Rochester Institute of Technology
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Thesis/Dissertation Collections
8-1-1995
Minimum aberration two-level split-plot designs
Peng Huang
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Recommended Citation
Minimum Aberration Two-Level Split-Plot Designs
by
Peng Huang
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Applied and Mathematical Statistics
Approved by:
Prof.
Joseph
o.
Voelkel
(Thesis Advisor)
Prot. _S_i9_n_a_tu_r_e_II_le_9_ib_le
_
Prof. _D_a_n_ie_I_La_w_r_e_n_ce
_
GRADUATESTATffiTICS
COLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
TABLE
OF CONTENTS
ABSTRACT
PAGE
1
INTRODUCTION
PAGE
1
RELATION BETWEEN SPLIT-PLOT DESIGNS
AND
FRACTIONAL
FACTORIAL DESIGNS
PAGE 2
MINIMUM ABERRATION SPLIT-PLOT DESIGNS
PAGE 7
SUMMARY
PAGE 18
APPENDIX
PAGE
19
REFERENCES
PAGE 21
ABSTRACT
In
this thesis
a certain aspect offractionated
two-levelsplit-plotdesigns
is
associated witha subset of
the
2"'kfractional
factorial
designs.
The
concept ofaberrationis
then
extendedto
these
split-plotdesigns
in
orderto
comparedesigns.
Finally,
two
methods arepresentedfor
constructing
two-level
minimumaberrationsplit-plotdesigns.
1. INTRODUCTION
In
multifactor experimentsin
whichit is
not practicalto
run allthe
factors in
acompletely
random
order,
a split-plotdesign
canbe
usedasanefficienttool.
Usually
a split-plotdesign
is
obtained
by
combining
two
separatedesigns,
onefor
the
wholeplotand onefor
the
subplot.In
this thesis the
concept of aberrationis
usedas away
ofselecting
"good"fractional
factorial
split-plot
designs
withtwo
levels.
It
willbe
seenin
this
paperthat
a good split-plotdesign
is
usually
the
one wherethe
subplotdesign is
constructedby
using
allthe
factors involved in
the
subplot and some
factors
from
the
whole plot.Section 2
ofthis
paperwillestablishthe
relationbetween
split-plotdesigns
andfractional factorial designs.
This
relationis best
seenthrough the
generating
matricesby
whichsplit-plotdesigns
are well represented.Through
this
relation,
the
concept of minimum aberration
for fractional factorial
designs
canbe
usedto
study
split-plotsplit-plot
designs.
Using
the
first
method,
one can useminimum aberrationfractional factorial
designs
to
constructminimum aberration split-plotdesigns.
In
orderto
comparetwo
split-plotdesigns,
the
aberrationfunction is defined
sothat
comparing
two
split-plotdesigns
is
equivalentto
comparing
the two
values ofthe
aberrationfunction.
This
aberrationfunction,
linear
integer
programming,
and someknown
properties onthe
wordlengthpatternareusedto
presentthe
second method.
The
second methodwillessentially
reducethe
problem offinding
aminimumaberration split-plot
design
to
alinear
integer
programming
problem wherethe
aberrationfunction is
to
be
minimized.2.
RELATION BETWEEN
SPLIT-PLOT
AND FRACTIONAL FACTORIAL DESIGNS
The
defining
relation of a 2"'kfractional
factorial design D
canbe
representedby
the
generating
matrix(Franklin,
1984),
and withoutloss
ofgenerality
the
generating
matrixG
can
be
writtenin
the
form
G=(I
C),
whereI
is
the
kxk
identity
matrix,
andC
is
akx(n-k)
matrix with
its
elements equalto
0
or1.
Now
weapply
this
notationto two-level
fractional factorial
split-plotdesigns.
Suppose
that
wehave
nt
factors in
the
wholeplotwithfractionation
elementk\
andn2
factors in
the
subplotwithfractionation
elementk2.
Then
there
are 2"l~tltreatment
combinations
in
the
wholeplot and 2i"l+"2)~(-k,+kl)treatment
combinationsfor
the total
and
G2
be
the
generating
matrix of a 2"1~kldesign
D2
for
the
subplot.Then
Gi
=(IiCi)
andG2
=(I2
C2),
whereIi
is
the
kixki
identity
matrix,
Ci
is
ahx(m
-h)matrix,
I2
is
the
k2xk2
identity
matrix,
andC2
is
ak2x(n2
-ty matrix.The
design
matrixfor
the
totalsplit-plotdesign
canbe
generatedthrough the
generating
matrixki
ni-kjk2
ri2-k2G
=fGl
l=f1'
Cl
0l
2)ki
0)
lo
G2J
lo3
04
I2
Cjk2
'
where all
the
elementsin
O,
,02
,03,
04
are zero.To
illustrate
the
aboveidea,
let
usconsideranexample.(Our
exampleis
too
smalladesign
to
normally
be
used,
but it
succinctly
illustrates
somekey
ideas.).
Suppose
there
are3
factors,
numbered1, 2,
and3,
in
the
wholeplotwithfractionation
element1,
and4
factors,
numbered4, 5, 6,
and7,
in
the
subplot withfractionation
element2. Then
wehave
a
23'1
design for
the
whole plotand a24'2design
for
the
subplot.Let
the
23'1design
in
the
whole plot
be
Di
withgenerating
relation1=123
andthe
24'2design for
the
subplotbe
D2
with
generating
relation1=46=567. Then
the
corresponding generating
matricesfor
Di
andD2
arerespectively
G!
=(l
1
1)
(\
0
1
(ft
and
G2
=U
1
1
\)
M
=12 3 4 5 6 7
(-\
-1 +1 -1 -1 -1 +1"*-1 -1 +1 +1 -1 +1 -1
-1 -1 +1 -1 +1 -1 -1
-1 -1 +1 +1 +1 +1 +1
+1 -1 -1 -1 -1 -1 +1
+1 -1 -1 +1 -1 +1 -1
+1 -1 -1 -1 +1 -1 -1
+1 -1 -1 +1 +1 +1 +1
-1 +1 -1 -1 -1 -1 +1
-1 +1 -1 +1 -1 +1 -1
-1 +1 -1 -1 +1 -1 -1
-1 +1 -1 +1 +1 +1 +1
+1 +1 +1 -1 -1 -1 +1
+1 +1 +1 +1 -1 +1 -1
+1 +1 +1 -1 +1 -1 -1
L+l
+1 +1 +1 +1 +1 +VNote
that
althoughM
is
the
design
matrixofthe
design
27'3=2(3+4)'a+2)with
the
generating
relation
1=123=46=567,
this
fractional factorial design
is
not equivalentto the
abovesplit-plot
design.
The
reasonis
that the
split-plotdesign
consists ofboth
atreatment
design
(here
a27'3)
and an error-controldesign,
whichdefines how
the treatment
combinationsare randomized and
blocked. (The italicized
phrases arethose
usedby
Flinkelmann
andKempthorne
(1994).).
We only
considerthe
case wherethe
error-controldesign for
the
whole plot units are
completely randomized,
andthe
error-controldesign for
the
subplotunitsare
completely
randomizedfor
eachtreatment
combinationin
the
whole plot units.Since
the
sequelonly
discusses
the treatment-design
aspects ofthe
split-plotdesign,
wewill
be
ableto
regardthese
split-plot restrictions ofthe treatment
design
ascreating
anSince 1=123=46=567 is
the
generating
relationofthe
27'3design,
wecaneasily
obtainits
generating
matrix as:G
=12 3 4 5 6 7
(\
1
1
0
0
0
6\
0
0
0
10
10
^0000111/
which
follows
immediately
from
(1)
andthe
expressionsfor
Gi
andG2
.The
above examplehas
shown usthe
motivationfor
using
agenerating
matrixfor
asplit-plot
design.
This
example alsotells
ushow
the
fractional
factorial 2
"design
andthe
split-plot
design
are related.This
relation canbe
readily
extendedto
the
following
generalcase.
Equivalence 1. Given
a split-plotdesign
with/
factors
andk\
fractionation in
the
wholeplot and
n2
factors
andk2
fractionation in
the subplot,
supposethegenerating
matrixG
ofthis
split-plotdesign is
givenin (1).
Let
j)=2<-"i+"2)'(kl+kl)be
the
fractional factorial design
with
the
generating
matrixG.
Then
the
split-plotdesign
is
equivalentto
this
fractional
factorial
2("l+"2)~(*1+*2)design,
in
the
sensethat their treatment
designs
areidentical.
This
equivalence willbe
extendedin
a useful way.Let
us reconsider ouraboveexample.
The
design
Di
for
the
wholeplot andthe
design
D2
for
the
subplot wereconstructed separately.
That
is
to say, the
generatorsin
Dt
werethe
wordsformed only
from
the
letters
(factors)
1, 2,
and3
in
the
wholeplot,
andthe
generatorsin
D2
werethe
words
formed
only
from
the
letters
(factors)
4,
5,
6,
and7
in
the
subplot.If
weinstead
fix
generating
relation1=146=567,
then,
asin
the above, this
modified split-plotdesign
has
the
generating
matrixG*=
12 3 4 5 6 7
(\
1
1
0
0
0
0^10
0
10
10
Kp
o
o
o
i
i
v
and
this
split-plotdesign
can nowbe identified
withthe
design
27'3having
the
generating
relation
1=123=146=567,
in
the
sensethat
theirtreatment
designs
areidentical.
Comparing
the
original split-plotdesign
withthe
modifiedone,
wefind
that the
seconddesign
is better
than the
first
one.In
the
seconddesign
no single-factor effectis
confoundedwithany
othersingle-factor effect
the
defining
relationsis
1=123=146=567=2346=123567=1457=23457
but
in
the
originaldesign
the
single effect4
is
confoundedwiththe
single effect6.
The
idea
ofusing
D2*,
whosegeneratorsinclude
the
letter
1
from
the
wholeplot,
is
mentioned
in
Cochran
andCox
(1957).
D2*is
still referredto
as a subplotdesign,
andthroughout the
remainder ofthe
thesis,
"subplot
design"is
usedin
the
extended senseand,
for
simplicity'ssake,
is denoted
by
D2.
The
generating
matrixG,
then,
ofa split-plotdesign
withnt
factors
andk\
fractionation in
the
wholeplot andn2
factors
andk2
fractionation
in
the
subplotwillhave
the
form
kj
m-kik2
n3-k2G
ri,
c,
o,
o^*,
(2)
Here,
Ii, I2,
Ci,
C2,
O,
,02
arethe
matrices withthe
samestructuresasin
(1),
Bi
andB2
are matrices withelements
0
or1, (Ii
d),
denoted
by Gu
representsthe
generating
matrixof
the
whole plotdesign,
and(Bi, B2, 12, C2),
representsthe
generating
matrixofthe
subplot
design.
The
G
in
(2)
is
a natural generalization ofthe
G in (1). We
writeacorresponding
extension ofEquivalence 1
asfollows.
Equivalence
2.
Given
a split-plotdesign
with;
factors
andki
fractionation in
the
wholeplot, n2
factors in
the
subplot,
andk2
fractionation
in
a subplotdesign,
supposethat the
generating
matrixG
ofthis
split-plotdesign
is
givenin
(2).
If
D=2(n,+'*M*1+*2)is
the
fractional factorial
design
withthe
generating
matrixG,
then the
split-plotdesign
is
equivalent
to this
fractional factorial
design,
in
the
sensethat their treatment
designs
areidentical.
3. MINIMUM
ABERRATION
SPLIT-PLOT DESIGNS
3.1.
Definition
To
construct a split-plotdesign
ofform
(2),
it is
naturalto
first
selecta whole plotdesign
Di
withgood properties.For example,
the
designs displayed in
Table 12.15
ofBox,
Hunter
andHunter
(1978)
canbe
used sincethey
are of minimum aberration.Now,
then,
design.
Hence
a natural questionis how
the
subplotdesign
is
obtained,
givena specificwhole plot
design.
In
fact,
this
questionis
equivalent, using Equivalence
2,
to
asking
how
to
constructBx, B2,
C2
in
(2),
givend,
sothat the
corresponding fractional factorial
2("l+^)"(t,+*2)
design
is
good under some criterion.We
solvethis
problemby
using
the
concept of minimum aberration about
fractional factorial designs
introduced
by
Fries
andHunter
(1980),
extending
this
idea
to
minimum aberration split-plotdesigns.
For
convenience,
wefirst
introduce
the
following
notation.Notation:
Given
akxxm
matrixGi=(Ii Q),
the
symbol 2("1+',2M*1+*j)(Gi)
willbe
usedto
denote
afractional factorial
2<-"l+"lMkl+k2)design
whosegenerating
matrixis in
the
form
of(2)
withthe
prescribed matrixCi,
andthus
by
Equivalence
2,
represents a generalizedsplit-plot
design.
Now
consideradesign
D=2(n,+'*)"(*1+*2)(Gi). The
number ofletters
in
a wordappearing
in
the
defining
relation ofthe
design
D
is
calledthe
wordlengthofthe
word.Let
A(D)
be
the
number of words oflength i in
the
design D. The
wordlength pattern ofthe
design
D
is
defined
asW(D)=(Ai(D),
A2(D),
...An+B2
(D)).
For example,
the
split-plotdesign
D corresponding
to the
generating
matrixG*in Section
2
has
defining
relation1=123
=146=567=2346=123567=1457=23457, and soW(D)=(0,0,3,2,
1,1,0).
Definition. Suppose
that
D'andD"aretwo
2("i+'*M*'+t2)(Gi)
designs.
Let
rbe
the
smallest
i
suchthatA;(D')
*A;(D").
Then
D'is
saidto
have less
aberrationthan
D"if
design
2("l+'hy<-kl+kl)(Gi)
is
saidto
have
minimum aberrationif
no otherdesign
2
i+z -(*i+*2)(q^
jjas
jess
aberrauona
split-plotdesign
withgenerating
matrixG
in
(2)
is
said
to
have
minimumaberrationif its corresponding
2(-"l+"l)~(-kl+k2)(GO
design
has
minimumaberration.
Remark
1.
Our
definition
of minimum aberration 2<-"l+"iy(kl+kl)(GO
designs
is
similarto the
definition
of minimum aberration2"'kdesigns
(see Chen
andWu,
1991).
However,
it
should
be
notedthat
allthe
designs
2<-"l+"2y(kl+k2)(GO
already
have
h
generatorsdescribed
by
Gi=(Ii
C{).
Thus,
finding
aminimumaberration 2i"i+"2>~{ki+k2)(GO
design
requires achoice of
only
k2
generators.Remark 2.
Designs
with more shorterwordlengthwords areinferior
to those
withmorelonger
wordlength words.For
example,
considertwo
Resolution IV
designs,
oneof whichis
a minimum aberrationdesign.
When
it is
safeto
assumethat the
effectsofthree-factorinteractions
andhigher
orderinteractions
willbe
zero,
thenthe
minimum aberrationdesign
willestimate more unconfounded
two-factor
interactions
than the
otherResolution IV
design.
Remark
3. The
class of2("1+^)"(fcl+tj)(GO
designs
for
a givenGi
are a subset ofthe
class of2"'k
3.2
Finding
minimum aberration split-plotdesigns
We
examinetwo
methodsto
obtain a minimum aberration split-plotdesign.
Method
I,
based
on remark3,
finds
suchdesigns from
available minimumaberrationfractional
factorial designs.
Method
II
employslinear integer
programming.Method I
For any
nthe
minimum aberration2"'kdesigns for
k<4
and=5
are givenin
Chen
andWu
(1991),
andChen
(1992),
respectively.Also for
somenand somek>6
the
minimumaberration 2"'k
designs
are givenin Franklin
(1984).
All
ofthe
minimumaberrationfractional factorial designs
with resolutionIII
orhigher
and most ofthe
minimumaberration
fractional factorial designs
with resolutionII
canbe
usedto
construct minimumaberration split-plot
designs.
For
this
purpose,
wefirst
wantto
get a simplifiedversionofform (2).
Let
gy be
atypical
element ofG
of(2),
whichlies in
the
ith
row andjth
column.If
one nonzeroelement
gij
is in
Bi,
then
adding
the
jth
row ofG
to the
ith
rowwithreduction modulo2,
to
replacethe
ith
row,
willreducethe
elementg
to
0.
Therefore,
performing
similar rowoperations
for
allthe
non-zeroelements ofBi
willreduceBi
to
O3,
a matrix with allelements equal
to
0,
andthus the
following
new matrixis
obtained:ki
m-ktk2
m-kifl,
C,
O,
o2
*,
l03
B3
I2
cj k2
Since
the
sum oftwo
rows with reduction modulo2
representsthe
product ofthe two
generators
corresponding
to these two
rows,
it is
seenthat the
abovematrixis
also agenerating
matrix ofthe
split-plotdesign corresponding
to the
G in (2).
Switching
somecolumns of
the
above matrixwillgive usthe
following
matrixki
fo
ni-kj m-fe(\
o,
c,
o^i*.
lo3
I2
B3
Cj
k2*which,
ofcourse,
is
agenerating
matrix ofthe
same split-plotdesign
exceptfor
relabeling
of
the
factors.
Now
let
us use an exampleto
showhow
to
obtain minimum aberration split-plotdesigns from
minimum aberrationfractional factorial designs.
The strategy
here is
to
rearrange
the
matrixto
be
ofthe
form
(3),
whosetwo
key
features
arethat the
left
portionis
the
identity
matrix ofdimension
ki+k2
andthe
upperright
portionis
akixfn^k^
matrixof
O's. For convenience,
wewillusethe
symbol[i]
<-[j]
to
denote
the
elementary
rowoperation
that the
positions ofthe
ith
row andjth
row of a matrixG
have
been
switched.The
following
fractional
factorial
29'4design
withgenerating
relation1=12346=12357
=2458=3459
is
a minimum aberrationdesign
(see
Chen
andWu,
1991). The
generating
matrix of
this
design
is
7 23456789
r\
1
1
1
0
1
0
0
6\
1110
10
10
0
G
=0
10
110
0
10
^0
0
1
1
1
0
0
0
v
Let
usperformthe
following
elementary
operations:67
8912345
^10
0
0
11110^
0
10
0
1110
1
0
0
10
0
10
11
K.0
0
0
1
0
0
1
1
\)
Thus
comparing
(I
C)
withform
(3),
one gets*i=l, fc=3,
nr*i=4(so
/=4+#/=5),
andG
(switch
columns)-* =(IC).
nr-kf^l
(so
n2=1+^=4).
Therefore,
the
minimum aberration split-plotdesign
representedby
(I
C)
is easily
stated asfollows:
factors
1, 2,
...,9;
whole plotdesign
2n,~kl=25'' with
factors
1, 2, 3, 4,
and6,
andgenerating
relation1=12346;
subplotdesign
2"2~kl=24'3 withfactors
5, 7, 8,
and9,
andgenerating
relation1=12357=2458=3459. Of
course,
one canmake a
relabeling
ofthe
factors
(for
example,
just
switching
the
positions offactors
5
and6
)
to
getthe
following
minimum aberration split-plotdesign
statedin
a somewhatstandard
form: factors
1, 2,
...,
9;
whole plotdesign
2"1"*1=25"7
with
factors
1, 2, 3, 4,
and5,
andgenerating
relation1=12345;
subplotdesign
2"2~k2=24'3
with
factors
6, 7, 8,
and9,
and
generating
relation1=12367=2468=3469.
Note
that
anothertwo
minimum aberration split-plotdesigns
canbe derived from
the
above
29'4
design
as well.Observe
that the
columnunderfactor
1
in (I
C)
has
two
0
elements.
We
wantto
"move"them to the
upperright hand
corner and so performthe
following
elementary
operations:(I
C)
(switch
columns)-*6 7 8 9 2 3 4 5 1
(\
0
0
0
1
1
1
0
\\
0
10
0
110
11
0
0
10
10
110
U)
o
o
i
o
i
i
i
o;
([3]<->[l],
[4]<-*[2])-*
(switch
columns)-*6 7 8 9 2 3 4 5 1
(0
0
1
0
1
0
1
1
&\
0
0
0
10
1110
10
0
0
1110
1
\0
1
0
0
1
1
0
1
IJ
896723451
(\
0
0
0
1
0
1
1
0^1
0
10
0
0
1110
0
0
10
1110
1
;o
o
o
i
i
i
o
i
V
= G*
Comparing
G*with
form
(3),
onehas
h=2, fe=2,
/=4+i=6, n2=\+k2=3,
andaminimumaberration split-plot
design is
obtained asfollows: factors
1, 2,
...,9;
whole plotdesign
2,-*,=2-2
^h factors
2, 3, 4, 5, 8,
and9,
andgenerating
relation1=2458=3459;
subplotdesign
2"2~kl=23'2with
factors
1, 6,
and7,
andgenerating
relation1=12346=12357.
Now,
the third
row ofC
has
two
0
elements,
and we wantto
"move"themto the
upperright
hand
corner.To
do
so,
we performthe
following
operations:(IC)
(switch
columns)-*(
[3]
<->[1]
)->
67 8913245
(\
0
0
0
1
1
1
1
&\
0
10
0
1110
1
0
0
10
0
0
111
lo
0
0
1
0
1
0
1
IJ
6 7 8 9 13 2 4 5
(0
0
1
0
0
0
1
1
A
0
10
0
1110
1
10
0
0
11110
^000101011;
(switch
columns)-*876924513
(\
0
0
0
1
1
1
0
&\
0
10
0
10
111
0
0
10
110
11
K.0
0
0
1
0
1
1
0
\)
Therefore,
asbefore,
one obtains a minimum aberration split-plotdesign
asfollows:
factors
1, 2,
...,9;
whole plotdesign
2"l~kl=24"1
with
factors
2,
4,
5
and8,
andgenerating
relation
1=2458;
subplotdesign
2"2-*2=25"3
with
factors
1, 3, 6, 7,
and9,
andgenerating
relation 1=12357=12346=
3459.
Generally
speaking,
a minimum aberrationfractional
factorial
2"'kdesign
withresolution
III
orhigher
often gives us atleast
two
different
minimum aberrationsplit-plotdesigns.
In
fact,
let G
be
agenerating
matrix of such a2"'k
design.
Switch
the
columns ofG
to
get agenerating
matrixin
the
form
(I
C).
If
one column(row)
ofC
has
k\
(r)
zeros,
then the
elementary
operations ofswitching
rows and columns of(I
C)
asin
the
aboveexamplewill
lead
to
agenerating
matrixG*of a minimum aberrationsplit-plotdesign,
where
there
are nfactors
in
total,
n-k+kr\
(n-k-r+1)
factors
andfa
(I)
fractionation in
the
wholeplot,
andk-ki+l
(k+r-\)
factors
andk-k\
(-1)
fractionation in
the
subplotdesign. The
generating
relations ofthe
whole plot and subplotdesigns
ofthe
minimumaberration split-plot
design
canbe
writtendown
directly
from G*.
Of course,
the
collectionof
the
wordsappearing
in both
generating
relations,
along
withI,
constitutethe
generating
relation ofthe
originalfractional factorial
2"'k
design.
It
shouldbe
notedthat
quite often one canobtaina minimum aberration split-plot
design from
a minimumaberration
fractional factorial
2"'kdesign
with resolutionn,
andonly in
the
rare casewhereall
the
elementsofC
are equalto
1
can one notdo
so.(Of course,
suchdesigns
by
themselves
have little
usein
practice.)
Method
II
(An
algorithm)
To
get analgorithmfor constructing
the
minimum aberration split-plotdesigns,
wefirst look
at someknown
properties ofthe
wordlength patterns.As
in
the
case ofunrestricted
fractional factorial
designs,
wecansafely
assumein
the
following
thateach ofthe
w;+n2
letters
in
a 2(-"l+n2)~<-kl+kl)(GO
design
must appearin
the
defining
relation.From
Chen
(1992),
the
wordlength pattern(Ab A2,
...,Ai+(l2
)
of a2(*+n2)-(*'+*2)
(GO
design
must
satisfy
(i)
Z
A
=2k-1
, where
k=kj
+k2
.(ii)
Z
Aa-i
=2k'1or
0
.(iii)
Z
iA;
=n
2M,
wheren=nj+n2
.(iv)
Z
i2A
>2*"2[
n2+
q2(2*-l)
+2qr
+r],
where=q(2*-l)
+rwithn=n,+n2.(v)
Z
i2A
is divisible
by
2*'1.
(vi)
Z
iA2i
is
divisible
by
2*'3 .Of
course,
if
welet
(ai,
a2,
...,aB[
)
be
the
wordlengthpatternofthe
wholeplotdesign
Di,
then
we musthave
(vii) Aj
>a,-,
j=l,
2,
...,m;
Aj
>0,
j=W;+l,
/+2,
...,n;+ n2.Given
the
wordlengthpatternW=(Ai, A2,
...A,^
)
of adesign
d=2(^+"2)"(*'+*2)(GO,
define
the
aberrationfunction
f(D)
=f(W)
=A,
+K-i2k+A..222*+... +A^""1*
,
where
n=n,+n2
andk=k,
+k2.
Let
W'=(Ai', A2',
...,A^
)
be
the
wordlength pattern ofanother
design
D'. Then
the
judgment
aboutwhich ofthe two
designs is better
canbe
made
through the
aberrationfunction.
Precisely,
wehave
Theorem,
(a)
W=W'if
andonly if
f(D)
=f(D').
(b)
D
has
less
aberrationthan
D'if
andonly if
f(D)
<f(D').
A
proof ofthe theorem
is
givenin
the
Appendix.
The
basic
algorithmis
now set asfollows.
Step
1. Find
a subplotdesign
D2
sothat the
corresponding
split-plotdesign
Do=2(Bl+2)-(t1+t2)
^
is
Ukely
tQ
have
minimum aberration.Set
B=0.
Step
2. Let
W=(Ai, A2,
...,Ai)
withn=ni+n2.Find
the
solutionto the
following
linear
integer
programming
problemminimize
f(W)
subject
to
(i),
(ii),
(iii), (iv), (vii),
(4)
B<f(W)<f(Do).
If
there
is
nosolution, then
D0
has
minimumaberration,
so stop.If
there
is
asolution,
proceed
to the
next step.Step
3.
Check
conditions(v)
and(vi)
for
the
solutionW
to
(4).
If
W does
notsatisfy
(v)
or(vi),
then
setB=f(W)
and goback
to
step 2. If
the
solutionW
satisfies(v)
and(vi)
andcorresponds
to
adesign
D,
then this
design
D
has
minimumaberration,
so stop.If
the
solution
W
satisfies(v)
and(vi)
but does
not correspondto
adesign,
simply
setB=f(W)
and go
back
to
step 2.
In
the
abovealgorithm,
the
linear integer programming
methodis
usedto
find
the
possible wordlength
pattern,
which enablesthe
aberrationfunction
to
achieveits
minimumvalue.
Thus,
minimum aberrationis
obtained.(For
moredetails
onlinear integer
programming,
seeWalukiewicz
(1991).)
Understanding
ofthe
algorithmmay
be
aidedby
the
following
example.For
the
examplein Section
2,
wehave
sevenfactors
1,
2, 3,
4,
5, 6,
7. The
wholeplotdesign is
Di=25";with
three
factors
1, 2,
and3,
anddefining
relation1=123.
The
subplotcontains
four factors
4, 5,
6,
and7. We
wantto
constructaminimum aberrationsplit-plotdesign
with16 runs;
that
is,
we needto
construct a minimum aberration2(3+4)'(,+2)(GO
design,
whereGi
is
the
generating
matrix ofDi. In Section 2
weused,
asa subplotdesign,
D2*=2*"2
with
factors
4, 5,
6,
and7
andgenerating
relation1=146=567. Now
wecombineD!
andD2*to
form
aninitial
split-plotdesign
J)0=2(3+4)'(1+2}
(d)
withgenerating
relation1=123=146=567. The
solutionto the
linear
integer
programming
problem(4)
withB=0
is
W=(0, 0, 2, 3, 2, 0,
0). This W
satisfies(v)
and(vi)
and correspondsto
adesign
D
=2<3+4>-<I+2>(Gi)
withgenerating
relation1=123=146=2567.
Therefore,
this
split-plotdesign
D
has
minimum aberration.The
subplotdesign
is
24'2with
four factors
4,
5,
6,
and7,
andgenerating
relation1=146=2567.
Note
that the
above minimumaberrationsplit-plotdesign
D
cannotbe
obtainedby
using
Method
I
sincethe
minimum aberrationfractional
factorial
27'3
design has
resolutionIV
(see
Chen
andWu,
1991).
4.
SUMMARY
The
collection of split-plotdesigns
generatedby
the
matrices ofform
(2)
or(3)
is
anextension of
the
collection ofcommonly
used split-plotdesigns
representedby
the
matricesof
form (1).
The
representation of split-plotdesigns in
the
matrices of(2)
clearly
indicates
that the treatment-design
aspect of split-plotdesigns
canbe
treated
in
the
sameway
asin
the
case offractional factorial designs.
In
particular, the
concept ofaberration canbe
applied,
whichallows usto
comparedesigns
in
a natural way.We
have
presentedtwo
methods
for
constructing
minimum aberration split-plotdesigns.
Method I
canbe
done
by
hand
quiteeasily,
whileMethod II
requiresinteger
programming.Most
ofthe
results presentedin
this
paper canbe
readily
extendedto the
case ofs-level split plot
designs,
where sis
a prime.The definition
of minimum aberrations""*
fractional
factorial
designs
canbe found in
Franklin
(1984)
orChen
andWu
(1991).
The
obviousmodificationof
the
definition
givenin
this
paperwilllead
to the
definition
ofminimum aberrations-levelsplit plot
designs.
Note
that
for
any
s"'kfractional factorial
design,
its
generating
matrixcanbe
writtenin
the
form G=(I
C),
whereI
is
the
kxk
identity
matrix andC
is
akx(n-k)
matrix withits
elementsfrom
the
Galois
field
G(s)={0,
1, 2, 3,
...,s-1}
(See
Chen
andWu
(1991).),
andthat
for
s-levelsplit plotdesigns,
the
elements of
d, C2,
Bi, B2
andB3
in
the
generating
matrices(1), (2),
and(3) belong
to
G(s). Therefore it is
straightforward
to
useMethod
I.
Also
notethat the
aberrationfunction
nowis
f(D)=f(W)=An
+A,.i
sk+
A,.2
s2*+ ...+
Ai
s(n'v*,
andthe
theoremstatedin Section
3
ofthis
paperis
true
in
the
general case.However,
it
seemsto
usthat
enoughconditions about
the
wordlengthpatternsof s-levelfractional factorial designs have
notbeen found. Hence it is hard
to
use methodII.
APPENDIX
Proof
ofTheorem
2: For
(a)
the
necessity
is
obvious.So let
us considerthe
sufficiency.Suppose
f(D)
=f(D');
that
is,
(ArAi')2*
+ (A2-A2')2-2*+ ... +
(Afl-An')2"B*
=0,
We
wantto
show
W=W'. It
sufficesto
showAi=Ai'.
In
fact,
if
Ai*Ai\
wehave
|
Ai-Ai'|>l. Therefore
2-*<|(A,-A1')2l
=
\(A2-A2')T2k+...+(An-An')Znk\
<(2fc-l)2-2t+...+(2*-l)2-ni
<(2*-l)2-2*(l+2-*
+
2-2t+...
)
=
Tk.
This
contradictionshowsthat
Ai=Ai'.
Now
considerthe
necessity
part of(b).
If D
has less
aberrationthan
D',
thenthere
existssome m such
that
Ai=Ai',
A2=A2\
...,Am.1=Am.i',
Am<Am'.
Therefore
n
-nk_ /a f a \*-mk , X"* / A r a \o-ifc (f(D')-f(D))2-',*
=
(Am'-Am)2-m*+
2
(Ai'-A)2-'i=m+\
>2~mk-^
(2k-l)2"i* i=m+l>2""*
-(2*-l)(2"(m+1)*
+2<m+2)k+...
)
=
0.
For
the
sufficiency
part of(b),
supposef(D)<f(D').
We
wantto
showthat
D
has less
aberration
than
D'. If D
andD'have
the
samewordlengthpattern,
thenby
(a), f(D)=f(D'),
which
is impossible.
If
D'has less
aberration,
then,
similarto the
necessity
partin
(b),
onehas
f(D')<f(D),
whichis
alsoimpossible.
Therefore D
musthave
less
aberrationthanD'.
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G. E.
P.,
Hunter,
W. G.
andHunter,
J.
S.
(1978).
Statisticsfor
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York:
John
Wiley.
Chen,
J.
(1992).
Some
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minimumaberration
designs.
Ann.
Statist.,
20,
2124-2141.
Chen,
J.,
andWu,
C. F.
J.
(1991). Some
results ons""*
fractional factorial designs
withminimum aberration or optimal moments.
Ann.
Statist.,
19,
1028-1041.
Cochran,
W.
G.,
andCox,
G.
M.
(1957).
Experimental
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Franklin,
M. F.
(1984).
Constructing
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W.
G.
(1980).
Minimum
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(1994).
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