简易计算百分之五十终点方法(A-simple-method-of-estimating-fifty-

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THEAMERICANJOURNALOFHYGIENEVOL.27MAY,1938No.3ASIMPLEMETHODOFESTIMATINGFIFTYPERCENTENDPOINTSl'2BYL.J.REEDANDH.MUENCH(ReceivedforpublicationOctober26th,1937)Anendpointinbiologicalproceduressuchastitrationofseraorvirusesisusuallytakenasthedilutionatwhichacertainproportionoftestanimalsreactordie.TheadvantagesofusingthedilutionbywhichhalftheanimalsareaffectedhavebeensetforthbyGaddum(1)onthebasisofpharmacologicaltests.Thisendpointislessaffectedbysmallchancevariationsthanisanyother;theworstinthisrespectbeingthe100percentpointsofrequentlyused.Thebestmethodofdeterminingtheendpointistheuseoflargenumbersofanimalsatdilutionsnearthevaluefor50percentreaction,andtheinterpolationofthecorrectvaluebytheproceduresdescribedbyGaddum.Butthisisscarcelypracticalinagreatdealofimmuno-logicalwork,wherethetitermayvarybetweenwidelimitsinanygiventest;whilehighlyrefinedstatisticalproceduresarehardlyjustifiedinviewoftheuncontrolledvariablesinvolved.Tocoverawiderangeofpossiblevariationsintiter,itisgenerallynecessarytousealargenumberofsmallgroupsoftestanimalsatdifferentdilutions.Thefollowingmethodutilizes,atleastinpart,thelarge,totalnumberinvolved.ItisessentiallytheonegivenbyReed(2),somewhatsimplifiedandadaptedtorapiduse.Theeffectisthatofusing,atthetwocriticaldilutionsbetweenwhichtheendpointlies,largergroupsofanimalsthanwereactuallyincludedatthese1ThismethodhasbeenusedintheestimationofendpointsattheYellowFeverLaboratoriesoftheInternationalHealthDivisionofTheRockefellerFoundation,NewYorkCity.2FromtheDepartmentofBiostatistics,TheJohnHopkinsUniversitySchoolofHygieneandPublicHealth,andtheInternationalHealthDivisionofTheRocke-fellerFoundation.493494L.J.REEDANDH.MUENCHdilutions.Byincliningtoequalizechancevariations,themethodtendstodefinethepointmorenearlythanwouldbepossibleifitweresimplyinterpolatedbetweenthetwobracketingresults.Forpurposesofillustrationtheresultsofahypotheticaltitrationofaprotectiveserumaregivenintable1.SixmicearetestedateachTABLE1TitrationofahypotheticalprotectiveserumDilutiona1:1248163264128256Alive6665642201Deadc001024465TotalAlived3226201595311Deade001137111722Percentmortality00562558799496dilutionandtheresults,innumberssurvivinganddying,areenteredincolumnsbandc.Itisassumedthatamousesurvivingatagivendilutionofserumwouldhavesurvivedatalowerdilution.Columnbisthereforeaddedfromthebottom,andthesubtotalforeachdilutionisenteredincolumndastheaccumulatednumberofmicesurvivingatthisplushigherdilutions.Conversely,amousedeadatagivendilutionwouldhavediedatanyhigherone.Columncisaddedfromthetop,andthesubtotalsincolumnegivethecumulativenumbersofmicedyingatgivendilutionsplusthoselower.Percentagemortalityiscalculatedfromcolumnsdandeandisenteredincolumn/.Forexample,ofatotaloftwelvemicetreatedwithadilutionof1:16,ninesurvivedatthisorhigherdilutionswhilethreediedatthisorlowerones.Thecalculatedmortalityrateistherefore25.percent.Figure1illustratesthemethod.Thetwolinesrepresentcolumnsdandeandmeetattheassumedpointof50percentmortality,inthiscase,aboutthree-quartersofthedistancefromdilution1:16to1:32.Itisobviousthatthepositionoftheendpointwouldnotbechangedbyanindefinitenumberofdeathsathigherdilutions,orofsurvivalsatlower.TheaccidentaloccurrenceofsurvivalsatMETHODOFESTIMATINGFIFTYPEECENTENDPOINTS49510Micesurviving(Col.d,tablel)Micedyin(Col.e,table11281:25681632DilutionsFIGURE1.Accumulationofdeathsandsurvivals.highdilutions,orofdeathsatlow,wouldshiftthepointifoneortheotherpredominated.Thisshiftislessforthe50percentpointthanforanyother,butitshouldbeeliminatedasfaraspossible.Sincehighlysusceptibleorrefractoryanimalsareoccasionallyen-counteredinmosttypesoftest,itistobeexpectedthattheseacci-dentalsurvivalsanddeathswilloccur,attimes,atpracticallyanydilution.Theyshould,ingeneral,tendtocanceleachotherifanequalnumberofdilutionsistakenoneachsideoftheendpoint.Thismaybeaccomplishedbyabridgingtheoriginalresults,ashasbeendoneintable2.TABLE2Abridgedtabulationoftheresultsoftable1Dilutiona1:8163264128Aliveb64220Deadc02446TotalAlived148420Deade0261016Percentmortality0206083100496L.J.REEDANDH.MUENCHSince,intable1,1:32wasthedilutionnearesttheendpoint,thisistakenasthecentralvalueofthenewtable,thetwoaboveandtwobelowbeingretained.Threeorfourdilutionsoneachsidecouldbeusedaswellandwouldbepreferableifmortalityincreasedslowlywithdilution.Columnsd,e,and/areconstructedexactlyasbefore,butarenowbasedonthesmallernumberofresults.CalculationofresultsThe50percentpointevidentlyliesbetween1:16and1:32,butnearerthelatter.Itisassumedthat,sincethemortalityat1:32is40percentabovethatat1:16,whilethe50percentpointis30percentabove,theendpointis30percent/40percentorthree-10050.2A.6.8FIGURE2.Graphicalinterpolationoffinaltiter.1.0quartersofthedistancefrom1:16to1:32.Theformulafortheproportionatedistanceoftheendpointabovethedilutiongivingnextbelow50percentmortalityis:50percent—(mortalityatdilutionnextbelow)(mortalitynextabove)—(mortalitynextbelow)ProportionaldistanceMETHODOFESTIMATINGFIFTYPERCENTENDPOINTS497Sincedilutionsareincreasingonalogarithmicscale,itisnecessarytoobtainthefinalreadingasfollows:Loga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