Indicator polynomial
function
for multilevelfacto
rialdesigns
GiovanniPistone
DIMA T –P
olitecnicodi T orino
–giovanni.pistone@polito.it
&
Maria PieraRogantin
DIMA–
Universit`a diGenova
–rogantin@dima.unige.it GROST A TVI
Menton
The description
of fractional
factorial designs
using the
polynomial
representations of
their indicator
functions has
been introduced
for
binary designsin
Fontana, R.,
Pistone, G.
and Rogantin,
M.
P . (2000).
Classification of tw o-level
factorial fractions,J.
Statist.
Plann.
Inference 87(1), 149–172.
andgeneralized toreplicates
by
Y e, K.Q.
(2003).
Indicator functionand
itsapplication int
wo-level factorial
designs,
TheAnnals ofStatistics
. Inp ress.
Fo ra similar
approach seealso:
T ang, B.and
Deng,L.
Y.(1999).
MinimumG -aberration 2
for nonregular
fractinal
factorial designs,The
Annalsof Statistics
27(6), 1914–1926.
Herew egeneralize
tomultilevel factorial
designs, representing
the
levelsof afacto
asthe r elementsof
themultiplicative groupof
complex
roots ofunit
y,generalizing allthe
prop ertiesalready
known for
binary
designsto thecase
wherethe numbers
oflevels are
prime numbers.
Complexco dingfo
rfull factorial
designs
Notations:
- number mthe
offacto rs
n - thenumb j
erof levelsof
eachfacto r,j
,. =1 ..
, ,m
n with ap j
rimenumb er.
We code
the levels n
of a factor
with A the
complex solutions
ofthe equationζ
=1: n
k ω
=exp
2π i k n
for 0,. k=
..
−1 ,n
- [k ] theresidue n
ofk mod
expecially n;
,fo rinteger
(ω h, ) k
=ω h [hk
]
n
The mapping
n Z 3
↔ k exp
2π i k n
is a group
isomorphism on
the
The full
factorial design
D, as
a subset
of
m C with
= N
Q m j=1 j n
points, isdefined
by thesystem
ofequations
n ζ
j
−1 j
=0 for
1,. j=
..
,m
Afraction isa
subset F⊆
D;
allthe fractionsa
reobtained by
adding
equations( generatingequations
)to restrictthe
setof solutions.
Example
0 ω
0 ω
0 ω
0 ω
0 ω
1 ω
1 ω
1 ω
0 ω
2 ω
2 ω
2 ω
1 ω
0 ω
1 ω
2 ω
1 ω
1 ω
2 ω
0 ω
1 ω
2 ω
0 ω
1 ω
2 ω
0 ω
2 ω
1 ω
2 ω
1 ω
0 ω
2 ω
2 ω
2 ω
1 ω
0 ω
Theclassical
4−2 3 regular III
fractionis definedb
y
3 j ζ
−1
=0 for
1,. j=
..
,4
togetherwith thegenerating
equations
1 ζ
2 ζ
3 ζ
−1
=0 and
1 ζ
2 2 ζ
4 ζ
−1
=0
Sucha representation
isclassically termed“multi-
plicative”notation.
Inour approach
theequations
are definedon
thecomplex field
. C Thereco
ding
mapsb etw
eenthe residueclass
group
3 Z andthe
field . C
Responses
Acomplex response
onthe f design
Dis a
-valuedfunction C defined
D. on Itis
arestriction Dof to
acomplex polynomial.
L= -
n
(α α=
,. 1
..
,α ): m
j α ,. =0
..
j ,n
−1
=1 ,j ,.
..
,m
o
X - thei i
-thcomp onentfunction,
mappinga designp
oint(
1 ζ ,.
..
m ,ζ )
intoits component i-th
i ζ . Frequently calledfacto
r.
- Interactionterms
or monomialresp
onses
α X
withα α =(
,. 1
..
m ,α α ),
∈L X :
:=X α
1 α 1
··
α ·X
m m
α X D3 :
1 (ζ ,.
..
,ζ ) m
7→
α ζ
1 1
··
α ·ζ
m m
- Theterm
α X haso
k rder ifin them
-tupleα therea
rek non-null
values:
α X isan interactionof
order k
(inthe binary
casethe order
isequal tothe
degree)
Responses onthe
design
- Eachresp
onsef isrep
resentedas anunique
linear combinationof
constant,effects, interactions:
f=
P
θ α∈L
X α
, α
θ
∈ α
C
- Meanvalue
f of D, on
D E (f E ):
(f D 1 N )=
P
∈D ζ
f(
ζ)
- Acontrast isa
response suchthat f
D E (f )=
0.
- Two
responses andg f
are orthogonal
Dif on
D E g) (f
=0.
Theset ofall
responses isa
complexHilb ertspace
withscala rp
roduct
<f ,g
>=
D E g) (f
Basicp roperties
connectingthe algebra
withthe Hilbert
structure:
1.
α X
β X
[α =X
−β
,where ]
[·]
denotesthe modulo
operation extended
toL
;
2.
D E
0 (X )=
1,and
D E
α (X )=
0fo
6= rα
0;
3.
D E
α (X
β X )=
D E
[α (X
−β
)= ]
1 ifα
=β
0
6= ifα
β
Responses onthe
fraction
Algebraic methods
allow to
find bases
of the
vectorial space
of re-
sponses F. on
We denotesuch
basesb yEst (F), τ
whereτ isan
ordering onthe
terms,
andb (F) yC
theset ofall
(complex)functions identifiedon
F:
Est ( τ
F)
=
n β X
∈M :β
o
and (F) C
=
X
∈M β β θ
β X ,
β θ
∈ C
The vector
space (D) C
of the
responses D on
can be decomposed
into tw o orthogonal sub-spaces:
the space
(F) C of
the identifiable
responses F on
andthe spaceof
thenull responses
F, on see
Galetto F.,
Pistone G.
and Rogantin
M.
P . (2003).
Confounding revisited
with
commutativecomputational algebra,
J.Statist.
Plann.
Inference Inp ress.
Indicator function
Theindicato rfunction
of F F isa particula
rresp onsesuch
that
F(
= ζ)
1
∈F ifζ
0
∈D ifζ F r
Ap olynomialF
isan indicator
functionof somefraction
Fif andonly
2 ifF
−F
=0 D. on
Itis represented
as
F=
X α∈L
b X α α
Notethat b
=b α [−α
because ]
isreal F valued.
Meanvalue ofa
response on f
F
F E (f 1 )=
#F
X
∈F ζ
f(
= ζ) 1
#F
X
∈D ζ
F(
ζ) f(
= ζ) N
E #F (F D
f) .
α IfX
β andX are simpleo
rinteraction termsthen:
E 1.
(X F
)= α N #F
α b
E 2.
(X F
X α
)= β F E
[β (X
−α
)= ] N
b #F
−α [β ]
Contrastsand orhogonalities
F on
- Acontrast F on
isa response
suchthat f
F E (f )=
0.
- Two
responses andg f
are orthogonal
F on
F ifE g) (f
=0.
Mainresults: Orthogonalities
Generalization of Fontana, Pistone,
Rogantin (2000)
to the
case of
mixedfractional factorial
designs.
Impo rtantstatistical
featuresof thefraction
canb eread outfrom
the
form ofthe
polynomial representation
ofthe indicator
function.
1.
Asimple termo
ran interactionterm
α X isa contraston
Fif and
onlyif b
=b α [−α
=0 ]
.
2.
Two simple
or interaction
terms
α X X and
are β
orthogonal F on
ifand onlyif
[α b
−β
=b ]
−α [β
=0 ]
;
3.
X If is α
a contrast
then, for β any
γ and such
α= that
−γ [β ]o r
[γ α=
−β ],
β X iso rthogonalto
γ X .
Mainresults: T yp
esof fractions
1.
F If F and
are 0
complementary fractions
b and and α
0 b are α
the
coefficients of
the respective
indicator functions,
b then
=1 ∅ 0 −b
∅
b and
= α 0 −b
α
2.
Lb Let ethe setof
indices
∈L α suchthat
b
6= α
0and l= let
#L.
Fis Then regular
ifand onlyif
thenon-zero coefficients
are ofthe
form:
b
= α
k ω
. l
Regular fractions(p
roof )
Hb Let ea sub-groupof
#H= Land h.
LetΩ be n
theset ofthe
roots ofthe
unit:
Ω
= n
{ω ,. 1
..
,ω }. n−1
Lete H→ :
Ω be n
ahomomo rphism:
e([
α+
=e β]) (α )e (β )
α LetX
−e (α ),
∈H α ,b ethe systemof
definingequations ofa
regular fraction
F.
ThenX (A α
)=
e(
for α)
∈H allα ifand
onlyif
∈F A . Thatis:
X
α∈H α (X
(A
−e ) (α (X ))
(A α
−e ) (α )) h =2
− X
α∈H
e(
α)
α X (A )
! , =0
1 h X
α∈H
e(
α)
α X (A
−1 )
=0 ifand
onlyif
∈F A
Thefunction
1 G=
h
P e( α∈H
α)
α X isan indicator
function,b ecauseG
2 =G D. on
Infact ahomomo eis
rphismand:
2 G 1 =
2 h X
α∈H
X
∈H β
e(
α) e(
β)
[α X
+β
= ]
1
2 h X
∈H α
X
β∈H
e([
α+
β])
[α X
+β
= ]
1
2 h X
∈H γ
(γ he
γ )X
=G.
ThenG isthe
indicator functionof
F, Hequals Land to
b
= α e(
,fo α)
rall
∈H α .
Viceversa,let 1 F=
l X
α∈L
e(
α)
α X
be theindicato
rfunction F. of
We have
X
α∈L α (X
−e (α (X ))
−e α
(α ))
=2 1 l
1 − l X
∈L α
e(
α)
α X
!
=
0 F on
2l D on
F r .
Inconclusion
α X
=e (α
∈L ),α ,is theset
ofthe definingequations
F. of
Theset Lis
closedfo r[
.]
;in n
factif
∈L α,β ,then
[α X
+β
=e ]
(α )e (β )=
e([
α+
β])
Example
We consideras
examplethe classical3
−2 4 III
fraction,considered befo
re.
Itsindicato rfunction
is:
= F 1
9
1+
2 X
3 X
4 X
2 2 +X
2 3 X
2 4 X
1 +X
2 X
2 3 X
2 +X
2 2 X
3 X
1 +X
2 2 X
4 X
2 +X
2 X
2 4 X
1 +X
3 X
2 4 X
2 +X
2 3 X
4 X
We checkthat
itis aregula
rfraction.
Generationof fractions
Thefollo wingresult
shows how
toderive algebraic
equationsdescribing
theindicato rfunction
ofa fractionwith
agiven statisticalp
roperties.
Theco efficientsb
ofthe α
indicator functionof
Fa rerelated
according
to:
b
= α X
[β +γ ]=α β b b
γ
Bibliography
- Fontana, R.,Pistone,
G.and Rogantin,M.
P . (2000).
Classificationof tw
o-level
factorial fractions,J.
Statist.
Plann.
Inference 87(1), 149–172.
- Galetto,
F., Pistone,
G.
and Rogantin,
M.
P . (2003).
Confounding revisited
withcommutative computationalalgeb
ra,J.
Statist.
Plann.
Inference Inp ress.
- Pistone,
G., Riccomagno,
E.
and Wynn,
H.
P . (2001).
Algebraic Statistics:
ComputationalCommutative Algebra
inStatistics ,Chapman&Hall.
- T ang, B.and
Deng,L.
Y.(1999).
MinimumG -aberration 2
for nonregular
fracti-
nalfacto rialdesigns,
TheAnnals ofStatistics
27(6), 1914–1926.
- Wu, C.F.
J.and Hamada,M.
(2000).
Experiments ,John
Wiley&
SonsInc.,
New Y o rk.
Planning, analysis,
and parameter design
optimization, A Wiley-
IntersciencePublication.
- Y e, K.
Q.
(2003).
Indicator function
and its Application
in tw o-level
factorial
designs,The Annalsof
Statistics . Inp ress.
