| Recently,the literature on failure time data analysis has been growing fast.By failure time data,we mean data that represent times to certain events with positive random variables.There exist many examples of the event referred to as the failure or survival event in our life,such as death,the onset of certain disease,the failure of a mechanical component of a machine,or learning something.One key feature of failure time data is censoring,and there exist many literatures on right censoring.In this paper,we focus on interval-censored data and competing risk data,which are more challenging than right censoring.The analysis of interval-censored failure time data has attracted considerable at-tention in recent years.In this paper,we consider two types of interval-censoring including case ? interval-censored failure time data and case ? interval-censored fail-ure time data that commonly occur in practice.By case ? interval-censored failure time data,we usually mean that instead of being observed exactly,the failure time of interest is known or observed only to belong to an interval(Deng and Fang,2009;Huang,1999:Sun,2006;Sun,Feng,and Zhao,2015).In other words,only partial in-formation about the failure time is available and it is apparent that such data include right-censored data as a special case.One important and commonly asked question is to test the existence of stratum effects in the stratified Cox model that specifies that the subjects in different strata have different,time-dependent effects rather than some constant effects on the hazard function of the failure time of interest.Among others,Sun and Yang(2000)discussed this problem and gave a nonparametric test procedure for the case of right-censored failure time data.In this paper,we developed a nonpara-metric test procedure for the stratum effect with interval-censored data using the idea in Sun and Yang(2000).We will first discuss the test procedure for right-censored data in Sun and Yang(2000).Consider a survival study that involves n independent subjects from q strata.For subject i in the kth stratum,let Tki and Zki denote the failure time of interest and a vector of covariates,respectively,i=1,...,nk,k=1,...,q,where n1+...+nq=n.Suppose that given Zki,Tki follows the stratified proportional hazards model?ki(t,Zki)=?k(t)e?'Zkt,(1)where ?k(t)denotes an unknown baseline hazard function for the subjects in stratum k and ? the vector of regression parameters.Note that here for the notation simplicity,we assume that the covariates have the same effects for the subjects in different strata and the method given below is still valid if they have different effects.In the following,we will focus on the testing of the hypothesis H0:?1(t)=?2(t)=…=?q(t).It is apparent that if HO is true,the Tki's follow the standard proportional hazards model(Cox 1972;Kalbfleisch and Prentice,2002).To test Ho,in this part,we will first assume that one observes right-censored failure time data given by {Nki(t),Yki(t);i=1,...,=1,k...,q},where Nki(t)and Yki(t)denote the counting process and the at-risk indicator process corresponding to subject i from stratum k,respectively.Define Nk(t)=?i=1nk Nki(t),N(t)=?k=1q Nk(t)and Yk(t,?)=?i=1nk Yki(t)exp{?'Zki).Also define Jk(t)=I(?i=1nk Yki(t)>O)and J(t)=I(?k=1q ?i=1nk Yki(t)>0).Then it is common to estimate the cumulative hazard function ?k(t)=?0t ?k(s)ds for stratum k by and the common cumulative hazard function ?0(t)under the hypothesis H0 by with ? replaced by an estimator.The estimators above are commonly referred to as Breslow estimators(Breslow,1972,1974).To test the hypothesis H0,let ?p denote the partial likelihood estimator of ? under HO and ? the longest follow-up time on all subjects in the study.Then Sun and Yang(2000)proposed to use the test statistic Wr(?p)=(W1r(?p),…,Wqr(?p))',where Wkr(?)=?0?L(t)Yk(t,?){d?kr(t,?)-Jk(t)d?0r(t,?)}with L(t)being a nonnegative bounded predictable weight process.It is easy to see that the test statistic above represents the estimated cumulative differences between the cumulative hazard functions in general and under the hypothesis HO and it is expected to be close to zero if HO is true.In the following,we will generalize it to the interval-censored data situation.Suppose that one observes interval-censored data given by {(Lki,Rki];i=1,…,nk,k=1,,q},where Lki<Tki ?Rki.That is,each failure time is known only to be between Lki and Rki and for the exact observation,we will assume that Lki=Rki.In this situation,it is apparent that both ?p and the Wkr(?)'s are not available and thus so the test statistics described above.In the following,we will assume that the interval-censoring is non-informative(Sun,2006).To construct the test statistic for HO,note that we can treat both Lki and Rki as failure time variables and thus define the corresponding counting processes and at-risk processes NkiL(t)and YkiL(t)for Lki and NkiR(t)and YkiR(t)for Rki,respectively.For each pair(k,i),define NkiT(t)=1/2{NkiL(t)+NkiR(t)},YkiT(t)=1/2{YkiL(t)+YkiR(t)},which can be regarded as the counting process and at-risk process for Tki.Note that here for right-censored observations,we will treat Rki to be right-censored at Lki,and it is easy to see that for right-censored data,the NkiT(t)'s for YkiT(t)'s reduce to the Nki(t)'s for Yki(t)'s,respectively.Let pdenote the maximum likelihood estimator of? under H0(Sun,2006).Then motivated by the test statistic Wr(?p),we propose to employ the test statistic W(?)=(W1(?),…Wq(?))'.Here Wk(?)=?0T L(t)YkT(t,?){d?k(t,?)-JkT(t)d?0(t,?)},where L(t)is defined as above,and YkT(t,?),JkT(t),?k(t,?)and ?0(t,?)are defined as Yk(t,?),Jk(t),?kr(t,?)and ?0r(t,?)with replacing the Nki(t)'s and Yki(t)'s by the NkiT(t)'s and YkiT(t)'s,respectively.Theorem 1 Under the null hypothesis HO and mild regularity conditions,the dis-tribution of n-1/2W(?)can be asymptotically approximated by the multivariate normal distribution with mean zero and the covariance matrix n-1?(?)=n-1(?k1k2(?))q×g given in Chapter 2 as n ?? and nk/n ?qk,where qk>0 and ?k=1q qk=1.For the statistic W(?),note that ?k=1q Wk(?)0.These results suggest that the testing of the hypothesis HO can be performed by using the test statistic U(?)=(W1(?),…,Wq-1(?))D(?)-1(W1(?),…,Wq-1(?)))'based on the ?2-distribution with the degrees of freedom q-1,where D(?)denotes the matrix ?(?)with the last row and column removed.Similarly to weighted log-rank test(Peto and Peto,1972;Andersen et al.,1982;Fleming and Harrington,1991),it is needed to include the weight process to apply the above test procedure.If focusing on a paxticular time period for the comparison of the underlying hazard functions,we can include the weight process(Klein and Moeschberger,2003;Fleming and Harrington,1991;Andersen et al,1993).In addition,in some cases,we can improve the tests by choosing appropriate weight process,such as the G?,?1 test(Fleming and Harrington,1991;Buyske,Fagerstrom and Ying,2000;Yang and Prentice,2010).For the test pro-cedure above,we adopt the weight process L(1)(t)?1,L(2)(t)=n-1 ?k=1q ?i=1nk YkiT(t)and L3(t)=1-L(2)(t),which is proposed by Zhang(2006)and applied by Zhao and Sun(2011)and Zhao et al.(2011).It is easy to see that the weight process L(1)(t)weights everything equally.The weight processL L2(t)is proportional to the number of subjects under observation,so it gives more weights to the early survival differences.Contrary to L(2)(t),the weight process L(3)(t)emphasizes the late survival differences.It is easy to prove that all three weight functions satisfy the conditions of Theorem 1.Note that although the idea used above is the same as in Sun and Yang(2000),the implementation and the derivation of the asymptotic distribution are much more complicated and difficult than these with the case of right-censored data.For example,with the latter situation,the partial likelihood estimator ?? can be easily obtained independent of the cumulative hazard function ?0(t),while to obtain the maximum likelihood estimator ?,one has to deal with or estimate A0(t)together.By current status data,we usually mean that each study subject is observed at a particular observation time,and an indicator of whether the event has occurred no later than the observation time.Test for the existence of stratum effects in the stratified Semi-parametric models is one important and commonly asked question.Among others,Sun and Yang(2000)considered nonparametric tests for stratum effects of right-censored failure time data from the Cox model.Fan et al.(2019)discussed nonparametric test for stratum effects of interval-censored data under the Cox model.Then,we will discuss nonparametric test for stratum effects of current status data under additive hazards model using the idea in Sun and Yang(2000).Firstty,We will consider the test procedure without informative censoring.Sup-pose that given Zki,Tki follows the additive hazards model?kit(t,Zki(s),s ?t)=?k(t)+?'0Zki(t),(2)where ?k(t)is an unspecified baseline hazard function for the subjects in stratum k and?0 is a vector of unknown regression parameters.For the failure times T'kis of interest,we will be assumed that each subject is observed only once at time Cki.and one observes current status data given by {(Cki.?ki=I(Tki?Cki),Zki(t),t?Cki);i=1,...,nk,k =1,...,q}.In practice,Cki may depend on covariates too.For this,we assume that given{Zki(s),s ? t},they follows the proportional hazards model?kic(t,Zki(s),s?t)=?c(t)e?'0Zki(t),(3)where ?c(t)is an unspecified baseline hazard function and ?0 is a vector of unknown regression parameters as ?0.As discussed in Sun and Yang(2000),to simplify the notation,we suppose that the covariates have the same effects in models(2)and(3)for the subjects in different strata and the test procedure given here is still valid if they have different effects.In the following,we consider the null hypothesis H0:?1(t)?2(t)=…=?q(t).To test H0,for each ki,define Nkic(t)=I{Cki ?min(t,Tki))and Ykic(t)=I{Cki?t).Then Nkic(t)is a counting process with the intensity process?ki(t,Zki(s),s ?t)=e?'0Zki(t)-?'0Zki*(t)?c(t)e-?0t?k(s)ds=?k(t)e?'0Zki{t)-?'0Zki*(t),(4)(Lin et al.,1998),where Zki*(t)=?0tZki(s)ds,and ?k(t)=?c(t)e-?0t?k(s)ds.Note that equation(4)is the Cox proportional hazards model.Let(?,?)denote the partial likelihood estimator of(?0,?0)under HO(Lin et al.,1998;Feng et al.,2015).Then motivated by the test statistic Wr(?p),we propose to use the test statistic W(?,?)=(W1(?,?),...,Wq(?,?))'.Here Wk(?0,?0)=?0TL{t)Ykc{t,?0,?0){d?k{t,?0,?0)-Jkc(t)d?0(t,?0,?0)},where L(t)being a nonnegative bounded predictable weight process,and Ykc(t,?0,?0),Jkc(t),?k(t,?0,?0)and ?0(t,?0,?0)are defined as Yk(t,?),Jk(t),?kr(t,?)and ?0r(t,?)with replacing the Nki(t)'s,Yki(t)'s,? and Zki's by the Nkic(t)'s,Ykic:(t)'s,(?0,?0)and(-Zki*(t),Zki(t))'s,respectively.Theorem 2 Under the null hypothesis H0 and mild regularity conditions,it is shown that n-1/2W{?,?)has an asymptotically multivariate normal distribution with mean 0 and the variance-covariance matrix n-1?(?,?)=n-1(?k1k2(?,?))q×q given in Chapter 3,as n?? and nk/n?qk,where qk>0 and ?k=1q qk=1.Let W0(?,?)denote the first k-1 components of W(?,?),and D(?,?)the ma-trix ?(?,?)with the last row and column removed.Then the hypothesis H0 can be performed by using the test statistic ?0=W0(?,?)TD-1(?,?)W0(?,?)based on the?2-distribution with q-1 degrees of freedom.This is because ?k=1q Wk(?,?)=0.To employ the test procedure above,one needs to choose the weight function L(t).For this,a simple and natural choice is L(1)(t)?1.Some other choices include L(2)=n-1 ?k=1q ?i=1nk YkiT(t)and L(3)(t)=1-L(2)(t).Now we consider the test procedure with informative censoring.Suppose that the observation time is related to the underlying survival time,which is referred to as information censoring(Ma et al.,2015;Chen et al.,2012;Li et al.,2017a).In this case,instead of model(2)and(3),one may consider that Tki follows the stratified additive hazards model?kit(t,Zki(s),s ?t)=?k(t)+bki(t)+?'0Zki(t),(5)and Cki follows the stratified proportional hazards model?kic(t,Zki(s),s?t)=?c(t)e?'0Zki(t)+bki(t),(6)where random effect bki(t)'s are independent and identically distributed,given bki(t)'s and Zki{t)'s,T'kis and C'kis are independent.In the following,we also consider the hypothesis HO.We note that Nkic(t)is a counting process with the intensity process?ki{t,Zki(s),S?t)=e?'0Zki(t)-?'0Zki*(t)?c(t)e-?0t?k(s)dsEb(ebki(t)-?0tbki(s)ds)=?k(t)e?'0Zki(t)-?'0Zki*(t),(Feng et al.,2018;Zhang et al.,2005),where ?k(t)=?k(t)Eb(ebki(t)-?0tbki(s)ds),and'E'b means the expectation with respect to the random effect bki{t)'s.Note that the equation above says that ?ki(t,Zki(s),s ?t)satisfies the Cox proportional hazards model.Using the similar procedure above,we can obtain that the hypothesis H0 can also be performed by using the same test statistic ?0=W0(?,?)TD-1(?,?)W0(?,?)based on the ?2-distribution with q-1 degrees of freedom,where(?,?)is the partial likelihood estimator of(?0,?0)under HO.Competing risks data arise when an subject can experience two or more types of event and the occurrence of one type of event precludes the occurrence of other types of events.In the following,for convenience,we assume that there are only two types of failure,where Type 1 represents the failure type of interest and Type 2 includes all other competing risks.Among others,Li and Yang(2016)developed simultaneous inference procedures for joint analysis of both cause-specific hazard of Type 1 failure and the all-cause hazard and joint analysis of both cause-specific hazard of Type 1 failure and the cause-specific hazard for other failure types.However,nonparametric testing procedures for joint testing of both cause-specific hazard and the all-cause hazard,or joint testing both cause-specific hazard of Type 1 failure and the cause-specific hazard for other failure types for competing risk data in the semi-parametric model,such as Cox model or additive hazard model,does not seem to be available.So,our primary purpose is to develop nonparametric testing procedures for joint testing of both cause-specific hazard and the all-cause hazard,and joint testing both cause-specific hazard of Type 1 failure and the cause-specific:hazard for other failure types for competing risk data in the Cox model,using the idea discussed in Li and Yang(2016).Firstly,We will first consider the tests for cause-specific hazard.Consider q d-ifierent groups of independent subjects in a survival study with total sample size n.For subject%in group k,let Tki,Dki,Cki and Zki denote the failure time of inter-est,the type of failure,the censoring time,and a vector of covariates,respectively,i=1,…,nk,k=1,…,g,where n1+…+nq=n.Assume that within group k,(Tki,Dki,Cki,Zki)are independent and identically distributed and that Cki and Tki are independent given covariates Zki.The k groups are allowed to have different censor-ing distributions.For group k(k=1,…,g),we observe a right censored competing risks failure time data {(Xki,?ki,Zki),i=1,...,nk},where Xki=min{Tki,Cki)and?ki=DkiI(Tki ?Cki).For subject i in group k(k=1,…,g),suppose that the cause-specific hazard function(Prentice et al.1978)for Type 1 failure has the form?1ki(t|Z)=?1k(t)e?'1Zki(1),(7)where ?1k(t)is the unknown baseline cause-specific hazard function,?1 is the vector of unknown regression parameters,and Zki(1)dsenotes the vector of covariates which may be the same as or a part of Zki.To simplify the notation,in model(7),(12)and(17),we suppose that the covariates have the same effects for the subjects in different group(Sun and Yang,2000;Fan et al.,2019),and the test procedure given below is still valid if they have different effects.In the following,consider the following null hypothesis H0:?11(t)=?12(t)=…?1q(t)for all 0<t<?·(8)For the case of right censored data instead of the competing risk data,Fan et al.(2019)proposed a nonparametric test procedure.Then motivated by Li and Yang(2016),the nonparametric test procedure in Fan et al.(2019)for right censored failure time data can be applied to test HO by treating all other competing risks as independent right censoring(Li and Yang,2015;Tsiatis,1975;Prentice et al.,1978;Lindkvist and Belyaev,1998).Specifically,let Njki(t)=I{Xki?t,Dki=j)be the counting process of corresponding to subject i of type j failures in group k,and Yki(t)=I{Xki?t}be the at risk indicator process corresponding to subject i in group k,k=1,2,…,q.Define Yk(t,?)=?i=1nk Yki(t)exp(?'1Zki)and Jk(t)=I(?i=1nk Yki(t)>0).Let ?1 denote the maximum likelihood estimator of ?1 in model(7)under HO(Kalbfleisch and Prentice 2002).The test statistic for(8)is defined as W1(?1)=(W11(?1),…W1,q-1(?1))',(9)here where L1(t)is a nonnegative bounded predictable weight function that converges in probability to some deterministic function l2(t)as n?? and ? is the longest follow-up time on all subjects in the study.It can be shown that under the null hypothesis(8),n-1/2W1(?1)can be asymptotically approximated by the multivariate normal distribu-tion with mean zero and the covariance matrix n-1D1(?1)=n-1(?k1k2(?1))(q-1)×(q-1).Furthermore,where ?k1l=1 if k1=l and 0 otherwise,and ?k2l=1 if k2=l and 0 otherwise.This leads to an asymptotic ?2 test with the degrees of freedom q-1 for(8)based onSecondly,We will first consider the joint tests for both cause-specific hazards.Suppose that the Cox proportional cause-specific hazards model(7)holds for Type 1 failure.In addition,suppose that for subject i in group k(k=1,…,q),the cause-specific hazard function(Prentice et al.,1978)for Type 2 failure has the form where ?2kl(t)is the unspecified baseline cause-specific hazard function,?2 the vector of unknown regression parameters,and Zki(2)the vector of covariates which may be the same as or a part of Zki.Consider the following joint null hypothesis Let ?2 be the maximum likelihood estimator of ?2 in model(12)under H0(Kalbfleisch and Prentice,2002).Let be the nonparametric test statistic for H0:?21(t)=?22(t)=…?2q(t)for all 0<t<?(Fan et al.,2019).In the above formula,where L2(t)is a nonnegative bounded predictable weight function that converges in probability to some deterministic function l2(t)as n? ?.It can be shown that under the null hypothesis H0:?21(t)=?22(t)=…?2q(t)for all 0<t<?,n-1/2W2(?2)can be asymptotically approximated by the multivariate norual distribution with mean zero and the covariance matrix n-1D2(?2)given in Chapter 4.Theorem 3 Let W1(Let W1(?1)and W2(?2)be defined by(9)and(14).Under the null hypothesis(13)and mild regularity conditions,n-1/2(W1(?1)',W2(?2)')' has an asymptotic multivariate normal distribution with mean 0 and the covariance matrix n-1?(1)(?1,?2)=n-1(?k1 k2(?1,?2))2(q-1)x2(q-1)given in Chapter 4 as n?? and nk/n ?qk,where qk>O and ?k=1q qk=1.These results allow one to construct an asymptotic ?2 joint test with the degrees of freedom 2q-2 for(13)similar to those for(8)based onThirdly,we will discuss the joint tests for cause-specific hazard and all-cause haz-ard.Assume that the proportional cause-specific hazards model(7)holds.In addition,assume that for subject i in group k(k=1,…q),the proportional all-cause hazards function has the form where ?.k(t)is an unknown baseline hazard,?.the vector of unknown regression pa-rameters,and Zki(3)the vector of covariates which may be the same as or a part of Zki.Consider the following null hypothesis H0:?11(t)=?12(t)=…?1q(t)and ?.1(t)=?.2(t)=?.q(t)for all 0<t<?.(18)Let p.be the maximum likelihood estimator of ?.in model(17)under HO(Kalbfleisch and Prentice 2002).Let be the nonparametric test statistic for HO:?.1(t)=?.2(t)=?.q(t)for all 0<t<?(Fan et al.,2019).Here where L.(t)is a nonnegative bounded predictable weight function that converges in probability to some deterministic function l.(t)as n??.It can be shown that under the null hypothesis H0:?.1(t)=?.2(t)=?.q(t)for all 0<t<?,n-1/2W(?.)can be asymptotically approximated by the multivariate normal distribution with mean zero and the covariance matrix n-1D.(?.)given in Chapter 4.Theorem 4 Let W1(?1)and W(?.)be defined by(9)and(19).Under the null hypothesis(18)and mild regularity conditions,n-1/2(W1(?1)',W.(?.)')' has an asymptotic multivariate normal distribution with mean 0 and the covariance matrix n-1?(2)(?1,?.)=n-1(?k1k2(2)(?1,?.))2(q-1)×2(q-1)given in Chapter 4 as n?? and nk/n? qk,where qk>0 and ?k=1q qk=1.Then the hypothesis(18)can be tested by using the statistic U1.(?1m?.)=(W1(?1)',W.(?.)')?(2)(?1,?.)-1(W1(?1)',W.(?.)')',(21)which has an asymptotic ?2 distribution with 2q-2 degrees of freedom.Theorem 5 Let W1(?1),W2(?2)and W.(?.)be defined by(9),(14)and(19).If Zki(1)=Zki(2)=Zki(3),?1=?2=?.and L1(t)=L2(t)=L.(t),Then under the null hypothesis(13)and(18),and mild regularity conditions,U12(?1,?2)=U1.(?1,?.)a.s.,as n ?? and nk/n? qk,where qk>O and ?k=1q qk=1,that is,the test statistic of the joint tests for cause-specific hazard and all-cause hazard is almost surely equal to the test statistic of the joint tests for both cause-specific hazards.Under the conditions of the theorem above,the joint tests for cause-specific hazard and all-cause hazard is equivalent to the joint tests for both cause-specific hazards.We can see that the conditions of the theorem are strict since it is rare that the covariates have the same effects for both cause specific hazards and all cause hazard.However,we can find some connections between the joint tests for cause-specific hazard and all-cause hazard and the joint tests for both cause-specific hazards. |
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