alexa Group Sequential Survival Trial Designs Against Historical Controls under the Weibull Model | Open Access Journals
ISSN: 2155-6180
Journal of Biometrics & Biostatistics
Like us on:
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Group Sequential Survival Trial Designs Against Historical Controls under the Weibull Model

Jianrong Wu* and Xiaoping Xiong

Department of Biostatistics, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA

*Corresponding Author:
Jianrong Wu
Department of Biostatistics
St. Jude Children’s Research Hospital
262 Danny Thomas Place, Memphis
TN 38105, USA
Tel: (901) 495-2850
Fax: (901) 495-4585
E-mail: [email protected]

Received date: July 04, 2014; Accepted date: August 04, 2014; Published date: August 11, 2014

Citation: Wu J, Xiong X (2014) Group Sequential Survival Trial Designs Against Historical Controls under the Weibull Model. J Biomet Biostat 5:209. doi:10.4172/2155-6180.1000209

Copyright: © 2014 Wu J, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are are credited.

Visit for more related articles at Journal of Biometrics & Biostatistics

Abstract

In this paper, two parametric sequential tests are proposed for historical control trial designs under the Weibull model. The proposed tests are asymptotically normal with properties of Brownian motion. The sample size formulas and information times are derived for both tests. A multi-stage sequential procedure based on sequential conditional probability ratio test methodology is proposed for monitoring clinical trials against historical controls.

Keywords

Brownian motion; Group sequential trial; Historical control; Information time; Sample size; Time-to-event; Weibull distribution

Introduction

Randomized clinical trials are the gold standard for comparing a new therapy to a standard treatment. However, when randomization is not feasible because of ethical concerns, patient preference, or regulatory acceptability, comparing data from patients receiving a new therapy to those from patients previously treated by standard treatment (historical control) is an alternative. If patients enrolled in the current trial are similar to those in the historical study, clinical trials with a historical control improve the reliability of testing results of single-arm phase II trials by including the variation of the null parameter, which is usually estimated from historical data. Compared with randomized phase III trials, clinical trials with a historical control require a much smaller sample size, and are therefore easier to conduct and save time and patient resources [1].

Despite the practical and statistical issues associated with historical control trials [2-6], they have been appropriately used in many clinical practices. Sample size calculations to design such trials have been discussed by Makuch and Simon [7] for binary endpoints and by Dixon and Simon [8] and Emrich [9] for exponential survival endpoints. These methods have been widely used in oncology trial designs. However, Korn and Freidlin [10] reported that these popular methods do not preserve the power and type I error when considering the uncertainty in the historical control outcome data. Recently, several studies have discussed sample size calculations for historical control trials by taking into account the uncertainty in historical control outcome data [11-13].

Clinical trials with historical controls are often monitored by preplanned interim analyses to stop accrual if patients in the current trial have poorer outcomes than those in the historical control. The monitoring of clinical trials with historical controls poses a statistical problem of comparing two outcomes in a situation wherein data from the current study are sequentially collected and compared with all data from historical controls at each interim analysis. Few studies have discussed the monitoring of clinical trials against historical controls. For example, Chang et al. [11] proposed a two-stage design for binary outcome and Xiong et al. [12] developed a multistage group sequential procedure for monitoring historical control trials with binary, continuous, and survival endpoints.

In this study, we propose a multistage group sequential procedure to de-sign survival trials against historical controls under the Weibull model. In Section 2, two sequential parametric tests are proposed for the trial design under the Weibull model. In Section 3, formulas for the number of events required for the current study are derived. In Section 4, a multistage group sequential procedure based on the sequential conditional probability ratio test (SCPRT) by Xiong [1] is proposed. In Section 5, simulation studies to calculate the empirical power and type I error of the proposed tests are described. In Section 6, an example is given to illustrate the proposed methods. The discussion and concluding remarks are given in Section 7.

Sequential Test Statistics

Two parametric sequential test statistics are discussed in this section to provide group sequential design of survival trials against historical controls under the Weibull model. Assume that the failure time variable Tj of a subject from the jth group follows the Weibull distribution with a common shape parameter κ and a scale parameter ρj, where j=1 for the historical control group and j=2 for the current study group. That is, Tj has survival distribution function

equation

and hazard function

equation

The shape parameter κ indicates the degree of acceleration (κ>1), constant (κ=1), or deceleration (κ<1) of the hazard over time. In a cancer trial, the median survival time is an intuitive endpoint for clinicians. The median survival time of the jth group for the Weibull distribution can be calculated as equation. Therefore, the Weibull survival distribution can be expressed as

equation

The one-sided hypotheses of a historical control trial defined by median survival times can be expressed as a two-sample test of the following:

equation

For notational convenience, we convert the scale parameter ρj to a hazard parameter equation. Then the survival distribution is equation with hazard function equation in which κ is taken as a known constant. In this case, the above hypotheses on median survival times are equivalent to

equation

where the hazards ratio equation

Now, suppose there are n1 subjects on the historical control group and let Ti1 and Ci1 denote, respectively, the failure time and censoring time of the ith subject of the historical control group. Further assume that during the accrual phase of the current trial, n2 subjects are enrolled in the study, and let Ti2 and Ci2 denote, respectively, the failure time and censoring time of the ith subject of the current study group, with both being measured from the time of study entry Yi2. We assume that the failure time Tij is independent of the censoring time Cij and entry time Yi2, and {(Tij,Cij,Yi2); i=1,···, nj} are independent and identically distributed. When the current study data are examined at calendar time t ≤ τ, where τ is the study duration, we observe the time to failure equation and the failure indicator equation for the historical control, and the time to failure equation and the failure indicator equation for current study group up to time t. We assume that survival data remain the same (no further follow-up) for the historical control during the process of the current trial, whereas survival data are updated for the current study from one look to the next in the trial. On the basis of the observed data equation at interim look time t, the observed likelihood function is proportional to ([14], Chapter 3)

equation

where equation is the total number of events of the historical control; equation is the cumulative follow-up time of historical controls penalized by the Weibull shape parameter κ; equation is the total number of events observed in the current group by time t; and equation The maximum likelihood estimates of λ1 and λ2 can be derived as

equation

with variances equation respectively. Therefore, under the null hypothesis, the Wald statistic of the log-hazard ratio log(δ) at calendar time t is given by

equation      (1)

which has approximately a standard normal distribution. To derive the group sequential design, let

equation        (2)

then under the alternative of δ=λ12>1, the statistic equation is approximately normal with mean log(δ)V(t) and variance V(t) and has an independent increment structure, where equationThe above results can be derived from Tsiatis et al. [15], who reported similar results for general parametric survival models. Because

equation                               (3)

where D1 is the total number of events in the historical control and equation is the number of events in the current study up to time t. Thus, equation is approximately a Brownian motion with drift parameter equation and information time equation where D(τ) is the value of D(t) at t=τ.

Sprott [16] showed that the distribution of equation and equation in small samples is much more closely approximated by a normal distribution. Then equation and equation are approximately normal with mean equation and variance estimate equationand equation [17]. Therefore, the test statistic

equation

is an approximately standard normal distribution under the null hypothesis.

Let

equation

then under the alternative, the statistic equation is approximately normal with mean equation and variance V(t), where equationand U(t) has an independent increment structure. Because

equation                          (4)

equation is approximately a Brownian motion with drift parameter equation and information time equation

Sample Size for Fixed Sample Test

Because historical control data are obtained from previous trials, sample size n1 and total number of events D1 for the historical control group are known. Therefore, we only need to calculate the sample size for the current study for a fixed sample test at the end of the study. On the basis of the test statistic Z(t) at t=τ , under the null hypothesis,

equation

has an approximately standard normal distribution. To calculate the power under the alternative δ=λ1/ λ2(> 1), Z(τ) is an approximately normal distribution with mean log(δ)D1/2(τ) and unit variance. Therefore, given a significance level α, the power (1−β) of the Z(τ) test under the alternative is given by

equation

where Φ(⋅) is the standard normal distribution function and equation Thus, the number of events required for the current study based on the Z(τ) test can be calculated by

equation                           (5)

where δ=(m2/m1)κ and D1 is the total number of events observed in historical control data. Therefore, the sample size for the current group is given by

equation

where p2(τ) is the probability of a subject from the current group having an event during the study. Similarly, the number of events required for the current study based on the S(τ) test can be calculated by

equation                    (6)

and the sample size is given by equation

To calculate the number of subjects required for the study, we need to calculate p2(τ), the probability of a subject in the current group having an event during the study. Typically, we assume that subjects are accrued over an accrual period of length ta with an additional followup period of length tf. A subject enters the study at time u, the entry time is uniformly distributed on [0,ta], and no subject is lost to followup during the study. Then the probability of a subject having an event during the study under the Weibull model can be calculated by [18]

equation                            (7)

Therefore, given the design parameters δ (or κ), m1,m2, α, β, tf and ta, the number of subjects n2 required for the current study can be calculated by equation and using the formula in equation (5) or (6).

In designing an actual trial, given the accrual time ta, calculating the sample size is often impractical because it may be not possible to enroll the total number of subjects as planned in the given accrual duration. It is more practical to design the study starting with the accrual rate r and then calculate the required accrual time ta. This can be accomplished under the Weibull model assumption. First, the integration in the probability formula (7) can be simplified by approximation, using the Simpson rule

equation (8)

Then, combining the sample size formula based on equations (5) or (6) with equation (8), we can define a root function of the accrual time ta

equation  (9)

Now the accrual time ta can be obtained by solving the root equation root(ta)=0 numerically in Splus using the uniroot function. The total sample size required for the current study is approximately n2=[rta]+, where [x]+ denotes the smallest integer greater than x.

Once the number of events or sample size is calculated for the fixed sample test, we can calculate the information time at the planned calendar time t for the interim analysis by t* = D(t) / D(τ ). For example, if we plan K interim analyses at calendar time tk, k=1,...,K, then the information time at calendar time tk can be calculated byequation where D(t) is given by equations (3) and (4) for Z(t) and S(t), respectively. After some simplifications, the information time t* = D(t)/ D(τ) can be rewritten as

equation                                                            (10)

where I = D2(t)/ D2(τ) is the information time for the current study and R=D2(τ)/D1 is the ratio of the number of events of the current study to the historical control for the Z(t) test, and R =δ2/3D2(τ)/D1 for S(t) test. This is called the transformed information time [12]. Because D1 is known from historical control data, thus, under the Weibull model, the information time t* can be obtained by calculating D2(t)=n2p2(t), which is the expected number of events in the current study up to time t, where p2(t)=P(Δ12(t)=1) can be calculated as

equation                    (11)

where equation When t=τ, equation (11) is identical to equation (7).

For a maximum information trial where the trial continues until a pre- specified number of events D2(τ) observed for the current study, the information time at the kth look planned at number of events D2k for the current study can be calculated by equation where Ik=D2k/D2(τ), R=D2(τ)/D1 and equation for Z(t) and S(t), respectively.

Group Sequential Procedure

In this section, we will apply an SCPRT procedure [1] to the test statistics Z(t) and S(t). The SCPRT has two unique features: (1) the maximum sample size of the sequential test is not greater than the size of the reference fixed sample test; and (2) the probability of discordance, or the probability that the conclusion of the sequential test would be reversed if the experiment were not stopped according to the stopping rule but continued to the planned end, can be controlled to an arbitrarily small level [12]. Furthermore, the power function of the SCPRT is virtually the same as that of the fixed sample test [1]. The SCPRT boundaries derived in our study have analytical solutions. All these features make the SCPRT attractive and simple to use.

Now we apply the SCPRT to the test statistic equation which is a Brownian motion in information time equation on [0, 1], and drift parameter equation for Z(t) and equation. Therefore, the conditional density equation is the normal density of equation Let equation be the critical value of B1 to reject the null for the fixed sample test. Then the conditional maximum likelihood ratio for the stochastic process on information time t* (see, [1,19]) is

equation

Taking the logarithm, the log-likelihood ratio can be simplified as

equation

which has a positive sign if equation and a negative sign if equation Suppose kth interim looks are planned at calendar time tk, k=1,...,K. Then on the basis of the SCPRT procedure presented above, the lower and upper boundaries for * tk B at the kth look are given by

equation            (12)

for k=1,...,K, where equation is the information time at the kth look at calendar time tk. The a in (12) is the boundary coefficient, and it is crucial to choose an appropriate a for the design such that the probability of conclusion by the sequential test being reversed by the test at the planned end is small but not unnecessarily too small. The larger the a, the smaller is the discordance probability and the wider apart are the upper and lower boundaries, making it harder for the sample path to reach boundaries and stop early and resulting in larger expected sample sizes. Thus, an appropriate a can be determined by choosing an appropriate discordance probability [1,19]. The nominal critical p-values for testing H0 are

equation             (13)

The observed p-value at the kth look is

equation

The stopping rule for monitoring the trial can be executed by stopping the trial when, for the first time, equation (accept H0 and stop for futility) or equation (reject H0 and stop for efficacy). Because Z(tk) or S(tk) has the same asymptotic distribution as the equation under the null hypothesis, the observed p-value at the kth stage can be calculated from the test statistic Z(tk) or S(tk) by applying all observations up to stage k.

Simulation Studies

In this section, we conducted simulation studies to compare the power and type I error of the proposed parametric test statistics Z(t) and S(t) under various scenarios. In the simulations, the survival distribution of the jth group was taken as equationwhich is the Weibull distribution with shape parameter κ and median survival time mj, j=1,2, where j=1 and j=2 represent the historical control and current study, respectively. The shape parameter κ was taken as 0.5, 1, and 2.0 to reflect cases of decreasing, constant, and increasing hazard functions, respectively. We assume a median time-to-event m1=3.4657 and a sample size n1=140 for the historical control. The null hypothesis was set to H0 : m1=m2, and the hazard ratio δ=(m2/m1)κ under the alternative was taken as 1.5-2.0. Furthermore, we assumed that subjects of the current study were recruited with a uniform distribution over the accrual period ta=4 (years) and followed for tf=1 (years), and no subject was lost to follow-up during the study period τ=ta + tf=5. Therefore, a subject was censored at calendar time t if his/her event time was longer than t−u, where u is the time when the subject entered the current study.

In Table 1, the sample sizes required for the current study were calculated by equations (5) and (6) for test Z(t) and S(t), respectively. Furthermore, in each design parameter configuration, 100,000 observed samples of censored event times were generated from the Weibull distribution to calculate the test statistics under the null or alternative hypothesis. The nominal significance level and power were set to 0.05 and 80%, respectively. Two simulation studies were done. The first simulation was done to study the empirical type I error and power for the fixed sample tests. The second simulation was done to study the empirical type I error and power for a two-stage SCPRT design at calendar times t1=3 and t2=5. The simulated empirical type I errors and powers in various scenarios for the fixed sample tests and two-stage SCPRT tests are summarized in Tables 1 and 2, respectively. The results of the fixed sample tests showed that the S(τ) test needs a slightly larger sample size for a small δ and smaller sample size for a large δ compared with the Z(τ) test. The simulated empirical type I errors and powers were close to the nominal levels for the S(τ) test, and the Z(τ) test was somewhat overpowered for a large δ. For the two-stage design S(t) had adequate empirical power and type I error whereas the Z(τ) test was conservative and under-powered for a large δ in the first stage. Overall, the test statistic S(t) performed better than Z(t) and is recommended for use in the trial design. By the way, to show if the sample size formula (5) and information time (10) developed for the Z(t) test also work for the non-parametric log-rank test L(t), the empirical type I errors and powers were simulated for the log-rank test too (Tables 1 and 2). The results showed that both sample size formula (5) and transformed information time (10) worked well for the logrank test. A rigorous derivation of these results for the log-rank test will be the future research.

Design   δ =1.5 δ =1.6 δ =1.7
κ Test *n2 α 1 - β n2 α 1 - β n2 α 1 - β
0.5 Z(t) 262 0.052 0.795 152 0.051 0.804 108 0.049 0.817
  S(t) 285 0.051 0.799 149 0.05 0.802 100 0.051 0.804
  L(t) 262 0.053 0.795 152 0.051 0.804 108 0.049 0.818
1 Z(t) 305 0.053 0.793 170 0.05 0.807 118 0.049 0.813
  S(t) 344 0.052 0.799 168 0.05 0.802 111 0.052 0.806
  L(t) 305 0.053 0.793 170 0.05 0.807 118 0.049 0.813
2 Z(t) 367 0.054 0.795 191 0.053 0.803 130 0.05 0.811
  S(t) 445 0.049 0.801 195 0.051 0.8 124 0.051 0.802
  L(t) 367 0.054 0.794 191 0.053 0.802 130 0.05 0.81
Design   δ =1.8 δ =1.9 δ =2.0
κ Test n2 α 1 - β n2 α 1 - β n2 α 1 - β
0.5 Z(t) 84 0.049 0.823 70 0.047 0.833 60 0.046 0.84
  S(t) 75 0.05 0.806 61 0.051 0.813 51 0.051 0.815
  L(t) 84 0.049 0.823 70 0.048 0.834 60 0.047 0.842
1 Z(t) 92 0.048 0.823 75 0.046 0.828 65 0.046 0.838
  S(t) 82 0.051 0.807 66 0.05 0.812 55 0.051 0.814
  L(t) 92 0.049 0.823 75 0.047 0.828 65 0.047 0.839
2 Z(t) 99 0.048 0.817 81 0.047 0.825 69 0.047 0.833
  S(t) 90 0.051 0.805 71 0.051 0.806 59 0.051 0.812
  L(t) 99 0.048 0.815 81 0.048 0.824 69 0.048 0.832

Table 1: Sample size and simulated empirical type I error (α) and power (1-β) based on 100,000 simulation runs for the Weibull distribution for fixed sample Z(t) test, log-rank test L(t) and S(t) test with a nominal type I error of 0.05 and power 80% (one-sided test).

        Type 1 error Power
Design κ Test At kth interim look k = 1 k = 2 total k = 1 k = 2 total
d=1.5 0.5 Z(t) Empirical 0.0067 0.0457 0.0524 0.3659 0.4293 0.7952
    S(t) Empirical 0.0082 0.0435 0.0517 0.4104 0.3887 0.7992
    L(t) Empirical 0.0066 0.046 0.0526 0.3545 0.4402 0.7947
      Nominal 0.0068 0.0435 0.0503 0.3686 0.4311 0.7997
  1 Z(t) Empirical 0.0052 0.0484 0.0535 0.2874 0.5062 0.7936
    S(t) Empirical 0.0066 0.0454 0.0521 0.3397 0.4594 0.799
    L(t) Empirical 0.0058 0.0483 0.0541 0.2674 0.5245 0.7918
      Nominal 0.0055 0.045 0.0505 0.2912 0.5083 0.7995
  2.0 Z(t) Empirical 0.0037 0.0503 0.054 0.185 0.6096 0.7946
    S(t) Empirical 0.0054 0.0448 0.0502 0.2541 0.547 0.801
    L(t) Empirical 0.0043 0.0498 0.0541 0.1655 0.6257 0.7912
      Nominal 0.0045 0.0463 0.0508 0.1997 0.5994 0.7991
d =1.7 0.5 Z(t) Empirical 0.0037 0.0457 0.0494 0.2611 0.556 0.8171
    S(t) Empirical 0.0059 0.0455 0.0514 0.3054 0.4989 0.8043
    L(t) Empirical 0.0041 0.0457 0.0497 0.2708 0.5467 0.8176
      Nominal 0.0054 0.0451 0.0505 0.2851 5144 0.7995
  1 Z(t) Empirical 0.0023 0.047 0.0493 0.1612 0.6522 0.8134
    S(t) Empirical 0.0052 0.0479 0.0531 0.2259 0.5806 0.8065
    L(t) Empirical 0.0029 0.0466 0.0495 0.1753 0.5245 0.7918
      Nominal 0.0045 0.0462 0.0508 0.2049 0.5943 0.7992
  2 Z(t) Empirical 0.0014 0.0486 0.0499 0.0607 0.7496 0.8104
    S(t) Empirical 0.0046 0.0477 0.0523 0.1366 0.704 0.8031
    L(t) Empirical 0.0024 0.0486 0.051 0.0858 0.7227 0.8085
      Nominal 0.0043 0.0471 0.0514 0.1259 0.6726 0.7985
d= 1.9 0.5 Z(t) Empirical 0.0022 0.0452 0.0474 0.2041 0.629 0.833
    S(t) Empirical 0.0051 0.0465 0.0516 0.2732 0.5402 0.8135
    L(t) Empirical 0.003 0.0449 0.0479 0.2326 0.6015 0.8341
      Nominal 0.005 0.0455 0.0505 0.2574 0.542 0.7994
  1 Z(t) Empirical 0.0011 0.0453 0.0464 0.0947 0.733 0.8276
    S(t) Empirical 0.0046 0.0472 0.0518 0.1883 0.6239 0.8122
    L(t) Empirical 0.003 0.0449 0.0479 0.2326 0.6015 0.8341
      Nominal 0.004 0.0465 0.0509 0.1803 0.6187 0.799
  2 Z(t) Empirical 0.0002 0.047 0.0472 0.0067 0.8168 0.8235
    S(t) Empirical 0.005 0.0484 0.0533 0.1005 0.704 0.8045
    L(t) Empirical 0.0017 0.0468 0.0485 0.0513 0.771 0.8224
      Nominal 0.0044 0.0473 0.0516 0.1084 0.6899 0.7983

Table 2: Simulated empirical type I error and power of the two-stage SCPRT designs based on 100,000 simulation runs for sequential Z(t), log-rank L(t) and S(t) tests with a nominal type I error of 0.05 and power 80% (one-sided test).

An Example

Between January, 1974 and May, 1984, the Mayo Clinic conduct a double- blind randomized trial in primary biliary cirrhosis (PBC), comparing the drug D-penicillamine (DPCA) with a placebo (Fleming and Harrington, 1991). PBC is a rare but fatal chromic liver disease of unknown cause, with a prevalence of about 50-cases-per-millian population. The primary pathologic event appears to be the destruction of interlobular bile ducts, which may be mediated by immunologic mechanisms. A total of 65 had died among 158 patients treated with DPCA. The median survival time was 9 years. Suppose a new treatment is now available and investigators want to design a new trial using Mayo Clinic patients treated with DPCA as the historical control group. The survival distribution of DPCA data were estimated by Kaplan-Meier method and the Weibull model. The Weibull distribution fitted the survival distribution well with shape parameter κ=1.22 and scale parameter ρ=11.8−1. Thus to design the study, we can assume that the failure time of a patient on the current study follows the Weibull distribution with shape parameter κ=1.22 and median survival time m2. Let δ=(m2/m1)κ be the hazard ratio, where m1 is the median survival time of the historical control. Our aim is to test the following hypotheses:

equation

with significance level of α=0.05 and power of 1-β=90% to detect an alternative δ=1.714, which is calculated from by increasing 5 years median survival times of the historical control (m1=9) to the current study (m2=14). Given type I error α=.05, power of 90%, number of deaths of the historical control D1=65, effect size δ=1.714, and the Weibull shape parameter κ=1.22, the number of events required for the current study for the Z(τ) test is calculated by

equation

The number of events required for the S(τ) test is calculated by

equation

which is 54 events too. Assume that the lengths of accrual and followup for the current study are ta=5 and tf=3, respectively, and the study duration is τ=8. Then the probability of having an event during the study for a subject on the current study can be calculated by numerical integration

equation

where κ=1.22 and m2=14. Thus the number of patients required for the current study is n2 = D2(τ)/p2(τ)=54 / 0.1985 = 273. Suppose that the test statistic S(t) will be used to monitor the trial, and 3 interim looks are planned at calendar times t1=4, t2=6 and t3=8 years. Then the transformed information times can be calculated by

equation                  (14)

where equation and equation with equation and

equation

where equation Thus, the information time calculated by equations (14) and (15) is t*=(0.436, 0.773, 1), the lower and upper boundaries calculated by equation (12) are (a1,a2,a3)=(-0.425, 0.307, 1.645) and (b1,b2,b3)=(1.859, 2.236, 1.645), respectively, and the nominal critical p- values calculated by equation (13) are equationand equation for the lower and upper boundaries, respectively. To monitor the trial at kth interim look, the survival data collected up to calendar time tk from the current study combined with all data of the historical control to calculate the sequential test statistic S(tk) as described in Section 2, and the observed p-values

equation

At kth stage, we stop the trial for futility if equation and stop the trial for efficacy if equation The operating characteristics of the sequential test S(t) for this example are given in Table 3.

At kth interim look k = 1 k = 2 k = 3 total
Type I error
Empirical of S(t) 0.0028 0.0047 0.0422 0.0496
Nominal 0.0024 0.0046 0.0436 0.0506
Power        
Empirical of S(t) 0.171 0.2994 0.3886 0.8389
 Nominal 0.1204 0.2533 0.4257 0.7994
Probability of stopping under null
Empirical of S(t) 0.2574 0.3907 0.3519 1
Nominal 0.2626 0.3916 0.346 1
Probability of stopping under alternative
Empirical of S(t) 0.1756 0.315 0.5094 1
 Nominal 0.1315 0.28 0.5885 1

Table 3: Operating characteristics of the three-stage SCPRT design for test statistic S(t)based on 100,000 simulation runs under the Weibull distribution with uniform censoring distribution on [tf ,ta+ tf], and nominal type I error of 0.05 and power 80% for the example in Section 6.

Acknowledgements

We proposed two parametric sequential tests for group sequential trial de-sign against historical controls. Simulation results showed that the empirical power and type I error of the S(t) test are close to those of the nominal levels, and it outperforms the Z(t) test. Hence, we recommend using the S(t) test for historical control trial designs under the Weibull model. We derived transformed information times t*=(1+R)I/(1+RI) for both test statistics Z(t) and S(t). It is simple and convenient to use the transformed information time t* to derive the sequential monitoring rule for the historical control trial design based on the SCPRT procedure. With this monitoring procedure, data from the current study are sequentially collected and com-pared with data from the historical control. This allows investigators to monitor the trial at any calendar time of enrollment or at a pre-specified number of events of an interim look. The number of events required for the current study can be calculated by a simple formula. Therefore, the study design is much simpler than that of the method for survival data proposed by Xiong et al. [12], in which information times of the sequential test statistic are random and depend on data instead of being predetermined. The maximum sample size of the sequential test is the same as that for the fixed sample test and the group sequential boundaries have analytical solutions. Therefore, the proposed group sequential procedure is effective and simple to use. For the study design purpose, we need the number of events from the historical control data only. However for the trial monitoring and final data analyses, we need full failure time data from the historical control study to calculate the sequential test statistic Z(tk) or S(tk). In practice, the historical control data are often available from previous trials done by the same institution or by the same sponsor. If there is no such historical control data available from the same institution, then we need to extract the relevant data from published literatures. Recently, Guyot et al. [20] have proposed a method to reconstructing the survival data from published Kaplan-Meier survival curves. Thus designing survival trials with historical controls are feasible by using control data from published literatures.

Finally, even though the sample size formula (5) and transformed information time (10) were derived for the Z(t) test under the Weibull model, our simulation results showed that they also work well for the nonparametric log-rank test under the proportional hazard models. A rigorous derivation of these results for the log-rank test will be the future research

Acknowledgment

This work was supported in part by the National Cancer Institute (NCI) support grant P30CA021765-35.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Recommended Conferences

Article Usage

  • Total views: 11451
  • [From(publication date):
    August-2014 - Jun 26, 2017]
  • Breakdown by view type
  • HTML page views : 7706
  • PDF downloads :3745
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2017-18
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

 
© 2008-2017 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
adwords