INTEHACl'lON p,;t'j TU FTJI,CTI,,_ ':
Institute of Stati6ti~~
Mimeograph Series No. 92~
.June, 197)+
PRO'I'ECTING MAIN EFFECT MODELS AGAINS'l' 'IT'wO-F'ACTOR
INrrEHACTION BIAS IN F'RAf';TIONS OF'
2
l'
FACTORIAL DESI G.NS
.j(.
(;.
y.,
.'
,/'1..•
i~.~
~18..nC:iOL
an~l
R. J.
l"1on:~Joe
Department of Statistics, Nortb Carolina State Univel'Lli ty, Raleigh, North
Carolina
-2-
PJ3S 'mAC l'
Minimum bias estimation is applied to the classical
situatioI"l where tbe factors arc quali taUve.
2
r
experimenta~
A true experimental
design
Illode:~
for a
.
factorlal design which consists of the mean, main effects, and two-factor
interactions is approximated by a function containing only the estimated mearJ apd
the estimated main effects.
cri tex·la.
Ell'e
For this true model and approximating function,
developed which indicate \vhen regular and irregular fractional
factorial designs aLloVl the bias dl,J.e to the presence of two-factor irte.ractiom:
to
bf;
m:i
!,~mj;7,ccL
relatior~\'lLile
Regular J'racti on;; Q...t'c con;3tructed by specifyi ng an identi tv
irregular fractions are constructed by combining regular fractions.
For reguLar fractions to allo',·' tbe existence of the minimum bias estimator, it is
necessary and sufficient that they be resolution IV designs with at least 21'
design points.
Resolution:j:V regUlar fractions satisfy the Box-Draper condi bar,
in wbich case the minimum bias estimator becomes the ordinary least sqUa.l;es
estimator'.
Irregular fractions tha.t allow minimum bias estimators may be
constructed by caliliining regular fractions whicb have resolution less than IV.
IrregUlar fractions that: are constructed in this manner are cataloged foro) (;?
and 3(2
r-j·
) designs vThen r
=
3,
L},
5, and 6.
r-2,
For seven or more factors,
irregUlar fractions for 3 (21'-2) and 3 (21'-3) designs that allo-\'i" nti nimUlfi ill <1;,
estimators and satisfy the BOX-Draper condi tjOtl, ean be constructed by comtli [ling
,
.
., 1'-2
. ,,1'-3
the resolution IV regular f.ractlons 0:t:2
and c:
designs.
)
-31,
When a true model;
11 '"
of experimentation (~1:~2)
x
't3
- l- 1
E:
INTRODUCTION
+ x/B.'. ts approxJmated by"
-2-0:::
y
=
x'b
-1-1
over a region
H, discrepancies between the approximate and true
from both the sampling error (~.~., variance error), and the i n-
response stem
adequacy of the approximating function to represent exactly the true response
model (:!:.. ~., bias error).
In
1959, Box and Draper, (ED L [3 J considc;red both
the variance error and the bias error by stUdying designs that rninirnj.ze the mean
square error (MSE) of
the bus term
:y
dOJUinate~:
avr~raged over thce: region E.
Becau:,;: the (·0'1t.cibutlon of
tilE, contribution of the varj_ance tc'rm in the average £vISE,
Box and Draper proposed that an experimenter (1) estimate
e.l
squares methods (OL8), and (2) find an experimental design R-l(the -bias term.
T'
_f .X*
= (-l(-.
X :X*)
1 2
by ordinal'y least
C
R which minimizes
. a matrix of the n-des1gn
.
f' * a
18
poin t s oR,
necessary and sufficient condi Lion for a design to minimize the bias L"r-m
18
the
-1
-L
BD condi ti on: H I-I
=, M M
T.<There the regi onal moments are II. . =: Or x.x~dx
lJ
JR-l-Jll 12
ll 12
with 0-
1
""
SRd~
and the design moments are M
ij
-1
-J(._)t
n ~! X. for i
;L
J
and j = 1, 2.
and Hader, (KMH), [81~lere able to find smaller
average MSE tl;J.an is obtained using the BD method by proposing that an experimenter (1) estimate ~l so as to minimizes the bi as term, and (2) find an
experimental design R* which minimizes the variance term.
A necessary and
sufficient condi tioll for the existence of a IIlinimum bias estimator (l\ffiE) is that
the experi.mental design R* allow estimation of the linear combiuation of paraThe KIvlH method has been applied to the
polynomial model [8J, ttle exponential model [121, the mixtu:c(,; problern [llJ, and
rational functions
[4J.
In each of' these applJcations, the apI'l'oximating
function is a low order polynomial wi th variables define:iovcl' a continuous space.
'1'h1s :paper is concerned with applying the;; KMH mEothod to the claDsical
experimental design si tuation where the rae tors each have a fin:i. te flu.enbEr of
··4levels.
For this situation, the region of experimentation R consists of a finite
number of lattice points.
DERIVATION OF THE ESTIMA'L'OR
2.
Let the region of experimentati on R be the set of points representing all
combinations of the levels of the experimental variables.
points.
'Tl(x )J/.
X
'-1
'I -Ill
[
1
'
1
x'l.x'
l .-'l
J'
2 '
.
= 1,
l
,- [ ~.l~~J .
a matri x of' fuJ.l col1..un:--l r3.nk IJ'. X
.. o.~.~
--
p X 1 vector of
pa.J.·muetel'~'; ~2
Lj a
1<;:
..
"There
..... ,
2,
,
to
~,
Xl~l
'J'he true model over R is expressed as: ] ""
••• J
R is a set of m lattice
"'! I
x: 1
-m
'{-
)!21n;~.J ; ,
--
[x
. X ] (l
"-.1: 2 ~ t::.l
d
>< 1 vector of parmueten;; and
~
'1.'0 facilitate the derivation of the MEE, the matri.x Xl is assumed to have full
column rank (Xl can ahmys be reparameterized to full co1wnn rank).
Let the estimates of the p parameters in
~l
be the terms in
£1
of the
approximating function:
.)i:-
is chosen to
A linear transformation of the n X 1 vector of observations 1.
)(-
estimate ~l (:!:.~., £.1 = T'y. ), where an asterisk ()(.) indi cates the vectors \'lhich
are associ.ated
\'li
th the experimental design R* •
If the observed !'esponse at the
i
..
]
:i Illay be
1
.1'
~~
~
= cr
2
and
:4: j, then the nor-mali.zed, sununed mean squ.are error (J =- SMSE)
expressed as the swn of' a bias ter'm and a var:i.ance term.
Term + Varianoe Term
=B
B
+ V, where
A
_E....
m (E[;Y(!l'
2 2:i=l
mCi
and
\
poi.nt in R is Y(x.)
-" '11 (x.
) -r e,1 wbere
-1
-1
1, 2, "', m and the random vari.able e. has E(e.):= u> .E.(e.)
E{ei~J) = 0 for i
of'
. th
1-
n - T1 (~i_ )} 2, ,
J
Bias
11
j
-5The KMH approach is used to achieve acceptably small. levels of J.
e$timator
~l
First an
is selected so as to minimj.ze the bias term B; and then an experiment-
a1 design R* with accepta'Lly small Val'i3.{lCe term V 'is chosen to satisfy other
design cri teria desired by the eXIJer·jmenV;r.
Since the second derivative of' the bias term B with respel:t to E(£1)
involves a positive definite matrix) minimu.lll B if, achieved by set.ting the first
d~rivative
equal to zero to get:
In ttl'": fini te
r,~g:i.on
expected value of
121
B, the
re~;iorw.L moment,~
are
is:
E(T'l* )
]1 / ]
-)(-
·k
'fIX
t:.
Therefore; the matrix T is the solution to the matrix equation T'X
If the generalized inverse of a matrix S is denoted by S
-)(-
then
j
= u.
~
necessary
and sufficient condition for the solution of the matrix equation is
G (X' *X* ) -Xi *X* "" G (i.~., ~ must be estimable).
When this necessary and
sufficient condition is satisfied, the minimum bias estimator which gives smallest
variance for a fixed design is:
~4B
* * _X' *Yx
G(X t X )
•
Substituting the MBE into the bias term, the minimum value of B J"S:
The value of B
is the same for every de;:;igt!
MB
iL!
the class of des:i.gns that alia"
the existence of the IvffiE:.
If the OL8 est:i.mator
~1S
value of the bias term 13 is:
n
-1-1
~LS ~ ~ ~2(M12MllH11M1IM12
+}f
). r:;
- ')2 t:.). .
L...
~
::T
13
L8
does not have a
(;OnS1~ant
value ov(-;r aD d·')s
and is at least as
larg~
as
-6-When designs sati;;f'y tbe BD condi l;iotl J
estimabi lity condition
those
desigt~s
becau~:;e
tlley alBo ;';atisfy the KMH
E: (Q1S) "" [1: M~;Ml.2J~
iE~
'VC'ct,Ol'J
the MBE be-
the sum of the OI,S es timator and
another COmpOl1E:ut \'Ifacb :18 tlh"; produ.c:t uf th,:' Iliatl'i,:es conts.i ned
condi hon and a k X 1
Therefore,
of all df':signs that
de~signs,
For tbis sUbclas;.; of
In general, th(': MBE
comes the 018 estimator.
[I: H~.iHL::,I:.
;;u.~:;clb.sS
tbat sati.[;fy tJ:ie BIi (:crcditi con an, a
satisfy the m.ininmln l):i as condi tion.
0=
ltl
the BD
~.~:,
.,
M~)11
)
-.1......1
..... 1-
)i;
and
F "" [1
Let R'
expressed
c""
(,H-1H
11 12
-)(.
Xi FX
D
C'
2
'rhen the variance ter-m for the ME.E can te
the sum to tl/lO traces:
~s
'fhe first trace is the variance term of the OL8 estimator and the second
of a noq-negative definite matrix.
V
MB
~
V '
LS
trac~
Since both trace terms are non-negative,
Equality holds it' and only if ttw MEE become:s the OLS estimator.
When an experimenter is using designs that satisfy ttc BD
th2 BD arlll KMH method. gi y,:, the OLS estimatoT end
term V and bias term B.
tj-;ereforr~)
~ond:ition,
cit-;si2)1~; \~tl.i..Le
both
i-tie same v Bl':i Bnc:_'
However .• using the KMH methcd, t.he experimenter
advantage of choosing from a larger class of'
ha~;
the
,:1,11J. miulrrrizing the
bias term B, and thus obtain greater design flexibi liLy to mer:t other cri teria
l'llus·tratt",.,j-
is
w'letl
ti-le·>
t·"l.'"
1_0
1-"
-_
-,·t,'1
lmcr1.
l.L
"""'''Y'"
CXl)I~
....
,-,1.,1
rJlc.~L"':~
p,
'::/
_ J.t....
d..:.....-
f'a-·j'"V'}"
,,1
'._ -_ .....· _ c
-'-
.'L";,,,·';',-n
U\:._,_~
... .:;.").J
corp:;ists of thF:: mean, the main f::ffect;;, unO. ti>' tvo-1'ccto.!' ic,te::action:.;
approximat.ed by a f'utlcti on
main effects.
c:ontainitl~J;
ool.y tll," ,,;::tiJIlo.tE'd
Hkar'
uy,'·.t-,
hd.1.LJ..1
i~;
and the es timated
·-7 ~
t·
2
3·
FACTORIAL EXPERIMENTS
To illustrate the notation to be u;;ed; n a general
consider an
vii tb
exampl,~
l'
'c: fa2tOl's,
"--
_
'D =
before reparallleterization is
(~
1'--'
110 1
z"c 1 -' z
r
i
1 0 1 0 1
I
'1,1
11
100
I
IT1?1
01.0
II
} __,
the true model
-".~,
I
Tn
L c."."
de~ign
c. '.Jr:
~
J 0 0
~
factorial experiment,
factorial
d.
';: = -1 2-:'
1 1 0 0..1-' "
10110
For
y'
Y
I
f
Ii
o
l
j
I
J
0 1
j :'22J
where in a 21' factorial design,
.
(i
19
th
:=
,
1, 2, ... ; r) at the g-- level, g
c=
Ij 2; and (T
J
1.1i th
4.J g
-che
. th f actor,
1-
) gh the two-factor inter-
jj
f th . th
ttl ,
th
ac t',1' on o'
e 1 - and- .J-tactors at the g-and-,nth
- level.
(z. ,) h associated
,
denotes the overa.LL inf,an; 'f.
fJ.
The column vector
t,(Je (T .. )
pa.rameteris derived by- !ilultiplying the
, lJ gh
corresponding elements of the main effect column vectors z.
-J.e;
aed z 'h'
-~)
'This
elementwise multiplication of ,,'ectors is called a Hadamard product [10] and wi 11
be denoted by z.
-lg
for the 2
z
-0
2
-
:=
(z .. ) ','
-lJ gn
factorial example may
C!__ -_e"
'" 1
z'l
-J1
<;)
[)e
Using the Hadamard p{'oduct, the true response
wri tten as
8.
linear combi na.tion of thf: vector
a vector of ones) and the vectors associated wi ttl the mai n effects;
.11
/.1Z
- t ; ' '2
~O
~i=l
'-'=
I: 2
g=l
T z
ig-ig
In the true model} the Zl matri,x is rCDaram(,t,;cized to the matrix Xl of full
colwnn rank,
of x
-·i
~~
Z
-i.l
:J.i'o.c a 2
-
'Z
r
factorial experim'"Lt, tbe G,'thcgo'1aL n-,parameterization
"
=-i 2'
.1.
is denoted by a plus one (1'1) and the;
l()(J
l"'V(c'l
Li
Ch"tlOled by
,-!,
minus one ( .. ])
y.
The set of all 2- reparametcrlzed colurHc ,'HetO.":
experiment is denoted by the set FD(
, . " . . ' . ,. . ' . ,. x
1
-
~
\oJ
•••
0
,;"
v
~
-"-",
~.l"
t·
:f'U:C
8
l
fa.ctorial
interaction colwnn vectors are the Hadamard pl'oducts of mai n effect column vectors.
The group of colwnn vectors in the matrix X~
j ,;
r
a sut'set of the FD(2 ) column
vectors consisting of the mean column "'lector anj the main effect colwnn vectors,
Let the parameter F,
Then the repara-
1
~
r-l +1 +1 ",
!
1 +1
.. " I~
1
l
I
,
-1 +1. I
!
i
L~ 2
i
J
1 -1 -1 \
'-_'_.1
For a general 21' factorial experiment,
as a. lineae combination of the vectors ~Oj x , and (~_j iii (') ~jh)J for g '" 1, 2 and
h
1, 2:
==
'Y'\
'I
==
~x
-0
+ L::
l'
i==l
F x
i-i
-I 2,,1' -1
'i==1
L:: r
j=i+l
L:::2
\ ,')
I
'f
)
(z
g l ~h=l \ :,j 'gh -ig
(j
z
)
(3.1)
-jh'
'l'he parameters in the approximating function few Lhe sl~cotld order model in (3.1)
are estimated from a fraction of the points in a complete 21' factorial design.
The construction of these fractional factorial designs is described in the next
section.
3.1
Factional Factorial Designs
The second order model in 0.1) L; aFf)j'oximated by, first order fmlction,
1.
X
*1E. l
consisting of an estimated mean aud
c~stimated
maie.
efL~cts:
+
*,
+.-}:;-
~ 1)
is formed from a subset of elemellts -in tl,e
••• J
L~
I'
a set of 2
X
l.
-r';
c.:ci'ce;:;pOi'iiil-r;g.:<;LuJnrl
"
>I
~~ ~~I}
vector in the
there
column 'leeton; ,,(bleb is called a f:euctional factorial design
* ~*l' "') ~r;
* ~l* (:) ~2)
*
FFD (n) == (!O'
cor re spond:i. Of!;
.!..~.,
l'
to the FD(2 ) set.
nts is n :s: m, the colwnn
Since the :Iumber of I:;X: erimental
Fractional factorial
conc:tru(~tE~d
designs are
by spu:; fying linear l'elationsh5ps that the colwnn vectors
of FFD(n) must satisfy.
Two column vectors x
for a > O.
)C
and
r .*
a)
1'he alias scalar.
Fractional factorial
in FFD (n) art' ,~;ai j
iE;
taY~erl
to be
-j_.
tc (:e ali ased if x
1 In
*
x·
+- 8.'f.. ,
:=
factorial experiments.
designs Hi this paper are cotJstr'ucted OJ specifying the
;;(
~O
:=
~~).
The column vect,)):'s form an :iclentt
rdaLo[; cet (IDR) and this set is
denoted by setting "I" equal to tLe l.etters l'epresenUng Uk factors or interaction of factors tLat are aliased with x .
--0
[1m' ex.':unple) the IDR, I
+ ABC
0=
CDE .- - ABDE means the foLlowing coLumn!ectors jn FFD(n) ace aliased:
= -J(
x-
1(-
* r:> ~3
x"
~l r:> -c
0=
~Oj
i(.
7-'
column vector ttl an IDR
2'
c'
oO
',"c
)i-
+;,
CJ ~2
c:
*-
...l',_
"*
x(_
<:)
~4
Each
~O'
called a viOrd and the Hadamard product of any two words
forms a word that is also in the IDE.
are not the Hadamard
70:
~-C/ and x
6
x'? 0 "'v
-j
~oroducts
A subset of "rords (called generators) that
of each other, generate the remaini ng words by
taktng the Hadcilllard product of all possHJle l'(1!llc}nation8 of ttl"cS2 generators.
From a set of s-generatoI's> 2"" - ~ " 1 \/oTd:, 'cd." Lc gEr:,',~'a1e':i (,)
28
["1
-
1, vmrdc; lnue
't~'
ID1~.
= 2.): - s
( l'
,.
\ __ .::::.... ,"i
K~'
1)' > t
-
'j2
'1 '.. ",
~-- d~.
;,LcL.l
3, 5, .. ,) Ivhich
drc
called ilTegulai.' !,jacLJ,iJnt:;;,
for the main effects of an irregular fracticll vii. U: i\:,
-<
by combining the main effect colwnn vectors "I' K :c'c:'2:U.rcE
IDR i'ami
]~y.
total of
Th::refore}
t:t on·,
fractions can be combiued to form ('1'e':1o: onaJ fs.cturIa', dcs.i )),m~"rL th
points (K =
Ll
'!
11
Regular
~, K(2
r
-
s
)
The colUlr,n vectors
'"
.POlfl'.:,S
ctre.
(~ons
true ted
let-ions from the
i3:3,rne
Regular and irregular fractions
that can be estimated from the design.
interactjcr:~,
effects, two-factor
CLEt'-;ified by Uk order of the effects
aTi:'
IT' 1lU.:U.n (:ffeeL; are called first order
an: called ,';2cond order effects, etc., and those
effects that are assUIueci lc be zero or- near zero are called negligible" then a
fractional factorial
de~;ign
[7J if:
is of' coven or- oud resolution
Ordel' of
Es timable ECfect;;
Resolution
Ore1er of
Negligible Effects
2t + 1
?t
----_.
__._---
t
l'esclu'~i(,
for regular fractions, the
the smal1es t v-rord
j
'[{
11
m
t he IDE,
(l
-IX/X = m-1m.l..
l' 1
•..
<:)
r
e
z,
-Jfl
[U'p
1, and the matrix
i
Since the inter-
l' -0'' "
)
z 1 ':l
-.1.1. <i) -z 2' .
1 ' - ,'-
-.Lg
'-lg
a 21' fiC.ctorial
(Z .
j
action column vectors (z, ".
(z.
In
the identity matrix of
p
.. p
-1
H1,2 = m (~fO' -x."
l.
gr~ater
0.£
cf the Ln:ctioil is equa.l to the length of
The regional moments fa ..' the second or'dc:l' 'node}
!
" 1
) are related to tile calum,] vectors in FD(J) by;
rx
z 1) = (_l)g+h
tJ x. + \/ - l,h+1
.)
~~L-f
- j l'
--i-J
. (7+1
(_lIb
\,
-.'
x
·~·i
+
1 \g+h]
( _ "J
-J/ 1+,
d
each element in the matrix HC0 can be founJ using tne p1
(:'8
of' Hudmndyd
-Lt:.
Therefore, j f 1:>1e estimator .....
b,.L ii, ,;11e f-irt onler approximating
products.
. b·'e a ".J..;.c,-,
"'~r'
f 'unc t'l o n .('J:
J . 2)'J.S to
a des j
;{-
p~n
X"1
::IrJ.~~"L
::30
/)
E(Fp )
F'
P
-+- 11'
/4
+
I,P,-1 I; ...
:...Ji::.::l g=l
r
1/4 z:: j ::::[.J+l
\'
~
.~
1
2
1.
j
h~1-1
"
i]
1
g"'-l.
,.,
,)
If an experjmental de;;;ign a11oc'l::" a
p
1, 2, .,', 1; So that
t!1E:
expect'~d
I,'a::'ues OJ
tb,':
cctc,l' 0:' observed
"')
responses
x..* ,
for every p.
HOWe'/f'T}
'C,:
x
('!..''';i )
co
..':t:
'
11
·x
.;+ \
'*
a/x~
every i
b
-p'
(x~
-l
-- -"'(J
x
\:) x, ')
c~,rery
p.
tite vectors a arul b ,
0
:~
1)
~'~~
l,
b'x
0 for every i. awl,j,
there are no linear
i'ur
--
These
~? ,
,..
0
)
for' q
---r~-(j
=
*
j
r;LLLH'S
ndcn'.:Les itl\'cLvjug mai:, ,:,f't'e<.
aI
.f.';
')
'- >
LL';dd
By equating
=
1, 2,
*
x.)
= 0
--p'
"<.
0) !.-),
and
,= .t:/]
!
expected values) tbe fOl.Lu\vingi.cicritJ tieci
... , r can be derived;
x
r: (~I
from the experirnenta.l deE,i
/ *
(X
'-'.1.
\:)
-cl
for
••• >
to the condition that
t. col',iJnn vectors in the subte' [LlU;ractions
from the FF'D(n) sCot.
T'llis C'otlcLitiou
a fractional factorial desigr~
(:J
If a fe'act-tonal factorial
.i.S
a
L:2::c:S.'~
l>o.,'cs;)lutLm
de~',igL
ry
V
iU;;J
suflic:Lent condition for
[9].
aLlei'vS the existence of a MBE, then the
design must be a resoluticG IV de
An example of a fract:i.on&.L factorial dc,:31g ' l that a.llow:::; the exi,stence of a
lviBE and has a m:inimum number or po:Lnts is a.
IDR, I=::+ ABeD.
}+-l
,~
r.[EcS],(:')l
that :i.E' defined by the
The converse of 'l'rleorem )01 is [Jut true in general, but for
regular fract10ns tile
onver3e ic tr-ue.
THEOREM 3.2
Ii' a regular Lac-Liot! i::) a
re~o.LHti.c"l
J.'''s.i
'!
+
h
:",'
exis tenee of a MBE,
•
Proof:
, i':(I_\
'. '} ,i t
ttl Ee'
'~lJ
c ondi ti on
holds.
Because regula.r frac ti ons
the lV1BE becomes the
OL~)
()J'
t::3t1l1uL()l'.
Y-I<~C
the ED cond1tioo,
·
t·
- ,
1
f'
I ..P K-l"egu 1 ar _rae
t ·].OtlS are '-ccm'...ule,i, WI 1 TregU._al.
,-
cOOf~tructed.
points can be
, ,
.
C I.
(1'5.:'.(;lO'1 "JlL.rl
rl
K / _r -:) ,
='\.2
)
f'rcl.ction car: be
constructed hy combining three
IDR family; X
=C'
+ AD '" + BCD'" + ABC.
AD
Fraction 1:
I
Fraction 2:
I
Ar
Fraeti on
I
AD
'i
/
:
-+
'j
~)
Bel)
+ ABC
J~CI)
r
ABC
B( ,~_~)
.;.;::
ABC
,11,(:
.~'r",
'bon
I DB.
.1 2
-'-_._-fr'-"-
3
i
-
AD
-+
-
BCD
+ - +
ABC
+
j-
-
For this eXaJnp1e, four different ;sets of three IDR
l~,
could ha'Je been chosen to
,.}3
construct the irregUlar fraction.
In general, which of the (~ ) possible regUlar
fractions from the same IDR famile are used to conEitruct all irregular fraction
does not affect the existence or non-exL.i. tence of a HDK,
variance term V, or the value of the; Li a,; term
However, :in order to have a
fractions, Addeilnan's method
COil;,}
ell
t.,!€, va.1:l1.e of tbe
l~.
:3t,-,n1: !Ili:,thod of COl!: tr'ucting
j
rregular
i~~ !Lv~d to ;;hoose th, ·1j~f0r(:nt regu.I.a.r L:dJ;t;O~lf'
from an IDR family.
K is an odd number, so AddeJJoan'
f',
m(,tho(i a,; ignf: an odd ',:,1lllkJ of th'" K a.l.t
H.E:
scalars equal to -1 for IDE won},c: of' (",.. i,) Le
sc~.tars
equal to -1 for
IDI~
words ('1' f>n;, ]ength.;;.
"V
combining resolution IV, regular
fL',g.I;t,j
uw:.
-15THEOREM 3.3
If an irregular fraction iE, cODf,tructed by combining reg"cllar f'racti Gns of
resolution IV, then the design allows the existence of' a MBEo
Proof:
As in Theorem 3.2, the assumptions are shown to imply that the BD condi bon holds,
For seven or more 'factors there always exist resolution IV, 2'r - 2 and 21'-3
Therefore, Theorem). 3 guarantees that I're can always combi.ne
regular fractions.
.
these regular fractlons to
allow MBE.
, . ( . .1'-2 \
. ' ' ~r-3 ,
I\,cl
,DLd KU::
.)
"jY'.1:>'(~':>.i.:r
tOY.'I!l
Hegular fractions which
resolu.tiotl less thacl IV may also be
havf~
combined to form irregular fractions that allow Iv1BE.
the estimabili ty of
~
LcactioG'-' ,'J'ncr,
To demonstrate this fact,
_,
,r-2)
was examined by using a computer program tor 3 (e::
and
3 (21'-2) irregular fractions constructed from .regular fract.ions of resolution IV
or less for r ::: 3, 4, 5 and 6.
The generators of the defining IDR family were selected systematically
according to their word length and by how many letters each generator has in
common with the other generators.
For the 3 (21'-2) irregUlar, fractions, there are
no designs that allow a ABE if r :.: 3, and
that allow a MBE for r :s; 5.
listed
of a MBE' are
.
_
designs for r
=
s
1'J-tuple (N = 2
l' n
1.0
3 (~{-3) irTt:gtdar fraction~o exbt
Those .Lrregnla-:· f'ractioL::'-rhich alhjvi the
. 6--3 )
' , . e'-l' gr ' '.' 1 i tl 'I'a·r'·,lo ;:)
"'a"-'l
.L
I.J
e .]_ ..t'OI' /-~ (0
~
Qt~"
r l:.' o.!lC.,
_.. "
~
4, 5 and 6.
]j
UH~
val.:.ll~S
The only design in the two tables where the
6- '-)
0
design (4,4,4).
()f
,;ted in t.he tablec;.
two variance terms are equal, t.hen the f.\'lBE and the OLS
valm~,,;
For
t.he; variance term V
If the values of the
l;,~;t.imaLor
are equivalent.
of the, vanarlce terms are equal
For tti:·; design, (:a.cr-, word lengt.h :in U'j(:~ cl::;~
fining IDR family is four, so by Tkleon:cHl 3.,j, trJis
oCLdi ti on 0
!I
- 1) of the lengths oj' the words in the d(~fini ng lDR. fanli ly.
for the MBE and the OL8 estimator are
C.:
. r·· 2,
1'(·1'
. J . , .-~/ ('-J
\ r..
Each design in T~bles 1 and 2 is classified by an
each desi.gn that allows the existence of a MBE,
is for the 3(2
existetlCt~
dt~.:j gr,
8.1,;(', Bati:::fi
Cf.i
Uw BD
t~l:tima.tor
The advantage of using the !'iDE rathf!l' than the OLS
6
for the irregular 3(2 -2) fraction (:2,4,6).
Let cI (' :=- 5MBE for the OL3 estimator,
,L1.-)
and
i'~.
1
,
-i
-..l.
- .1 +1
I -1
+.1.
+1
·-1
-1
+1
+1
-1
+1
-1
••.J.
if
+1
i:3 illustrated
J
A
I
r
1
!,
,
+1
MBE is:
By graphing J L8 - cI
MB
versus the non-negative quadratic form ~'A~£/()
2
(Figure
AL
one can see that the OL8 estimator should be used only if ab'Aab has a value less
than 1/8 the sampling variance CJ2.
The size of ab'Aab
negligibili ty of the two-factor interactions.
situation, the maximum increase in ,J
approximately
5%
MB
J--'0
MB
a measure of the non-
Even if the MBE is used in this
over Jr,8 is only
of the variance term V .,.'
always used, the increase in .J L8 over J
H;
0.375, which is
HOIvevcr, if t.hc" OLS "c:timutoI' is
can be qui te large-: and .in fact L;
fl(':;t.
bounded.
3.3
Example
All of the designs that a.llow a HBE in.Sect.ion jar" f.::r
model (3.1) being approximated by a first or-lifer functi on
knowledge is avai,lable about some of tr}e hro-fac
t,o..~' .1
'1.
C;. :?).
nt'c"ra.ct.ioCl
full second order
If ::dii tjonal
t'Tml~
so tlley C':111
be eliminated from the true model (:3.2), them desoign c ; U'lat 0.11')1"'- fimE may exist
where they did not exist using the ful1. .':econd order mode.l.F'or example,
an experiment is conducted using three chemicalb
concentrations and it is known that th(: ,·I:f"IlLi.cal
(!'2"
'·)UP.iJOSl~
a, b, ~ld c) puch at two
-1':)-
" a " and "b".
i, j and k
The true model is assumed to be TJ
1, 2, but the chemist wanb; to use thp.
=
n + a.
l
+
bj
-I-
Ck
IJ.
.=
ijk
+
8.
i
,;impJ.,~r
+ bj
-I-
c
-I-
(ab
\j
for
approximating function
basing the estimators on a six point defdgn.
second order model there are no
k
For the full
3 (23 - 2 ) desigw; that allow a MBE, but by
eliminating two of the two-factor interactions a MBE does exist for each of the
following irregular fractions:
Design (1, 2,~) )
Design (1, 1,2)
l',
,.
IDR
Fraction
1 2 )
A
-I- -I-
-
AD
-I-
-
-
c
-I- -
-I-
Be
-
-I-
-
AC
-I-
Ii'ra~:tion
IDE
1. ;?-
f'
3
"
':rac:ti0n
1 2 . -,
AC
-I-
3.
The SMSE for these three designs are listed in Table
The individual values of J are plotted for each of the three designs in
Figure B versus the non-negative quadratic form
ab/A~~/
?
(J-.
Adopting the SM8E
cri terion to choose the "best" of the three designs, for minimum bias estimation,
only the variance term V (~.~., intercept) of the J
MB
' s need be considered since
all three designs yield the same bias term B (~o~., tbe J
slopes ).
MB
Hnes ltave identi cal
Because the smalles t variance term V for the ,J vb-' sis 5.2';;0 for the
11
design (1,2,3 ) it is the optimal choice which is contrary to the si tuatiocl resulting when OL8 estimation is used.
variance term V than the J
for the MBE.
L8
The
d,e>'i:nl
(]_" 2 :3) ha"~:> a ~m"C_...L'leY'
. . . . 8MSE J IJS for
..
5"
~_..
- 1-.>
0
for the designs (1,1,2) and (2,2,2), as iS31~,o true
However, the bias term B (~o!::", tb.e hloI'c) for de:.;;ign (1,2,3) L;
larger than the bias term B Jor de.signs (1,1,2) and
C2, 2, 2).
In the presencc? of
a non-negligible AB interaction, the larger bias "'(erm (1.)'5 times ",-s la:tge) would
make the design (1,2,3) less appealing than e.i ther of thE': otber two designs whee.
OL8 estimation is used.
.~16-
4.
SUMMAEY AND CONCLUSIONS
The KMH method of minimum bias estimation has been applied to the classical
experimental
des~:gn
model with factors having a finite number of levels.
For
this situation, the M13E gives the minimum bias term 13 for the SMSE which, in
general, is smaller than the bias term 13 using the OL8 estimator.
The variance
term V for the MBE is greater or equal to the variance term V for the OL8 estJmator,
and equality occurs if and only if the two estimators are equal.
The MBE does be-
come the OL8 estimator on the subc lass c:Jf' designs sat:! [dyi ng tbe Box-DTetptcr
condi tion v[hich are contained in the class of all de.c;igns allowing the existence
of a minimum bias estimator.
For 2
r
factorial experiments, factional factorial designs were investigated
when a second order model was approximated by a firs t order function,
A
sufficient condition for designs to allow the existence of a M13E is that the
fractional factorial design must be of resolution IV.
For regular fractions,
,
resolution IV is also a necessary condi tion for the design to satisfy the 13oxDraper condition and therefore, allow a
~rnE.
Irregular
fract~ons
that are con-
structed by combining resolution IV, regUlar fr·actions abo saUsfy the BoxDraper condition.
However, examples of irregular fractions constructed .from
regular fractions of resolution less than DJ which allow the existence of a MBE
6
c
are given for 3(2 - 3 ) irregular fractions in Table 1 and 3(2· -2) irregular
fractions in Table 2 for r
= 4, 5,
and
6.
-17TABLE 1
3 (26 - 3 ) irregular fractions
=r=-
.....----..---'--_.
! - - - - - - -----.........·-'f
Sq~
Factor
No. of
Design
Points
6
24
(1,1,1,2,2,2,3 )
No
A,B,C
24
(1,1,2,2,3,3,4)
No
A,B,CD
24
(1,1,2,3,4,4,5)
No
A,B,CDE
24
(1,1,2,4,5,5,6)
No
A,B,CDEF
24
(1,2,2,2,"3,3,3)
No
A,BC,ClJ
24
(1,2,2,3,3,4,5)
No
A,BC,DE
24
(1,2,3,3,3,4,4)
No
A,BC,CDE
24
(1,2,3,3,4,5,6)
No
A,BC,DEF
24
(1,2,3,4,4,5,5)
No
A,BC,CDEF
24
(1,3,3,4,4,4,5)
No
A, BCD, DE.F'
24
(2,2,2,2,2,2,4)
No
AB,BC,CD
24
(2,2,2,2,4,4,4)
No
AB,BC,DE
24
(2,2,2,3,3,4,4)
No
AB,BC,CDE
24
(2,2,2,3,5,5,5)
No
AB,BC,DEF
24
(2,2,2,4,4,4,6)
No
AB,BC,CDEF
24
(2,2,2,4,4,4,6)
No
AB, CD, EF
24
(2,2,3,3,3,3,4)
No
AB,CD,ACE
24
(2,2,3,3,4,5,5)
Yes
AB, CD,AE:"
9· 750
7·500
24
(2,2,4,4,4,4,4)
Yes
AB,CU,ACKF
8.000
cr, )00
24
(2,3,3,3,3,4,6)
Yes
jlJ3 J
BCD, AEF
10.125
'7. ~~50
24
(2,3,3,3,4,4,5)
Yes
AB,CDE,ACF
9·375
'7.250
24
(3,3,3,3,4,4,4)
No
DeSign_~
I
Generators
Min. I c,.J.L. . The IDR
Bias ii
Famt ly
Min. Bias Geast
Variance ; Variance
Term
Term
----
I
ABC) eDE, ADF
- - -.._-----------
-18-
TABLE 2
. 1'-2
3 (2
Factor
_ _.--+-
4
12
12
12
12
12
12
I ....
Of
Design
Points
I
Design
+---_
(1,1,2)
.No
.No
(1,2,3)
(1,3,4)
(2,2,2)
(2,2,4)
(2,3,3)
(l} l) 2)
24
24
24
24
24
24
24
6
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
l
. ~_
Yes
No
Yes
Yes
,
2)l
MlH.
Bias
(1,2,5)
(1,3,4)
(1)
, , '+, 5)
"'----_.
i\Io
No
[;8
-:!.es
(2,2,2)
No
(2,2,4)
(2,3,3)
(2,3,5)
(2,4,4)
(3,3,4)
Yes
Yes
Yes
Yes
Yes
(1,1,2)
(1,2,3)
(1,3,4)
(1,4,5)
(1,5,6)
(2,2,2)
(2,2 4)
(2,3,3)
(2,3,5)
(2,4,l..r)
(2,4,6)
No
No
Yes
Yes
Yes
No
Yes
Yes
Ye~~
Yes
Yes
6
and
aSl
.' .:-7=--=r:="'c~'-'==~·
MIn.:. Bl
Lea. ~t.'. Sq.
Va1'lance
Van.ance
"=;=-'-==f'==.~.-.
=f=---.,'''-
i
No.
4, 5,
) irregular fractions for r =
(JE~'l~rato~'~
Uf The IDE
Family
_ ...
..
'
I
I
Term
'l'e:nn
6,750
5·250
I
.J_..
.
.__
A, B
A, Be
A) BCD
AB, Be
AB, cn
AB,BCD
"(.
j\"
8
ii, Be
.A, .DCD
1(08'{1)
6.000
A, BeDE
AS, BC
AB, CD
AB, BCD
AB, CDE
Jill, BCDE:
ABC, CDE
A, B
A, BC
A, BCD
A, BCDE
A, BCDEF
AB, Be
AB, CD
AB ' BCD
.
('rr;'
~.-.LJ-'--J
jG,
AB, BCD.E:
.fill , C~D:E~.bl
lLS, ECO.8F
(2,5,~;)
C') , 3 , I)
j.
Yes
Ye,s
A,J~C
(3,3,6)
(3,4,5)
(4,4,4)
Yes
ABC, DEI'
y CiS
ABC, CD,S.F
Ye;.;
ABeD,
J
CDE
CJJEJ:i1
1~:'5
t.J .,
6·7:;0
'(.OOO
(). ')O~)
8.125
6·750
6,250
6. 0
6.375
'[.000
250
6.000
'J. 000
7·
r7
I
(
8'75
·875
p
\..)"
c'.J •
?
0
"i
OJO
'(":0
r/
b ..
·
, noo
·
!
,
i ,.
_~~[)o
'7. 250
'7. ~2-::IO
.
;-/
I
,
"(
r
I
r"i
6
1
;
i
~
7
~h~;j
(
: .' ,.
;
-,
I
I
'r
~_.;
,
'. '
..... -.
(,.,1\),-j
17
-----_.-.---,
~----------------'---"'--"-------'-'--'-"~----"'-------------
-19TABLE 3
Li~F;
+ c
k
+ (ab)
true mode 1 :
)L
(1,1,2)
SMSE
rJ
Least Squares
JI'C'
lJ')
Minimum Bias
J
l~eaf.~t
(;1/2c/:')ab / Aab
-+-
4·500
-+-
5.625
MB
Sqv.. aren
"
(3/u~) ab/Aa~
Minimum Dias
(2,2,2)
+ b.
J
E:l.
l
ij
Design
(1,2,3)
+
2
(10/:)(5 ),9-0 1 Aab + 5.500
Leas t Squares
')
Minimum Bias
,J
MB
('3/ (J_t....)
ab 1 Aab +
5.62')
-20-
15 -
10
5
-----r--·----r
2
-1
Figure A
J
LS
-
J
MB
for the
3(26-2 )
design
(2,4,6)
-21-
L
.~
..I
/'
/
25 ..
/
,/
'"
10
" "-, ,
.::::::::.
Figure B
The SMSE's of
T1 1, J'k
= JJ, +
3(~-2)
designs for the true model:
a. + b. + c + (ab) ..
1
J
k
.
lJ
..
f\
~ 1-,
. ?
.'
~
IT ~
[lJ
Addelman, S.
"Irregular fractions of the 2
Technometrics
[2J
Addelman, S.
3
n
factorial experiments".
(1961), 479 - 496.
"Recent developments in the design of factorial experiments".
Journal of the American Statistical Association 67 (1972), 103 - 111.
,
DJ
r
Box, G. E. P. and-Dx:oaper N. R.
surface design".
"A basis for the selection of a response
Journal of the American Statistical Association 54
(1959), 622 - 654.
[4J
Cote, C., Manson, A. R., and Hader R. J.
general regressi on model';,
"Minimum "bias approximation of a
Juo.rnal of the Americal Statistical
Association 68 (1973), 633 - 638.
[5J
Graybill, F. A.
New York:
[6]
Graybill, F. A.
An Introduction To Linear Statistical Models.
McGraw-Hill Book Company, Inc.
John, P. W. M.
1961.
Introduction To Matrices With Applications In Statistics.
Belmont, California:
[7]
Volume I.
Wadsworth Publishing Company, 1969.
Statistical Design And Analysis of Experiments. New York:
The Macmillan Company, 1971.
[8J
Karson, M. J., Manson, A. R. and Hader, R. J.
"Minimum bias estimation and
experimental design for response surfaces",
[9J
Margolin, B. H.
"Results on factorial designs of resolution IV for the 2
n m
and 2 3 series".
[lOJ
Margolin, B. H.
'l'echnometrics 11 (1969),
Technometrics 11 (1969), 431 - 444.
"Resolution IV factional factorial desi.gns",
Journal of
the Royal Statistical Society B, 31 (1969), ~)14 - 52:3.
[llJ
Paku, G. A., Manson, A. R., and Nelson, L. A.
The Mixture Problem.
Carolina:
1971.
Minimum Bias Estimation In
Mimeograph Series No. '7'57, Raleigh, North
Institute of Statistics, North Carolina State University,
n
-23-
[12J
Parish, R. G.
Order.
Minimum Bias Approximation of M02-e1 s by Polynomials of Low
Mimeograph Series No. 627"
Raleigh, North Carolina:
Insti tute
of Statistics, North Caroliha state University, 1969.
[13J
Rao, C. R.
Generalized Inverse of Matrices And Its_Applications.
John Wiley and Sons, Inc.
New York:
1971-
[14 J Rhode, C. A. "Generalized inverse of partitioned matrices ".
Journal Society
of Industrial Applied Mathematics 13 (1965), 10:33 - 1035.