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CHAPTER 13Survival Data13.1 IntroductionIn many follow-up trials,the primary response variable is a subjects survival timeor the time to occurrence of a particular event such as relapse or remission ofsymptoms.These are elapsed times measured from an origin pertinent to eachsubject,often the date of entry into the study.Although time is typically measuredon a chronological scale of days,weeks or months,this need not always be so.Insome epidemiological studies cumulative exposure plays the role of survival time,in a veterinary study of dairy cattle the equivalent measure might be cumulativemilk production,or in a reliability strength test it could be the cumulative stressor load applied.We shall phrase the methods of this chapter in terms of“survival”or“failure”time measured from entry to a study.However,the same methods canbe applied quite generally for any time to event response.An individuals survival time may be censored if failure has not occurred whenthat individual is last seen or when the data are analyzed.In Section 3.7,weintroduced the basic concepts of survival models,including the survival functionS(t)and hazard rate h(t)of a response time distribution.We also describedthe two-sided,group sequential log-rank test for testing equality of two survivaldistributions.Suppose two sets of subjects,A and B,have respective hazard rateshA(t)and hB(t),then the log-rank test is used to test the null hypothesis H0:hA(t)=hB(t)for all t 0,or equivalently SA(t)SB(t),and is particularlypowerful against“proportional hazards”alternatives where hB(t)hA(t)forsomeconstant.InSection3.7wefollowedthesignificancelevelapproachwhichrequiresequalincrementsininformationbetweensuccessiveanalysesiftheTypeIerror rate is to be attained accurately.In a survival study,information is roughlyproportional to the total number of observed failures and increments are usuallyunequal and unpredictable.The error spending approach of Chapter 7 is thus anattractive alternative,as illustrated in the example of Section 7.2.3.In Section 13.2,we shall review the group sequential log-rank test describedin Section 3.7,adding a modification to accommodate tied survival times.Thistest is further generalized in Section 13.3 to permit the inclusion of strata.InSection 13.4,we describe how to adjust the comparison of survival distributionsfor the effects of covariates through use of the proportional hazards regressionmodel of Cox(1972)and we explain how to construct RCIs for a hazard ratio,in Section 13.5.These methods are illustrated by a case study in Section 13.6.In Section 13.7,we show how successive Kaplan-Meier(1958)estimates of thesurvival function can be used as a basis for group sequential tests and repeatedconfidence intervals concerning survival probabilities and quantiles.c?2000 by Chapman&Hall/CRC13.2 The Log-Rank TestConsider the problem of testing the equality of survival distributions SA(t)andSB(t)for two treatment arms,A and B,based on accumulating survival data.Ateach analysis we observe a failure or censoring time for each subject,measuredfrom that subjects date of entry or randomization as defined in the study protocol.Let dk,k=1,.,K,denote the total number of uncensored failures observedacross both treatment arms when analysis k is conducted.Some of these timesmay be tied and we suppose that d?kof the dkfailure times are distinct,where 1 d?k dk.We denote these distinct failure times by 1,k 2,k.0 and?=1,.,L.The stratifiedlog-ranktest can beusedinthissituationandisparticularlypowerfulagainst alternatives in which the hazard rates in the two treatment arms areproportional within each stratum with a common hazard ratio in each case,i.e.,hB?(t)=hA?(t)for all t 0 and?=1,.,L.The test is described in thenon-sequential settingin a number of textbookson survival analysis,for example,Klein&Moeschberger(1997,Section 7.5).Generalizing the notation of Sections 3.7 and 13.2,we let d?k,k=1,.,Kand?=1,.,L,denotethe number ofuncensored failuresinstratum?observedwhen analysis k is conducted.Considering only those subjects in stratum?,wesuppose d?kof the d?kfailure times observed at analysis k are distinct and denotethese times by 1,?k 2,?k.0,(13.5)where h0(t)represents an unknown baseline hazard function and=(1,.,p)Tis a column vector of parameters to be estimated.The log-linearlink function between covariate values xijand the hazard rate hi(t)is not asrestrictive as might seem at first,since interactions and transformed covariatesmay be included in the vector x.Entry of individuals to a study will usually bestaggered and survival,measured as elapsed time from entry to the study,can besubject to competing risk censoring,which we shall assume to be non-informative(see Klein&Moeschberger,1997,p.91)as well as“end-of-study”censoring atinterim and final analyses.We use the same notation as in previous sections.For k=1,.,K,letdkdenote the total number of failures observed by the time the kth analysis isconducted.Supposing for now that there are no ties,we denote the survival timesof the subjects observed to have failed at analysis k by 1,k 2,k.0,(13.8)the baseline hazard rate,h0?,depending on the stratum?but the vector ofregression coefficients,remaining constant across strata.If there are L strata,the log partial likelihood is a sum of L terms of the form(13.6)computedseparately within each stratum.Differentiating this with respect to produces asum of L terms in the score equations corresponding to(13.7).For further details,see Klein&Moeschberger(1997,Section 9.3).The score equations(13.7)must be solved numerically and many statisticalcomputer software packages now include procedures to do this.The informationmatrix associated with?(k)is Ik(?(k),whereIk()=22Lk().This matrix and its inverse are routinely provided along with?(k)in the output ofstatistical software packages.It can be shown(see the references in Section 13.8)that the sequence ofestimates?(1),.,?(K)has,asymptotically,the standard multivariate normaljoint distribution given by(11.1).Hence,the sequence of standardized vectorstatisticsI1(?(1)1/2?(1),.,IK(?(K)1/2?(K)has an asymptotic joint distribution that is the multivariate analogue to thecanonical joint distribution of(3.1).The remainder of this section describesimportant applications of this result.c?2000 by Chapman&Hall/CRC13.4.2 A Two-Sample Test for Equality of Survival Distributions withAdjustment for Covariates.We can now derive a generalization of the group sequential log-rank testincorporating adjustment for covariates.Suppose survival distributions aremodeled by the Cox regression model(13.5)or its stratified version(13.8)andxi1is a binary treatment indicator,so that the scalar coefficient 1represents thetreatment effect.Our goal is to construct a group sequential test of H0:1=0in the presence of confounding variables xi2,.,xipfor each subject i.Themaximum partial likelihood estimate of 1at stage k is(k)1and the variance ofthis estimate is approximated byvk=Ik(?(k)111,(13.9)the(1,1)element of the inverse of the information matrix Ik(?(k);see Jennison&Turnbull(1997a,Section 5.2).An alternative variance estimator,which is alsoconsistent under H0and can be more accurate in small samples is obtained byreplacing?(k)in(13.9)with the value of that maximizes(13.6)when 1isconstrained to be zero.Defining the standardized statisticsZk=1(k)/vk,k=1,.,K,(13.10)the results stated at the end of Section 13.4.1 imply that Z1,.,ZKhave,approximately,the canonical joint distribution(3.1)when we equate with 1and Ikwith v1k.Note that if p=1,so the only regression term is for the treatment effect,theassumed model is identical to the proportional hazards model of Section 13.2,orthe stratified proportional hazards model of Section 13.3 in the case of a stratifiedCox regression model.The statistics(13.10)are then asymptotically equivalentto the standardized log-rank statistics and either statistic may be used to define agroup sequential test.The effects of covariates on the observed information add to the difficulty oforganizing interim analyses to occur at pre-specified information levels,and theerror spending method of Chapter 7 is the obvious choice for constructing testingboundaries.If randomization and similar patterns of survival on the two treatmentarms lead to balance in covariate values between treatments throughout a study,information for 1will be roughly one quarter of the number of observed failures,as in the log-rank test.Thus,in designing a study to reach a target informationlevel Imax,it might be decided to enroll a total of 4Imax/subjects,where is the estimated probability that any one subject will be observed to fail by thetime of the last scheduled analysis.Alternatively,a somewhat higher sample sizecould be chosen to protect against errors in estimating the baseline failure rate,thelevel of competing risk censoring or the relation between number of failures andinformation.c?2000 by Chapman&Hall/CRC13.4.3 Factorial DesignsAnother application of the results of Section 13.4.1 is in making inferences abouta linear combination of parameters of the form dT=d11+.+dpp.Thiscould be desirable if the x-variables represent factors in a factorial design withmultiple treatments,an increasingly common feature of clinical trials(Natarajanet al.1996).The vector d=(d1,.,dp)Tmight then be chosen so that dTrepresents a main effect,a linear trend,or possibly an interaction.We can test thehypothesis H0:dT=0 by defining the sequence of test statisticsZk=dT?(k)Ik,k=1,.,K,where Ik=dTI1(k,?(k)d1.This sequence has,approximately,thecanonical joint distribution(3.1),given I1,.,IK,and group sequential tests canbe constructed in the usual way.Note,however,that in a factorial trial involvingmultiple treatments,more than one main effect may be designated a primaryendpoint to be monitored.We discuss group sequential tests for multivariateendpoints in Chapter 15.13.5 Repeated Confidence Intervals for a Hazard RatioConsider again the problem of comparing the survival distributions for twotreatment groups with adjustment for covariates,as described in Section 13.4.2.We can use the arguments of Section 9.1 to construct RCIs for 1,the log hazardratio between treatments.These RCIs are obtained by inverting a family of groupsequential tests of hypotheses H0:1=?1,where?1ranges over the real line.The RCI for 1is the set of values of?1for which H0is currently accepted.A group sequential test of H0:1=?1can be based on the standardizedstatisticsZk(?1)=(k)1?1)/vk,k=1,.,K,where(k)1and vkare as defined in Section 13.4.2.A sequence of RCIs for 1withoverall confidence level 1 is then given by(k)1 ck()vk),k=1,.,K,(13.11)where ck(),k=1,.,K,are critical values for a group sequential two-sidedtest with Type I error probability based on standardized statistics Z1,.,ZKwith the canonical joint distribution(3.1)for information levels Ik=v1k.If anerror spending approach is used,each ck()is obtained as the solution of(7.1)with kequal to the Type I error probability allocated to analysis k.RCIs for thehazard ratio e1are found by exponentiating the limits in(13.11).RCIs can beconstructed in a similar fashion for linear combinations of regression parameters,as described in Section 13.4.3.Small sample accuracy may be improved by tailoring the estimate of thevariance of(k)1to each hypothesized value?1.This is done by replacing?(k)in(13.9)with the maximum partial likelihood estimate of subject to theconstraint 1=?1,thereby defining a new variance estimate,vk(?1)say.Thec?2000 by Chapman&Hall/CRCkth RCI is then?1:(k)1?1 ck()vk(?1).Computation of this RCI is less straightforward since a separate vkmust be foundfor each value of?1considered,and the constrained estimation of variance is notso readily available in standard software packages.In principle,the critical valuesc1(),.,cK()also depend on?1,since they are defined for the particularinformation sequence Ik=vk(?1)1,k=1,.,K;however,it should usuallysuffice to calculate the ck()using the single sequence of information levelsIk=v1kwith vktaken from(13.9)for k=1,.,K.It is instructive to examine more closely the case p=1 when the onlyregression term,x1,is a binary treatment variable taking the value 0 for treatmentA and 1 for treatment B(or equivalently 1 for A and 2 for B).We then have atwo-sample comparison under the proportional hazards model of Section 13.2 orunder a stratified proportional hazards model,as in Section 13.3.For simplicity,we consider the non-stratified proportional hazards model and suppose there areno ties;the stratified case is similar but with sums over strata in the formulae.We write =1so =log()if,as in Section 13.2,denotes the hazardratio of treatment B to treatment A.Using the notation of Section 13.2,the partiallikelihood for the data observed at analysis k isdk?i=1?riA,kriA,k+eriB,k?iA,k?eriB,kriA,k+eriB,k?iB,k.(13.12)The value(k)maximizing this expression is the maximum partial likelihoodestimate,(k)1in our previous notation for the Cox model.We can calculate(k)asthe root of the score equation Sk(0)=0,whereSk(0)=dk?i=1?iB,ke0riB,kriA,k+e0riB,k?(13.13)is the score statistic for testing H0:=0at analysis k,obtained as the derivativewith respect to of the logarithm of(13.12)at =0.The information for when =0is the second derivative of the log partial likelihood,Ik(0)=dk?i=1e0riA,kriB,k(riA,k+e0riB,k)2.By definition,Ik(k)1=vkand,hence,the RCI sequence(13.11)can also bewritten as(k)ck()Ik(k)1/2),k=1,.,K.(13.14)Tailoring the estimated variance of(k)to each hypothesized value of gives thealternative sequence0:(k)0 ck()Ik(0)1/2,k=1,.,K.However,the sequence(13.14)is simpler to compute and Harrington(1989)notesthat it is sufficient,in large samples,to base the RCI at analysis k on the singleinformation estimate Ik(k).c?2000 by Chapman&Hall/CRCIf one wishes to avoid solving the score equations to find the maximum partiallikelihood estimates(k),k=1,.,K,a sequence of RCIs can be constructedfrom the standardized log-rank statistics Zkand associated information levelsIkgiven by(13.1)and(13.2)for the unstratified proportional hazards model orby(13.3)and(13.4)for the stratified model.These Ikare the same as Ik(0),k=1,.,K,in our current notation.Also,Zk=Sk(0)/Ik(0)where Sk(0)isthe log-rank score statistic for testing H0:=0,referred to as Skin Section 3.7.Under the assumption that the Zkfollow the canonical joint distribution(3.1)given information levels Ik,we obtain the RCIs for:(Zk/Ik ck()/Ik),k=1,.,K,(13.15)where the constants ck()are calculated using the observed values of I1,.,IK.These RCIs are reliable if is close to zero but,when the true value of is awayfrom zero,one should expect the intervals(13.14)to attain their nominal coverageprobabilities more accurately.Substituting Zk=Sk(0)/Ik(0)into(13.15)along with the approximationIk(0)dk/4,which is appropriate if the numbers at risk on the two treatmentarms remain roughly equal and is close to zero,we obtain the RCIs for:?4 Sk(0)dk2ck()dk?,k=1,.,K.(13.16)This formula,first given in Jennison&Turnbull(1984,Eq