zberg (~~~~) use a more general insult to compute a n the determinant of 0 or t if the corresp or not. ricer
ore /
N-p
ng pairs of design points
is at most p are wnsidere
of design points is at least the nu that p=N=2’-‘*<2p is selected if one exists.
number of model parameters.
factors that are ch
(0
43
b
ab
c
ac bc ubc
321
L.M. Moore / M&singpa& of designpoints
328
he following theorem formulates the value of $x2 for the comple%zfirst cx second order models in terrrns of k(& xi) = kr, 2. Let xi and xi denote two factorial de&n points for n two-level factors r the first order model, p = n + 1 and =
1
l+nGk.
) For the second order model,
=l+n++n(n-1)
and
= l+n-2k+in(n-1).2k(n-k) = 2k2-%(n+ lijk+p.
(5)
The proofs follow from straightforward counting arguments. are n + I- k values of + 1 contributing to x; and k values of -1 so e conrtribution corresponding to the first order model portion of the second he two factor interaction terms of the second order model - 1 and the remaining (+n(n- 1) -k(n - k)) values are
Theorem 2 is easily extended to a complete t-th order model by counting roof. In this case, the number of arguments similar to t parameters in the model is
he inner product between two factorial design points for a t-th order model is ([ .] denotes the gregtest ir;,>trfikpicfio@
ovide the investigative ts~k for exa factorial design when a first o
ore /
ng pairs
of design
points
329
for design point pairs results are state
k is an integer in [p - + further, ifJ for so e pair of design p-#N, the upper or lower values on from omission of is singular for the first oramermoaeL It is clear that a
with either zero or on
om for error will
n two level factors e with the fact that a
columns define am orthogona!
Table 1 Znwr* fractions for fiist order models, b0UPldsOn k fOr design point pah, n -~~
N=2n-r+
4=22
24-Q
k=2 lSkS4
5
25-Q
2SkS4
6
26-3=r$
k=3or4
7
27-44
k--4
a=23
28-4,
23-l
ased on
=4
16
and exhrtpies
Examples
lSkS8
{I = ABC), a saturated design (i) (I = ABED) , singui~ if any foldover pair is lost, such QS(I) and abed. (ii) {I = ABC), singularif any pair such that
k= 1 is lost, such as a and ad. t=CDE), singularif abcde and abe are lost shce k= 2, or if abe and cde are lost since k =4; nonsingularfor k = 3, such as ac and e lost. (I=ABCD=ABEF=CQEF=ACE=BDE= BCF = ADF), a 6 point subdesign is sing&r. (I=ABCDEFG=ABC=DEFG=ADE= BCFG = BCQE = AFG = BDG = ACEF = ACDG=BEF=ABEG=CDF=CDE= ABDF~ , satur$t~. ~I=ABCD=E~GH=ABCDEFGH=ABEF= CDEF = ABGH = CDGH = ACFH = BDFH = ADEG = BDEG = BCEH = ADEH = BCFG = ADFG~, singuIarif any fo~dov~ pair is Iost.
argo~in (~9~9) and bb (~9~$), resolution IV factored des two level factors must con a mi~mum of n foldover design point pairs clear in this case that the orthogonal. John (1979) showed that loss of a design leaves a singufar foldover pair U from a minimal 2n point resolution er modes, or a resolution sign. This coincides ince from (?), for a N= oint orthogonal fact design, k is an integer in [I, .I?]. n this case, if k(x;,~$) = 1 or n then omission of oints leaves a singular subde attention to IF+ fractional riai designs for 3 s 92zs 8, Table f lists the size of the minimal fraction for a first order model and, from (7), t and lower airs. last co~um
ore / Missing pairs of design points
L.M
41
The
331
range of values of k =
Let J”(k) denote the quadratic determined by (5). n terms off, the sAtions to inequality (2) are determined from the quadratic inequality
(8)
-(N-p)sf(k)sN.-p.
TO solve this inequality, it is only necessary to look at the two e
f(k) = N-p
or, equivalently, k2- (n +
pnd
fW= -(N-p)
or, equivalently, kt - (n + I)k + +
The following theorem states the solutions to ine equations.
332
fraction. owever, as stated previously, N=2n”* is the smallest fraction size required to model the p = It+ n + ~~(~ - 1) par~eters of the second order model, so we examine the possibility that (11) might hold for some values of II. In this situation the roots, r3 and r4, of equation (10) are real and on truerange of ~(~~,~~).The fo~~owimg lemma specifies such that (11) holds.
for
e~~~iity (11) hoi~s if sore integer t,
the integer n, the number of
factors,
issuch that
J22’-l
(12a)
n=2’“‘-f*
(12b)
or , the inequality(11) is strict whifein case (12b) equality he&&in (11). . consider PI= 2’+s where 0 s ss 2’0 1; loea92’ is the largest power of two in n and s is the residue, Also n 22, so t 2 1e The smallest possible fraction size, N, is evaiuated in the foilswing two cases: case 1: if 2’~n
is a decreasing f~~nctionof n, IA case 2 where N = 22t’ II d(2’+l- 1)=0, so d(n) > 0, that is (11)’ does not hoId, for
owever, for case (12b), n = ;Zt+r- 1, the situation is such t as stated, 2n--r’case 1 where N==
LX
Moore / Mihing pairs of design points
333
The range (12a) is narrow, of 1 es of t for which an integer value, n, falls in this range. s of 8 cOmputt%izedsearch for the firsi: few values in the sequence of integers rr that satisfy (12a) for seme integer t are listed in Table 2. For values oft <: 15 e is table, the range (12a) containe Thus, for values of n < from the table, there is no int (12a) is satisfied and, therefore, (11) does not hold as well. his indicates that case of Theorem 4 generally holds except for the few v s of n listed in the table. ere is a continuing sequence of vaiues for t > I5 for tains an integer and thus a continuing sequence of values of n > ) of Theorem 4 is true, but, as is evident even from the value t = 12 with corresponding value of n = 5792, the corresponding numbers of two level factors considered, by virtue of their magnitude, are of little practical interest. In fact, PI= 90 two level factors is of marginal interest for modeling the p =4!X%=2W-78 parameters of the second order model; however, if inequality (8) corresponding to ic “__ cnlved .n. - QO I_ *Y - _-; then the- range -----Y- of -- values of k is seen to be null. This indicates that a resolution V orthogonal 2RI-‘* fraction does not exist. Further, in Section 5 below, it will be demonstrated that a minimal resolution V orthogonal fraction does not exist for n =22. ) is an integer Theorem 4 indicates that the range of possible values for k in [Q,Qi -whereQ yIIu , finA r2 are mats of equation (9): j(k) = N - p that .k(x;,xi) is an integer in {1,2,3, . . . , n} so it is of interest whether or not provides further restriction on the range of possible values of k f (0) = f (n + 1) = p > N-p, it is apparent that the integers in [rr, rz] {1,53,...#). In fact it can be shown that the roots rl and r2 lie strictly within (1, n) except for values of n such that, for some integer t, either ii is in (J222t-+j4+22t+i-~)++)~
(I 3a)
n =2*.
(
or
Table 2 Values of n such that (10) has distinct real roots, 33and 1-4,as determinedby t for which Mis in the range (12a) t
L.M. Moore / Missing pairs of design points
ike the range (Da), the range (Ha) is narrow with length less than 3. or any integer in the ran (Isa), the set of roots p1an is known apriori that k(x;, the solution range [l,n]. for these select values of n there is little benefit in consi heorem 4, on the possible val
r n=2’
than the set of int
roots q and r2 (and likewise the possible values of in an orthogonal 2;1-‘* tionally, the number of parameters in the second order model (p), the minim Table 3 The size IV= Par* of a minimal fraction for modeling the p parameters of the second order model based on n two-level factors, n s 25, and the possible values of k(xi,xijS the number of level changes between a pair of design points in such a design, if oqe exists n
P
4 5 6 7 8 9 10 11 12 13 14 15 16 17 10
11 16 22 29 37 46 56 67 79 92 106 121 137 154 1-* LIL 191 211 232 254 277 301 *e&f 3&W
;; 20 21 22 23 25
N=2n-r+
error d.f. N-P
16,24-o l&2”-’ 32=26-l 32=27-2a 64=2*-2
5 0 10
a4=29-3 ~~2’0-4 128-2”
-4
*2*,212-5 128 = 2’3-6 i28~2’~-’ 128=2’5-*
256=216-* 25#j=p7-9 25~-2’*-
10
256~219’‘1 256 = 2*O-
I2
25fj=2”+‘3 25Q‘-222512&3-14 51&&4-‘5 515!&“-‘6
14 a
3 27 18 8 61 49 36 22 7 119 102 -&I 65 45 24 2 235 211 186
&,2
range
{L23,4} 2 or 4 (1,23,...,6) (3,4,5) {1,23,...,8} {53,4, . ...8) {3,4,5,...,8) (h33,w 11) (53,4,..., 11) {3,4,5 ,..., 11) (4,5,6+, 11) (67,89,10) (1,53,..., 16) {53,4, l=., 16) (3,4,5,..., 16) *[4,5,6,..., 16) (6,7,8,-0.9 15) {7,8,9 ,..., 15) 9 or 14 (A23,...,23)
L.M. Moore / A&sing pairs of designpoints
number of design points in the fraction (N=Y-‘*), dom (N-p) are listed.
335
and the error degrees of free-
4.2 Values of k such that we now investigate the existence of integer solutions, rl, r2, r3, and r4, to the quadratic equations (9) and (IO). These are of interest because k(lse;, ily integer valued and if a pair of design points or 4, solving (9) or (10) is almitted from the desi maining subdesign, Xr , is singular. Solutions to (9) or (10) are integer valued if the discriminant of the respective equation is 8 s~-m-wl intmmar . Thaf is! tbc soktions t-r and pi to equation w_uurru AUa+ya.. * uL .-, (9) are integer-valued if and only if B = 2N- n2 - 3 is a squared integer.
(14)
Equation (10) has integer solutions, r3 and r4, if and only if D = (n + 1)2- 2N is a squared integer.
(15)
Again, limiting consideration specifically to orthogon 2”-‘” fractional factorial designs for a complete second order model, we will first consider those values of n for which r3 and r4 exist and are integer-valued. From Lemma 1s the roots rS and pj of (10) are real if and only if, for some integer t, n is an integer in (12a), which is a subcase of case 1 in which iV=22r, or R =2’+’ so that N=22r+1. If n =21+1 then I3 = 0 and r3 = p4=&z + 1) which is an integer solution of (10). From Lemma 1, iP n is in the interval (12a), (d2 2’ - I, d2 2’- +), for some integer t, then r3 and r4 are distinct real roots of equation (10). It can be shown that distinct integer roots r3 and r4 occur only for n =2 and n = 5. ror n =2, p = 4 = a2 = N so, as expected, no less t the complete 22 factorial is required for the second order model and k(xi, xi) = or 2 for all design point pairs. For n = 5 etermines two level factors, p = 16= iv= 25-1, and as an example { saturated orthogonal fraction for modeling a mean, dn effects, and all two factor interactions. It is immediately clear that loss of a pair of design points would result in a sing&*: subdesign, but this result is so evident since equations (9) an (10) are identical with roots k = 2 or 4 have for a design point pair in a saturate model.
L.M. Moore / Missing pairs of design points
336
an orthogonal A/= 2”’ point resolution V fraction exists then any N- 2 point subdesign is still resolution ecial case 2: ro= 2’+ ’ - 1. his is also the case in which t rding to the above discu r n = 3 two level factors, the corn eters of the second order that loss of any pair of design points results in a singular design for the second order model. r other values of n inspection of the discriminant value ing theorem sum arizes the cases when omission of a pa pair from a resolution V orthogonal fraction leaves a singular subdesign. if t=l,
so that n=3.
be an orthogonal N= 2n-rQpoint fractional “factorialdesign for a second order model based on n two-!evel factors. (a) Ifn= 2, 3 or 5 then any N- 2 point subdesign is singular. it, xi and Xi, omitted from the =2c9’-1, tz2, then a leaves a singular su (c) if, for nrr6 and n?t2’+‘--1, a squared integer then a ir, x; and xi, ves a singular subdesign, ) =rl or r2 where rl and r2 are the integer roots of equation (9), f(k)=N-p. otherwise, an N- 2 point subdesign of is nonsingular for the second order model. .
The results of a direct search of those cases fo er so that the roots rl and r2 are inte those special cases described above where n = 2’+ ’ - 1 for 5r: 2 where rl and r2 are not in teger are listed for n s 1 n addition to comparable infor mation to tk: ble 4 lists the integer values of k, being rlr r2, r3
ing section.
ore /
ng pairs of design points
337
Table 4 The values of n I 100 for which a 2”-‘* orthogonal resolution V fraction, if one exists, is singular if a particular pair of design points is omitted P
n
N=p-r+
error d.f.
kt.2 range
M-P 2 3 5 6 7 10 15 22 30 31 37 43 58 63 91
4 7 16 22 29 56 121 254 466 497 704 947 1712 2017 4187
4=22 8=23 16=2s-’ 32=26-r 32=27-2a 64=2’0-4 128,2’s-8 256=222-14a 512,230-21 512=231-u 1()24=237-27 1024=243-33 2@gj=258-4’ 2048 = 263 - 52 8192=29’-‘*
If klv2= is lost then subdesign is singuiar l
0
WI
k=l
1 0 10 3 8 7 2 46 15 320 77 336 31
(1,531 2 or 4 (1,53,...,6) {3,4,5) {3,4,5,...,8} {6,7,&9,10) 9 or 14 (10~11,1~...,21) {13,14,15, . . . . 19) (6,7,&...,32) 515,16,17, . ...29) (16,17,18,...,43} (27,2&29 ,..., 37) (1,911
k= 1, 2 or 3
5
or2
k=2 or 4 k=l or6 k=4 k=3 k=$ k=9
or 8 or 14
k= 10 or 21 k=l6 k=6 or 32 k= 15 or 29
k=16 or 43 k=32 k=l or 91
a A 27-2 orthogonal fractional factorial design does not exist, and nonexistence of a 222- l4 fraction will be demonstrated in Section 5.
L.M. Moore / Miming pairs of design points
338
to the group structure. or n = 7 and n = 10, such constraint n elements of a defining contrast subgroup can be used to show that a resolution fraction cannot exist. For example, for n = 7 a resolution V quarter of a 2’ factorial must be g contrast subgroup
are interaction effects of length five or more. owever, in this case cessarily of iength less than fi that the resolution V pro ne of argument does not w r n = 22, but the following exploits the restrictions on k(lc;,xi) listed 1~4 to show that a minim tion does not exist for n =22 two level factors. or n =22 two level factors, a second order model contains p =254 6 = 222-14 design points are required for om Table 4, k=9 or 14 for every pair of ; and xi in a resolution V 222- I4 fraction, if one exists. This result can be used to show that a resolution V 222-14 fraction does not exist as follows. Assume a resolution V 222-*4 fraction exists. Then, there is a defining contrast subgroup which determines ZIfraction of treatment combinations which form a of the group of 222treatment combinat ere are 256 = 222- l4 g the treatment combination (1). other than (l), must be such that plies that the string length ’ is nine or fourteen for than (1). Say t elements have a string length o string length of fourteen. Then, the cumulative 9r c 14(255- t) = 3570 - 9. so, every letter must appear in ex half the treatment combinations in 110~ssince the set of elements o hat do not include the letter a a), is a subgroup of that is distinct from . It is necessary that so that the main effect A is estimable. The remaining elements of all contain the Thus, each of the 22 let -14-’ of the elements o 22- 14- 1 his implies that the cumulati 1.
et
fb
ss
ore /
ng pairs of design points
ges
between factorial design
er of factors, it is
is va
co
ts
a
3. am Uvi. Herzberg (1979). The robustness and optimal&y of response surface designs.
ctors at two levels
P-4 plans robust against linear an
340
L.M. Moore / Missingpairs of design points
John, P.W.M. (1966). On identity relationships for 2n-r designs having words of equal length. Ann. Math. Statist. 37, 1842-1843. John, P.W.M. (1971). Statistical Design and Analysis of Experiments. Macmillan, New York. John, P.W.M. (1979). Missing points in 2” and 2n-k factorial designs. Rchnometrics 2 = fiw*hof experimental design in engineering. Technical Report # 9, Center J&n, P.W.M. (1984).Th,p ,r,.... for Statistical Sciences, University of Texas, Austin, TX. Margolin, B.W. (1969). Orthogonal main effects plans permitting eStimi3tiOn Of all tWO-factor S for the 2n3m factorial series of designs. Technometrics , 747-762. (1985). Ordering the Points in Pactorial Experiments to Protect Against Early Termination. Ph.D. Dissertation, The University of Texas, Austin, TX. Plackett, R.L. and J.P. Burman (1946).The design of optimum mn!tifactorial experiments. Biometrika 33, 305-325. Smith, D. and D.D. Schmoyer (1982). First-order ‘interruptible’ designs. 55-58. Steinberg, .M. and WG. Hunter (!984). Experimental design: Review and metrics , 71-97. W&l?i SR !1968), Non-orthogonal designs of even resolution. Technometrics