Survival Analysis Using R

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1、SurvivalAnalysisUsingS/RSlidesfÄurdenNachdiplomkursinangewandterStatistikanderETHZÄurichProfessorMaraTablemanDept.ofMathematics&StatisticsPortlandStateUniversityPortland,Oregon,USAtablemanm@pdx.edu21.August2006Chapter1RationaleforSurvivalAnalysis²Time-to-eventdatahaveasprincipalend-

2、pointthelengthoftimeuntilaneventoccurs.Theeventiscommonlyreferredtoasafailure.²Censoring:Afailuretimeisnotcompletelyobserved.²SurvivalAnalysis:Thecollectionofsta-tisticalproceduresthataccommodatetime-to-eventcensoreddata.1Example:AMLstudyBelowarepreliminaryresults(1977)fromaclinical

3、trialtoevaluatethee±cacyofmaintenancechemotherapyforacutemyelogenousleukemia(AML).Afterreachingastatusofremissionthroughtreatmentbychemother-apy,thepatientswhoenteredthestudywereassignedrandomlytotwogroups.The¯rstgroupreceivedmain-tenancechemotherapy;thesecond,orcontrol,groupdidnot.

4、Theobjectiveofthetrialwastoseeifmaintenancechemotherapyprolongedthetimeun-tilrelapse.GroupLengthofcompleteremission(inweeks)Maintained9,13,13+,18,23,28+,31,34,45+,48,161+Nonmaintained5,5,8,8,12,16+,23,27,30,33,43,45The+indicatesacensoredvalue.2²Seriousbiasinestimatedquantities,which

5、lowersthee±cacyofthestudy.a.Throwoutcensoredobservations.b.Treatcensoredobservationsasexact.c.Accountforthecensoring.η=medianµ=meanηµηηµµ2325.1283138.552.6aabcbc-0.0050.0000.00520304050weeksinremission3BasicDe¯nitions&IdentitiesTher.v.TdenotesfailuretimewithcdfF(¢)andpdff(¢).cdfF(¢)

6、:ZtdF(t)F(t)=P(T·t)=f(x)dxand=f(t)0dtThatis,byde¯nitionofderivative,F(t+¢t)¡F(t)P(tt)=1¡F(t)=f(x)dxtAtt=0,S(t)=1anddecreasesto0astincreasesto1.WethuscanexpressthepdfasdS(t)f(t)=¡:dt4Hazardfunctionh(¢):P(tt)f

7、(t)h(t)=lim=¢t!0+¢tS(t)dS(t)=dtdlog(S(t))=¡=¡S(t)dtOfcourse,h(t)¸0atalltimest.CumulativehazardfunctionH(¢):ZtH(t)=h(u)du=¡log(S(t))0Att=0,H(t)=0andincreasesto1astincreasesto1.Hence,therelationshipS(t)=exp(¡H(t)):5Thehazardfunctionh(t)¸0²speci¯estheinstantaneousrateoffailureatT=tgive

8、nthattheindividuals