Treatment selection in a randomized clinical trial via covariate-specific treatment effect curves.

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Article

Treatment selection in a randomized clinical trial via covariate-specific treatment effect curves

Statistical Methods in Medical Research 0(0) 1–18 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0962280214541724 smm.sagepub.com

Yunbei Ma1 and Xiao-Hua Zhou2,3

Abstract For time-to-event data in a randomized clinical trial, we proposed two new methods for selecting an optimal treatment for a patient based on the covariate-specific treatment effect curve, which is used to represent the clinical utility of a predictive biomarker. To select an optimal treatment for a patient with a specific biomarker value, we proposed pointwise confidence intervals for each covariate-specific treatment effect curve and the difference between covariate-specific treatment effect curves of two treatments. Furthermore, to select an optimal treatment for a future biomarker-defined subpopulation of patients, we proposed confidence bands for each covariate-specific treatment effect curve and the difference between each pair of covariate-specific treatment effect curve over a fixed interval of biomarker values. We constructed the confidence bands based on a resampling technique. We also conducted simulation studies to evaluate finite-sample properties of the proposed estimation methods. Finally, we illustrated the application of the proposed method in a real-world data set. Keywords Predictive biomarker, covariate-specific treatment effect curve, time-to-event outcome, pointwise confidence interval, simultaneous confidence interval, varying coefficient

1 Introduction Treatment selection based on patient’s personal information is one of the highest concerns in modern medicine. By comparing the treatment effects, we hope to be able to identify who are more likely to benefit from a given treatment and to provide the most effective treatment to an individual or a subpopulation of patients. 1

School of Statistics, Southwestern University of Finance and Economics, Chengdu, Sichuan, China HSR&D Center of Excellence, VA Puget Sound Health Care System, Department of Biostatistics, University of Washington, Seattle, USA 3 Beijing International Center for Mathematical Research, Peking University, Beijing, China 2

Corresponding author: Xiao-Hua Zhou, HSR&D Center of Excellence, VA Puget Sound Health Care System, Department of Biostatistics, University of Washington, Seattle, WA, USA. Email: [email protected]

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Statistical Methods in Medical Research 0(0)

Song and Pepe1 proposed a graphical method, the selection impact curve, to perform treatments selection based on a single continuous-scale biomarker. In their setting, large values of the biomarker, which are used in the future for selecting patient’s treatment, were assumed to be associated with better performance of treatment A versus B. This is, however, a strong assumption, because sometimes the relative effect of treatment A versus B on the patient’s outcome may not be a monotone function of the biomarker. In this paper, we represent the impact of the biomarker on the treatment effects by the covariate-specific treatment effect (CTE) curve and don’t require the monotonicity assumption on the biomarker. The CTE curve, a graphical plot of the treatment effect as a function of the biomarker value, can visually display the treatment effect on the patient’s outcome as a function of the biomarker value. When survival time is used to measure treatment effects, most of the current statistical methods are based on a comparison of estimated survival curves between a treatment and control group, stratified by the biomarker value. One main limitation of these subgroup approaches is that these approaches require dichotomization, which is artificial and may lose important information. Bonetti and Gelber2,3 developed an improved subgroup based method, called subpopulation treatment effect pattern plot (STEPP), which graphically displays the patterns of treatment effect across overlapping intervals of the biomarker values and allows simultaneously multiple dichotomizations. This method allows for a variety of treatment effect definitions.4 Bonetti et al.5 carried out several simulation studies for STEPP and found out that the STEPP method is effective for large sample sizes, but have poor behavior when sample sizes are smaller than 500. Bonetti et al.5 then proposed an alternative and better permutation-based inference. However, the STEPP method still requires a specification of the number of patients per subpopulation and the largest number of patients in common (or overlapping) among consecutive subpopulations. How to choose these numbers has an important impact on the results of the analysis.5 Unfortunately, to our knowledge, no effective method has been developed to choose these numbers yet. Furthermore, since the STEPP method allows an overlap of subgroups, interpretation of the treatment effects may be difficult when treatment appears to be effective in a subpopulation, but not in another adjoint subpopulation. In this paper we propose a new method for treatment selection for future patients based on CTE curves. This article is organized as follows. In Section 2, we first introduce the mathematical definitions of the CTE curves. Then, we illustrate how to use the CTE curves to select an optimal treatment for a future individual patient and a biomarker-defined subpopulation of patients who are most likely benefited from the optimal treatment. In Section 3, we propose pointwise confidence intervals and simultaneous confidence bands for the CTE curves and their differences. We propose a resampling (RS) method for constructing the confidence bands. We also provide theoretical justification on the proposed confidence bands. We report the simulation study results and the analysis of a real data example in Sections 4 and 5, respectively. We list conditions and technical lemmas in Appendix 1. We give detailed proofs of the asymptotic results of Section 3 in a technique report.

2 CTE curves and treatment selection The CTE curve evaluates the treatment effect at each value of the predictive biomarker. In this section, we describe the CTE curves for a time-to-event outcome based on a casual framework. We first give some necessary notations and definitions. For i ¼ 1, . . . , n, we let Ti(k) be the failure time of the ith individual that would be observed if the individual had received the kth treatment, where k ¼ 1, . . . , K, and Ti(0) be the event time that would be observed if the individual had received the control. Let Vi denote the biomarker value measured



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at the baseline for the ith individual. We first define the potential conditional hazard rate functions as Pðt 5 Ti ðkÞ t þ tjTi ðkÞ 4 t, Vi ¼ vÞ , t Pðt 5 Ti ð0Þ t þ tjTi ð0Þ 4 t, Vi ¼ vÞ ð0Þ ðtjvÞ ¼ lim t!0 t

ðkÞ ðtjvÞ ¼ lim

t!0

k ¼ 1, . . . , K,

In most medical research with a survival outcome, the treatment effect is usually represented by the logarithm of a hazard ratio (HR). Hence, we define the CTE curve at a fixed time t as follows ðkÞ ðtjvÞ k ðv; tÞ ¼ log ð0Þ , k ¼ 1, . . . , K ð1Þ ðtjvÞ where k(v;t) is an unknown smooth function of v for any fixed t 2 ð0, Þ. We may also be interested in the treatment effects on those patients whose biomarker values belong to a fixed interval [a, b]. In this case, we need to evaluate CTE curves over a fixed interval ½a, b, k ðv; tÞ, v 2 ½a, b, at time t. Let Zi ¼ ðZi1 , . . . , ZiK ÞT denote a vector of the indicators for indicating which treatment the ith subject actually receives. Here Zik ¼ 1 if the ith subject receives treatment k, and Zik ¼ 0 otherwise, where i ¼ 1, . . . , n, and k ¼ 1, . . . , K. Here K > 1. Let ek be a K-dimensional vector with the kth element being 1 and other elements being 0 and 0K be a K-dimensional vector with all elements being 0, where k ¼ 1, . . . , K. To select the optimal treatment for a future patient by comparing treatment effects at time t0, we just need to compare the CTE curves. Without loss of generality, we take Figure 1 (K ¼ 3) as an example, where we assume that the range of the biomarker is [0,1] and the time point is fixed at t0. From Figure 1, we see that, given a fixed time point t0, all CTE curves, 1(v;t0), 2(v;t0), and 3(v;t0), are below 0. Therefore, we can say that all the treatments are more efficient than the control. Furthermore, we also see that 3 ðv; t0 Þ 5 2 ðv; t0 Þ 5 1 ðv; t0 Þ 5 0 when v 2 ½0, a, that 2 ðv; t0 Þ 5 3 ðv; t0 Þ 5 1 ðv; t0 Þ 5 0 when v 2 ða, b, and that 1 ðv; t0 Þ 5 2 ðv; t0 Þ 5 3 ðv; t0 Þ 5 0 when v 2 ðb, 1. Hence, we would assign future patients with the biomarker values in [0,a], (a, b], and (b,1] to receive Treatment 3, Treatment 2, and Treatment 1, respectively. In the rest of this paper, we focus on a special case that bðv; tÞ bðvÞ in a multiple-treatment randomized clinical trial. This assumption holds under many well-known models, such as Cox’s model and varying-coefficient Cox’s model. Consequently, we assume the following varyingcoefficient proportional hazard regression model for event time ðtjZi , Vi Þ ¼ 0 ðtÞ exp bT ðVi ÞZi þ gðVi Þ ð2Þ where bðÞ ¼ ð1 ðÞ, . . . , K ðÞÞT denotes the vector of the CTE curves. This model has been studied by Fan et al.6 and Cai et al.7 They proposed to estimate coefficient functions by local partial likelihood and established their pointwise asymptotic normalities. Their results, however, can only be used to construct pointwise confidence intervals but cannot be used to construct simultaneous confidence bands. To select the optimal treatment for a future individual patient with a specific biomarker value, say v0, ideally we compare each of the CTE curves at v0 with the zero line and compare every two different CTE curves at v0. Since we do not know the CTE curves, we need to estimate them. To take into account variability associated with the estimated CTE curves, we propose the following method to this comparison problem. First, by constructing a confidence interval for k(v0), we assess whether



−0.0

treatment 3

treatment 2

−1.0

−0.8

−0.6

−0.4

treatment 1

−0.2


CTE ( v; t0 )

4

a 0.0

0.2

b 0.4

0.6

0.8

1.0

biomarker v

Figure 1. The CTE curves for different treatment arms: The solid, dashed, and dotted curves are the CTE curves for Treatment 1, 2, and 3, respectively.

the kth treatment is better than the control, where k ¼ 1, . . . , K. Then by constructing a confidence interval for k1 ðv0 Þ k2 ðv0 Þ, where 1 k1 < k2 K, we can assess whether the k1th treatment is better than the k2th treatment. From those confidence intervals, we can identify the best optimal treatment(s) for this patient with the specific biomarker value, v0. We construct those confidence intervals, based on a local partial likelihood estimation method, proposed by Fan et al.6 We give details of those methods in Section 3. Furthermore, rather than a specific biomarker value v0, we may be more interested in selection of the optimal treatment for a biomarker-defined subpopulation, say ½a, b V, of future patients. Hence we need to consider the alternative comparison problem between CTE curves over [a, b]; that is, we need to compare k ðvÞ, v 2 ½a, b , for k ¼ 1, . . . , K. Unlike pointwise comparisons, here we need to construct confidence bands instead of confidence intervals. More generally, we focus on a linear combination of CTE curves over [a, b], say lT bðvÞ, v 2 ½a, b , where l is a K-dimensional vector and ðvÞ ¼ ð1 ðvÞ, . . . , K ðvÞÞT . Note that, if we let the kth component of l be 1 and the remaining components being 0, the confidence band for lT bðvÞ over v 2 ½a, b assesses whether the kth treatment is better than the control among patients whose biomaker values are within [a, b]; if we let the k1th component of l be 1, the k2th component of l be –1, and the other components be 0, the confidence band for lT bðvÞ over v 2 ½a, b assesses whether the k1th treatment is better than the k2th treatment among patients whose biomarker values are within [a, b]. We propose a new RS-based method to construct such confidence bands. We give details of the RS-based method in Section 3.

3 Pointwise and simultaneous confidence intervals for the CTE curves We first introduce the local partial likelihood for bðÞ, proposed by Fan et al.6 Then, we use this local partial likelihood to estimate the CTE curves, k(.), where k ¼ 1, . . . , K. For any given point v0,



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by Taylor series expansion, we obtain that bðvÞ bðv0 Þ þ b0 ðv0 Þðv v0 Þ þ ðv v0 Þ and gðvÞ gðv0 Þ þ g0 ðv0 Þðv v0 Þ gðv0 Þ þ dðv v0 Þ

ð3Þ

where ¼ b0 ðv0 Þ, and d ¼ g0 ðv0 Þ. Let Ti and Ci denote the failure time and censoring time for the ith individual, where i ¼ 1, . . . , n. We let Xi ¼ minðTi , Ci Þ denote the observed time for the ith individual. Let i be a failure indicator, which equals 1 if Xi < Ci and 0 otherwise. We assume that the censoring times are independent of the failure times conditional on the covariates. Let be a fixed time value indicating the end of the study. Let RðtÞ denote the set of the individuals at risk prior to time t and assume that the observations are independent. Using equation (3) for the data around v0 and utilizing a kernel function, we obtain the following logarithm of the local partial likelihood X n 1X Kh ðVi v0 Þi T Zi þ T Zi ðVi v0 Þ þ d ðVi v0 Þ log Kh ðVj v0 Þ ‘n ð, , d; v0 Þ ¼ n i¼1 j2RðXi Þ ð4Þ exp T Zj þ T Zj ðVj v0 Þ þ d ðVj v0 Þ where Kh ðÞ ¼ Kð=hÞ=h is a symmetric kernel function and h is a bandwidth. Suppose that equation ^ , ^ d^ Þ. Then b (4) is maximized at ð, b ¼ ^ is a local linear estimator of the coefficient function bðÞ at ^ The estimated the point v0. An estimator of g0 ðÞ at v0 is simply the local slope d^ðv0 Þ, i.e., g^ 0 ðv0 Þ ¼ d. ^ can be obtained by integrating g^0 ðv0 Þ with respect to v0, using the Trapezoidal rule.8 For curve gðÞ the purpose R v of ensuring the identifiability of g(), we set g(0) ¼ 0 without loss of generality. Hence, g^ ðv0 Þ ¼ 0 0 g^ 0 ðvÞdv.

3.1

Pointwise asymptotic properties and confidence intervals of the CTE curves

To obtain asymptotic properties of the maximum partial likelihood estimator b bðvÞ, we need some notations on the counting process. Let Ni ðtÞ ¼ IðTi t, i ¼ 1Þ and Yi ðtÞ ¼ IðXi tÞ. Define F t,i to be the failure, censoring and covariate information up to timeR t, and let Mi ðtÞ ¼ Ni ðtÞ R Yi ðtÞðtjZi , Vi Þ be the F t,i - martingale. Let i ¼ vi KðvÞdv and i ¼ vi K2 ðvÞdv, i ¼ 1, 2, . . . : Let b0 ðÞ and g0() be the true functions of bðÞ and g(), respectively. Denote ðu, z, v0 Þ ¼ PðX ujz, v0 Þ exp bT0 ðv0 Þz þ g0 ðv0 Þ For k ¼ 0, 1, 2 and l ¼ 0, 1, 2, we define akl ðu, v0 Þ ¼ fðv0 Þl E ðu, Z, v0 ÞZk jV ¼ v0 , Z akl ðu, v0 Þd0 ðuÞ akl ðv0 Þ ¼ 0

where f() is the density of V and Zk ¼ 1, Z and ZZT for k ¼ 0, 1, and 2, respectively.



6


Proposition 1 (Theorem 1 and Theorem 2 of Fan et al.6) Under the conditions in Appendix 1, we can obtain the following results. (1) Uniform consistency As n ! 1 P

sup jðb bðvÞ b0 ðvÞÞj ! 0 v2V

(2) Asymptotic normality For each v0 2 V

L pffiffiffiffiffi 2 nh b bðv0 Þ b0 ðv0 Þ h2 b00 ðv0 Þ ! N 0, 0 1 ðv0 Þ 2

Here b00 ðv0 Þ is the second derivative of bðvÞ at v0, and Z ð v0 Þ ¼ 0

fa20 ðu, v0 Þ a10 ðu, v0 ÞaT10 ðu, v0 Þa1 00 ðu, v0 Þ0 ðuÞ du

The following Corollary 1 is a straightforward result of Proposition 1. Corollary 1 Under the conditions of Proposition 1, for each v0 2 V and any given K-dimensional vector l

o L pffiffiffiffiffin T 2 nh l b bðv0 Þ b0 ðv0 Þ h2 lT b000 ðv0 ÞÞ ! N 0, 0 lT 1 ðv0 Þl 2 Note that Fan et al.6 have given consistent estimators

of the asymptotic biases and covariance of db d b b d b bðv0 ÞÞ ¼ bðv0 Þ and Cov bðv0 Þ , respectively; that is, h2 biasð bðv0 Þ, denoted by bias db b000 ðv0 Þ þ op ð1Þ and nh Covð bðv0 ÞÞ ¼ 0 1 ðv0 Þ þ op ð1Þ. (Definition and consistency of

d b d b bias bðv0 Þ and Cov bðv0 Þ are shown in the proof of Theorem 1 in the Supplementary Materials

2 2

(Available at http://smm.sagepub.com/)). We can then construct a two-side 100ð1 Þ% pointwise confidence interval for lT bðv0 Þ as follows h

i1=2 Tb T db T 2 b l bðv0 Þ l biasðbðv0 Þ c1 =2 l ^ bðv0 Þ l ð5Þ db bðv0 ÞÞ ¼ Covð bðv0 ÞÞ and c1 =2 is the 100ð1 =2Þ quantile of the standard normal where ^ 2 ðb distribution.

3.2

Simultaneous asymptotic properties and confidence bands of the CTE curves

In general, the confidence bands, based on the strong approximation theorem (Lemma 5 in Appendix 1), are not very accurate.10 In this section, we propose a more accurate new RSbased confidence band of a CTE curve, which is based on the idea of the RS



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techniques.11,12 We first give some notations. For any function b(v) and any matrix BðvÞ ¼ ðbij ðvÞÞK , we define !1=2 K X K X b b b jjbðvÞjja ¼ sup jbðvÞj and jjBjja ¼ jjbij jja : v2½a,b

i¼1 j¼1

For each v0, we let Z i ¼ ðZTi , ZTi ðVi v0 Þ=h, ðVi v0 Þ=hÞT , i ¼ 1, . . . , n. Let 0 ðv0 Þ ¼

T T T 0 0 b0 ðv0 Þ, hb0T ðv0 Þ, hg00 ðv0 Þ be the true value at v0 and ^ðv0 Þ ¼ b bT ðv0 Þ, hb b T ðv0 Þ, hb g0 ðv0 Þ be the corresponding estimator. We define Snk ðt, 0 ðv0 ÞÞ ¼

n 1X Yi ðtÞ exp 0T ðv0 ÞZ i þ g0 ðv0 Þ Kh ðVi v0 ÞZ k , i n i¼1

n 1X Yi ðtÞ exp bT0 ðv0 ÞZi þ g0 ðv0 Þ Kh ðVi v0 ÞZ k S nk t, b0 ðv0 Þ ¼ i n i¼1

for k ¼ 0, 1, 2. Note that, for each fixed v0, if nh5 ! 0, we have the following result

pffiffiffiffiffi

pffiffiffiffiffi ^ðv0 Þ, v0 Un ðv0 Þ nh ^ðv0 Þ 0 ðv0 Þ nhI1 n where

Sn1 ðt, 0 ðv0 ÞÞ Kh ðVi v0 Þ Z i dMi ðtÞ, Sn0 ðt, 0 ðv0 Þ 0 8

2 9 > > = > n i¼1 0 ; :Sn0 t, ^ðv0 Þ Sn0 t, ^ðv0 Þ Un ðv0 Þ ¼

n 1X n i¼1

Z

Now for each fixed v0, we define

3 S n1 t, b bð v 0 Þ

5dNi ðtÞGi , Kh ðVi v0 Þ4Z i U~ n ðv0 Þ ¼ n i¼1 0 S n0 t, b bð v 0 Þ 8

2 9 > > = > n i¼1 0 ; :S n0 t, b bðv0 Þ S n0 t, b bð v 0 Þ n Z 1X

2

where {Gi, i ¼ 1, . . . , n} is a random sample from the standard normal distribution and independent of the data ðXi , Zi , Vi , i Þ, i ¼ 1, . . . , n . This RS method is similar to bootstrap in terms of the sample space, convergence, etc. The sample space for U~ n ðv0 Þ is conditional on the data, whereas that 12 6 of Un(v0) is unconditional. Based onp the ffiffiffiffiffi similar arguments pffiffiffiffiffi as in Lin et al. and Fan et al., we can show that conditional on the data, nhU~ n ðv0 Þ and nhUn ðv0 Þ have the same limiting distribution,



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Nð0, ðv0 ÞÞ, for any fixed v0 and for almost all realizations of data ðXi , Zi , Vi , i Þ, i ¼ 1, . . . , n . Here ðv0 Þ ¼ diagð0 ðv0 Þ, 2 ðv0 Þ=2 Þ with a22 ðv0 Þ a12 ðv0 Þ ðv0 Þ ¼ aT12 ðv0 Þ a02 ðv0 Þ Following the same argument in Fan et al.,6 for each fixed v0, we can also show that In ð ^ðv0 Þ, v0 Þ and I~n ðb bðv0 Þ, v0 Þ have the same limiting matrix, ðv0 Þ ¼ diagððv0 Þ, ðv0 ÞÞ, in probability. Let

e ðv0 Þ ¼ l T I~1 b ð Þ, v ð6Þ U~ n ðv0 Þ b v Q 0 0 n

T Then, conditional on the data ðXi , Zi , Vi , i Þ, i ¼ 1, . . . , n , letting l ¼ lT , 0Tpþ1 , we have the following theorem. Theorem 1 Assume that the conditions in Appendix h ¼ nd , 1=5 5 d 5 1 2=s, s 4 2, we have the following result

1

hold.

If

½a, b , and

b 0 0 1 1 pffiffiffiffiffi e nhQðvÞ 1=2 B B C C supP@ 2 logððb aÞ=hÞ dn A 5 xjfXi , Zi , Vi , i g, i ¼ 1, . . . , nA @h T d b i1=2 x l Cov bðvÞ l a 0 0 1 1

b T b l ðvÞ bðvÞ b 1=2 B 0 B C C P@ 2 logððb aÞ=hÞ dn A 5 xA ¼ op ðn" Þ @ h

i 1=2 T 2 b l ^ bðvÞ l a

for some " 4 0. Here dn ¼ f2 logððb aÞ=hÞg

1=2

(R ) ðK0 ðvÞÞ2 dv 1 þ log 40 f2 logððb aÞ=hÞg1=2

Since we further assume that h ¼ nd , 1=5 5 d 5 1 2=s, s 4 2, the bias term is negligible and db Theorem 1 shows no bias anymore. Note that ^ 2 ðb bðv0 ÞÞ ¼ Covð bðv0 ÞÞ=nh. Hence, Theorem 1 demonstrates that conditional on the data fðXi , Zi , Vi , i Þ, i ¼ 1, . . . , ng, the conditional T ^2 b e distribution of jjQðvÞ=½l ðbðvÞÞl 1=2 jjba can be used to approximate the unconditional distribution T b of its counterpart, jjl ðbðvÞ bðvÞÞ=½lT ^ 2 ðb bðvÞÞl 1=2 jjba . Moreover, Theorem 1 also justifies the validity of the following confidence band: For a given data set ðXi , Zi , Vi , i , i ¼ 1, . . . , n , we generate fGi , i ¼ 1, . . . , ng, M times. For each time, we calculate h

i1=2 e i Þ= lT ^ 2 b e i Þ, where i ¼ 1, . . . , n, and then Q ¼ max jQðV j. We use the 100ð1 Þ% bðVi Þ l QðV 1in

percentile, qen , of these Qs. Then, we obtain the following RS-based confidence band h

i1=2 Tb T 2 b , a v0 b l bðv0 Þ q~n l ^ bðv0 Þ l


ð7Þ


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3.3 Test for the constant treatment effect A test for the constant treatment effect is identical to the test for the null hypothesis that H0: supv2V jlT bðvÞ Cj ¼ 0, where C is an unknown constant and needs to be estimated. P R P bðv0 Þdv0 ¼ 1n ni¼1 lTb bðVi Þ. Note that Under H0, we estimate C by C^ ¼ 1n ni¼1 V Kh ðVi v0 ÞlTb R P n 13 1 it is easy to show that i¼1 V Kh ðVi v0 Þdv0 ¼ 1. Then according to Theorem 3 in Yin et al., n pffiffiffi ^ P nðC CÞ ! 0 as n ! 1. According to the algorithm of the RS technique, a RS version of the p-value, p, for H0 can be ^ T ^ 2 ðb bðvÞÞl 1=2 jjba in Qs. And bðvÞ CÞ=½l calculated by letting p be the inverse-quantile value of jjðlTb reject H0 if p 0.05, for the given significant level 0.05. In the following simulation studies, we evaluate the finite-sample performance of the confidence bands and the hypothesis testing procedures.

4 Simulation studies In this section, we carried out several simulation studies to assess the finite-sample properties of the proposed confidence bands. We generated failure times from the hazard regression model, ðtjV, Z1 , Z2 Þ ¼ 0:6t2 exp 1 ðVÞZ1 þ 2 ðVÞZ2 þ V2 . We then generated the biomarker value, V, from a uniform distribution over [0,1]. We focused on the performance of the confidence bands for 1(v), 2(v), and 1 ðvÞ 2 ðvÞ over [0,1]. Let b1 and b2 be i.i.d. Bernoulli random variables with the probability of being 1 as 0.3 and 0.5, respectively. We constructed covariates Z1 and Z2 as Z1 ¼ b1 and Z2 ¼ ð1 b1 Þ b2 . We generated the censoring time C from an exponential distribution with mean cc V, where cc was chosen to yield an approximately censoring rate of 25%. The number of the simulations was chosen to be 500, and the number of random samples for the RS technique was chosen to be 1000. To evaluate the proposed method, we considered three cases: (i) 1 ðvÞ ¼ 2 ðvÞ ¼ 2 sinð3vÞ; ðiiÞ 1 ðvÞ ¼ 1 expðvÞ, 2 ðvÞ ¼ expðvÞ; ðiiiÞ 1 ðvÞ ¼ expðvÞ, 2 ðvÞ ¼ 2 sinð3vÞ. For each case, we constructed 95% confidence bands based on our proposed RS technique. We also calculated their coverage probabilities and the average widths of the confidence bands, respectively. We used the empirical bias method14,15 to calculate the optimal bandwidth hôpt for asymptotic ^ ^ 1=51=4:95 to be the bandwidth for the RS technique; that is properties, and then let hRS ¼ hopt n d h^RS ¼ Op n with d ¼ 1/4.95. The detail algorithm of hôpt is supplied in Section 1 in the Supplementary Material (Available at http://smm.sagepub.com/). Furthermore, to evaluate the impact of bandwidth selection on the confidence bands, we used three different bandwidth values, given by h ¼ h^RS =1:5, h^RS , and 1:5h^RS . Results based on the sample size n ¼ 500 are presented in Table 1. From Table 1 we saw that, when the sample size was large enough (n ¼ 500), results based on the RS techniques performed very well and were not sensitive to the selection of the bandwidth. We next report simulation results in Table 2 to investigate the performance of our proposed methods under different sample sizes, n ¼ 100 and n ¼ 200. Table 2 showed that the RS technique still performed well when the sample size decreased to n ¼ 100. Hence, the RS technique is also stable when constructing the confidence bands of the CTE curves for small size real data. We further evaluated our methods for testing the constant treatment effect. Let ^ ¼ supv2½0,1 j1 ðvÞ 2 ðvÞ Cj ^ ¼ 0, we calculated p– l ¼ ð1, 1ÞT , for H0 : supv2½0,1 jlT bðvÞ Cj values based on our proposed method to test the null hypothesis H0. To compare with the STEPP method, we also test the null hypothesis H0 via the permutation-based STEPP test



10


Table 1. Coverage probabilities (average widths) of confidence bands given significant level ¼ 0:05 and censoring rate 25% based on 500 simulations and 500 sample sizes. Setting (i)

Setting (ii)

Setting (iii)

1(h^ RS ¼ 0:079)

1(h^ RS ¼ 0:109)

1(h^ RS ¼ 0:109)

h^ RS 0.945 (0.890)

h^ RS =1:5 0.930 (1.113)

1:5h^ RS 0.918 (0.726)

h^ RS 0.951 (0.780)

2(h^ RS ¼ 0:079)

h^ RS =1:5 0.946 (0.964)

1:5h^ RS 0.938 (0.657)

2(h^ RS ¼ 0:109)

h^ RS 0.944 (0.957)

h^ RS =1:5 0.918 (1.206)

1:5h^ RS 0.936 (0.781)

2(h^ RS ¼ 0:079)

h^ RS 0.952 (0.862)

h^ RS =1:5 0.940 (1.089)

1:5h^ RS 0.915 (0.679)

h^ RS 0.957 (0.861)

h^ RS =1:5 0.952 (1.100)

1:5h^ RS 0.946 (0.708)

h^ RS 0.945 (0.867)

h^ RS =1:5 0.941 (1.074)

1:5h^ RS 0.935 (0.691)

h^ RS 0.953 (0.687)

h^ RS =1:5 0.938 (0.846)

1:5h^ RS 0.951 (0.557)

h^ RS 0.940 (0.821)

h^ RS =1:5 0.946 (1.062)

1:5h^ RS 0.956 (0.684)

h^ RS 0.951 (1.085)

h^ RS =1:5 0.936 (1.391)

1:5h^ RS 0.902 (0.863)

1 2 h^ RS ¼ 0:104

1 2 h^ RS ¼ 0:104

1 2 h^ RS ¼ 0:009

Table 2. Coverage probabilities (average widths) of confidence bands given significant level ¼ 0:05 and censoring rate 25% based on 500 simulations under 100 and 200 sample sizes. n ¼ 100

n ¼ 200

(i)

1

2

1 2

1

2

1 2

h^ RS CP AW

0.124 0.941 (3.557)

0.124 0.946 (2.711)

0.208 0.950 (1.227)

0.094 0.948 (1.563)

0.09) 0.944 (1.470)

0.17) 0.954 (0.877)

n ¼ 100

n ¼ 200

(ii)

1

2

1 2

1

2

1 2

h^ RS CP AW

0.213 0.958 (1.636)

0.213 0.949 (1.911)

0.208 0.947 (1.955)

0.158 0.955 (1.111)

0.158 0.954 (1.239)

0.163 0.950 (1.211)

n ¼ 100

n ¼ 200

(iii)

1

2

1 2

1

2

1 2

h^ RS CP AW

0.208 0.940 (2.047)

0.139 0.954 (2.501)

0.15) 0.923 (2.341)

0.158 0.951 (1.342)

0.099 0.949 (1.400)

0.11) 0.927 (1.793)



Ma and Zhou

11 Table 3. True Type I error for the null hypothesis H0 based on 100 simulations.

Setting (i)

Setting (ii)

n

RS

STEPP

100 200 500 100 200 500

0.05 0.07 0.05 0.05 0.05 0.06

0.08 0.08 0.07 0.06 0.09 0.08

associated with the HR5 based on 2000 permutation times (by using stepp package in R). Given significant level 0.05, the true Type I error rates based on sample size: n ¼ 500, 200, and 100, and 100 simulations are reported in Table 3. In Table 3, the number of subjects in each subpopulation n1 and the number that determines the largest number of subjects common among consecutive subpopulations n2 are chosen to be same as those in Bonetti et al.5 That is n1 ¼ 0:3n, n2 ¼ 0:4n. Table 3 showed that the true Type I error rates of our methods are closer to the given significant level 0.05. Hence the RS technique has better performance.

5 Example We illustrated the application of the proposed methods in a real-world clinical study on the role of c-myc in selecting the optimal treatment for patients with colon cancer.16 In this clinical trial, patients with colon cancer could be treated by surgery alone or surgery plus chemotherapy. Surgery alone is less invasive and less expensive than surgery plus chemotherapy. It is desirable to identify the patients who might benefit more from surgery plus chemotherapy based on their biomarkers. Based on a study conducted by the Eastern Cooperative Oncology Group, Augenlicht16 suggested that the c-myc oncogene may be a predictive biomarker for patients with colon cancer. Using a subset of the cases from this clinical trial, Li and Ryan17 found that there might be an interaction between the c-myc oncogene expression level and the treatments on overall survival and disease progression-free survival. We applied the proposed methods to this data set to assess the role of c-myc oncogene in predicting patient’s response to treatment and distinguish sensitive patients from nonsensitive patients based on their c-myc levels, which ranged from 0.37 to 4.64. We gave a scatter plot for c-myc levels in Figure 2. Disease progression-free survivals of a total of 124 patients who were randomized to receive surgery alone, surgery plus chemotherapy, or other treatments were measured. We let Vi be the cmyc oncogene expression level of the ith patient, Zi1 and Zi2 be the indicator of receiving other treatments and the indicator of receiving surgery plus chemotherapy, respectively. We considered the following model ð8Þ ðtjZi , Vi Þ ¼ 0 ðtÞ exp 1 ðVi ÞZi1 þ 2 ðVi ÞZi2 þ gðVi Þ We fitted the model (8) to the progression-free survival. The 95% confidence limits and 95% confidence bands were calculated using the proposed methods. The confidence bands were constructed over [0.37,3.81], which approximately covered 98.4% of data. Gaussian kernel was employed, and the optimal bandwidth hôpt was found to be around 0.46 for all the three CTE



12


1

2

c−myc

3

4

C−myc oncogene expression level

0

40

20

60

80

100

120

Figure 2. Scatterplot of c-myc levels.

Table 4. The p-values for the tests for constant treatment effect.

RS

Other treatments v.s. Surgery only

Surgery plus chemotherapy v.s. Surgery only

Surgery plus chemotherapy v.s. Other treatments

0.004

0.004

0

curves of the three treatments. Furthermore, we found that h^RS ¼ hôpt n1=51=4:95 0:46, which was also used for the RS technique. In Table 4, we report the p-values for testing the constant treatment effect based on our proposed RS technique. From Table 4, we saw that all null hypotheses of constant treatment effects are rejected. From the CTE curves in Figure 4, we also found out that the treatment effects are not constants. Hence we may conclude that the treatment effects between any two of surgery only, surgery plus chemotherapy, and other treatments do vary with c-myc oncogene. For a comparison purpose, we also analyzed this data set using the sliding-window STEPP method (by using stepp package in R). Due to the small sample size of this example, the whole data can only be divided into a few subpopulations. The STEPP plots based on n1 ¼ 35, n2 ¼ 30 and estimated CTE curves with their confidence bands are presented in Figures 3 and 4, respectively. There were two horizontal lines in Figures 3 and 4. The top horizontal line corresponds to zero, and the bottom horizontal line corresponds to 1.33, where 1.33 is a common clinical log HR significant level for colorectal cancer. From this figure, we saw that the STEPP method was not informative and has several limitations. First, we can see that the confidence bands based on the STEPP method are too wide to tell the difference between treatments. Second, the treatment effects at extreme values of the biomarker (c-myc) could not be identified via the STEPP method because STEPP can only assess the average treatment effect for a subgroup of patients. To illustrate these points, we took Figures 3(b) and 4(b) as an example. Based on the clinical significant level



−3 −2 −1

0

1

(a)

13

2

Ma and Zhou

0.37 1.21

0.87 1.24

0.91 1.29

0.37 1.24

0.9 1.31

1.02 1.52

1.07 1.68

1.13 1.8

1.23 1.95

1.29 2.45

1.36 3.25

1.44 3.62

2

0.94 1.37

−3 −2 −1

0

1

(b)

0.47 0.87 1.24 1.32

1.14 1.7

1.22 1.8

1.29 2.08

1.35 2.7

1.4 3.73

1.52 3.18

2

1.01 1.38

−3 −2 −1

0

1

(c)

0.96 1.32

1 1.38

1.06 1.45

1.09 1.75

1.2 2.08

1.24 2.43

1.3 3.81

Figure 3. Figures (a), (b), and (c) are estimated treatment effects via sliding window STEPP method with their confidence bands based on n1 ¼ 35 and n2 ¼ 30: (a) treatment effect of other treatments over surgery only, (b) treatment effect of surgery plus chemotherapy over surgery only, (c) treatment effect of surgery plus chemotherapy over other treatments. Here, solid and dashed lines are estimated curves and 95% confidence bands, respectively.

¼ 1:33, Figure 4(b) shows that the treatment effect of surgery plus chemotherapy versus surgery is only significant for patients whose c-myc values are larger than 2.8 (^2 ð2:8Þ 1:33). However, due to the small sample size in the interval [2.8, 3.81] (i.e. there were only six subjects in this interval), the STEPP method could not be used in this interval. Therefore, the STEPP method cannot estimate the treatment effect in the subgroup defined by the biomarker values in [2.8, 3.81], while the CTE curve method could. It is worth to notice that when the level of c-myc exceeds 3, j1 ðvÞ 2 ðvÞj is very large (over 5, see Figure 4(c)). This is because all the patients with c-myc larger than 3 and treated by ‘‘surgery plus chemotherapy’’ survive, and all the patients with c-myc larger than 3 and treated by ‘‘other treatments’’ fail. That is the hazard rate ratio appears ‘‘01’’ type, which means the log HR tends to be 1. Figure 4(a) shows an opposite situation, where the hazard rate ratio appears ‘‘10’’ type and the log HR tends to be 1. We further focused on the CTE curves over [2.8,3.81]. Results are presented in Figure 5. The figure on the left-hand side represents the treatment effect of surgery plus chemotherapy versus surgery only, and the figure on the right-hand side represents the treatment effect of surgery plus chemotherapy versus other treatment. The decisions then can be made based on different



14

Statistical Methods in Medical Research 0(0) (a)

12 10

β (v)

8 6 4 2 0 −2 1

1.5

2

2.5

3

3.5

4

4.5

1

1.5

2

2.5

3

3.5

4

4.5

1

1.5

2

2.5

3

3.5

4

4.5

(b) 2 0 β (v)

−2 −4 −6 −8 −10

(c) 0

β (v)

−5 −10 −15

Figure 4. Figures (a), (b), and (c) are estimated CTE curves: (a) CTE curve for the treatment arm of other treatments, (b) CTE curve for the treatment arm of surgery plus chemotherapy, (c) the difference of CTE curves of surgery plus chemotherapy and other treatments. Here, solid, dashed, and dotted lines are estimated curves, 95% confidence limits, and 95% confidence bands, respectively.

clinical purpose. First, from Figure 5, we saw that the CTE curves in both the figures were under the horizontal line 1.33. Hence, based on the clinical significant level ¼ 1:33, we might conclude that surgery plus chemotherapy was the optimal treatment for patients whose c-myc values were larger than 2.8. However, this conclusion does not take into account variability associated with the estimated CTE curves.



Ma and Zhou

15 CTE curve of surgery plus chemotherapy 2

β(v)

0 −2 −4 −6 −8 2.8

3.0

3.2

3.4

3.6

Difference of CTE curves of surgery plus chemotherapy and other treatment 0

β(v)

−5 −10 −15 2.8

3.0

3.2

3.4

3.6

Figure 5. Confidence bands of CTE curves over a subinterval of c-myc [2.8,3.81]. Here, solid, dashed, and dotted lines are estimated curves, 95% confidence limits, and 95% confidence bands, respectively. (a) CTE curve of surgery plus chemotherapy, (b) difference of CTE curves of surgery plus chemotherapy and other treatment.

Using the confidence bands in Figure 5, we saw that for a c-myc-defined subgroup of patients, [3.4, 3.81], all the upper confidence bands were under the horizontal line 0. Hence, if we chose the clinical significant level ¼ 0, we could conclude that the treatment of surgery plus chemotherapy is the optimal selection for the c-myc-defined subgroup of patients with c-myc within [3.35, 3.81]. If we wanted to increase the clinical significance level to 1.33 ( ¼ 1:33), we observed that all the upper confidence bands were under the horizontal line 1.33 over the range [3.55,3.81] of c-myc values. Therefore, if we changed the clinical significant level from 0 to 1.33, only patients with c-myc values within [3.55, 3.81] should be treated by surgery plus chemotherapy.

6 Discussion In this paper, we have proposed pointwise confidence interval and confidence bands for a partial region of the CTE curve, which gives a graphical representation of the impact of a biomarker on the causal effect of one treatment versus another treatment on patient’s survival time. A confidence interval for the CTE curve can be used to select the best treatment for one individual patient with a given value of the biomarker, and a confidence band for a partial CTE curve allows us to find the best treatment for a subgroup of predefined patients, based on their biomarker values. The proposed methodology may be used in fast-growing personalized medicine, which recognizes genetic variability of patients and tailors a treatment to an individual patient, based on patient’s biomarker values. In this paper, we assume that k ðv, tÞ ¼ k ðvÞ. We are currently working an extension to the setting when k ðv, tÞ 6¼ k ðvÞ.



16


Acknowledgements Dr Ma’s work was partially supported by the National Natural Science Foundation of China (NSFC) (No. 11301424), and Dr Zhou’s work was supported in part by U.S. Department of Veterans Affairs HSR&D Research Career Scientist Award (RCS 05-196).

References 1. Song X and Pepe MS. Evaluating markers for selecting a patient’s treatment. Biostatistics 2004; 60: 874–883. 2. Bonetti M and Gelber RD. A graphical method to assess treatment-covariate interactions using the cox model on subsets of the data. Stat Med 2000; 19: 2595–2609. 3. Bonetti M and Gelber RD. Patterns of treatment effects in subsets of patients in clinical trials. Biostatistics 2004; 5: 465–481. 4. Lazar AA, Cole BF, Bonetti M, et al. Evaluation of treatment-effect heterogeneity using biomarkers measured on a continuous scale: Subpopulation Treatment-Effect Pattern Plot (STEPP). J Clin Oncol (Stat Oncol Ser) 2010; 28: 4539–4544. 5. Bonetti M, Zahrieh D, Cole BF, et al. A small sample study of the STEPP approach to assessing treatmentcovariate interactions in survival data. Stat Med 2009; 28: 1255–1268. 6. Fan J, Lin HZ and Zhou Y. Local partial likelihood estimation for lifetime data. Ann Stat 2006; 34: 290–325. 7. Cai J, Fan J, Jiang J, et al. Partially linear hazard regression for multivariate survival data. J Am Stat Assoc 2007; 102: 538–551. 8. Hastie T and Tibshirani R. Generalized additive models. London: Chapman & Hall, 1990. 9. Fan J and Zhang W. Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand J Stat 2000; 27: 715–731.

10. Hall P. On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J R Stat Soc Ser B 1993; 55: 291–304. 11. Tian L, Zucker D and Wei LJ. On the Cox model with time-varying regression coefficients. J Am Stat Assoc 2005; 100: 172–183. 12. Lin D, Fleming T and Wei LJ. Confidence bands for survival curves under the proportional hazards models. Biometrika 1994; 81: 73–81. 13. Yin G, Li H and Zeng D. Partially linear additive hazards regression with varying coefficients. J Am Stat Assoc 2008; 103: 1200–1213. 14. Ruppert D. Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J Am Stat Assoc 1997; 92: 1049–1062. 15. Fan J and Huang T. Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 2005; 11: 1031–1057. 16. Augenlicht L. Low-level c-myc amplification in human colonic carcinoma cell lines and tumors: A frequent, p53 independent, mutation associated with improved clinical outcome in a randomized multiinstitutional trial. Cancer Res 1997; 57: 1769–1775. 17. Li Y and Ryan R. Inference on survival data with covariate measurement error – an imputation approach. Scand J Stat 2006; 33: 169–190.

Appendix 1: Assumptions and technical lemmas We list the following assumptions in the appendix for our results. (1) For s > 2, EjZj j2s 5 1, j ¼ 1, . . . , K. (2) The biomarker V has a compact support V, in which fv(v) is continuous, and infv2V " fv ðvÞ 4 0 for some " 4 0, where V " ¼ fv : infv0 2V jv v0 j "g. (3) For any v 2 V " , ðvÞ is nonsingular, ðvÞ is positive definite, and the elements in ðvÞ are continuous on the compact support V. R (4) EðZ2s 0 ðtÞdt 5 1 and, jj fv ðvÞjjV 5 1, j jV ¼ vÞ is bounded for v 2 V, j ¼ 1, . . . , K: where jj jjV is the sup-norm of a function on V. (5) Functions () and g() have continuous second derivatives on the compact support V. (6) The kernel function K() is a bounded, R symmetric densityR function and uniformly continuous. Furthermore, KðxÞ ! 0 as x ! 1, jKðxÞjdx 5 1 and K2 ðxÞdx 5 1: (7) The conditional probability PðX ujz, vÞ is equi-continuous in the arguments (u, v) on ½0, V " . (8) Let Z s k ðu, , v0 Þ ¼ f ðv0 Þ E PðX ujz, v0 Þð, yÞRð yÞk jv ¼ v0 Kð yÞdy



Ma and Zhou

17

T where k ¼ 0, 1, 2, Rð yÞ ¼ ZT , ZT y, y , and ð, yÞ ¼ exp T Rð yÞ þ T ðvÞZ : We suppose that for k ¼ 0, 1, 2, s k ðu, , vÞ is bounded away from 0 on the product space ½0, C1 V", where C1 2 R2Kþ1 ; that is inf inf inf s k ðu, , vÞ 4 0

u2½0, 2C1 v2V "

and sup EjZj2 expfT Z þ gg 5 1 ðT ,gÞ2C

2

where C2 2 RKþ1 (9) We have nh= log n ! 1 as n ! 1, and nh5 is bounded. We then present several lemmas first as follows. Lemma 1 to Lemma 4 are needed for the proofs of Theorem 1. Given t 2 ½0, , let ð1 ðtÞ, V1 Þ, . . . , ðn ðtÞ, Vn Þ be an i.i.d. random sample from (x(t),V). We assume that V and the kernel function K() satisfy the conditions in Appendix 1. We further assume that x satisfies the following three conditions: (1) for an s > 2, supt2½0, EjðtÞjs 5 1; (2) the function s2(v) isbounded away from first derivative on [0,1], where 2 ðvÞ ¼ R 2 0 for v 2 ½0, 1 and has R a bounded s E 0 ðtÞðtÞdtjV ¼ v ; and ð3Þ supx supt2½0, j yðtÞj fðyðtÞ, xÞdy ¼ cx 5 1, where f(y(t),x) is the joint density of (x(t),V). Let n Z X 1 i ðtÞKh ðVi vÞdMi ðtÞ mðvÞ ¼ 1 ðvÞ f1=2 ðvÞ n i¼1 0 For the process m(v), we have the following Lemma 1 Lemma 1 Suppose Assumptions 6–7 hold. If h ¼ nb , for some 0 5 b 5 1 2=s, we have n pffiffiffiffiffi

o 1 P ð2 log hÞ1=2 nh1=2 jjmðvÞjj d 5 x ! exp 2 expfxg n 0 0

ð9Þ

2 below. Lemma 2 Suppose Assumptions 6–7 hold. If h ¼ nb for some 0 5 b 5 1 2=s, we have n pffiffiffiffiffi

o 1 P ð2 log hÞ1=2 nh1=2 jjm ðvÞ E m ðvÞ jj d 5 x ! exp 2 expfxg n 0 0 Lemma 3 For each v0 2 V, let n V i v0 l 1X Yi ðtÞ exp 0T ðv0 ÞZ i þ g0 ðv0 Þ Kh ðVi v0 Þ Zk , i n i¼1 h n Vi v0 l 1X nkl ðt, 0 ðv0 ÞÞ ¼ Yi ðtÞ exp bT0 ðv0 ÞZi þ g0 ðv0 Þ Kh ðVi v0 Þ Zk i n i¼1 h nkl ðt, 0 ðv0 ÞÞ ¼


ð10Þ


18

Statistical Methods in Medical Research 0(0) for k ¼ 0,1,2, l ¼ 0,1,2. If Assumptions 1–7 hold, we have ðt, 0 ðv0 ÞÞ akl ðt, v0 Þjj10 ¼ Op sup jjnkl

t2½0,

logð1=hÞ nh

1=2 !

and sup jjnkl ðt, 0 ðv0 ÞÞ akl ðt, v0 Þjj10 ¼ Op

t2½0,

logð1=hÞ nh

1=2

! þh2

where k ¼ 0, 1, 2, l ¼ 0, 1, 2. Lemma 4 If Assumptions 1–7 hold, then ! 2 1 1=2 @ ‘n ð ðv0 Þ; v0 Þ logð1=hÞ ^ 0 Þ 0 ðv0 Þjj1 þh2 þ jj ðv ðv0 Þ ¼ Op 0 ¼ ðv0 Þ @ @ T nh 0

ð11Þ

^ 0 Þ and 0(v0), for any v0 2 V. where ðv0 Þ is between ðv Lemma 5 is the asymptotic distributions of the normalized maximum deviations of lTb bðvÞ from lT b0 ðvÞ Lemma 5 Assume that the conditions in Appendix 1 hold. If ½a, b V, h ¼ nd , and 1=5 d 5 1 2=s, where s > 2, then we have the following results 9 8 0 1

b T > > d b = > ; : T 2 b ^ l bðvÞ l a

The proofs of Lemma 1 to Lemma 5 and Theorem 1 are presented in the Supplementary Material on the website (Available at http://smm.sagepub.com/).


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