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HierarchyprobabilitycostanalysismodelincorporateMAIMSprincipleforEPCprojectcostestimation4.Hierarchyintegratedprobabilitycostanalysis(HIPCA)modelsforEPCcostestimation.Inthissectionweintroducehierarchyprobabilitycostanalysis(HIPCA)methodology,whichincorporatesallaforementionedconceptsfordeterminingthetotalprojectcost(TPC)ofEPCprojects.OurobjectiveistodevelopanoptimalbutrealisticTPCforagivenprobabilityofsuccess(PoS)thatweassumehasbeenspecifiedbyallocatingthebaselinebudgets,andmanagingcontingency,basedonthedesiretowintheprojectandrisktolerance.4.1.CorrelationcoefficientanditsfeasibleverificationOncehistoricaldataisavailable,twodifferentmeasuresareusedtoreflectthedegreeofrelationbetweencostelementsinliterature.Thefirstoneisanordinaryproduct-moment(Pearson)correlationcoefficientandthesecondisarank(Spearman)correlationcoefficient.Anon-parametric(distribution-free)rankstatisticproposedbySpearmanin1904asameasureofthestrengthoftheassociationsbetweentwovariables(Lehmann&D’Abrera,1998).TheSpearmanrankcorrelationcoefficientcanbeusedtogivearealestimate,andisameasureofmonotoneassociationthatisusedwhenthedistributionofthedatamakePearson’scorrelationcoefficientundesirableormisleading.Whileitmaybedifficulttojustifyuseofaspecificnumericvaluetorepresentthecorrelationbetweentwocostelements,itisimportanttoavoidthetemptationtoomitthecorrelationaltogetherwhenaprecisevalueforitcannotbeestablished.Suchanomissionwillsetthecorrelationinquestiontotheexactvalueofzero;whereaspositivevaluesofthecorrelationcoefficienttendtowidenthetotal-costprobabilitydistributionandthusincreasethegapbetweenaspecificcostpercentile(e.g.,70%)andthebest-estimatecost.Thatistosay,thecontingencycouldbelarger.Therefore,usingreasonablenon-zerovalues,suchas0.2or0.3,generallyleadstoamorerealisticrepresentationoftotal-costuncertainty.Subjectivejudgmentalsofindsapplicationinspecifyingthecor-relationsbetweencostelementsqualitatively.Tothisrespect,researcherscansubjectivelychoosetwogroupsofcorrelationstoassessstrong,moderate,andweakrelations:{0.8,0.45,0.15}(Touran,1993)and{0.85,0.55,0.25}(Chau,1995).Othermorerecentscholarsexplain,simply,‘‘asaruleofthumb,wecansaythatcorrelationsoflessthan0.30indicatelittleifanyrelationshipbetweenthevariables.’’Reasonablecorrelationvaluesintherange0.3–0.6shouldleadtomorerealisticcostestimatesthantheoverlyoptimisticvaluesassumingindependenceortheoverlypessimisticvaluesassumingperfectcorrelation(Kujawskietal.,2004).Matrixtheoryimpliesthatacorrelationmatrixwillnothaveanynegativedeterminantsinreallife.Whenacorrelationmatrixisusedinsimulation,animportantrequirementistoensureitsfeasibility,whichrestrictsthematrixtobepositivesemi-definiteregardlessofitstype(product-momentorrank)orthewayitisestimated(historicalorsubjective)(Lurie&Goldberg,1998).Beingpositivesemi-definitemeanstheeigenvaluesofthecorrelationmatrixmustbenon-negative.Thatistosay,internalconsistencycheckingbetweencostelementsisnecessaryforcostestimation.Intheliterature,ithasfrequentlyoccurredthatthecorrelationmatrixisnotpositivedefiniteasindicatedbyRanasinghe(2000).Thisisparticularlyanissuewhenthenumberofdimensionsincreasesbecausethepossibilityofhavinganinfeasiblecorrelationmatrixwillgrowrapidlyasthedimensionincreases(Kurowicka&Cooke,2001).Touran’sapproachwastoreduceallthecorrelationsslightly(say0.01)andrepeatuntilthecorrelationmatrixbecomesfeasible(Touran,1993).Thisapproachoverlooksthepossibilityofincreasingsomecorrelationswhilereducingothers.Ranasinghe(2000)developedacomputerprogramtoiterativelycalculateandlisttheboundsofeachcorrelationtomakethematrixpositivesemi-definite.Theprogramthenaskstheestimatortochangetheoriginalvaluesandwaituntiltheprogramre-checksthefeasibilityandnewbounds.Thisprocesscontinuesuntilreachingthefeasibility.Thisapproach,however,maybetimeconsumingduetoitsiterativenature.Yang(2005)developedanautomaticproceduretocheckthefeasibilityofthecorrelationmatrixandadjustitifnecessary.Itiscomplicatedanddifficulttounderstandduetodecomposingthecorrelationmatrixintoadiagonalvectoroftheeigenvalues,andnormalizationofthediagonalelementstoensureunitdiagonals.Here,weadvocatethatCrystalBallcanbeadoptedtoconducttheeigenvaluetest,onthecorrelationmatrixtouncoverthisproblem.TheprogramwarnstheuseroftheinconsistentcorrelationsasFig.2.Adjustingthecoefficientsallowstheusertoensurethatthecorrelationmatrixisatleastnotdemonstrablyimpossible.Asimpleapproachtousingthecorrelationalgorithmintheprogramistoadjustthecoefficientspermanentlyafterwritingdownwhattheywereoriginally.InthiswaytheanalystwillfindoutafterthesimulationwhatCrystalBallhadtodotothecoefficientstomakethempossible.Thisisaminimaltestanddoesnotensurethatthecorrelationcoefficientsare‘‘right’’inanysense.Afterexaminingwhattheprogramneededtodo,theriskanalyststillmusttakeresponsibilityforthecoefficientsactuallyused.4.2.DilemmaforPCAmethodologyTheonlypointvaluefromindependentconstituentdistributionsthatcanbeaddedtoobtainthecorrespondingstatisticalpointvaluefromthesumoftheconstituentdistributionsisthemeanvalue.Therefore,task-levelcontingenci