Two-stage optimal designs with survival endpoint when the follow-up time is restricted
Abstract
Background
Survival endpoint is frequently used in early phase clinical trials as the primary endpoint to assess the activity of a new treatment. Existing two-stage optimal designs with survival endpoint either over estimate the sample size or compute power outside the alternative hypothesis space.
Methods
We propose a new single-arm two-stage optimal design with survival endpoint by using the one-sample log rank test based on exact variance estimates. This proposed design with survival endpoint is analogous to Simon’s two-stage design with binary endpoint, having restricted follow-up.
Results
We compare the proposed design with the existing two-stage designs, including the two-stage design with survival endpoint based on the nonparametric Nelson-Aalen estimate, and Simon’s two-stage designs with or without interim accrual. The new design always performs better than these competitors with regards to the expected total study length, and requires a smaller expected sample size than Simon’s design with interim accrual.
Conclusions
The proposed two-stage minimax and optimal designs with survival endpoint are recommended for use in practice to shorten the study length of clinical trials.
Keywords
Clinical trials Exact variance One-sample log-rank test Restricted follow-up Simon’s two-stage designAbbreviations
- ESS
Expected sample size
- ETSL
Expected total study length
- PET
Probability of early termination
- TIE
Type I error
Background
A multiple-stage design is often preferable in early phase clinical trials to investigate the activity of a new treatment. Such design is able to protect patients better as compared to the traditional one-stage design by allowing a trial to be stopped earlier when the new treatment is indeed ineffective. For this reason, early stopping for futility is always allowed in these trials. Among multiple-stage designs, a two-stage design is widely used in phase II clinical trials whose sample size is relatively smaller than that in the following phase III trial to confirm the effectiveness of the new treatment(s).
When the outcome is binary (e.g., response VS non-response), Simon’s two-stage minimax and optimal designs are widely used in practice [1, 2, 3, 4, 5, 6, 7, 8]. When the required number of patients in the first stage are enrolled, a trial generally has to be suspended temporally to allow these patients completing the treatment schedule. After that, data analysis is performed to make the decision whether a trial proceeds to the second stage or not, based on the result from the first stage. This suspension during the clinical trial could lead to a longer study time as compared to the modified Simon’s two-stage design with interim accrual [9]. Recently, adaptive version of Simon’s two-stage design has been proposed to improve the flexibility of trials [3, 4, 10, 11, 12]. In such trials, the second stage sample size depends on the outcome from the first stage.
In some other trials (e.g., cytostatic therapies), a survival endpoint is served as the primary outcome to measure the activity of a new treatment. Feldman et al. [13] reviewed seven single-arm phase II trials for patients with refractory germ cell tumors, and recommended a 12-week progression-free survival as compared to the commonly used response rate, to test the activity of novel agents. For such trials, a multiple-stage design with survival endpoint would be appropriate for use in practice. Lin et al. [14] proposed group sequential designs for a trial with survival endpoint by deriving the asymptotic joint distribution of the Nelson-Aalen estimates at different time points. Base on Lin et al.’s work, Case and Morgan [9] developed a two-stage optimal design evaluating survival probabilities with restricted follow-up. They proposed two-stage optimal designs with the smallest expected duration of accrual or the smallest expected total study length. Later, Kwak and Jung [15] proposed a new two-stage optimal design based on the one-sample log-rank test without follow-up restriction. Power of their proposed design was computed under the average of the cumulative hazard function under the null hypothesis and that under the alternative hypothesis. In addition, the asymptotic variance estimate of the one-sample log-rank test was used in type I error rate and power calculation. Recently, Belin et al. [16] proposed a two-stage design based on the design setting as in Kwak and Jung [15], but having restricted follow-up as in Case and Morgan [9].
For a trial with a survival endpoint as the primary outcome, the survival probability at the clinically meaningful follow-up time is often the parameter of interest, (e.g., the survival probability at 1 year). We develop a new single-arm two-stage optimal design by using the one-sample log-rank test with exact mean and variance estimates [17, 18]. A trial is allowed to be stopped in the first stage due to futility to protect patients when the treatment under investigation is indeed ineffective. Although exact mean and variance estimates of the one-sample log-rank test are used for sample size calculation, the joint distribution of the test statistic for the first stage and that for the two stages combined is assumed to asymptotically follow a bivariate normal distribution. For this reason, the actual power of the identified study design may not be guaranteed [19]. We propose adjusting the nominal power level in design search to guarantee that the new designs meet the power requirement. The proposed two-stage minimax and optimal designs with survival endpoint are compared with the design by Belin et al. [16] and Simon’s two-stage designs with or without interim accrual.
The rest of this article is organized as follows. In Section Methods, we present the type I error rate and power calculation for a two-stage design with survival endpoint by using the one-sample log-rank test, and provide a detailed search method for two-stage minimax and optimal designs. In Section Results, we compare the performance of the new proposed two-stage designs with the existing Belin’s design with survival endpoint and Simon’s two-stage design with binary endpoint. At the end of that section, we revisit two trials to illustrate the application of the proposed two-stage designs with survival endpoint. Lastly, we provide some comments in Section Discussion.
Methods
When a new study is assumed to have a different failure rate as historical data, the HR is then calculated as \(\Delta =\frac {\lambda _{0}^{k_{0}}}{\lambda _{1}^{k_{1}}} \times \frac {k_{1} t^{k_{1}-1}}{k_{0} t^{k_{0}-1}}\), where k_{0} and k_{1} are the shape parameter under the null hypothesis and that under the alternative hypothesis, respectively.
Simon’s two-stage designs with binary endpoint
In Simon’s two-stage optimal designs, a trial is allowed to be stopped in the first stage when the number of responses is insufficient. Suppose X_{1} and X are the number of responses out of n_{1} and n participants from the first stage and the two stages combined, respectively. The sample size in the second stage is n_{2}=n−n_{1}. The null hypothesis is rejected when X_{1}>r_{1} and X>r, where r_{1} and r are the critical values for the number of responses from the first stage and both stages, respectively.
The resectable pancreatic cancer clinical trial with S_{0}(t_{c}=1)=35%, and S_{1}(t_{c}=1)=50% to attain 90% power at the significance level of 10%
Survival endpoint | Simon’s design, interim accrual | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
The proposed method | Belin | No | Yes | ||||||||||
n _{1} | n | c _{1} | c | E S S _{0} | E T S L _{0} | E S S _{0} | n | E S S _{0} | E T S L _{0} | n | E S S _{0} | E T S L _{0} | |
Minimax | 44 | 73 | 0.240 | -1.281 | 61.3 | 3.1 | 59.3 | 4.0 | 72 | 69.8 | 3.5 | ||
Optimal | 41 | 79 | -0.085 | -1.279 | 58.7 | 2.9 | 59.1 | 69 | 53.2 | 3.6 | 81 | 67.4 | 3.2 |
The results of Simon’s two-stage designs with interim accrual are presented in Table 1. As compared to the traditional Simon’s two-stage design without interim accrual, the modified design with interim accrual requires a shorter ETSL_{0} but a larger ESS_{0}.
Two-stage optimal designs with survival endpoint when the follow-up time is limited
In a two-stage design with sample sizes of n_{1} in the first stage and n_{2} in the second stage, the maximum possible sample size in the study is n=n_{1}+n_{2}. Given the patient accrual rate of θ, the accrual time for the first stage is t_{1}=n_{1}/θ. When the trial goes to the second stage, the total accrual time of the study is t_{a}=n/θ, and the total study time for all patients to complete the study is t=t_{a}+t_{c}.
where ϕ and Φ are the probability density function and the cumulative distribution function of the standard normal distribution, and ρ_{0} is the correlation coefficient estimate between Z_{1} and Z under the null hypothesis, see Appendix for the detailed formula for ρ_{0}. The actual power of the study can be computed similarly with ρ_{0} being replaced by the ρ estimate under the alternative hypothesis.
Optimal design search
Similar to the search for Simon’s two-stage design, the two-stage optimal design with survival endpoint has to be searched over all the possible sample sizes (n_{1} and n) and critical values (c_{1} and c), given the design parameters (α,β,t_{c},S_{0}(t_{c}),S_{1}(t_{c}),θ).
Although the exact variances of Z_{1} and Z are available for use in sample size determination, the exact joint distribution of Z_{1} and Z is not that straightforward. For this reason, we utilize the limiting distribution of (Z_{1},Z) in searching for the two-stage optimal design for a study with the design parameters (α,β,t_{c},S_{0}(t_{c}),S_{1}(t_{c}),θ), then use a simulation study to calculate the actual TIE and power of the optimal design. The following three steps are used to search for the two-stage minimax and optimal designs.
Step 1: Given the total sample size n, the range of the first stage sample size n_{1} is from 1 to n−1. The critical value c_{1} from -0.3 to 1.6 with an increment of 0.005 is used in the design search. Similar to Kwak and Jung [15], the range of c_{1} is chosen based on the simulation studies for all the configurations studied in this article. The range of c_{1} is modifiable in the software program for design search.
For each combination of n_{1} and c_{1}, the critical value c can be determined as the largest c value such that TIE(c)≤α from Eq. (3). Power of the study is then computed by using Eq. (4) in Appendix. If power is above the nominal level, this set of sample sizes and critical values, (n_{1},c_{1},n,c), is saved as a candidate for the optimal two-stage design. Among all the sets satisfying the power requirement, the one with the smallest ESS_{0} is the optimal two-stage design when the total sample size is n, and it is denoted as B(n)=(n_{1},c_{1},n,c) whose expected sample size is ESS_{0}(n).
Step 2: The design search starts with a relatively small n (e.g., 5) with an increment of 1, and B(n) could be a empty set when n is small. The two-stage minimax design is the one with the smallest n, n_{minimax} such that B(n) is not empty. The optimal two-stage design is the one with the smallest ESS_{0}. The search may be stopped at n_{u} when its ESS_{0}(n_{u}) is 10% more than the smallest ESS_{0} from the identified optimal designs with n from n_{minimax} to n_{u}: ESS_{0}(n_{u})≥110%× min{ESS_{0}(n):n_{minimax}≤n≤n_{u}}.
Step 3: Once the minimax and optimal two-stage designs are identified from Step 1 and Step 2, we use a simulation study to calculate the actual TIE and power based on 100,000 simulations. We find that the actual TIE of the optimal design B(n)=(n_{1},c_{1},n,c) is always guaranteed, while power may not be preserved in some cases. If the simulated power of the two designs meet the nominal levels, they are the final two-stage minimax and optimal designs. Otherwise, we search for the designs again with the power nominal level being increased by 1%, (α,β−1%) in Step 1 and Step 2 again. This process is stopped when both minimax and optimal two-stage designs meet the power requirement.
Results
We first compare the proposed two-stage minimax and optimal designs with survival endpoint when the follow-up time is restricted, with the designs developed by Belin et al. [16] (referred to as Belin’s design). They developed a two-stage optimal design as a modification of the design by Kwak and Jung [15] by adding restricted follow-up in the study design [9]. In Belin’s design, power of the study is computed at the average of the cumulative hazard functions under the null and the alternative, that is less than the cumulative hazard functions under the alternative at which value the actual power should be computed. This leads to an decreased effect size in sample size calculation; thus, the computed sample size may be over-estimated. As a result of the over-estimated sample size, the actual power is often above the nominal level.
Comparison between the proposed two-stage minimax and optimal designs with survival endpoint and Belin’s two-stage optimal design with survival endpoint, when the follow-up time is restricted to the clinically meaningful follow-up time t_{c}=1 year
Minimax design | Optimal design | Belin | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Power | θ | n _{1} | n | c _{1} | c | E S S _{0} | n _{1} | n | c _{1} | c | E S S _{0} | n | E S S _{0} |
90% | 15 | 28 | 52 | -0.10 | -1.64 | 39.1 | 26 | 56 | -0.30 | -1.64 | 37.5 | 53 | 42.3 |
95% | 15 | 36 | 65 | -0.09 | -1.64 | 49.5 | 33 | 70 | -0.29 | -1.64 | 47.3 | 65 | 52.6 |
90% | 30 | 30 | 52 | 0.30 | -1.64 | 43.6 | 30 | 55 | -0.04 | -1.64 | 42.2 | 53 | 44.6 |
95% | 30 | 40 | 65 | 0.19 | -1.64 | 54.3 | 40 | 69 | -0.20 | -1.64 | 52.2 | 65 | 54.8 |
90% | 50 | 34 | 52 | 0.51 | -1.64 | 46.5 | 32 | 54 | 0.32 | -1.63 | 45.7 | 52 | 47.0 |
95% | 50 | 44 | 65 | 0.46 | -1.64 | 58.2 | 42 | 68 | 0.17 | -1.64 | 56.7 | 64 | 57.5 |
Simulated TIE and power of the proposed two-stage minimax and optimal designs in Table 2
Minimax design | Optimal design | ||||
---|---|---|---|---|---|
Power | θ | TIE | Power | TIE | Power |
90% | 15 | 0.040 (0.036,0.044) | 0.907 (0.901,0.913) | 0.037 (0.033,0.041) | 0.903 (0.898,0.909) |
95% | 15 | 0.041 (0.037,0.045) | 0.957 (0.953,0.961) | 0.038 (0.035,0.042) | 0.955 (0.951,0.959) |
90% | 30 | 0.040 (0.037,0.044) | 0.911 (0.905,0.916) | 0.039 (0.035,0.043) | 0.910 (0.904,0.916) |
95% | 30 | 0.042 (0.038,0.046) | 0.959 (0.955,0.963) | 0.040 (0.036,0.044) | 0.958 (0.954,0.962) |
90% | 50 | 0.041 (0.037,0.045) | 0.911 (0.905,0.916) | 0.040 (0.037,0.044) | 0.909 (0.903,0.914) |
95% | 50 | 0.042 (0.038,0.046) | 0.960 (0.956,0.963) | 0.041 (0.037,0.045) | 0.959 (0.955,0.963) |
Comparison between the proposed two-stage minimax design with survival endpoint and Simon’s two-stage minimax design with binary endpoint with or without interim accrual, when α=5%, β=20%, and the shape parameter k=0.5 in the Weibull distribution
Simon’s two-stage minimax designs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Survival endpoint | No interim accrual | Interim accrual | |||||||||
S_{0}(t_{c}) | S_{1}(t_{c}) | n _{1} | n | E S S _{0} | E T S L _{0} | n _{1} | n | ESS_{0}(%) | ETSL_{0}(%) | ESS_{0}(%) | ETSL_{0}(%) |
0.1 | 0.2 | 37 | 63 | 50.5 | 2.5 | 45 | 78 | 60.6 (17%) | 3.8 (35%) | 74.3 (32%) | 3.3 (26%) |
0.1 | 0.25 | 19 | 33 | 26.2 | 2.5 | 22 | 40 | 28.8 (9%) | 3.5 (30%) | 37.5 (30%) | 3.1 (21%) |
0.1 | 0.3 | 11 | 21 | 15.6 | 2.3 | 15 | 25 | 19.5 (20%) | 3.8 (39%) | 24.5 (36%) | 3.3 (30%) |
0.6 | 0.7 | 87 | 162 | 126.6 | 3.2 | 139 | 142 | 139.2 (9%) | 4.0 (20%) | 184.5 (31%) | 3.9 (19%) |
0.6 | 0.75 | 33 | 70 | 49.4 | 2.8 | 30 | 62 | 43.8 (-13%) | 3.6 (20%) | 55.7 (11%) | 3.1 (9%) |
0.6 | 0.8 | 17 | 39 | 26.0 | 2.6 | 13 | 35 | 20.8 (-25%) | 3.1 (16%) | 28.5 (9%) | 2.8 (5%) |
Comparison between the proposed two-stage optimal design with survival endpoint and Simon’s two-stage optimal design with binary endpoint with or without interim accrual, when α=5%, β=20%, and the shape parameter k=0.5 in the Weibull distribution
Simon’s two-stage optimal designs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Survival endpoint | No interim accrual | Interim accrual | |||||||||
S_{0}(t_{c}) | S_{1}(t_{c}) | n _{1} | n | E S S _{0} | E T S L _{0} | n _{1} | n | ESS_{0}(%) | ETSL_{0}(%) | ESS_{0}(%) | ETSL_{0}(%) |
0.1 | 0.2 | 26 | 72 | 45.1 | 2.2 | 30 | 89 | 50.8 (11%) | 3.3 (35%) | 67.6 (33%) | 3.0 (27%) |
0.1 | 0.25 | 15 | 37 | 24.0 | 2.2 | 18 | 43 | 24.7 (3%) | 3.1 (29%) | 34.9 (31%) | 2.8 (22%) |
0.1 | 0.3 | 10 | 23 | 15.0 | 2.2 | 10 | 29 | 15.0 (0%) | 3.1 (29%) | 21.6 (30%) | 2.8 (21%) |
0.6 | 0.7 | 66 | 179 | 109.2 | 2.7 | 53 | 173 | 91.4 (-20%) | 3.3 (18%) | 124.0 (12%) | 2.9 (9%) |
0.6 | 0.75 | 27 | 76 | 46.1 | 2.6 | 27 | 67 | 39.4 (-17%) | 3.2 (18%) | 53.9 (14%) | 2.9 (10%) |
0.6 | 0.8 | 15 | 41 | 25.1 | 2.5 | 11 | 43 | 20.5 (-23%) | 3.1 (17%) | 28.9 (13%) | 2.8 (7%) |
Examples
We revisit the cancer trial discussed by Case and Morgan [9] in “Simon’s two-stage designs with binary endpoint” subsection to investigate the effectiveness of a combination of Gemcitabine and external beam radiation for patients with resectable pancreatic cancer. The clinically meaningful follow-up time is assumed to be 1 year, t_{c}=1. The survival probability under the null and the alternative are S_{0}(1)=35%, and S_{1}(1)=50%, respectively. The survival function follows an exponential distribution. This trial is designed to attain 90% power at the significance level of 10%. We compute the detailed two-stage designs with survival endpoint, including sample sizes and critical values for each stage in Table 1. The ESS_{0} of the new design is slightly larger than that of Simon’s design, but much smaller than that of Simon’s design with interim accrual. The ETSL_{0} of the new design is always shorter than that of Simon’s designs with or without interim accrual, and the study time saving is substantial.
We also consider a second clinical trial evaluating the activity of a combination of irinotecan and cisplatin for patients with refractory or recurrent non-small cell lung cancer [20]. The response rates are 10% and 25% under the null and the alternative hypotheses. Suppose the clinically meaningful follow-up time is 1 year. For Simon’s two-stage optimal design when α=5% and β=20%, the maximum possible sample size is n=43 and the expected sample size under the null hypothesis is ESS_{0}=24.7, see Table 5 for the case with S_{0}(t_{c})=10% and S_{1}(t_{c})=25%. The proposed new two-stage optimal design with survival endpoint needs a slightly smaller ESS_{0} as 24.0, and can save the expected total study length by almost 1 year (2.2 VS 3.1 from Simon’s design). A 95% two-sided confidence interval of the response rate was reported in the original research article by Takiguchi et al. [20]. The hypothesis is one sided in both Simon’s design and the proposed design. Therefore, a 90% two-sided confidence interval for the response rate or the survival rate should be reported when α=5%.
Discussion
In the design search process, we search for the minimax and optimal designs when both designs have power above the nominal level. In practice, when one type of design is of interest (e.g., the two-stage minimax design), we would suggest searching for the design such that power of this particular type design is above the nominal level. The written R program computes the designs to have both the minimax design and the optimal design meet the nominal power level, which is available upon request from the first author.
Conclusions
The commonly used Simon’s two-stage design has to suspend the enrollment temporally after n_{1} patients enrolled in the first stage [5, 11, 21, 22, 23, 24, 25, 26, 27, 28]. The research team has to wait a while (t_{c}) until all n_{1} patients complete the study. The calculated test statistic from the first stage is then compared to the pre-determined critical value to make a go or no-go decision to the second stage. Meanwhile, the proposed two-stage designs with survival endpoint do not have to suspend the trial, thus the comparison between the proposed design with Simon’s two-stage design with no interim accrual is not very appropriate. Due to the popularity of Simon’s two-stage design, we include this design as reference. Simon’s two-stage design with interim accrual is a reasonable competitor for the proposed two-stage design with survival endpoint.
Appendix
Test statistics of Z _{1} and Z
Mean and variance estimates of W _{1} and W under the null hypothesis
Mean and variance estimates of W _{1} and W under the alternative hypothesis
where \(\tilde {c_{1}}=\frac {\sigma _{01}}{\sigma _{11}}\left (c_{1}-\frac {\omega _{1} \sqrt {n_{1}}}{\sigma _{01}}\right)\), and \(\tilde {c}=\frac {\sigma _{02}}{\sigma _{12}}\left (c-\frac {\omega _{2} \sqrt {n_{2}}}{\sigma _{02}}\right)\).
Notes
Acknowledgment
We would like to thank Dr. Jianrong Wu and Dr. Lisa Belin for sharing their R codes with us. Authors would like to thank Associate Editor and two referees, for their valuable comments and suggestions that helped to improve this manuscript.
Funding
Shan’s research is partially supported by grants from the National Institute of General Medical Sciences from the National Institutes of Health: P20GM109025. Zhang’s work is supported by the Zhejiang Provincial Natural Science Foundation of China (grant no. LY19F020003) and the National Natural Science Foundation of China (grant no. 61672459).
Availability of data and materials
Not applicable. This is a manuscript to develop novel statistical approaches, therefore, no real data is involved.
Authors’ contributions
The idea for the paper was originally developed by GS. GS and HZ computed the required sample size for a two-stage design with a survival endpoint. GS and HZ drafted the manuscript and approved the final version.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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