[[abstract]]This paper illustrates a martingale method of constructing repeated significance tests for a multi-dimensional parameter in survival analysis by reducing it to tests for one-dimensional parameters. This illustration is made with parametric survival data with staggered entry. A simulation study is included to indicate its numerical performance.[[notice]]補正完

Chang, I-shou

This paper illustrates a martingale method of constructing repeated significance tests for a multi-dimensional parameter in survival analysis by reducing it to tests for one-dimensional parameters. This illustration is made with parametric survival data with staggered entry. A simulation study is included to indicate its numerical performance.補正完

Tamkang University Institutional Repository

Statistica Sinica 9(1999), 913-922REPEATED SIGNIFICANCE TESTS OF A MULTI-PARAMETERIN SURVIVAL ANALYSISI-Shou Chang, Chao A. Hsiung and Leng-Cheng HwangNational Central University, Academia Sinica, Taipei and Tamkang UniversityAbstract: This paper illustrates a martingale method of constructing repeated sig-nificance tests for a multi-dimensional parameter in survival analysis by reducing itto tests for one-dimensional parameters. This illustration is made with parametricsurvival data with staggered entry. A simulation study is included to indicate itsnumerical performance.Key words and phrases: Multi-dimensional parameter, orthogonal estimating func-tions, repeated significance tests, survival analysis.1. IntroductionThe purpose of this paper is to illustrate a martingale method of constructingrepeated signiﬁcance tests for a multi-dimensional parameter in survival analysisby reducing it to tests for one-dimensional parameters. We will make the illus-tration by considering parametric survival data with staggered entry times. Theparameter space is assumed to be two-dimensional for simplicity, although it canbe extended to higher dimensions without diﬃculty.For parametric survival data with staggered entry, we know the likelihoodscore process is a martingale relative to the calendar time ﬁltration. In casethe parameter space is one-dimensional, we can make a random time change toobtain a Brownian motion approximation to the likelihood score process, whichpaves the way to proposing repeated signiﬁcance tests of the parameter. (cf.Andersen, Borgan, Gill and Keiding (1993), p.397).The situation is diﬀerent in the multi-dimensional case. Although compo-nents of the likelihood score process are martingales, they are not orthogonal.The operating characteristics of sequential tests deﬁned in terms of these like-lihood processes would be diﬃcult to calculate. We will consider martingaletransforms of these components so as to get orthogonal martingale estimatingfunctions and then apply the strong representation theorem to obtain the stan-dard R2-valued Brownian motion approximation to them. With this preparation,we propose repeated signiﬁcance tests and study their asymptotic properties.We would like to remark here that, although the model of Tsiatis, Boucherand Kim (1995) involves a multi-dimensional parameter, the hypothesis they914 I-SHOU CHANG, CHAO A. HSIUNG AND LENG-CHENG HWANGconsidered concerns only the value of a one-dimensional parameter. In fact, theirwork can be extended with the method presented in this paper.This paper is organized as follows. Section 2 contains the theoretical develop-ments of the repeated signiﬁcance tests and Section 3 presents a numerical study.The numerical study indicates that our theory is satisfactory. We would like tonote that this method may also be useful in other models when the hypothesisof interest is multi-dimensional. One example is the case of paired survival datawith staggered entry (cf. Chang, Hsiung and Chuang (1997)). Another is thecase of Cox regression model with simultaneous entry.2. Repeated Significance Tests for Parametric Survival DataLet (Yj , Zj ,Xj , Cj) be an i.i.d. sequence of random vectors with Yj denotingthe entry time, Zj the covariate, Xj the survival time and Cj the censoring timeof the jth person in a clinical trial. Assume that Yj and Xj are independent; con-ditional on Yj and Zj, Xj and Cj are independent. Assume the hazard functionof Xj given Zj is of the form{λ(·, Zj , θ) | θ ∈ Θ} (2.1)for some open set Θ ⊂ R2.We are interested in proposing repeated signiﬁcance tests for the true pa-rameter p in the situation that the data available at calendar time t is{Yj ∧ s, Zj, (Yj +Xj ∧Cj)∧ s, 1[(Yj+Xj)∧s≤(Yj+Cj)∧s] | s ≤ t, j = 1, . . . , J}. (2.2)LetMj(t, θ) = 1[Yj+Xj ,∞)(t∧(Yj+Cj))−∫ t0λ(s−Yj, Zj , θ)1(Yj ,Yj+Xj∧Cj ](s) ds. (2.3)The assumption (2.1) implies that Mj(t, θ) is a calendar time martingale relativeto the probability for θ.In order to write down the log-likelihood for (2.2), we introdue the stochasticprocessesN tj(u) = 1[Xj ,∞)(u ∧ Cj ∧ (t− Yj)+),Htj(u) = 1(0,Xj∧Cj∧(t−Yj)+](u).We note that N tj(u) −∫ u0 λ(s, Zj , θ)Htj(s) ds is a martingale in u for each t > 0.Then, according to Chang and Hsiung (1988), the log-likelihood of the data (2.2)at time t isLJ(t, θ) =J∑j=1∫ t0log λ(u,Zj , θ) dN tj (u)−J∑j=1∫ t0λ(u,Zj , θ)Htj(u) du. (2.4)MULTI-PARAMETER REPEATED SIGNIFICANCE TESTS 915Assume that λ(t, Zj , θ) has bounded third derivatives in θ and is boundedaway from 0. Then, the likelihood score processUJ,l(t, θ)≡ ∂∂θlLJ(t, θ)=J∑j=1∫ t0∂λ∂θl(u− Yj, Zj , θ)λ(u− Yj, Zj , θ) 1(Yj ,∞)(u) dMj(u, θ) (2.5)is a calendar time martingale relative to the probability for θ. (cf. Chang andHsiung (1988), Tsiatis, Boucher and Kim (1995)). Here l = 1, 2.LetU˜J,1(t, θ) = UJ,1(t, θ),U˜J,2(t, θ) = UJ,2(t, θ)−J∑j=1∫ t0aJ(u, θ)∂λ∂θ1(u− Yj, Zj , θ)λ(u− Yj, Zj , θ) 1(Yj ,∞)(u) dMj(u, θ).(2.6)HereaJ(u, θ) =( J∑j=1( ∂λ∂θ1 (u− Yj, Zj , θ))( ∂λ∂θ2 (u− Yj, Zj , θ))λ(u− Yj , Zj , θ) 1(Yj ,Yj+Xj∧Cj ](u))·( J∑j=1( ∂λ∂θ1 (u− Yj, Zj , θ))2λ(u− Yj, Zj , θ) 1(Yj ,Yj+Xj∧Cj ](u))−1. (2.7)In this paper, we adopt the convention that 00 = 0. We note that aJ is well-deﬁned because of the Schwartz inequality for a mutual variation process. (cf.Elliott (1982), p.101). Let a(u, θ) be the limit of aJ(u, θ) as J tends to inﬁnity,whose existence is implied by the Law of Large Numbers.It follows from (2.6) and (2.7) that, relative to the probability for θ, themutual predictable variation process< U˜J,1(·, θ), U˜J,2(·, θ) >t=J∑j=1∫ t0∂λ∂θ1(u− Yj , Zj , θ)( ∂λ∂θ2 (u− Yj, Zj , θ)− aJ(u, θ) ∂λ∂θ1 (u− Yj, Zj , θ))λ(u− Yj, Zj , θ)·1(Yj ,Yj+Xj∧Cj ](u) du= 0,which says that U˜J,1(·, θ) and U˜J,2(·, θ) are orthogonal martingales relative to theprobability for θ.Let P J(φ) denote the probability for the parameter θ = p + φ√J. Applyingthe Martingale Central Limit Theorem, we get916 I-SHOU CHANG, CHAO A. HSIUNG AND LENG-CHENG HWANGProposition 2.1. Under P J(φ), 1√J( U˜J,1(t, p + φ√J )U˜J,2(t, p + φ√J ))converges to(G1(t)G2(t))asJ goes to infinity, where G1(·) and G2(·) are two continuous independent mean0 Gaussian martingales with variancesg1(t) = Ep∫ t0( ∂λ∂θ1 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du, (2.8)g2(t) = Ep∫ t0(( ∂λ∂θ2 (u− Y1, Z1, p))λ(u− Y1, Z1, p) − a(u, p)( ∂λ∂θ1 (u− Y1, Z1, p))λ(u− Y1, Z1, p))2·λ(u− Y1, Z1, p)1(Y1,Y1+X1∧C1](u) du, (2.9)respectively.Straight-forward calculations (cf. Chang and Hsiung (1988)) lead toProposition 2.2. Under P J(φ), as J goes to infinity,i) 1J∂∂θ1U˜J,1(t, p) converges to −g1(t),ii) 1J∂∂θ2U˜J,1(t, p) converges to −g12(t),iii) 1J∂∂θ1U˜J,2(t, p) converges to 0,iv) 1J∂∂θ2U˜J,2(t, p) converges to −g22(t),whereg12(t) = Ep∫ t0∂λ∂θ1(u− Y1, Z1, p) ∂λ∂θ2 (u− Y1, Z1, p)λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du,g22(t) = Ep∫ t0( ∂λ∂θ2 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du+Ep∫ t0a(u, p)∂λ∂θ1(u− Y1, Z1, p) ∂λ∂θ2 (u− Y1, Z1, p)λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du.We note that g22(t) also equalsEp∫ t0( ∂λ∂θ2 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du+Ep∫ t0a2(u, p)( ∂λ∂θ1 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du,and g2(t) also equalsEp∫ t0( ∂λ∂θ2 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du−Ep∫ t0a2(u, p)( ∂λ∂θ1 (u− Y1, Z1, p))2λ(u− Y1, Z1, p) 1(Y1,Y1+X1∧C1](u) du.MULTI-PARAMETER REPEATED SIGNIFICANCE TESTS 917Applying the Mean-Value Theorem, we get1√J( U˜J,1(t, p)U˜J,2(t, p))=1√J( U˜J,1(t, p + φ√J )U˜J,2(t, p + φ√J ))− 1J( ∂∂θ1U˜J,1(t, θ∗) ∂∂θ2 U˜J,1(t, θ∗)∂∂θ1U˜J,2(t, θ∗∗) ∂∂θ2 U˜J,2(t, θ∗∗))φ, (2.10)for some θ∗ and θ∗∗ lying between p and p + φ√J .It follows from (2.10), Proposition 2.1 and Proposition 2.2 that we haveTheorem 2.1. Under P J(φ), 1√J( U˜J,1(t, p)U˜J,2(t, p))converges weakly to Gφ(t)=(Gφ1 (t)Gφ2 (t)),where Gφ1 (t) − φ1g1(t)− φ2g12(t) and Gφ2 (t)− φ2g22(t) are two continuous inde-pendent mean 0 Gaussian martingales with variances g1(t) and g2(t) respectively.It follows from (2.6) that predictable variation processes are<1√JU˜J,1(·, p) >t = 1JJ∑j=1∫ t0( ∂λ∂θ1 (u− Yj, Zj , p))2λ(u− Yj, Zj , p) 1(Yj ,Yj+Xj∧Cj ](u) du, (2.11)<1√JU˜J,2(·, p) >t = 1JJ∑j=1∫ t0( ∂λ∂θ2 (u− Yj, Zj , p)− aJ(u, p) ∂λ∂θ1 (u− Yj, Zj , p))2λ(u− Yj, Zj , p)·1(Yj ,Yj+Xj∧Cj ](u) du. (2.12)Letτ˜J,i(s) = inf{t ≥ 0 |< 1√JU˜J,i(·, p) >t> s}. (2.13)Because < 1√JU˜J,i(·, p) >t converges almost surely to gi(t), we know τ˜J,i(s) con-verges almost surely to g−1i (s). From this together with Theorem 2.1 and thestrong representation theorem (cf. Pollard (1984), p.71), we getTheorem 2.2. Under P J(φ), 1√J( U˜J,1(τ˜J,1(t), p)U˜J,2(τ˜J,2(t), p))converges weakly to Bφ(t) =(Bφ1 (t)Bφ2 (t)), where Bφ1 (t) − φ1t − φ2g12(g−11 (t)) and Bφ2 (t) − φ2g22(g−12 (t)) are twoindependent standard Brownian motions.Theorem 2.2 is useful in providing repeated signiﬁcance tests for the param-eter θ. For example, for the hypothesisH0 : θ = p,918 I-SHOU CHANG, CHAO A. HSIUNG AND LENG-CHENG HWANGwe introduce the stopping timeTi=inf{t≥0 |< 1√JU˜J,i(·, p)>t≥m0, | 1√JU˜J,i(t, p)|≥di√<1√JU˜J,i(·, p)>t},(2.14)truncated as soon as < 1√JU˜J,i(·, p) >t≥ m for some m > m0. The repeatedsigniﬁcance test is to stop sampling at (T1 ∧ η1) ∧ (T2 ∧ η2) and reject H0 if andonly if T1 < η1 or T2 < η2, whereηi = inf{t ≥ 0 |< 1√JU˜J,i(·, p) >t≥ m}. (2.15)Here m0,m and di are given constants. Because of the independence guaranteedby Theorem 2.2, the signiﬁcance level and some of the powers can be approxi-mated by the related results for one-dimensional Brownian motions. (cf. Sieg-mund (1985), pp.73-81). In Section 3, some of the detailed computation will beillustrated.3. Numerical Studies3.1. Complete sequential testsThe distributions of Yj, Zj, Xj and Cj for this simulation study are describedas follows. Let Yj be uniformly distributed in the interval [0, b], Cj = ∞, thecovariate Zj = (Zj1, Zj2) and Zji be i.i.d. with P (Zji = 0) = P (Zji = 1) = 12 .Conditional on Zj , Xj has intensity exp(θ1Zj1 + θ2Zj2). The simulation resultspresented in the following are based on sample size J = 100.Let T′(d) = inf{t ≥ m0 | |Wµ(t)| ≥ d√t}, where Wµ is a Brownian motionwith drift µ. From Corollary 4.19 and Theorem 4.21 of Siegmund (1985), weknowP0(T′(d) < m) = (d− d−1)φ(d) log(mm0) + 4d−1φ(d) + o(d−1φ(d)), (3.1)Pµ(T′(d) < m) = 1− Φ[m12 (dm−12 − µ)] + {φ[m 12 (dm− 12 − µ)]/(µm 12 )}·(1 + o(1)), (3.2)where Pµ is the probability corresponding to Wµ, φ and Φ are respectively thedensity function and distribution function of a standard normal variable.We now consider the hypothesis H0 : θ = p = (0, 0). Let m0 = 0.05 and m =0.3, di be speciﬁed by (3.1) so as to make P0(T′(di)< m) approximately αi, fori=1, 2. Here α1, α2 are non-negative numbers and satisfy α1+α2−α1α2 = 0.05.With these constants, we will consider the tests deﬁned in (2.14) and (2.15).MULTI-PARAMETER REPEATED SIGNIFICANCE TESTS 919Straightforward calculations give a(u, p) = 12 , g12(t) =12g1(t) and g22(t) =53g2(t). Let µ1 =√J(θ1 + θ22 ), µ2 =53√Jθ2. Theorem 2.2 indicates thatP (θ1,θ2)(T1 < η1 or T2 < η2) (3.3)= P (θ1,θ2)(T1 < η1) + P (θ1,θ2)(T2 < η2)− P (θ1,θ2)(T1 < η1, T2 < η2)is approximatelyPµ1(T′(d1)< m) + Pµ2(T′(d2)< m)− Pµ1(T′(d1)< m)Pµ2(T′(d2)< m), (3.4)where P (θ1,θ2) is the probability corresponding to the parameter (θ1, θ2).When θ1 = θ2 = 0, (3.4) shows that the nominal signiﬁcance level of thetest is 0.05. For other values of θ1, θ2, we can use (3.1), (3.2) and (3.4) to ﬁndasymptotic power, denoted by β. We denote the empirical power (3.3) by β(0)for b = 0 and β(1) for b = 1. We recall that b = 0 is the simultaneous entry case.Tables 1-4 present, for diﬀerent values of θ1 and θ2, the empirical powers β(0)and β(1) by calculating the proportion of 10,000 replicates for which T1 < η1 orT2 < η2.Table 1. Signiﬁcance levels, empirical and asymptotic powers β(0) and β forα1 = α2 = 0.02532 and for various values of (θ1, θ2) and 	t.(θ1, θ2) \ 	t 1 0.1 0.01 0.001 0.0001 β(0.0,0.0) 0.0078 0.0284 0.0473 0.0472 0.0438 0.0500(0.1,0.0) 0.0233 0.0532 0.0787 0.0914 0.0858 0.0648(0.0,0.1) 0.0257 0.0578 0.0850 0.0924 0.0922 0.0982(0.1,0.1) 0.0552 0.1074 0.1341 0.1477 0.1460 0.1175Table 2. Signiﬁcance levels, empirical and asymptotic powers β(1) and β forα1 = α2 = 0.02532 and for various values of (θ1, θ2) and 	t.(θ1, θ2) \ 	t 1 0.1 0.01 0.001 0.0001 β(0.0,0.0) 0.0106 0.0294 0.0418 0.0461 0.0509 0.0500(0.1,0.0) 0.0248 0.0617 0.0810 0.0879 0.0897 0.0648(0.0,0.1) 0.0229 0.0536 0.0757 0.0848 0.0799 0.0982(0.1,0.1) 0.0523 0.1028 0.1327 0.1389 0.1369 0.1175To simulate empirical power β(0) and β(1), we need to simulate the scoreprocess and then check whether it has crossed the boundary from informationtime m0 to m. Since the score process changes its value continuously, we considert ∈ [0, 3] and divide the interval [0, 3] into equal spaced subintervals and onlycheck whether it has crossed the boundary at the endpoints of these subintervals.920 I-SHOU CHANG, CHAO A. HSIUNG AND LENG-CHENG HWANGTable 3. Signiﬁcance levels, empirical and asymptotic powers β(0) and β forα1 = 0.005, α2 = 0.04523 and for various values of (θ1, θ2) and 	t.(θ1, θ2) \ 	t 1 0.1 0.01 0.001 0.0001 β(0.0,0.0) 0.0093 0.0299 0.0428 0.0470 0.0473 0.0500(0.1,0.0) 0.0161 0.0391 0.0621 0.0692 0.0642 0.0540(0.0,0.1) 0.0313 0.0653 0.0826 0.0902 0.0919 0.1084(0.1,0.1) 0.0444 0.0833 0.1153 0.1211 0.1223 0.1151Table 4. Signiﬁcance levels, empirical and asymptotic powers β(1) and β forα1 = 0.005, α2 = 0.04523 and for various values of (θ1, θ2) and 	t.(θ1, θ2) \ 	t 1 0.1 0.01 0.001 0.0001 β(0.0,0.0) 0.0100 0.0303 0.0451 0.0486 0.0449 0.0500(0.1,0.0) 0.0199 0.0427 0.0666 0.0638 0.0683 0.0540(0.0,0.1) 0.0267 0.0572 0.0750 0.0791 0.0825 0.1084(0.1,0.1) 0.0428 0.0855 0.1005 0.1128 0.1109 0.1151In Tables 1-4, 	t is the length of the subintervals. The ﬁrst column is thevalue of the parameter (θ1, θ2), the 2nd, 3rd, 4th and 5th columns display theempirical powers β(0) or β(1) when 	t = 1, 0.1, 0.01, 0.001, 0.0001 respectively,and the last column is the asymptotic power β. These tables indicate that theempirical powers get close to the asymptotic powers as the subintervals 	t getssmaller, and (3.4) provides quite good approximation to the power of the test.Remark. With the previous discussion, we now indicate an additional advantageof our approach as follows.Given the signiﬁcance level α, and parameter values θ1, θ2 satisfying θ21+θ22 =1, for example, we can use (3.1) and (3.2) to choose α1, α2 and d1, d2 so thatthe test has a large power in the direction (θ1, θ2).3.2. Group sequential testsSince one typically performs group sequential tests in medical applications,we present a simulation study for such tests based on the work in Section 2 andSubsection 3.1. In particular, the data in this simulation study is the same asdescribed in the ﬁrst paragraph of Subsection 3.1.For given d1, d2,	t and K, we deﬁneT¯i = inf{t ≥ 0|t = k · 	t, k = 1, . . . ,K, | 1√JU˜J,i(τ˜J,i(t), p)| ≥ di√t}. (3.5)LetT¯′(d) = inf{t ≥ 0|t = k · 	t, k = 1, . . . ,K, |Wµ(t)| ≥ d√t}. (3.6)MULTI-PARAMETER REPEATED SIGNIFICANCE TESTS 921It follows from Theorem 2.2 that the signiﬁcance level and power of the groupsequential testP (θ1,θ2)(T¯1 ≤ K or T¯2 ≤ K) (3.7)is approximatelyPµ1(T¯′(d1)≤ K) + Pµ2(T¯′(d2)≤ K)− Pµ1(T¯′(d1)≤ K)Pµ2(T¯′(d2)≤ K), (3.8)where P (θ1,θ2), Pµ1 and Pµ2 are the same as described in Subsection 3.1.Let β¯(0) and β¯(1) denote respectively the empirical signiﬁcance levels andempirical powers of (3.7) for b = 0 and b = 1. Let β¯ denote the empiricalsigniﬁcance levels and empirical powers of (3.8).Tables 5-6 present β¯(0), β¯(1) and β¯ for K = 5, d1 = d2 = 2.68 and 	t =0.05 and for diﬀerent sample sizes. Both β¯(0) and β¯(1) are the proportion of10,000 replicates for which T¯1 ≤ K or T¯2 ≤ K. Since (3.6) involves really onlymultivariate normal distributions, β¯ is the proportion of 10,000 replicates forwhich T¯′(d1)≤ K or T¯ ′(d2) ≤ K. These simulation results indicate that (3.8)approximates (3.7) reasonably well.Table 5. Signiﬁcance levels and empirical powers β¯(0) and β¯(1) for J = 100and β¯ for various values of (θ1, θ2).(θ1, θ2) β¯(0) β¯(1) β¯(0.0,0.0) 0.0485 0.0465 0.0491(0.1,0.0) 0.0813 0.0847 0.0614(0.0,0.1) 0.0862 0.0726 0.0870(0.1,0.1) 0.1340 0.1287 0.1135Table 6. Signiﬁcance levels and empirical powers β¯(0) and β¯(1) for J = 200and β¯ for various values of (θ1, θ2).(θ1, θ2) β¯(0) β¯(1) β¯(0.0,0.0) 0.0499 0.0505 0.0506(0.1,0.0) 0.1053 0.1026 0.0771(0.0,0.1) 0.1095 0.0949 0.1333(0.1,0.1) 0.1971 0.1866 0.1898AcknowledgementsThanks are due to Y. C. Yao and Z. Ying for helpful conversations duringthe preparation of this work. In particular, the simpliﬁed proof for Theorem 2.2922 I-SHOU CHANG, CHAO A. HSIUNG AND LENG-CHENG HWANGusing strong representation was suggested by Ying. We are also grateful to thereferee/Associate Editor for their valuable suggestions on the numerical studies.This research is partially supported by the National Science Council ofR.O.C.ReferencesAndersen, P. K., Borgan, ∅., Gill, R. D. and Keiding, N. (1993). Statistical Models Based onCounting Process. Springer, New York.Chang, I.-S. and Hsiung, C. A. (1988). Likelihood process in parametric model of censored datawith staggered entry–Asymptotic properties and applications. J. Multivariate Anal. 24,31-45.Chang, I.-S., Hsiung, C. A. and Chuang, Y.-C. (1997). Applications of a frailty model tosequential survival analysis. Statist. Sinica. 7, 127-138.Elliott, R. Y. (1982). Stochastic Calculus and Applications. Springer, New York.Pollard, D. (1984). Convergence of Stochastic Process. Springer-Verlag, New York.Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.Tsiatis, A. A., Boucher, H. and Kim, K. (1995). Sequential methods for parametric survivalmodels. Biometrika 82, 165-173.Department of Mathematics, National Central University, Chung-Li, Taiwan.E-mail: chang@math.ncu.edu.twDivision of Biostatistics, National Health Research Institutes and Institute of Statistical Science,Academia Sinica, Taipei, Taiwan.E-mail: hsiung@nhri.org.twDepartment of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan.E-mail: lchwang@math.tku.edu.tw(Received September 1997; accepted September 1998)

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Repeated significance tests of a multi-parameter in survival analysis

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Repeated significance tests of a multi-parameter in survival analysis

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