$D$-optimal saturated designs: a simulation study

0.07 MB·

by  Roberto Fontana

Chapter 1
D-optimal saturated designs: a simulation study
RobertoFontana,FabioRapalloandMariaPieraRogantin
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Abstract In this work we focus on saturated D-optimal designs. Using recent re-
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sults,weidentifyD-optimaldesignswiththesolutionsofanoptimizationproblem

withlinearconstraints.Weintroducenewobjectivefunctionsbasedonthegeomet-
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E ric structure of the design and we compare them with the classical D-efficiency
M criterion. We performa simulation study. In all the test cases we observe that de-
signs with high values of D-efficiency have also high values of the new objective
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at functions.
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1 1.1 Introduction
v
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4 The optimality of an experimental design depends on the statistical model that is
8 assumedandisassessedwithrespecttoastatisticalcriterion.Amongthedifferent
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criteria,inthischapterwefocusonD-optimality.
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1 Widely used statistical systems like SAS and R have proceduresfor finding an
0 optimaldesignaccordingtotheuser’sspecifications.Proc OptexofSAS/QC[5]
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searchesforoptimalexperimentaldesignsinthefollowingway.Theuserspecifies
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anefficiencycriterion,asetofcandidatedesignpoints,amodelandthesizeofthe
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designtobefound,andtheproceduregeneratesasubsetofthecandidatesetsothat
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X thetermsinthemodelcanbeestimatedasefficientlyaspossible.
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RobertoFontana
DepartmentDISMA,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10127TORINO,Italy,
e-mail:[email protected]
FabioRapallo
DepartmentDISIT,Universita`delPiemonteOrientale,VialeTeresaMichel11,15121ALESSAN-
DRIA,Italy,e-mail:[email protected]
MariaPieraRogantin
Department DIMA, Universita` di Genova, Via Dodecaneso 35, 16146 GENOVA, Italy, e-mail:
[email protected]
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2 RobertoFontana,FabioRapalloandMariaPieraRogantin
There are several algorithmsfor searching for D-optimaldesigns. They have a
commonstructure.Indeed,theystartfromaninitialdesign,randomlygeneratedor
userspecified,andmove,in a finitenumberofsteps, to abetterdesign.Allofthe
search algorithms are based on adding points to the growing design and deleting
pointsfromadesignthatistoobig.Mainreferencestooptimaldesignsinclude[1],
[4],[7],[8],[9]and[11].
Inthiswork,weperformasimulationstudytoanalyzeadifferentapproachfor
describingD-optimaldesignsinthecaseofsaturatedfractions.Saturatedfractions,
orsaturateddesigns,containanumberofpointsthatisequaltothe numberofes-
timable parameters of the model. It follows that saturated designs are often used
in place ofstandarddesigns,suchas orthogonalfractionalfactorialdesigns,when
thecostofeachexperimentalrunishigh.Weshowhowthegeometricstructureof
a fraction is in relation with its D-optimality, using a recent result in [3] that al-
lowsustoidentifysaturateddesignswiththepointswithcoordinatesin{0,1}ofa
polytope,beingthepolytopedescribedbyasystemoflinearinequalities.Thelinear
programmingproblemisbasedonacombinatorialobject,namelythecircuitbasisof
themodelmatrix.Sincethecircuitsyieldageometriccharacterizationofsaturated
fractions, we investigate here the connections between the classical D-optimality
criterionandthepositionofthedesignpointswithrespecttothecircuits.
In this way the search for D-optimal designs can be stated as an optimization
problemwhere the constraintsare a system oflinear inequalities.Within the clas-
sical framework the objective function to be maximized is the determinantof the
information matrix. In our simulations, we define new objective functions, which
take into account the geometric structure of the design points with respect to the
circuitsoftherelevantdesignmatrix.Westudythebehaviorofsuchobjectivefunc-
tionsandwecomparethemwiththeclassicalD-efficiencycriterion.
Thechapterisorganizedasfollows.InSect.1.2webrieflydescribetheresultsof
[3]andinparticularhowsaturateddesignscanbeidentifiedwith{0,1}pointsthat
satisfyasystemoflinearinequalities.TheninSect.1.3wepresenttheresultsofa
simulationstudyinwhich,usingsometestcases,weexperimentdifferentobjective
functionsandweanalyzetheirrelationshipwiththeD-optimalcriterion.Concluding
remarksaremadeinSect.1.4.
1.2 Circuits and saturated designs
As described in [3], the key ingredient to characterize the saturated fractions of
a factorial design is its circuit basis. We recall here only the basic notions about
circuitsinordertointroduceourtheory.Forasurveyoncircuitsanditsconnections
withStatistics,thereadercanreferto[6].
GivenamodelmatrixX ofafullfactorialdesignD,anintegervector f isinthe
kernelofXt ifandonlyifXtf =0.WedenotebyAthetransposeofX.Moreover,
we denote by supp(f) the support of the integer vector f, i.e., the set of indices
j such that f 6=0. Finally,the indicatorvectorof f is the binaryvector(f 6=0),
j j
1 D-optimalsaturateddesigns:asimulationstudy 3
where(·)istheindicatorfunction.Anintegervector f isacircuitofAifandonly
if:
1. f ∈ker(A);
2. there is no other integer vector g ∈ ker(A) such that supp(g) ⊂ supp(f) and
supp(g)6=supp(f).
ThesetofallcircuitsofAisdenotedbyC ,andisnamedasthecircuitbasisof
A
A.ItisknownthatC isalwaysfinite.ThesetC canbecomputedthroughspecific
A A
software.Inourexamples,wehaveused4ti2[10].
Given a modelmatrix X on a fullfactorial design D with K design points and
p degrees of freedom, we recall that a fraction F ⊂D with p design points is
saturatedifdet(XF)6=0,whereXF istherestrictionofX tothedesignpointsinF.
With a slight abuse of notation,F denotesboth a fractionand its support.Under
these assumptions, the relations between saturated fractions and the circuit basis
C ={f ,...,f }associatedtoAisillustratedinthetheorembelow,provedin[3].
A 1 L
Theorem1.F is asaturatedfractionifandonlyif itdoesnotcontainany ofthe
supports{supp(f ),...,supp(f )}ofthecircuitsofA=Xt.
1 L
1.3 Simulationstudy
The theory described in Sect. 1.2 allows us to identify saturated designs with the
feasible solutions of an integer linear programming problem. Let C = (c ,i =
A ij
1,...,L,j =1,...,K) be the matrix, whose rows contain the values of the indica-
torfunctionsofthecircuits f ,...,f ,c =(f 6=0),i=1,...,L,j=1,...,K and
1 L ij ij
Y=(y ,...,y )betheK-dimensionalcolumnvectorthatcontainstheunknownval-
1 K
uesoftheindicatorfunctionofthepointsofF.InourproblemthevectorY must
satisfythefollowingconditions:
1. thenumberofpointsinthefractionsmustbeequalto p;
2. thesupportofthefractionmustnotcontainanyofthesupportsofthecircuits.
Informulae,thisfacttranslatesintothefollowingconstraints:
1t Y =p, (1.1)
K
C Y <b, (1.2)
A
y ∈{0,1} (1.3)
i
whereb=(b ,...,b )isthecolumnvectordefinedbyb =#supp(f),i=1,...,L,
1 L i i
and1 isthecolumnvectoroflengthK andwhoseentriesareallequalto1.
K
Since DY =det(V(Y))=det(XFt XF) is an objective function,it followsthat a
D-optimaldesignisthesolutionoftheoptimizationproblem
maximize det(V(Y))
subjectto(1.1),(1.2)and(1.3).
4 RobertoFontana,FabioRapalloandMariaPieraRogantin
Ingeneraltheobjectivefunctiontobemaximizeddet(V(Y))hasseverallocalop-
timaandtheproblemoffindingtheglobaloptimumispartofcurrentresearch,[2].
Insteadoftryingtosolvethisoptimizationprobleminthisworkweprefertostudy
different objective functions that are simpler than the original one but that could
generatethesame optimalsolutions.ByanalogyofTheorem1,ournewobjective
functionsaredefinedusingthecircuitsofthemodelmatrix.
ForanyY, we definethevectorb =C Y.Thisvectorb containsthenumber
Y A Y
of points that are in the intersection between the fraction F identified by Y and
the supportof each circuit f ∈C ,i=1,...,L. From (1.2) we know that each of
i A
theseintersectionsmustbestrictlycontainedinthesupportofeachcircuit.Foreach
circuit f,i=1,...,Litseemsnaturaltominimizethecardinality(b ) oftheinter-
i Y i
sectionbetweenitssupportsupp(f)andYwithrespecttothesizeofitssupport,b.
i i
Therefore,weconsideredthefollowingtwoobjectivefunctions:
• g (Y)=(cid:229) L (b−b );
1 i=1 Y i
• g (Y)=(cid:229) L (b−b )2.
2 i=1 Y i
From the examples analyzed in Sect. 1.3.1, we observe that the D-optimality
is reached with fractions that contain part of the largest supports of the circuits,
althoughthisfactseemstodisagreewithThm.1.Infact,Thm.1statesthatfractions
containingthesupportofacircuitarenotsaturated,andthereforeonewouldexpect
thatoptimalfractionswillhaveintersectionsassmallaspossiblewiththesupports
ofthecircuits.Ontheotherhand,ourexperimentsshowthatoptimalityisreached
withfractionshavingintersectionsaslargeaspossiblewithsuchsupports.Forthis
reasonweconsideralsothefollowingobjectivefunction:
• g (Y)=max(b ).
3 Y
AsameasureofD-optimalityweusetheD-efficiency,[5].TheD-efficiencyofa
fractionF withindicatorvectorY isdefinedas
1 1
E = D#F ×100
Y (cid:18)#F Y (cid:19)
where #F is the number of points of F that is equal to p in our case, since we
consideronlysaturateddesigns.
1.3.1 Firstcase. 24 withmaineffects and 2-way interactions
Letusconsiderthe 24 designandthe modelwith mainfactorsand2-wayinterac-
tions.ThedesignmatrixX ofthefulldesignhas16rowsand11columns,thenum-
ber ofestimable parameters.As the matrixX hasrank 11,we search forfractions
with 11 points. A direct computation shows that there are 16 =4,368 fractions
11
with11points:amongthem3,008aresaturated,andtherem(cid:0)ai(cid:1)ning1,360arenot.
Notice that equivalencesup to permutationsof factor or levels are not considered
here.
1 D-optimalsaturateddesigns:asimulationstudy 5
Thecircuitsare140andthecardinalitiesoftheirsupportsare8in20cases,10in
40cases,12in80cases.Formoredetailsreferto[3].Thisexampleissmallenough
foracompleteenumerationofallsaturatedfractions.Moreover,thestructureofthat
fractionsreducestofewcases,duetothesymmetryoftheproblem.
For each saturated fraction F with indicator vectorY we compute the vector
b ,whosecomponentsarethesizeoftheintersectionbetweenthefractionandthe
Y
support of all the circuits, F ∩supp(f),i=1,...,140, and we consider b−b .
i Y
Recallthatbisthevectorofthecardinalitiesofthecircuits.Thefrequencytableof
b−b describeshowmanypointsneedtobeaddedtoafractioninordertocomplete
Y
eachcircuit.AllthefrequencytablesaredisplayedintheleftsideofTable1.1,while
ontherightsidewereportthecorrespondingvaluesofD-efficiency.
Table1.1 Frequencytablesofb−bY forthe24designwithmaineffectsand2-wayinteractions.
table(b−bY) EY
1 2 3 4 5 68.29 77.46 83.38
5 15 50 60 10 192 0 0
5 18 48 55 14 1,040 0 0
5 21 46 50 18 960 0 0
5 24 44 45 22 480 0 0
5 27 42 40 26 0 320 0
5 30 40 35 30 0 0 16
Total 2,672 320 16
Forinstance, consideroneof the 192fractionsin the first row.Amongthe 140
circuits, 5 ofthem are completedbyadding1 pointto the fraction,15 ofthem by
adding2points,andsoon.Weobservethatthereisaperfectdependencebetween
theD-efficiencyandthefrequencytableofb−b .
Y
However, analyzing the objective functions g (Y), g (Y) and g (Y), we argue
1 2 3
thatthepreviousfindinghasnotrivialexplanation.Thevaluesofallourobjective
functionsaredisplayedinTable1.2.
From Table 1.2 we observe that both g (Y) and g (Y) are increasing as D-
2 3
efficiency increases. Notice also that g (Y) is constant over all the saturated frac-
1
tions.Thisisageneralfactforallno-m-wayinteractionmodels.
Proposition1.Forano-m-wayinteractionmodel,g (Y)isconstantoverallsatu-
1
ratedfractions.
Proof. WerecallthatC =(c ,i=1,...,L,j=1,...,K)istheL×Kmatrix,whose
A ij
rows contain the values of the indicator functions of the supports of the circuits
f ,...,f ,c =(f 6=0),i=1,...,L,j=1,...,K.Wehave
1 L ij ij
L L L
(cid:229) (cid:229) (cid:229)
g (Y)= (b−b ) = (b) − (b ) .
1 Y i i Y i
i=1 i=1 i=1
6 RobertoFontana,FabioRapalloandMariaPieraRogantin
Table1.2 Classificationofallsaturatedfractionsforthe24 designwithmaineffectsand2-way
interactions.
g1(Y) g2(Y) g3(Y) EY n
475 1,725 9 68.29 192
475 1,739 10 68.29 960
475 1,753 10 68.29 960
475 1,739 11 68.29 80
475 1,767 11 68.29 480
475 1,781 11 77.46 320
475 1,795 11 83.38 16
Total 3,008
ThefirstaddendumdoesnotdependonY,andforthesecondoneweget
L L K K L
(cid:229) (cid:229) (cid:229) (cid:229) (cid:229)
(b ) = c Y = Y c .
Y i ij j j ij
i=1 i=1j=1 j=1 i=1
Nowobservethata no-m-wayinteractionmodeldoesnotchangewhenpermuting
the factors or the levels of the factors. Therefore, by a symmetry argument, each
designpointmustbelongtothesamenumberqofcircuits,andthus(cid:229) L c =q.It
i=1 ij
followsthat
L K
(cid:229) (cid:229)
(b ) =q Y =pq.
Y i j
i=1 j=1
⊓⊔
InviewofProp.1,intheremainingexampleswewillconsideronlythefunctions
g andg .
2 3
1.3.2 Second case.3×3×4 withmaineffects and 2-way
interactions
Let us consider the 3×3×4 design and the model with main factors and 2-way
interactions. The model has p =24 degrees of freedom. The number of circuits
is 17,994.In this case the numberof possible subsets of the full design is 36 =
24
1,251,677,700.Itwouldbecomputationallyunfeasibletoanalyzeallthefra(cid:0)ctio(cid:1)ns.
Weusethemethodologydescribedin[2]toobtainasampleofsaturatedD-optimal
designs. It is worth noting that this methodologyfinds D-optimaldesigns and not
simply saturated designs.This is particularlyusefulin ourcase becauseallows us
tostudyfractionsforwhichtheD-efficiencyisveryhigh.Thesamplecontains500
designs,380different.
1 D-optimalsaturateddesigns:asimulationstudy 7
TheresultsaresummarizedinTable1.3,wherethefractionswithminimumD-
efficiencyE havebeencollapsedinauniquerowinordertosavespace.Weobserve
Y
that for 138 different designs the maximum value of D-efficiency, E =24.41 is
Y
obtainedforbothg (Y)andg (Y)attheirmaximumvaluesg (Y)=970,896and
2 3 2
g (Y)=24.
3
Table 1.3 Classification of 380 random saturated fractions for the 3×3×4 design with main
effectsand2-wayinteractions.
g2(Y) g3(Y) EY n
≤963,008 ≤21 22.27 37
962,816 21 23.6 7
962,816 22 23.6 12
963,700 22 23.6 34
965,308 22 23.6 46
966,760 22 23.6 9
967,676 22 23.6 6
970,860 24 23.6 91
970,896 24 24.41 138
Total 380
1.3.3 Thirdcase. 25 withmaineffects
Letusconsiderthe25 designandthemodelwithmaineffectsonly.Themodelhas
p=6 degreesof freedom. The numberof circuits is 353,616.As in the previous
casewe usethemethodologydescribedin [2]togetasampleof500designs,414
different.
The results are summarizedin Table 1.4. We observethat for194 differentde-
signs, the maximumvalue of D-efficiency,E =90.48is obtainedfor both g (Y)
Y 2
andg (Y)attheirmaximumvaluesg (Y)=11,375,490andg (Y)=6.
3 2 3
Table1.4 Classificationof414randomsaturatedfractionsforthe25designwithmaineffects.
g2(Y) g3(Y) EY n
11,360,866 6 76.31 31
11,342,586 6 83.99 9
11,371,834 6 83.99 126
11,375,490 5 83.99 54
11,375,490 6 90.48 194
Total 414
8 RobertoFontana,FabioRapalloandMariaPieraRogantin
1.4 Concluding remarks
The examples discussed in the previous section show that the D-efficiency of the
saturatedfractionsandthenewobjectivefunctionsbasedoncombinatorialobjects
arestronglydependent.Thethreeexamplessuggesttoinvestigatesuchconnection
inamoregeneralframework,inordertocharacterizesaturatedD-optimalfractions
intermsoftheirgeometricstructure.Noticethatourpresentationislimitedtosat-
uratedfractions,butitwouldbeinterestingtoextendtheanalysistootherkindsof
fractions.Moreover,weneedtoinvestigatetheconnectionsbetweenthenewobjec-
tivefunctionsandothercriteriathanD-efficiency.
Since the numberof circuits dramatically increases with the dimensionsof the
factorialdesign,boththeoreticaltoolsandsimulationwillbeessentialforthestudy
oflargedesigns.
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