Measuring scientific contributions with modified fractional counting

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Contents lists available atScienceDirect

Journal of Informetrics

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o i

Regular article

Measuring scientiﬁc contributions with modiﬁed fractional counting

Gunnar Sivertsen

^a

, Ronald Rousseau

^b,c

, Lin Zhang

^c,d,∗

aNordicInstituteforStudiesinInnovation,ResearchandEducation,Oslo,Norway

bUniversityofAntwerp,FacultyofSocialSciences,B-2020Antwerpen,Belgium

cCentreforR&DMonitoring(ECOOM)andDept.MSI,KULeuven,Belgium

dSchoolofInformationManagement,WuhanUniversity,China

a rt i c l e i n f o

Articlehistory:

Received28August2018

Receivedinrevisedform20March2019 Accepted20March2019

Keywords:

Scientiﬁccontributions Researchoutput Bibliometrics Fullcounting Fractionalcounting Harmoniccounting Modiﬁedfractionalcounting Sensitivityparameter Performanceindicators Rankings

LeidenRanking WebofScience NorwegianScienceIndex

a b s t ra c t

Wedevelopandproposeanewcountingmethodattheaggregatelevelforcontributions toscientiﬁcpublicationscalledmodiﬁedfractionalcounting(MFC).Weshowthat,com- paredtotraditionalcomplete-normalizedfractionalcounting,iteliminatestheextreme differencesincontributionsovertimethatotherwiseoccurbetweenscientiststhatmainly publishaloneorinsmallgroupsandthosethatpublishwithlargegroupsofco-authors.

Asanextrabenefitwefindthatscientistsindifferentareasofresearchturnouttohave comparableaveragecontributionstoscientificarticles.Wetestthemethodonscientistsat Norway’slargestuniversitiesandfindthat,atanaggregatelevel,itindeedsupportscompa- rabilityacrossdifferentco-authorshippracticesaswellasbetweenareasofresearch.MFC istherebyusefulwhenevertheresearchoutputfrominstitutionswithdifferentresearch profilesarecompared,ase.g.,intheLeidenRanking.Finally,asMFCisactuallyafamilyof indicators,dependingonasensitivityparameter,itcanbeadaptedtothecircumstances.

1. Introduction

Thestatistics,evaluation,andfundingofresearchisoftenbasedonabibliometricquantificationofthecontributionsof differentactors(authors,institutions,countries).Yet,countingmethodsnotonlyrepresentpurelybibliometricormathemat- icalproblems:theycan,moreover,stronglyaffectdecision-makingandresourceallocationinresearch.Ourstudyfocuseson oneofthemostwidespreadapplicationsofbibliometrics:methodsforcountingscientificarticles.Onanempiricalbasis,we askhowwellthetraditionalcountingmethodsrepresenttherealityofscientificcontributionsandweofferanewsolution, calledmodifiedfractionalcounting(MFC).

Themostwell-knownandwidespreadcountingmethodsbasedonarticledata,arefullcountingandfractionalized counting.Fullcountinggiveseachcontributingauthoronecredit,i.e.,ﬁveauthorsequalsﬁvecredits.Fractionalcounting assignsafractionofonecredittoeachauthor(Egghe,Rousseau,&VanHooydonk,2000;Osório,2018;Waltman&vanEck,

∗ Correspondingauthorat:CentreforR&DMonitoring(ECOOM)andDept.MSI,KULeuven,Belgium.

E-mailaddress:zhanglin[email protected](L.Zhang).

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Table1

Countingmethodsforscientiﬁcpublications.

Eachcontributorreceivessomecredit Somecontributorsmaynotreceiveanycredit Creditsarenaturalnumbers Complete,alsoknownasfullornormalcount Firstauthorcount;Majorcontributioncount Creditsmaybefractions,summingtoone Complete-normalizedfractionalcounting;

Harmoniccounting

Fractionalizedmajorcontributioncount Creditsmayevenbeirrationalnumbers

(neverlargerthanone),possiblysumming toanumberlargerthanone

Modiﬁedfractionalcounting(MFC) (Notrelevant)

2015;Waltman,2016).Attheaggregatelevel,organizationsorcountriesmaybecreditedaccordingtothenumberofauthors afﬁliatedwiththem,oronlyonceforeachuniquecontributingorganizationorcountry.

Bothfullandfractionalcountingmethodscanbeusefulastheyprovideinformationfromdifferentperspectives,e.g., participation(fullcounting)versuscontribution(fractionalcounting)(Moed,2005).Forseveralreasons,includingtheneed fornormalizedindicatorsacrossfieldsofresearch,fractionalcountingisoftenpreferredinprofessionalandscientificbiblio- metricsoperatingatanaggregatelevel(Waltman,2016).Fromanaggregateperspective,fractionalcountingaddsuptothe samenumberofarticlesasareinthedata,whichprovidesbalance,consistency,andprecisioninadvancedfield-normalized bibliometricmeasurements(Waltman&vanEck,2015).Fullcountingismorewidespreadattheindividuallevel.Theh- indexintroducedbyHirsch(2005)isagoodexampleoffullcounting.(Although,someh-indexvariationstakethenumber ofauthorsintoaccount,suchase.g.,(Chai,Hua,Rousseau,&Wan,2008)

Ourstudyaimsatimprovingcountingmethodsattheaggregatelevel.Thefocusistherebyonfractionalcountingand howitrepresentsscientiﬁccontributions.Wewill,however,alsopresenttheresultsfromfullcountingthroughoutthis studysinceweareinvestigatingtheeffectsofintermediatesolutionsbetweenfullandfractionalcounting.

Countingmethodsforscientiﬁcarticlescanalsobeclassiﬁedaccordingtowhetherornoteveryauthorreceivescredit.

CombiningtraditionalclassiﬁcationsystemsandthenewMFCsolutioncreatesthealternativesshowninTable1,whichis anelaborationofasimilartablepresentedanddiscussedinRousseau,Egghe,andGuns(2018).

Inthisstudy,weseearticles,andhowtheyarecounted,asrepresentationsofcontributionstoscientificwork,notjustas contributionstothescientificliterature.Accordingly,weagreethatacountingprocedurecanbeseenasanestimationmethod todeterminecontributionsofscientistsor,onahigherlevel,institutionsorcountries(Eggheetal.,2000).Therefore,weonly focusonmethodsthatprovidesomecredittoallauthors,eliminatingmethodsinthelastcolumnofTable1.Moreover, wedonotfocusonindividualauthors–whichcouldimplytakingtheorderinthebylineintoaccount–butonahigher aggregationlevel,suchasorganizationsorcountries,wherewefindthattheorderofauthorsdonotaffecttheresults.For thisreason,ourstartingpointisgivingequalcredittoco-authors.

Table1separatesbetweenso-calledcomplete-normalizedfractionalcounting,bywhicheachco-authorreceives1/N ofonecredit,andharmoniccounting,aseldom-usedvariantoffractionalcounting,wheretherankofco-authorsistaken intoaccounttoweighttheircontribution.Forourpurposes,theterm“fractionalcounting”generallyreferstocomplete- normalizedfractionalcounting,althoughwehaveincludedabriefanalysisofharmoniccountinginourresultsinSection 8.

Countingmethodsareimportantbecausetheyareknowntomeasureperformancedifferentlyandresultindifferent rankings(Aksnes,Schneider,&Gunnarsson,2012;Eggheetal.,2000;Gauffriau&Larsen,2005;Gauffriau,Larsen,Maye, Roulin-Perriard,&vonIns,2007; Martin,1994).Awell-knownexampleisprovided bytheCWTSLeidenRanking¹ – a worldwiderankingsystemforuniversitiesbasedonscientificcontributionandimpact.Thedefaultrankingmethodis complete-normalizedfractionalcounting.Theworld’slargestuniversitiesarerankedaccordingtothevolumeofscientific contributions.Bythiscountingmethod,ZhejiangUniversityandShanghaiJiaoTongUniversityarerankedthirdandfourth in2018.However,inswitchingtothefullcountingalternativebyunticking“Calculateimpactindicatorsusingfractional counting”,thetwouniversitiesarenowrankedintheoppositeorderasnumbersnineandten.Inourview,itisnotsufficient tosaythatthefirstmeasurementrepresentscontributionwhilethesecondrepresentsparticipation.Bothareperceivedand usedasindicatorsofscientificoutputattheaggregatelevel.Ourstudywilldemonstratethatthedifferentresultsmayjustas wellreflectdifferentresearchprofilesasdifferenttendenciestocontributetoresearch,beitatthelocallevelorworld-wide.

Asstatedinitially,thestartingpointandconcernofourstudyisthatbibliometriccountingmethodsnotonlymatter totheﬁeldofbibliometrics,theyalsomatterinreallifebecausetheyprovidefeedbackandincentivestoscientists.Full countingstimulatescollaborationinresearchandthepossibleadditionofmore(unnecessary)authorswhilefractional countingprovidesbalancedand precisedata,but itcanalsoactasadisincentivetocollaboration(Bloch&Schneider, 2016).Complete-normalizedfractionalcountinghasbeenusedasthebasisforofﬁcialstatisticsforalongtime,e.g.,inthe annual“Science&EngineeringIndicators”reportbytheNationalScienceFoundation(USA)²and“TheScience,Researchand InnovationPerformanceoftheEU”(SRIP)reportbytheEuropeanCommission.³Additionally,severalEuropeancountriesuse

1 http://www.leidenranking.com/

2 https://www.nsf.gov/statistics/2018/nsb20181/.

3 https://ec.europa.eu/info/sites/info/ﬁles/srip-report-full2018en.pdf.

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bibliometricindicatorsintheiruniversityfundingformulas(Jonkers&Zacharewicz,2015).Forexample,Flanders(Belgium) providesfundingattheinstitutionallevelbasedonthefullcountingofarticles(Debackere&Glänzel,2004),asdoesCroatia andEstonia(Debackere,Arnold,Sivertsen,Spaapen,&Sturn,2018).Denmark,Finland,Norway,andSwedenusecomplete- normalizedfractionalcountingtodetermineinstitutionalfundinglevels(Sivertsen,2016a;Vetenskapsrådet,2014).

ReturningtotheexampleofthetwoChineseuniversitiesgoingupanddownintheLeidenRanking,itisnosurprisethat bibliometriccountingmethodshavebeenquestionedononeofthemostinﬂuentialresearchpolicyblogsinChina.⁴China’s sizeasascienceproducerversustheUSdependsonthecountingmethodused.Interestingly,bothtypesofcountingmethods arequestionedintheblog:fractionalcountingforunderestimatingscientiﬁccontribution,fullcountingforoverestimating it.Consequently,theblog’sauthor,YishanWu,launchedachallenge:“Cansomeoneboldlyproposeanewintermediate countingmethodbetweenfullcountingandfractionalcounting?”

Inthisstudy,wedonotfollowthehabitofstatingthatfractionalcountingisprobablypreferableformostpurposes.

Instead,we admitthatthetwo alternativesmayseem confusingfromperspectivesoutsidebibliometrics,suchasthe policy-makingperspectiveortheperspectiveofindividualscientistswhoseetheirpublicationscountedasfractions.We acknowledgethatbibliometricsisnotmerelyusedtorepresentandmodelscientificliterature.Theuseofbibliometricsis widespreadasawaytorepresent,support,andassessreal-liferesearchactivitythathasbeen,orwillbe,reportedinthe scientificliterature.Justasreferencesareusedtostudycitationimpactortheinfluenceofresearch,publicationsareused tostudyoutputsofresearch,researchprofiles,collaboration,andsoon.Bibliometricsneedstotaketherealityofscientific workintoaccount.Hence,wecometothequestion:

Howdoesthefullandfractionalcountingofarticlesrepresentreal-worldcontributionstoscientiﬁcresultsatanaggregate level?

Anotherreasonforaskingthisquestionisthattheempiricalevidencerevealedinourstudyshowsthat:

•Withfullcounting,scientiﬁcﬁeldsanddepartmentswhosescientistsfrequentlypublishwithahighnumberofco-authors seemtocontributemore.

•Withfractionalcounting,ﬁeldsanddepartmentswhosescientistsmostlypublishaloneorwithasmallnumberofco- authorsseemtocontributemore.

Thesameobservationshavebeenmadeatanaggregatelevel(Aagaard,Bloch,&Schneider,2015;Piro,Aksnes,&Rørstad, 2013).Forus,thisleadstoanotherquestionthathasnotyetbeenaskedinthebibliometricliterature:

Canfractionalcountingmethodsbemodiﬁedsothattheirresultsleadtocomparableaveragecontributionsacrossallﬁelds andallco-authorshippractices?

Thisisthecorequestionofourstudy.WewillarguethatModiﬁedFractionalCounting(MFC)providesanafﬁrmative answertothisquestion.

2. Representingcontributionstoscientiﬁcworkinpublications

Asstatedintheintroduction,weseearticles,andhowtheyarecounted,asrepresentationsofcontributionstoscientific work,notjustascontributionstothescientificliterature.Inthissection,wepresentmorereasonsformodifyingawell- establishedfractionalcountingprocedure.Wethinkacountingmethodshouldnotonlytrytorepresentthedatainthebest possibleway,butalsobevalidwithregardtowhatismeasured:contributionstoscientificwork.

Doingresearchisnotthesameaswritinganarticle.Apublicationissimplya“formalized”representationofcontributions toscientificresults. Withincreasingcollaborationinresearch,andincreasingnumbersofauthorsperarticle,studying howtoreplacethetraditionalone-authormodelwithacontributormodeltoshowhowascientificworkiscreated,has becomeapressingissue(Cronin,2001;Rousseauetal.,2018,p.32).Weneedtobeopen-mindedtowardsfindingouthow collaborativeworkinscienceisactuallypracticedandthenrepresentedinthebibliographicalinformationofanarticle.

Althoughtheremaybegainsresultingfromresearchcollaboration,thesegainsalsocomewithcosts(Katz&Martin,1997).

Representingscientiﬁccontributionsasfractionsofarticlesdependingonthenumberofauthorsmaynotbeagoodsolution.

Thecontributionsofthecollaboratingresearchersmaybeoverlappingandnottheresultofdisjointactions.Collaboration initselfdemandscoordinatingthecontributions.Inbiomedicaljournals,suchasBMJ,⁵authorsareaskedtodeclaretheir rolesandresponsibilitiesintheresearchthattookplacebeforepublishing,notjusttheircontributionstothewriting.And evenwhenitcomestowritingandpublishingtheﬁnalversion,overlappingresponsibilitiesmaybetherule.Forinstance, the“RecommendationsfortheConduct,Reporting,Editing,andPublicationofScholarlyWorkinMedicalJournals”bythe InternationalCommitteeofMedicalJournalEditors(ICMJE),whichBMJcontributestoandfollows,⁶requirethateachauthor mustnotonlymakeasubstantialcontributiontotheworkbutalsoapprovetheﬁnalversionforpublicationandagreetobe

“accountableforallaspectsoftheworkinensuringthatquestionsrelatedtotheaccuracyorintegrityofanypartofthework areappropriatelyinvestigatedandresolved”.Inaddition,butnotmentionedasarequirementfromICMJE,cometheusual

4http://blog.sciencenet.cn/blog-1557-1122817.html.

5https://www.bmj.com/about-bmj/resources-authors/article-submission/authorship-contributorship.

6http://www.icmje.org/news-and-editorials/updatedrecommendationsdec2017.html.

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tasksoforganizingtheprojectformally,creatingtheinfrastructure,providingfunding,andreportingbacktothefunding organizations.

TherecommendationsoftheICMJEseemincompatiblewithfractionalcountingasabibliometricpractice.Itis,however, commonknowledgeandwell-documented(Maruˇsi ć,Boˇsnjak,&Jeronˇci ć,2011)thattheICMJErecommendationsareseldom followedcompletelyandoftenevendisregarded.Butthisisnotanargumentinfavoroffractionalcounting.Rather,these recommendationsremindusthatacollaborativeresearchprocesswouldnotbepossiblewithoutinteraction,influence,and agreementsamongthecontributors.Theseagreementshaveconsequencesforreceivingco-authorshipcredit.

Thecontributionsofscientiststoco-authoredpublicationsmaynotberealisticallyrepresentedbythefractiongiven bydividingthepublicationbythenumberofco-authors.ThisisthemotivationbehindourdesignandtestingoftheMFC method.Still,ourconcernisnothowindividualresearchersshouldbecreditedforpublications.Wefocusonconsequences ontheaggregatelevelofusingdifferentcountingmethodsforcomparingorganizationsorcountrieswithdifferentresearch proﬁles.

3. Data

Ourstudycombinestheuseoftwonationalleveldatabases.Oneofthemisthe2018NationalCitationReport(NCR)for Norway,deliveredbyClarivateAnalytics,whichcoversallpublicationsintheWebofScience(WoS)1981–2017withat leastoneauthor’saddressinNorway.Thisdatabaserecordsthetotalnumberofauthorsineachpublication,aswellasan author’spositioninthesequenceofauthors.Asastartingpoint,weselectedscientiﬁcpublications(articlesandreviews) from2011–2017.TheseconddatabaseistheNorwegianScienceIndex(NSI),asubsetoftheCurrentResearchInformation SysteminNorway(Cristin),withcompletecoverageofallpeer-reviewedscientiﬁcandscholarlypublicationoutputssince 2010,includingbooks,editedvolumes,andconferenceseries(Sivertsen,2018).

AllarticlesandauthorsinNCRwerematchedtothecorrespondingrecordsinNSI.Wecouldthenunambiguouslyrelate thebibliographicinformationineachWoSarticletorealpeople,departments,andinstitutionsinNorway.Wecouldalso comparethecoverageinWoSwitheachresearcher’sfullsetofpublications.Sinceweknowfromacomparisonofthe twodatabases(Sivertsen&Larsen,2012)thattheextentofcoverageinWoSdiffersbyﬁeldofresearch,wechosetorun thecalculationsforbothdatabasestoseehowthesedifferencesmayaffectthemeasurementofscientiﬁccontributions.

Althoughwestudycountingmethodsfortheaggregatelevel,wewantedtoensureappropriateconditionsforcomparing individuallevelscientiﬁccontributions.Hence,weselectedonlypublicationsbyauthorswhoare,orwere,employedat Norway’sfourlargestresearchuniversities:theUniversityofBergen,theUniversityofOslo,theNorwegianUniversityof ScienceandTechnology(Trondheim),andUiT–TheArcticUniversityofNorway(Tromsø).Researchersinallﬁeldsatall fouruniversitiesaregiventhesameresourcesintime(onaverage50percent)forperformingresearch.Thesamesolution waschosenforthesamereasonsinanearlierstudy(Aksnesetal.,2012).

Further,welimitedthedatasettopublicationsbyscientistswithatleasttwopublicationsineachofthedatabasesin eachhalf(one-yearoverlapping)oftheperiod2011–2017toensurethattheresearchershadcomparablepossibilitiesto contributetoscientiﬁcresultsduringthewholeperiod.Thiscriterionyielded4048uniqueresearchersintheWoS-based datasetand5551researchersintheNSI-baseddataset.Wetestedastrictercriterion(atleastonepublicationeachyear, yielding1410and2186researchers)andamoreinclusivecriterion(atleastonepublicationineachhalfoftheperiod, yielding5553and7211researchers).Theresultsofourcalculationsweresimilarforallthreeselections.Weselectedthe intermediatealternativeasmostrobust.

OurWoS-baseddatasethas44,405uniquescientificarticlespublishedby4048uniquepersons.Toclassifythepersonsby fieldofresearch,weusedtheNSI-classificationofpublications(84differentfields,e.g.,history;politicalscience;neurology;

geosciences,whichareaggregatedintofourmajorareasofresearch:humanities,socialsciences,healthsciences,natural sciencesandengineering)andallocatedeachscientisttotheﬁeldinwhichtheymostfrequentlypublish.Only1.3percent oftheresearchersarefromthehumanitiesand4.2percentfromthesocialsciencesinourWoS-baseddataset,while55.4 percentarefromthehealthsciencesand39.1percentfromthenaturalsciencesandengineering.Alittlelessthanonethird (32.4percent)arefemaleresearchers.Theresearcherswereonaverage52yearsoldinthelastyearofourstudy.Almosttwo thirdsoftheresearchersareprofessors(66.2percent),12.1percentareassistantprofessorsand7.8percentarepostdocs.

TheremainingauthorsarePhDstudents,scientistswithaPhDbutwithoutaformalpostdocappointment,orcolleagues fromresearchmanagementofﬁcesandtechnicalservices.

TheNSI-baseddatasetisanextensionoftheWoS-baseddataset.ItincludesallWoS-publicationsbutisextendedto peer-reviewedresearchpublicationsnotcoveredbyWoS.Ithasintotal75,271uniquepublicationsinjournals,conference proceedingsandbooksthathavebeenpublishedby5551uniquepersons.Amongthesepersons,9.5percentarefromthe humanities,10.6percentfromthesocialsciences,45.0percentarefromthehealthsciencesand34.9percentfromthe naturalsciencesandengineering.Astotheothervariables,gender(33.7percentfemales),age(52yearsoldonaverage)and position(64.8percentprofessors),theNSI-baseddatasetverymuchresemblestheWoSdataset.

4. Groupingscientistsaccordingtoco-authorshippractices

Therearepersistentandwell-knownproblemswithusingbibliometricdatatocomparescientiﬁccontributionsacross ﬁeldsofresearch,typesofinstitutions,andindividualscientists.Acknowledgingtheseproblems,weneverthelessmakean

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Table2

The4048scientistsinourWoS-baseddatasetdividedinto12groupsbasedonthemediannumberofauthorsintheirpublications.

Groupname Numberofresearchers Mediannumberofauthorsinpublications Averagenumberofauthorsinpublications

1 53 1 1.6

2 169 1.5-2 2.6

3 421 2.5-3 3.7

4 550 3.5-4 4.5

5 599 4.5-5 6.2

6 664 5.5-6 7.9

7 459 6.5-7 8.4

8 373 7.5-8 9.4

9 231 8.5-9 10.8

10 153 9.5-10 12.1

15 238 10.5-15 15.5

1000 138 15.5-3017 712.8

Table3

The5551scientistsinourNSI-baseddatasetdividedinto12groupsbasedonthemediannumberofauthorsintheirpublications.

Groupname Numberofresearchers Mediannumberofauthorsinpublications Averagenumberofauthorsinpublications

1 594 1 1.3

2 502 1.5-2 2.4

3 640 2.5-3 3.7

4 699 3.5-4 4.8

5 674 4.5-5 6.5

6 720 5.5-6 8.3

7 487 6.5-7 9.0

8 407 7.5-8 10.4

9 249 8.5-9 12.0

10 171 9.5-10 13.4

15 263 10.5-15 17.2

1000 145 15.5-3017 577.9

importantworkinghypothesisinouranalysis:Weassumethatatanaggregatelevel,scientistsindifferentfieldscontribute onaveragetothesameextenttoscientificresults.Nofieldofresearchisinitselfmore“important”thananother.

Thereasonforassumingthisasafirststep,isthatallbibliometriccountingmethods(seeTable1)basicallyreferonly tothebibliographicinformationaboutauthorsinapublication,andnotthetypeofpublicationorfieldofresearch.Asan example,fractionalcountingisbasedonnootherinformationthanthenumberofauthors.Still,thismethodiswidelyused tocompareorganizationsorcountrieswithdifferentresearchprofiles.IntheintroductionwereferredtotheLeidenRanking asanexample.Itis,inpractice,basedonthesameassumption:Atanaggregatelevel,scientistsindifferentfieldsonaverage contributetothesameextenttoscientificresults.

Theworkinghypothesis,whichwewillreturntobelow,allowsus,asafirststep,todividescientistsingroups,notby fieldofresearch,butbytheirtypicalco-authorshippractices.Accordingtoourassumption,itisthenumberofauthorsand notthefieldofresearchthatdeterminestheoutcomeofacountingmethod.Becauseofthisassumption,theusedcounting methodshouldgiveabalancedresultfordifferentco-authorshippractices.

Here,wedecidedtogroupresearchersaccordingtoco-authorshippracticesandlookattheirﬁeldofresearchafter- wards.However,groupingresearchersaccordingtoco-authorshippracticesisnot straightforward.Takeonescientist’s co-authorshippracticeacross15publicationsasanexample:soleauthor–1publication;twoauthors–7;threeauthors– 1;fourauthors–2;ﬁveauthors–1;tenauthors–1;16authors–1;withthelastpublicationhaving40authors.Duetothe extremedifferencesbetweenthesevalues,wechosetorepresentthetypicalco-authorshippracticeofeachscientistbyits median(2inthiscase),notbyitsarithmeticmean(6.5inthiscase).

Wenextdividedscientistsinto12groupsaccordingtothemediannumberofauthorsintheirbodyofpublications.The ﬁrst10groupsarenamedaftertheirmedian,e.g.,thescientistsinGroup“2havepublicationswithamedianof1.5–2 authors.Thelasttwogroupsare“15formediansof10.5–15and“1000formediansof15.5-3,017.Thislastgroupincludes scientistsidentiﬁedasregularcontributorsto“hyperormega-authorship”(Cronin,2001;Kretschmer&Rousseau,2001).

ThenumberofscientistsandtheaveragenumberofauthorsintheirpublicationsforeachgroupareshowninTable2 (WoS-baseddata)andTable3(NSI-baseddata).

Co-authorshippracticesareknowntovarybyfield.Withourmethod,wecandemonstratethattheyalsovarywithin fields.ThisisshownforfourdifferentfieldsinFig.1(WoS-baseddataset)andfourotherdifferentfieldsinFig.2(NSI-based dataset).Contrarytoawidespreadbelief,itisquiteclearfromourresultsthatfieldsofresearchdonotcarryauniquetypical co-authorshippractice.

Ourresultsindicatethatitisimportantthatcountingmethodsbalancebetweendifferentco-authorshippractices.It willnotbesufficienttofield-normalizeoronlycompareresearcherswithinonefieldofresearchatatime.Inthenext

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Fig.1. Thedistributionofresearchersamongco-authorshipgroupsinfourresearchﬁelds,usingtheWoS-baseddataset.

Fig.2.Thedistributionofresearchersamongco-authorshipgroupsinfourresearchﬁelds,usingtheNSI-baseddataset.

sections,wepresentthemethodsandresultsofcalculatingtheoutcomesofusingdifferentcountingmethodsonthetwelve co-authorshipgroupsofresearchers.

5. Methods

Inprinciple,thetermModiﬁedFractionalCounting(MFC)referstoanymethodthatappliesafunctiontoequalfractions.

Ourapproachisbasedonthegeneralobservationthatfullcountingtendstooverestimatethecontributionstomulti-authored publicationswhilefractionalcountingtendstounderestimatethecontributionstosuchpublications–seetheintroduction.

JustasYishanWuinhisblog,welookforanintermediatemethodbetweenfullcountingandfractionalcounting.

Asanintermediatemethod,thesquarerootinfractionalcountingalreadyhadprecedenceinanempiricalstudythat pavedthewayforachangetothepublications-basedperformanceindicatorusedtofundresearchorganizationsinNorway (Sivertsen,2016b).Thesquarerootofafractionisinterestingbecauseitneverexceeds1,butitaddsvaluetothecontribution ofeachauthorwithaneffectthatitdiminishesasthenumberofauthorsincreases.Thesedynamicscorrespondwelltoour ideasaboutaddedandoverlappingcontributionsinresearchcollaboration(seeSection2above),andalsotothepolicyneed foracountingmethodthatdoesnotprovideanincentivetoaddmore(unnecessary)authors.

Nevertheless,wechosetoseethesquarerootofthefractionasaparticularcaseofmoregenerallyapplyingasensitiv- ityparameterbasedonexponentiationthatresultsinacontinuumfromcomplete-normalizedfractionalcountingtofull counting.Wewantedtoperformcalculationsbyusingthewholerangeofpossibilities.

WerefertothenumberkasthesensitivityparameterinMFCmethods.ApplyingMFCusingak-throotisequivalentto givingeachauthorofapublicationwithNauthorsacreditequalto1/√^k

N.Whenk=1,itrepresentstraditionalcomplete- normalizedfractionalcounting.Whenk=2,thesquarerootisused,andwithk=3,thecubicrootisused.Highervaluesof kcomeclosertofullcounting.ThenotationMFC₁representstraditionalcomplete-normalizedfractionalcounting,MFC₂ appliesa squarerootandMFC₃ acubicroot.Inaddition,wealsoshowresultsforMFC₄ andMFC_8.Themathematical foundationandimplicationsofthismethodispresentedintheAppendix.

6. Resultsobtainedfromcalculatingandcomparingdifferentcountingmethods

Thereisnoobjectivebestchoiceforthesensitivityparameter.Wedecidedtoﬁrstcomparek=2(thesquareroot)with thetwotraditionalmethods,ask=2seemstobeareasonablecompromise,andthenafterwardsshowresultsforhigherk values.

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Table4

Averagecontributionstopublishedresearchofeachco-authorshipgroupmeasuredbythreedifferentcountingmethodsusingtheWoS-baseddatasetwith 4048researchers.

Group #Researchers Averageauthortotal MFC1 MFC2 Fullcounts

1 53 1.6 5.45 5.85 6.51

2 169 2.6 5.24 7.20 10.46

3 421 3.7 5.31 8.68 14.95

4 550 4.5 4.82 8.88 17.22

5 599 6.2 4.51 9.26 20.09

6 664 7.9 4.24 9.46 22.34

7 459 8.4 3.93 9.34 23.66

8 373 9.4 3.25 8.19 22.01

9 231 10.8 3.28 8.68 24.63

10 153 12.1 3.10 8.48 25.27

15 238 15.5 3.00 8.73 28.34

1000 138 712.8 1.99 8.49 139.49

Fig.3. Distributionsofresultsforco-authorshipgroups1–15basedonthelastthreecolumnsofTable4.

AsexplainedinSection4,thescientistsinourtwodatasetsaredividedinto12groupsbasedonthemediannum- ber ofauthorscontributing totheir publications.These groups arereferred to as“co-authorshipgroups”.The results derivedfromthethreedifferentcountingmethods⁷ appliedtothe4048scientistsintheWoS-baseddatasetareshown inTable4.

Thedistributionsforgroups1–15basedonthelastthreecolumnsinTable4areillustratedinFig.3.

Ourresultsgenerallysupporttheideathatfractionalcountingispreferabletofullcounting.Fullcountingoveresti- matesthecontributionsofscientistsinvolvedinpublicationswithmanyauthors.Traditionalcomplete-normalizedfractional counting(MFC₁)hastheoppositeeffect,butnottothesamedegree.MFC₂usesthesquarerootoffractionsforbetterbalance acrossdifferingco-authorshippractices.OverallcontributionsappeartobelowerinGroups1and2.However,ascanbe seeninTable2,thesetwogroupsaresmaller,andbelongmostlytothesocialsciencesandhumanities,ofwhichthereare fewscientistsintheWoS-baseddataset.WewillnowperformthesameanalysisusingtheNSI-baseddataset,whichhasa morecompletecoverageofthesocialsciencesandhumanities.

Theresultsderivedfromthethreedifferentcountingmethodsonthe5551researchersintheNSI-baseddatasetare showninTable5.

Thedistributionsforco-authorshipgroups1–15basedonthelastthreecolumnsinTable5areillustratedinFig.4.

ExtendingthedatasourcebeyondWoSgivessomewhatdifferentresults.Thegainformainlypublishingmulti-authored publicationsbyfullcountinglevelsout.MFC₂balancesbetterforgroups1–3,probablybecausemorepublicationsinthesocial sciencesandhumanitiesarenowincluded.However,apartfromthis,MFC₂doesnotleadtothesamebalancedcounting resultsasinFig.3whichwasbasedonWoS-dataonly.Thisobservationledustoinvestigatehighervaluesofthesensitive parameterk(seeSection5).

7MFC1=traditional(complete-normalized)fractionalcounting.MFC2=MFCbasedonthesquarerootsoffractions.Thecontributionsofeachresearcher wereobtainedbeforeaverageswerecalculatedforeachgroup.ThesameappliestoTable5.

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Table5

Averagecontributionstopublishedresearchofeachco-authorshipgroupmeasuredbythreedifferentcountingmethodsusingtheNSI-baseddatasetwith 5551researchers.

Group #Researchers Averageauthortotal MFC1 MFC2 Fullcounts

1 594 1.3 11.95 12.54 13.51

2 502 2.4 9.16 11.89 16.43

3 640 3.7 8.22 13.12 22.33

4 699 4.8 6.60 12.09 23.55

5 674 6.5 5.23 10.71 23.51

6 720 8.3 4.80 10.73 25.71

7 487 9.0 4.33 10.38 26.84

8 407 10.4 3.56 9.00 24.57

9 249 12.0 3.62 9.60 27.86

10 171 13.4 3.28 9.10 27.96

15 263 17.2 3.11 9.05 30.00

1000 145 577.9 2.59 12.58 169.12

Fig.4.Distributionsofresultsforco-authorshipgroups1–15basedonthelastthreecolumnsofTable5.

Table6

Resultsfordifferentsensitivityparametervalues(k),calculatedfortheWoS-baseddatainTable4.

Group MFC1 MFC2 MFC3 MFC4 MFC8 Fullcounts

1 5.45 5.85 6.14 6.31 6.50 6.51

2 5.24 7.20 8.62 9.48 10.40 10.46

3 5.31 8.68 11.31 12.98 14.82 14.95

4 4.82 8.88 12.28 14.51 17.01 17.22

5 4.51 9.26 13.54 16.46 19.84 20.09

6 4.24 9.46 14.42 17.91 22.02 22.34

7 3.93 9.34 14.75 18.64 23.31 23.66

8 3.25 8.19 13.32 17.08 21.66 22.01

9 3.28 8.68 14.49 18.84 24.22 24.63

10 3.10 8.48 14.47 19.06 24.82 25.27

15 3.00 8.73 15.49 20.87 27.80 28.34

1000 1.99 8.49 28.68 60.93 132.21 139.49

Table6showstheresultswhenapplyingdifferentvaluesofkintheWoS-baseddataset.Again,MFC1representstraditional complete-normalizedfractionalcounting,whileMFC₂appliesthesquareroot,MFC₃thecubicroot,andsoon.Asexpected, seethemathematicalfoundationsofMFCintheappendix,highervaluesofkcomeclosertofullcounting.

MFC2seemstogivethemostbalancedrepresentationofaveragecontributionsacrossdifferentco-authorshippractices.

Mega-orhyper-authorship(ourgroupdenotedas1000)hasbeenregardedasanexceptiontotherulethatbibliographic informationisareasonableapproximationofactualcontributionstoscientiﬁcwork(Cronin,2001).Itis,therefore,somewhat surprisingthatMFC2balancesevenwiththisgroup.

HighervaluesofthesensitivityparameterkdonotgivemorebalancedvariantsofMFCinourWoS-baseddata.Thisis moreclearlyvisibleinFig.5withtheresultsforco-authorshipgroups1–15fromTable6.

Table7showsthesameparametersasinTable6,butnowusingtheNSI-baseddataset.Withthismorecompletedataset extendingbeyondWoS,MFC₃ (usingthecubicrootoffractions)seemstobalancebetterthanthealternatives,although group1000isnowanexception.TheresultsaremoreclearlyseeninFig.6.

SofarwemayconcludethatModiﬁedFractionalCountingbalancesbetterbetweendifferentco-authorshippatterns thantraditionalcomplete-normalizedfractionalcountingorfullcounting.Morespeciﬁcally,MFC2(usingthesquarerootof fractions)seemsmostadequatewhenapplyingWoS-baseddatawhileMFC₃seemsadequateaswellwhenapplyingmore

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Fig.5. Resultsforco-authorshipgroups1–15forthedatainTable6.

Table7

Resultsfordifferentsensitivityparametervalues(k),calculatedfortheNSI-baseddatainTable5.

Group MFC1 MFC2 MFC3 MFC4 MFC8 Fullcounts

1 11.95 12.54 12.96 13.22 13.49 13.51

2 9.16 11.89 13.86 15.06 16.34 16.43

3 8.22 13.12 16.98 19.43 22.14 22.33

4 6.60 12.09 16.73 19.80 23.27 23.55

5 5.23 10.71 15.73 19.18 23.20 23.51

6 4.80 10.73 16.45 20.51 25.34 25.71

7 4.33 10.38 16.53 21.01 26.43 26.84

8 3.56 9.00 14.72 18.96 24.17 24.57

9 3.62 9.60 16.16 21.15 27.37 27.86

10 3.28 9.10 15.74 20.90 27.45 27.96

15 3.11 9.05 16.19 21.93 29.41 30.00

1000 2.59 12.58 39.84 79.25 160.99 169.12

Fig.6. Resultsforco-authorshipgroups1–15forthedatainTable7.

completenationalorinstitutionaldatarepresentingallpeer-reviewedresearchpublications.Thesepreliminaryconclusions stillneedtobecheckedbybringingouranalysisdowntothelevelofareasorﬁeldsofresearch.

7. Resultsfromapplicationsatthelevelofareasandﬁeldsofresearch

WedemonstratedinSection4thatfieldsofresearchdonotcarryauniquetypicalco-authorshippractice.Differentco- authorshippracticesoccurwithineachfield,howeverwithdifferentfrequenciesacrossfields.OurresultsfromSection6 indicatethatModifiedFractionalCounting(MFC)maygiveabalancedrepresentationofdifferentco-authorshippractices.In Section4,weestablishedaworkinghypothesisassumingthatatanaggregatelevel,scientistsindifferentfieldscontribute onaveragetothesameextenttoscientificresults.Givenourresultssofar,ourassumptionimpliesthatMFCmightalsogive abalancedrepresentationofscientificcontributionsacrossfieldsofresearch.Wewillnowinvestigatetowhatextentthis canbeconfirmed.

Sofar,wehaveusedtwodatasets,onebasedonWoSandanotherextendeddatasetbasedonNSI.Aswenowapproach thelevelofareasandfieldsofresearch,wemusttakeintoconsiderationthattheWoS-baseddatasetinitselfcannotgive abalancedrepresentationofthescientificcontributionsacrossallfieldsbecauseitisbiasedtowardsthehealthsciences andthenaturalsciencesandengineering(Sivertsen&Larsen,2012).Thisbiasismeasurablewithinourdata:Focusingon

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Fig.7.AveragecontributionstopublishedresearchofresearchersinfourmajorareasofresearchusingfourdifferentcountingmethodsandtheNSI-based datasetwith5551researchers.

Fig.8. Averagecontributionstopublishedresearchofresearchersintwelvesubﬁelds(threeineachoffourmajorareasofresearch)usingMFC2andthe NSI-baseddatasetwith5551researchers(atotalof1441researchersareincludedinthisﬁgure).

the4048scientistsincludedintheWoS-baseddataset,wecanmeasuretheproportionsoftheirpublicationsinNSIthat areincludedinWoS.Thesharesarehighestforresearchersinthehealthsciences(84percent)andthenaturalsciences andengineering(75percent).Researchersinthesocialsciences(40percent)andthehumanities(32percent)haveclearly lowersharesoftheirpublicationsinWoS.Therearealsodifferencesattheﬁeldlevelwithineachmainarea,e.g.between biology(86percent)andcomputerscience(28percent),andbetweeneconomics(65percent)andlaw(30percent).Wewill thereforeonlyapplytheNSI-baseddatasetinthefollowinganalysis.

TheresultsshowninFig.7canbecomparedtotheresultsinFig.6basedonthesamedatasetandourpreliminary conclusionsfromSection6: WhileMFC₃ seemstoprovideasomewhatbetterbalanceinthis datasetwhenlookingat co-authorshippracticesonly,MFC₂providesabetterbalancewhentakingtheareasofresearchintoaccount.Wecannow concludethatMFC2maybepreferablenotonlywhenusingWoS-baseddataasshowninSection6(whiledisregardingbiases inﬁeldrepresentation),butalsowhenusingmorecompleteinstitutionalornationaldatasetstocompareorganizationswith differentresearchproﬁles.

Finally,wetesthowMFC2worksonsubfieldlevel.Amongthe84fieldsintheNSIfield-classification,wehaveselected threefieldsineachofthefourmajorareasofresearchwhileensuringthatonlyfieldswithlargenumbersofresearchersare included,andalsothatbothhigherandloweraveragescoresaccordingtoMFC₂arerepresentedintheselection.Theresults areshowninFig.8.

TheresultsinFig.8ontheonehandconﬁrmthatMFC2seemsnottogiveanybiastowardsanyofthemajorareasof research.Ontheotherhand,thedifferencesbetweenthehighestandlowestscoresarelargeenoughtoindicateasomewhat differentresultfromwhatisshowninFig.7:Itisonlyattheaggregatelevel,atthelevelofmajorareaofresearch,thatMFC₂ (andMFCingeneral)canmeasurecontributionsinabalancedway.

Ourexplanationforthis differentresult,isthat ourworkinghypothesis,which assumesthataveragecontributions acrossfieldsareequal,cannotbeconfirmedatareal-worldlocallevel.WiththefourNorwegianuniversitiesasa case study,therewillalwaysbereasonswhyresearchersinsomefields(departments)contributerelativelymorethanothersto scientificpublications.Oneexampleofanexplanationforsuchobserveddifferences,isthatsomedepartmentsmayhave moreresourcesfromexternalfundingthanotherdepartmentsatthelocallevel.

Althoughourworkinghypothesisaboutequalaveragecontributionsevidentlycannotbeconfirmedbyusinglocalreal- worlddata,ithasbeenusefultodemonstratethat ModifiedFractionalCounting(MFC2 andMFC3)in generalreduces differencesinscientificcontributionsbetweenco-authorshippracticesandfieldsofresearchwheneverthesecontribu-

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Table8

Acalculationoffullcounting,complete-normalizedfractionalcounting(MFC1)anddifferentothervaluesofkonWoS-baseddataforthefourlargest Norwegianuniversities^a.

MFC1 MFC2 MFC3 MFC4 MFC8 FullCounts

Oslo 7432 10734 13573 15535 17875 18053

Bergen 4395 6389 8163 9431 10991 11112

Trondheim 5719 7722 9253 10224 11301 11379

Tromsø 2369 3329 4140 4690 5330 5378

Total 19915 28174 35129 39880 45497 45922

aOslo:UniversityofOslo;Bergen:UniversityofBergen;Trondheim:NTNU-NorwegianUniversityofScienceandTechnology;Tromsø:UiT-TheArctic UniversityofNorway.ThesameappliestoTables9–11.

Table9

TheresultsinTable8expressedaspercentages.

Oslo 37.3% 38.1% 38.6% 39.0% 39.3% 39.3%

Bergen 22.1% 22.7% 23.2% 23.6% 24.2% 24.2%

Trondheim 28.7% 27.4% 26.3% 25.6% 24.8% 24.8%

Tromsø 11.9% 11.8% 11.8% 11.8% 11.7% 11.7%

Total 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

Table10

Acalculationoffullcounting,complete-normalizedfractionalcounting(MFC1)anddifferentothervaluesofkonNSI-baseddataforthefourlargest Norwegianuniversities.

Oslo 14566 19279 23293 26035 29265 29509

Bergen 7824 10549 12914 14568 16565 16718

Trondheim 11268 14895 17685 19456 21416 21559

Tromsø 4622 6012 7166 7940 8833 8899

Total 38270 50735 61058 67999 76079 76685

tionsaremeasuredatanaggregatelevelwiththeuseofbibliographicdata.Itnowremainstodemonstratehowthecounting methodsworkattheaggregatelevelwithdatarepresentingrealinstitutions.

8. Resultsfromanapplicationatinstitutionallevel

Ourstudyfocusesontheuseofcountingmethodsforstatistics,evaluations,andfundingatanaggregatelevel.Wewant toimprovethecomparabilitywhenorganizationsorcountrieswithdifferentresearchproﬁlesarecompared.Therefore,we testedhowMFCworksataninstitutionallevelandacrossdifferentﬁeldsofresearch.

ThefourlargestNorwegianuniversitiesarerepresentedinourdata.WeusethesameWoS-based(4048researchers)and NSI-baseddatasets(5551researchers)asdescribedinDatasection,buttheresearchers’contributionsarenowaggregated totheinstitutionallevel.Table8(WoS-based)andTable10(NSI-based)showthenumbers,whileTable9(WoS-based)and 11(NSI-based)showhowthenumbersaredistributedinpercentagesamongthefouruniversities.Alltablescomparethe resultsoffullcounting(deduplicated,i.e.,aninstitutioncanonlygetonecreditfromthesamepublication,nomatterhow manyresearchersarefromthissameinstitution),fractionalcounting(thesumofauthors’fractionsperinstitution),andthe sumofauthors’MFCfordifferentvaluesofk.Asanexample,ifapublicationhasbeenco-authoredbyﬁveresearchersand twooftheseresearchersareafﬁliatedwiththeuniversityinfocus,theuniversityiscredited1withfullcounting,2/5=0.4 withfractionalcounting,0.63withMFC₂,andsoon.

WeseeinTable9thatfullandfractionalcountinggivesdifferentresults,particularlybetweentwooftheuniversities (theUniversityofBergenvs.theNorwegianUniversityofScienceandTechnology(Trondheim)).Oneexplanationisthat onlyoneofthetwouniversities(UniversityofBergen)hascontinuouslyparticipatedinCERN,theEuropeanOrganization forNuclearResearch,whichveryfrequentlypublishesarticleswithmorethan3000authors.Asmuchas18percentofthe publicationsfromUniversityofBergenarecontributedbyresearchersinco-authorshipgroups10,15and1000.Theseare thegroupswiththehighestmediannumberofco-authorsintheirpublications.Only5percentofthepublicationsfrom NTNUinTrondheimfallinthesehighco-authorshipgroups.

AdifferencebetweenTables9and11isthatUniversityofOslohasahighersharewithfullcountingthanwithfractional countinginTable9whiletheoppositeisthecaseforTable11.ThiscanbeexplainedbythefactthatUniversityofOslohas averylargeFacultyofHumanities.Asmuchas45percentofallpublicationsinthehumanitiesfromthefouruniversities arefromUniversityofOslo.Thehumanitieshaveco-authorshippracticesthatgainlargerweightwithfractionalcounting thanwithfullcounting.ThiseffectismoreclearlyseenwithaNSI-baseddataset,whichrepresentsthehumanitiesmore completelythanwithaWoS-baseddataset.

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Table11

TheresultsinTable10expressedaspercentages.

Oslo 38.0% 38.0% 38.1% 38.3% 38.5% 38.5%

Bergen 20.4% 20.8% 21.1% 21.4% 21.8% 21.8%

Trondheim 29.4% 29.4% 29.0% 28.6% 28.2% 28.1%

Tromsø 12.1% 11.8% 11.7% 11.7% 11.6% 11.6%

Total 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

Themainresult,however,thatcanbederivedfromTables8–11,isthatMFC₂andMFC₃representanintermediatesolution thatsitsbetweenfullandcomplete-normalizedfractionalcounting(MFC₁)attheinstitutionallevel.Weexplainthiswith theevidencegiveninSections6and7:MFC2andMFC3provideamorebalancedrepresentationofdifferentco-authorship practices.Suchpracticesvaryacrossﬁelds,whichisachallengewhenorganizationsorcountrieswithdifferentresearch proﬁlesarecompared.

Asnotedintheintroduction,harmoniccounting(Hagen,2008)isavariantoffractionalcountingthatconsidersthe sequenceofauthors.Whenapplyingharmoniccounting,authorsreceivecreditaccordingtotheirrankinthebyline.An authorrankedinthei-thplacereceivesacreditequalto:

1

i

1+¹₂+...+_N¹

whereNisthenumberofauthors.

Theneedforharmoniccountingisjustifiedbytheobservationthatthesequenceoftheauthorsisimportantinrepre- sentingindividualcontributionsinmostfieldsofresearch.Itis,however,difficulttoimaginethatoneuniversitywouldhave morefirstauthorsthananotherinalargedatasetlikeours.Hence,ourhypothesiswasthatusingharmoniccountingwould notmakemuchdifferenceatanaggregatelevel.Wetestedthisandfoundthatpercentagesharesamongthefourmajor Norwegianuniversitiesarecomparable(lessthanonepercentage-pointdifferent)toTable11(resultsnotshown)andthat thesedifferencestendtodecreasewhenapplyingMFC_2.Weconcludethatthevariantoffractionalcountingusedmakes littledifferenceattheaggregatelevel.

9. Discussion

Giventhattheshareofpublicationswithcross-institutionalandinternationalco-authorshipisgrowing,itistimelyto reflectonthevariouscountingmethodsforpublicationstogaugetheextenttowhichtheyrepresentreal-worldscientific contributions.Organizationsandcountriesarecomparedtodaywithcountingmethods–fullorfractionalcounting-that donotbalancewellbetweendifferentco-authorshippractices,andtherebybetweenthedifferentresearchprofilesthat organizationsandcountriesmayhave.Inaddition,itcanbedemonstratedthatthefractionalcountingmethodsdonot reflectoverlappingtasksandethicalresponsibilitiesthatcomewithteamworkandco-authorshipinscience.

Wehavedevelopedanewcountingmethodcalledmodifiedfractionalcounting(MFC).Themethodisanintermediate countingmethodthatsitsbetweentraditionalcomplete-normalizedfractionalcountingandfullcounting.Itisspecifically designedtosupportcomparabilityataggregatelevels,andwehavetestedthatitworks.Themethodisnotdesignedtobe usedattheindividuallevelwherebibliographicinformationcannotbeusedalonetounderstandindividualcontributionsto scientificwork.Ourfocusisoncountingmethodsattheaggregatelevelbecausetheyareusedforstatistics,evaluationsand strategicpurposesthatcanmakearealdifferenceinoverallpoliciesandfunding.Atthislevel,itcanbeimportanttounder- standwhytwocountingmethodsgivequitedifferentresultsintheLeidenRankingforthetwolargeChineseuniversitiesthat weusedasexamplesinourintroduction.Wehavedemonstratedthatneitherofthetwocountingmethodssufficientlytake differencesinresearchprofiles(andtherebyco-authorshippractices)intoaccount,andthatmodifiedfractionalcounting (MFC)mayprovideamorebalancedrepresentationofsuchdifferencesandmoreconsistentresults.

Althoughwehaveleftcitationindicatorsoutofthisstudy,weareawarethatcountingmethodsforarticlesmayhave implicationsfortheconstructionofnormalizedcitationindicators(Fairclough&Thelwall,2015; Perianes-Rodriguez&

Ruiz-Castillo,2015;Waltman&vanEck,2015).Weintendtoincludethisperspectiveinfurtherworkoncountingmethods.

Authorcontributions

GunnarSivertsen:Concievedanddesignedtheanalysis;Collectedthedata;Contributeddataoranalysistools;Performed theanalysis;Wrotethepaper.

RonaldRousseau:Contributeddataoranalysistools;Performedtheanalysis;Wrotethepaper.

LinZhang:Concievedanddesignedtheanalysis;Contributeddataoranalysistools;Performedtheanalysis;Wrotethe paper.

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Acknowledgments

LinZhangacknowledgessupportfromtheNationalNaturalScienceFoundationofChina(GrantNo.71573085)andthe SocialScienceScholarshipinHeNanProvince(Grant2018-YXXZ-10).GunnarSivertsenacknowledgesthesupportfrom theFORINNPOLprogramme,projectnumber256223,attheResearchCouncilofNorway.TheauthorsthankTimEngels (UniversityofAntwerp)andanonymousreviewersforhelpfulremarks.

AppendixA. MathematicalfoundationsandimplicationsofModiﬁedFractionalCounting(MFC)usinga sensitivityparameter

LetS

x₁,x₂,...,x_A_S

representthearrayofarticlesofscientistS;A_Sdenotesthetotalnumberofarticles(co-)authored byS.ThisarrayistermedthepublicationarrayofscientistS.Ifitisclearwhowrotethesearticles,orwhenitdoesnotmatter, wesimplywriteAinsteadofAS.ThecontributionofscientistS,asdeterminedbyMFCwithasensitivityparameterk,where k=1,2,...,isdeﬁnedas

MFC_k(S)=

A

j=1

1

k

x_j=

A

j=1

x_j

₋¹

⁄

^k

(1)

Notethatkactsasanindex.Ifk=1,MFCbecomescomplete-normalizedfractionalcounting;ifkextendstoinﬁnityMFC becomesfullcounting(consequently,wemaydenotefullcountingasMFC_∞).Recallthatrootsarespecialcasesofexponen- tiation.Here,weonlyconsiderexponentsoftheform1/k,withkbeinganaturalnumbernotincludingzero.Thek-throot ofanon-negativerealnumberxisuniqueanddenotedas√^k

x.

Ifxisanaturalnumberlargerthan1,thenthesequence

_√_k

x

kisstrictlydecreasingwithalimitof1.Consequently,the sequence

1/√^k x

kisstrictlyincreasing,alsowithalimitof1.Ifx=1,thenallk-throotsareequalto1.

ApplyingMFCusingak-throotisequivalenttogivingeachauthorofapublicationwithNauthorsacreditequalto1/√^k N.

Theresultingindicator,appliedtoscientistS,isdenotedasMFC_k(S).Hence,thepublicationasawholereceivesacreditof N/√^k

N.Ifkdecreasesto1(complete-normalizedfractionalcounting),thisvaluebecomesequalto1,whileifkincreases toinﬁnityitbecomesN.ThenumberkisreferredtoasthesensitivityparameterinMFCmethods.Whentalkingaboutthe indicatoritself,i.e.,whennotappliedtoascientist,thesimplernotationMFC_khasbeenused.

BeforegoingdeeperintosometheoreticalissuesaboutMFC,weﬁrstillustrateitsuseanditsconsequencesinthree differentcases.HypotheticaldataareprovidedinTableA1andtheresultswithourMFCmethodareillustratedinFig.A1.

TableA1

Anillustrativeexample.

Author1 Author2 Author3

#articles 3 #articles 3 #articles 2

#authors 3 5 6 #authors 6 50 1000 #authors 1 2

sensitivity sum sensitivity sum sensitivity sum

1 0.333 0.2 0.167 0.700 1 0.167 0.02 0.001 0.188 1 1 0.5 1.5

2 0.577 0.447 0.408 1.432 2 0.408 0.141 0.032 0.581 2 1 0.707 1.707

3 0.693 0.584 0.55 1.827 3 0.55 0.271 0.1 0.921 3 1 0.794 1.794

4 0.760 0.669 0.639 2.068 4 0.639 0.376 0.178 1.193 4 1 0.841 1.841

8 0.872 0.818 0.799 2.489 8 0.799 0.613 0.422 1.834 8 1 0.917 1.917

infinity 1 1 1 3 infinity 1 1 1 3 infinity 1 1 2

Fig.A1.ThecontributionofthreeauthorsdependingonthesensitivityoftheMFCkindicator.