2-level fractional factorial designs which are the

arXiv:0710.5838v1[stat.ME]31Oct20072-LEVELFRACTIONALFACTORIALDESIGNSWHICHARETHEUNIONOFNONTRIVIALREGULARDESIGNSROBERTOFONTANAANDGIOVANNIPISTONEAbstract.Everyfractionisaunionofpoints,whicharetrivialreg-ularfractions.Tocharacterizenontrivialdecomposition,wederiveaconditionfortheinclusionofaregularfractionasfollows.LetF=PαbαXαbetheindicatorpolynomialofagenericfraction,seeFontanaetal,JSPI2000,149-172.RegularfractionsarecharacterizedbyR=1lPα∈LeαXα,whereα7→eαisangrouphomeomorphismfromL⊂Zd2into{−1,+1}.TheregularRisasubsetofthefractionFifFR=R,whichinturnisequivalenttoPtF(t)R(t)=PtR(t).IfH={α1...αk}isageneratingsetofL,andR=12k(1+e1Xα1)···(1+ekXαk),ej=±1,j=1...k,theinclusionconditionintermofthebα’sis(*)b0+e1bα1+···+e1···ekbα1+···+αk=1.Thelastpartofthepaperwilldiscusssomeexamplestoinvestigatethepracticalapplicabilityofthepreviouscondition(*).ThispaperisanoffspringoftheAlcotra158EUresearchcontractontheplanningofsequentialdesignsforsamplesurveysintourismstatis-tics.1.IntroductionWeconsider2-levelfractionaldesignswithmfactors,wherethelevelsofeachfactorarecoded−1,+1.ThefullfactorialdesignisD={−1,+1}mandafractionofthefulldesignisasubsetF⊂D.Accordingtothealgebraicdescriptionofdesigns,asitisdiscussedin[7],[6],thefractionidealIdeal(F),alsocalleddesignideal,isthesetofallpolynomialswithrealcoefficientsthatarezeroonallpointsofthefraction.TwopolynomialsfandgarealiasedbyFifandonlyiff−g∈Ideal(F)andthequotientspacedefinedinsuchawayisthevectorspaceofrealresponsesonF.Thefractionidealisgeneratedbyafinitenumberofitselements.Thisfinitesetofpolynomialsiscalledabasisoftheideal.basesarenotuniquelydetermined,unlessveryspecialconditionsaremet.AGr¨obnerbasisofthefractionidealcanbedefinedaftertheassignmentofatotalorderonmonomialscalledmonomialorder.Ifamonomialorderisgiven,itispossibletoidentifytheleadingmonomialofeachpolynomial.Asfarasapplicationstostatisticsareconcerned,aGr¨obnerbasisischaracterizedbythefollowingproperty:thesetofallmonomialsthatarearenotdividedbyanyoftheleadingtermoftheDate:PresentedbyR.FontanaattheDAE2007Conference,TheUniversityofMem-phis,November2,2007.12R.FONTANAANDG.PISTONEpolynomialsinthebasisformalinearbasisofthequotientvectorspace.Ageneralreferencetotherelevantcomputationalcommutativealgebratopicsis[2].Theringofpolynomialsinmindeterminatesx1...xmandrationalco-efficientisdenotedbyR=Q[x1...xm].ThedesignidealIdeal(D)hasaunique‘minimal’basisx21−1,...,x2m−1,whichhappenstobeaGr¨obnerbasis.Thepolynomialsthatareaddedtothisbasistogeneratetheidealofafractionarecalledgeneratingequations.Anidealwithabasisofbi-nomialswithcoefficients±1iscalledbinomialideal.Indicatorpolynomialspolynomialsofafractionwereintroducedin[3],seealso[9].Anindicatorpolynomialhastheform(1)F=Xαbαxα,α=(α1,...,αm)∈{0,1},xα=xα11···xαddanditsatisfiestheconditionsF(a)=1ifa∈F,F(a)=0otherwise.Ifnecessary,wedistinguishbetweentheindeterminatexj,thevalueajandthemappingXj(a)=aj.Howtomovebetweentheidealrepresentationandtheindicatorfunctionrepresentation,isdiscussedin[5].Thedefinitionandcharacterization,fromthealgebraicpointofview,ofregularfractionalfactorialdesigns(brieflyregulardesigns)isdiscussedin[3],seealso[9].Inparticular,thelastpaperreferredtoconsidersmixedfactorialdesign,butthiscaseisoutsidethescopeofthepresentpaper.Orthogonalarraysasaredefinedin[4]canbecharacterizedinthepreviousalgebraicframework,see[9]and[1],asfollows.AfractionFwithindicatorpolynomialFisorthogonalwithstrengthsifbα=0if1≤|α|≤s,|α|=Pjαj.Thenotionofindicatorpolynomialcanbeaccommodatedtocaseswithreplicateddesignpointsbyallowingintegervaluesotherthan0and1toF,see[11].Insuchacase,weprefertocallFacountingpolynomialofthefraction.Asystematicalgebraicsearchoforthogonalarrayswithreplicationsisdiscussedin[1].ForsakeofeasyreferenceinSection5below,wequoteacoupleofspecificresultaboutorthogonalarrays.Infact,consideringm=5factorsandstrengths=2,itisshownin[1,Table5.2]thatthereare192OA’swith12pointsandnoreplications,andthereare32OA’swith12points,oneofthemreplicated.Thispaperisorganizedasfollows.InSection2thealgebraictheoryisreviewedandinSection3itisappliedtotheproblemoffindingfractionsthatareunionofregularfractions.InSection5theimportantcaseofPlackett-Burmandesignsisconsidered.2.RegularfractionsAccordingtothedefinitionsin[3]and[8]aregularfractionisdefinedasfollows.LetLbeasubsetofL=Zm2,whichisanadditivegroup.LetΩ2bethemultiplicativegroup{−1,+1}Definition2.1.LetebeamapfromLtoΩ2.Anon-emptyfractionFisregularifUNIONOFREGULARFRACTIONS3(1)L⊂Lsasub-group;(2)theequationsXα=e(α),α∈LdefinethefractionF,i.e.areasetofgeneratingequations.Insuchacase,eisagrouphomeomorphism.Otherknowndefinitionsareshowntobeequivalenttothisonebythefollowingproposition.Theorem2.1.LetFbeafraction.Thefollowingstatementsareequivalent:(1)ThefractionFisregularaccordingtodefinition2.1.(2)TheindicatorfunctionofthefractionhastheformF(ζ)=1lXα∈Le(α)Xα(ζ),ζ∈D.whereLisagivensubsetofLande:L→Ω2isagivenmapping.(3)Foreachα,β∈LtheinteractionsrepresentedonFbythetermsXαandXβareeitherorthogonalortotallyaliased.(4)TheIdeal(F)isbinomial.(5)Fiseitherasubgrouporalateralofasubgroupofthemultiplicativeg