Consistency of bootstrap procedures for the nonparametric assessment of noninferiority with random censorship

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1 Consistency of bootstrap procedures for the nonparametric assessment of noninferiority with random censorship Gudrun Freitag 1 and Axel Mun Institut für Mathematische Stochasti Georg-August-Universität Göttingen, Germany Abstract In this paper we consider general Hadamard differentiable functionals φλ R, Λ T ofthe cumulative hazard functions of two samples of randomly right censored data, which can be used for the nonparametric assessment of noninferiority. We prove the consistency of various bootstrap procedures as suggested in Freitag et al. [1] for the practical implementation of tests for this problem. Keywords: Noninferiority, Relative ris, Odds ratio, Hazard ratio, Hadamard differentiability, Wea convergence, Bootstrap 1 Introduction We consider the problem of showing that a new, test treatment T is at most irrelevantly inferior to an established standard, reference treatment R, with respect to the corresponding survival probabilities. As to the motivation and specific methodological concerns related to noninferiority trials, we refer to the special issue of Statistics in Medicine on this topic in 23 [2] and to a number of draft guidelines of the Committee for Proprietary Medicinal Products CPMP and the International Conference of Harmonization ICH, such as [3]-[7]. A survey on existing methods for the assessment of noninferiority with censored data can be found in Freitag [8]. This paper deals with the approach suggested in Freitag et al. [1], where the comparison of the two samples is performed nonparametrically, using discrepancy functionals of the two underlying cumulative hazard functions, such as the cumulative hazard ratio or the cumulative odds ratio. These can be assessed over a whole time interval within the follow-up period of a clinical trial. For this we assume the setting of two samples of failure times, which are subject to simple random right censoring. However, other censoring mechanisms could be treated as well cf. the discussion of this issue in Freitag et al. [1]. We denote the cumulative distribution function cdf of a failure time under the reference and test treatment by F R and F T, respectively. Throughout the following we assume that the F are continuous cdfs with Lebesgue densities f,survivor functions S =1 F, and continuous hazard functions λ = f /S, = R, T. Let T i, i = 1,..., n, be independent and identically distributed i.i.d. failure times according to F, = R, T. Further, let the corresponding censoring times U i be distributed 1 Address for correspondence: Gudrun Freitag, Institut für Mathematische Stochasti, Georg-August- Universität Göttingen, Maschmühlenweg 8-1, 3773 Göttingen, Germany. Tel: +49/551/391351, Fax: +49/551/ freitag@math.uni-goettingen.de. 1

2 according to G,whereU i is independent of T i, i =1,..., n, = R, T. The observations consist of pairs X i,δ i, where X i = T i U i =min{t i,u i are the observed failure times and δ i = I {Ti U i are the associated observable censoring indicators, i =1,..., n, = R, T. For estimating the underlying cumulative hazard functions Λ and distribution functions F,we will use the standard nonparametric Nelson-Aalen and Kaplan-Meier estimators, respectively, which shall be denoted by ˆΛ and ˆF, = R, T. For comparing the two samples of failure times nonparametrically, one can consider the difference of the survival functions, the cumulative odds ratio or the ratio of the survival functions, or the ratio of the cumulative hazard functions cf. Remar 2.3. We will deal with these in a unified framewor, using a general discrepancy functional φ = φλ R, Λ T of the underlying cumulative hazard functions. Note that the survivor function S can be written as S t = P t [1 dλ s], where t P denotes the product integral cf. Andersen et al. [9]. An analogous relation holds for the estimated survivor function Ŝ =1 ˆF, i.e. Ŝ t = P t [1 dˆλ s]. Using these notations, the difference functional can be written as dt = F T t F R t = P t [1 dλ R s] P t [1 dλ T s] = φ d Λ R, Λ T t. 1 Here the subscript d in φ d Λ R, Λ T indicates the concrete discrepancy functional d. Freitag et al. [1] argue that for the nonparametric assessment of noninferiority of the test treatment as compared to the reference treatment, it is of interest to assess such a discrepancy over a whole time interval, instead of only at a single time point. Thus, in terms of a general discrepancy functional φ, we suggest hypotheses of the form H φ : φλ R, Λ T t φ for some t [τ,τ 1 ] vs. K φ : φλ R, Λ T t < φ t [τ,τ 1 ], 2 where φ is a fixed irrelevance bound which has to be ined in advance. For absolute discrepancy measures such as the difference this will be a small positive value, whereas for relative discrepancy measures such as the cumulative odds ratio it will be a value slightly larger than 1. For testing the above hypotheses we will use the nonparametric estimator ˆφΛ R, Λ T =φˆλ R, ˆΛ T. In the following we will show the wea consistency of the bootstrap based methods suggested in Freitag et al. [1]. 2 Wea convergence An essential prerequisite is the wea convergence denoted by = of the two underlying cumulative hazard processes, n 1/2 ˆΛ Λ, = R, T,inthespaceD[τ,τ 1 ]ofcàdlàg functions on [τ,τ 1 ], which will be considered as equipped with the remum norm and the sigma field of open balls cf. e.g. Shorac & Wellner [1] or Gill [11]. Assumption 2.1 Suppose Λ t < for t τ 1,and1 1 F τ 1 1 G τ 1 < 1for = R, T. Under Assumption 2.1, it is well nown that, in D[τ,τ 1 ], n 1/2 ˆΛ Λ D = Z, = R, T, 3 2 D

3 where Z is a mean zero Gaussian process with Z = and covariance structure COVZ s 1, Z s 2 = D s 1 s 2, with D t = t λ s ds 4 1 F s1 G s cf. Theorem IV.1.2 in Andersen et al. [9]. The following result on the wea convergence of the discrepancy process follows immediately from the functional delta method for Hadamard differentiable functionals cf. e.g. Gill [11]. Lemma 2.2 Suppose that the functional φ : D[τ,τ 1 ] D[τ,τ 1 ] D[τ,τ 1 ] is Hadamard differentiable at Λ R, Λ T with derivative dφ ΛR,Λ T. Further, assume that Assumption 2.1 holds. Then, if n R,n T,n T /n R + n T ρ, 1, 5 we have that, in D[τ,τ 1 ], κn ˆφΛ R, Λ T φλ R, Λ T = D dφ ΛR,Λ T ρ Z R, 1 ρ Z T = W φ, 6 where κ n = n R n T /n R + n T and Z is ined in 3, = R, T. Thus, the limiting random element W φ is a mean zero Gaussian process, and at each t [τ,τ 1 ] we have asymptotic normality of W φ t with expectation zero and variance σφ 2 t, which depends on the underlying failure time and censoring distributions. Remar 2.3 In the following we consider several special cases of the functional φ. 1. In case of the difference functional d from 1 the Hadamard differentiability is given immediately, using Assumption 2.1, the continuity of Λ, = R, T, and Lemma of van der Vaart & Wellner [12], which gives the Hadamard differentiability of the product integral. The covariance structure of the limiting random element W d is given by COVW d s, W d t = 1 ρs T ss T td T s t ρs R ss R td R s t, where D, = R, T,andρ are ined in 4 and 5, respectively. Here the independence of the two processes Z, = R, T,wasused. 2. For the relative ris functional 1 P t [1 dλ rt T s] = F T t/f R t = = φ 1 P t r Λ R, Λ T t, [1 dλ R s] the additional assumption F R t >, t [τ,τ 1 ], is required. The covariance structure of the corresponding limiting random element W r is then given by COVW r s, W r t = 1 ρ S T ss T t F R sf R t D T s t ρ F T ss R sf T ts R t FR 2sF R 2t D R s t. 3

4 3. For the cumulative odds ratio functional, ot = F T t1 F R t 1 F T tf R t = t P [1 dλ T s]1 t P [1 dλ R s] 1 t P [1 dλ T s] t P [1 dλ R s] = φ o Λ R, Λ T t, we need the additional assumption F R t1 F T t >, t [τ,τ 1 ]. structure of the resulting limiting random element W o is then given by The covariance COVW o s, W o t = 1 ρ 4. For the cumulative hazard ratio, S R ss R t S T ss T tf R sf R t D T s t ρ F T sf T ts R ss R t S T ss T tf 2 R sf 2 R t D Rs t. ht =Λ T t/λ R t = φ h Λ R, Λ T t, it has to be assumed in addition that Λ R t >, t [τ,τ 1 ]. The covariance structure of the limiting random element W h is then given by 2.1 Suprema 1 COVW h s, W h t = 1 ρ F R sf R t D T s t ρ F T sf T t FR 2sF R 2tD Rs t. The hypotheses in 2 suggest to base a test on t [τ,τ 1 ] ˆφt, which has be pursued in Freitag et al. [1]. In order to proof the consistency of such a test, we use results by Raghavachari [13] on the convergence in distribution of Kolmogorov-Smirnov type statistics under the alternative. Following the lines of the proof given therein, we can derive the convergence in distribution of our general remum statistics. Lemma 2.4 Under the assumptions of Lemma 2.2 it follows that D κn ˆφt φt W φ t, 7 t K + φ with W φ from 6 and K + φ = {t [τ,τ 1 ]: s [τ,τ 1 ] φs =φt. The proof of Lemma 2.4 essentially mimics the proof of Theorem 1 in Raghavachari [13]. However, the details will be required in the proof of the subsequent Theorem 3.2, hence we give the complete proof here. Moreover, instead of a result on the empirical process cf. Lemma 1 in [13], a more general result on wealy convergent processes is required in the following. Proof: Let λ + = φt, 4

5 D n + = ˆφt, Z n + = ˆφt φt, t K + φ D n ++ = D n + λ + Z n +. P Thus, we have to show that κ n D n ++ asn R,n T, since the limiting distribution of κn Z n + is given by t K + W φ t cf. Lemma 2.2. φ As φt is a continuous, bounded function on [τ,τ 1 ], the set K + φ is a non-empty compact subset of [τ,τ 1 ]. From the wee convergence result of Lemma 2.2 and Theorem V.1.3 in Pollard [14] we have that for every ε, η >, there exists a partition τ = t <t 1 <... < t l = τ 1 such that κn lim P max ˆφt φt ˆφt i +φt i >ε <η, 8 n R,n T i t J i where J i =[t i,t i 1 fori =, 1,..., l 1. Choose those J i for which there exists a t J i K + φ, and denote these by J 1,..., J p and t i,..., t p, respectively p l. Thus, K + φ p J i=1 i = M p.let Mp c =[τ,τ 1 ] \ M p.thenwehave { κn P D n ++ >ε From 8 it follows, for sufficiently large n R,n T, { κn P ˆφt λ + Z n + >ε t M p { κn = P ˆφt λ + Z n + >ε { κn P ˆφt λ + Z n + >ε t M p { κn + P ˆφt λ + Z n. + >ε t Mp c { κn P max ˆφt φt+φt λ + Z i t J n + >ε i { κn P max ˆφt φt+φt φ t i ˆφt i +φt i >ε i t J i { κn P max ˆφt φt ˆφt i +φt i >ε i t J i <η. Further, on Mp c, φt is bounded above by a number ρ with <ρ<λ+. This follows from the continuity of φt, compactness of K + φ and K+ φ M p. Let ν>such that ν<λ + ρ. By the Gliveno-Cantelli Theorem we have, for sufficiently large n R,n T, ˆφt λ + < ˆφt λ + + ν<ρ λ + + ν< t Mp c t Mp c with probability one. Thus, { lim P κn ˆφt λ + Z + n R,n T n >ε =. t M c p 5

6 Hence, { κn lim P D n ++ >ε <η. n R,n T Since η was chosen arbitrary, P { κ n D ++ n >ε asn R,n T. 3 Bootstrap approximations The methods proposed in Freitag et al. [1] are based on bootstrapping. In the case of randomly right censored data, the so-called simple method suggested by Efron [15] can be used. According to Theorem 2.1 in Aritas [16], this method yields consistent estimators of the underlying distribution functions in the following sense. Theorem 3.1 For sample {R, T, a bootstrap sample Xj,δ j,j=1,..., m, is drawn from the pairs X i,δ i,i=1,..., n. The corresponding Kaplan-Meier estimator ˆF is then calculated. Then we have, as m,n, m 1/2 ˆF ˆF D = X, in D[τ,τ 1 ], for almost all sample sequences X i,δ i,i=1,..., n.herex denotes the limiting random element of the product-limit process n 1/2 ˆF F, and we have X = Z /S on [τ,τ 1 ] cf. 3 and Th.IV.3.2 in Andersen et al. [9]. An analogous result can be proved in the same way via martingale representations; cf. [16] for the bootstrapped Nelson-Aalen processes. This yields, as m,n, m 1/2 ˆΛ ˆΛ D = Z, in D[τ,τ 1 ], for almost all sample sequences X i,δ i,i=1,..., n. Plugging the bootstrap estimators ˆΛ, = R, T, into the inition of φt yields a bootstrap estimator ˆφ t. In Freitag et al. [1] two methods are proposed to solve the testing problem 2, the pointwise and the remum approach. Both methods require the wea consistency of the bootstrap approximations they use. This is ined via the convergence of the sequences of distributions of the random elements under consideration as follows. D Let L[X] be the distribution of a random variable in R, and pose {X n Rwith X n X in R. Further, let {Xm be a bootstrap version of {X n, and denote the conditional distribution, given {X n,byl. Then the sequence {L [Xm] is called wealy consistent for {L[X n ], if ρ P L [Xm], P L[X] form, n,whereρ P is the Prohorov metric cf. Gill [11], p Pointwise approach Here an upper 1 α-confidence bound φt 1 α for φt is constructed for each t [τ,τ 1 ]. Then the pointwise test of 2 consists of rejecting H φ if φt 1 α < φ for all t [τ,τ 1 ]. We 6

7 suggest using bootstrap confidence bounds for calculating φt 1 α. Thus, it has to be assured that the bootstrapped discrepancy process at t, κ n ˆφ t ˆφt is wealy consistent for the distribution of W φ t, which is normal with mean zero and and variance σφ 2 t cf. Lemma 2.2. Note that here the bootstrap sample sizes have been chosen as m = n, = R, T.Therequired result follows immediately under the assumptions of Lemma 2.2 and Theorem 3.1, applying the functional delta method for the bootstrap as given in Theorem 5 in Gill [11]. Based on this, there are several possibilities to construct φt 1 α, t [τ,τ 1 ] cf. Efron & Tibshirani [17] or Shao & Tu [18]. In Freitag et al. [1] the percentile and the bias-corrected accelerated percentile methods were applied. 3.2 Supremum approach Here the aim is to calculate an upper 1 α-confidence bound φ 1 α directly for the remum of the discrepancy process, i.e. for t [τ,τ 1 ] φt. Then H φ can be rejected if φ 1 α < φ. Thus, it has to be shown that the bootstrap approximation of the remum functional is wealy consistent for the limiting random element in 7. Note that, in contrast to the case φ =, there arise technical difficulties due to the unnown set K + φ. These can be circumvented by choosing the bootstrap sample sizes smaller than the original sample sizes, as stated in the following Theorem. Theorem 3.2 Suppose that the assumptions of Lemma 2.2 are satisfied. Further, assume that m is chosen such that m = on and m T /m R + m T ρ as m,n, = R, T.Then it follows that the bootstrap approximation κm ˆφ t is wealy consistent for the distribution of the limiting random element t K + φ Lemma 2.4. ˆφt W φ t from Proof: As in the proof of Theorem 5 in Gill [11] but now focussing on the cumulative hazard functions instead of the distribution functions, application of the almost sure construction by Sorohod, Dudley and Wichura cf. Shorac & Wellner [1] yields sequences ˆΛ D = ˆΛ with n 1/2 ˆΛ Λ Z a.s., n, Z D = Z, = R, T cf. 3. Then it follows for the bootstrapped Nelson-Aalen estimators, ˆΛ,that m 1/2 ˆΛ ˆΛ = D Z a.s., as n,m,= R, T. Another application of the almost sure construction theorem yields for the given sequences ˆΛ D new sequences ˆΛ = ˆΛ with m1/2 ˆΛ ˆΛ Z a.s., n,m,wherez D = Z, = R, T. Using the assumption m = on, = R, T, yields, given ˆΛ, = R, T, m 1/2 ˆΛ Λ = m 1/2 ˆΛ ˆΛ +m1/2 ˆΛ Λ = m 1/2 ˆΛ ˆΛ + m n 1/2 n ˆΛ Λ Z D = Z a.s., as n,m,= R, T. 7

8 Let ˆφ be the discrepancy functional applied to ˆΛ, = R, T and ˆφ the functional applied to ˆΛ, = R, T. Now we can use the Hadamard differentiability of φ and get, given ˆΛ, = R, T, κm ˆφ ˆφ = κ m ˆφ φ κm κ n κn ˆφ φ Then it follows that, given ˆΛ, = R, T, κm ˆφ t D = κ m D t K + φ dt ΛT,Λ R Z T, Z R D = W φ a.s., as n,m,= R, T. ˆφ t ρ P L [ κm a.s., as n,m,= R, T. ˆφ t φt κ m W φ t, a.s., as n,m,= R, T. ˆφ t φt To this end, repeat the arguments in the proof of Lemma 2.4 and use the fact that the second summand in 9 equals κm κn ˆφ t φt, κ n which tends to zero in probability. From this the wea consistency of the bootstrap approximation follows, i.e. ] [ ] ˆφ P t ˆφt, L W φ t t K + φ 9 The results of Lemma 3.2 give rise to the use of the hybrid bootstrap method for addressing the testing problem 2. To this end the upper 1 α-confidence bound for t [τ,τ 1 ] φt is constructed as follows. Denote the α-quantile of the distribution of t K + W φ t byw α.then φ we have κn P ˆφt φt w α 1 α. 1 Now, w α can be estimated by the α-quantile of B bootstrap replications κm ˆφ b t ˆφt, b =1,..., B. If [x b ] α denotes the α-quantile of a sample {x b,b=1,..., B, then [ κm ˆφ b t ˆφt ]α = [ κ m ˆφ b t ]α κ m ˆφt. Plugging this into 1 yields φt κn + κ m κn κm [ ˆφt κn ˆφ b t ] α = φ 1 α, which is the upper confidence bound for t [τ,τ 1 ] φt as suggested in Freitag et al. [1]. 8

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