Two-stage optimal designs based on exact variance for a single-arm trial with survival endpoints

Abstract

Sample size calculation based on normal approximations is often associated with the loss of statistical power for a single-arm trial with time-to-event endpoint. Recently, Wu (2015) derived the exact variance for the one-sample log-rank test under the alternative, and showed that a single-arm one-stage study based on exact variance often has power above the nominal level while the type I error rate is controlled. We extend this approach to a single-arm two-stage design by using exact variances of the one-sample log-rank test for the first stage and the two stages combined. The empirical power of the proposed two-stage optimal designs is often not guaranteed under a two-stage design setting, which could be due to the asymptotic bi-variate normal distribution used to estimate the joint distribution of the test statistics. We adjust the nominal power level in design search to guarantee the simulated power of the identified optimal design being above the nominal level. The sample size and the study time savings of the proposed two-stage designs are substantial as compared to the one-stage design.

1. Introduction

In an early phase clinical trial (e.g., a phase II trial) to assess the activity of a new treatment, a two-stage or multi-stage design is often used to allow a trial to be stopped earlier due to futility to protect patients when the new treatment is indeed ineffective. For a trial with binary endpoint, Simon’s two-stage optimal designs are widely used [1, 2, 3]. In other trials with survival endpoints (such as cytotoxic therapies to prevent the growth of tumors), we compare the progression-free survival (PFS) of a new treatment with historical data in a single-arm trial or the standard care in a randomized clinical trial [4, 5].

For a single-arm study with time-to-event endpoint, the one-sample log-rank test is commonly used to assess the activity of a new treatment by comparing its survival distribution with historical data [6, 7]. Finkelstein et al. [8] developed a sample size formula by using the one-sample log-rank test based on the number of events. Kwak and Jung [9] proposed a new sample size formula by using the average of the cumulative hazard function (CHF) under the null and the CHF under the alternative as the estimate of CHF in power calculation. These one-stage designs based on asymptotic variance generally do not preserve the nominal power level. For this reason, Wu [10] derived the exact variance estimate for the one-sample log-rank test in sample size calculation, and showed that the power of the one-stage design based on exact variance is always guaranteed.

A two-stage design is often preferable in early phase clinical trials to investigate the effectiveness of a new treatment and protect patients [11]. In the case that the new treatment is not as effective as expected, a two-stage design is able to stop the trial earlier in the first stage to avoid further treating more patients in an inferior treatment. In addition, a two-stage design can save sample size and study time substantially as compared to the traditionally used one-stage design. Kwak and Jung [9] developed two-stage optimal designs for a study with time-to-event endpoints which follows the exponential survival distribution. They computed the power of the study in a conservative manner by using an underestimated CHF. As a result of that, the empirical power of their computed two-stage designs should be higher. However, it is still under the nominal level. As pointed out by Whitehead [12], the power of a study based on normal approximation is often below the nominal level. We propose extending the one-stage design based on exact variance to a two-stage design setting. Two types of two-stage optimal designs are developed: a two-stage optimal design with the smallest expected sample size under H₀, and a two-stage minimax design with the smallest maximum possible sample size (n).

The rest of this article is organized as follows. In Section 2, we first introduce the one-sample log-rank test, then propose single-arm two-stage optimal designs with a detailed algorithm to search for optimal designs. We conduct extensive numerical studies to compare the performance of the proposed two-stage designs with the existing one-stage design by Wu [10], and the two-stage designs by Kwak and Jung [9] in Section 3. At the end of this section, an example from a real clinical trial is used to illustrate the application of the developed two-stage optimal designs. In the last section, we provide some comments and remarks for the proposed two-stage designs.

2. Optimal two-stage designs

In a trial to evaluate the activity of a new treatment when the outcome is the time-to-event endpoint, the one-sample log-rank test is commonly used for data analysis and sample size determination. Suppose T_i and C_i are the survival time and the censoring time for the i-th patient in a trial, i = 1, 2, ⋯, n, where n is the total sample size, T_i and C_i and are assumed to be independent from each other. Then, the observed time is X_i = min (T_i, C_i), and the failure indicator is defined as ζ_i = I(T_i ≤ C_i) for the i-th patient, where I(.) is the indicator function with I(T_i ≤ C_i) = 1 when T_i ≤ C_i and 0 otherwise.

Let S(t) and F(t) be the survival function of the survival time T and the censoring time C. In this article, we assume the survival distribution of T follows the Weibull distribution with the shape parameter k and the scale parameter λ, with the survival function and cumulative hazard function presented as

$$\begin{array}{ccc}S\left(t\right)={\exp }^{-{\left(t/\lambda \right)}^k}& \text{and}& \Lambda \left(t\right)\end{array}={\left(t/\lambda \right)}^k,$$ (1)

where S(t) = exp^−Λ(t). It is well known that the exponential distribution is a special case of the Weibull distribution with k = 1.

Let t_c be the clinically meaningful follow-up time to compare the PFS between a new treatment and the estimated historical survival rate S₀(t_c). Then the hypotheses of a study can be specifically expressed as

$$H_0:S\left(t_c\right)\le S_0\left(t_c\right)\text{against}{H}_a:S\left(t_c\right)>S_0\left(t_c\right),$$ (2)

which is equivalent to test the cumulative hazard function as

$$H_0:\Lambda \left(t_c\right)\ge {\Lambda }_0\left(t_c\right)\text{against}{H}_a:\Lambda \left(t_c\right)<{\Lambda }_0\left(t_c\right).$$ (3)

The survival function in Equation (1) can be alternatively written as

$$S\left(t\right)={\exp }^{-\log \left(2\right){\left(t/m\right)}^k},$$ (4)

where m is the median survival time. In a randomized phase II trial of Onartuzumab in combination with Erlotinib to treat patients with advanced non-small-cell lung cancer [13], the median survival times for the control arm (placebo plus Erlotinib) and the new treatment arm (Onartuzumab plus Erlotinib) were estimated as 3.3 months and 5.5 months, respectively.

Suppose n patients are enrolled in a study with the accrual rate of θ patients per unit (e.g, per year). The parameter of interest is the PFS rate at the clinically meaningful follow-up time t_c. A patient who survives at time t_c, is still on the study to be followed. Then, the total accrual time is t_a = n / θ and the total study time is t_s = t_a + t_c. Suppose N_i(t) = ζ_iI(X_i ≤t) and Y_i(t) = I(X_i ≥t) are the event process and the at-risk process, respectively. The one-sample log-rank test is widely used to test the aforementioned hypotheses in Equation (3), with the following test statistic:

$$L=\frac{O-E}{\sqrt{E}},$$

where $O={\sum }_{i=1}^n{\int }_0^{\infty }d{N}_i\left(t\right)$ are $E={\sum }_{i=1}^n{\int }_0^{\infty }{Y}_i\left(t\right)d{\Lambda }_0\left(t\right)$ are the observed number of events and the expected number of events, respectively. The one-sample log-rank test can be alternatively written as

$$L=\frac{W}{\widehat{\sigma }},$$

where $W=\left(O-E\right)/\sqrt{n}$ and $\widehat{\sigma }=E/n$, and ${\widehat{\sigma }}^2$ is the variance estimate of W.

In a two-stage design, a trial is allowed to be stopped for futility at the first stage (t₁). At the end of the first stage, the number of patients available for data analysis is n₁ = t₁θ. By using the available data at time t₁, we calculate the one-sample log-rank test statistic $L_1=W_1/{\widehat{\sigma }}_1$ for this stage. The observed time for the i-th patient at the end of the first stage is: X_i1 = min(T_i, C_i, t₁ − E_i), where E_i is the calendar time of the i-th patient enrolled in the study, 0 ≤ E_i ≤ t₁. When T_i ≤ C_i and T_i ≤ t₁ − E_i, an event occurs before the data analysis time t₁. When T_i > C_i or T_i > t₁ − E_i, the i-th patient is censored at time t₁ or time C_i + E_i for data analysis.

When L₁ is larger than the critical value of c₁, a trial proceeds to the second stage with additional n₂ patients enrolled, with a total number of patients n = n₁ + n₂. The time for the final data analysis is t_s = t_a + t_c. The observed time for the i-th patient is calculated as X_i = min (T_i, C_i, t_s − E_i). It follows that the one-sample log-rank test is then calculated as $L=W/\widehat{\sigma }$ by using data from all patients. When sample sizes (n₁ and n) are large enough, the joint distribution function of (L₁, L) asymptotically follows a bivariate normal distribution [9]. The detailed calculation for the actual type I error (TIE) rate and the power for a two-stage design with sample sizes and critical values (n₁, c₁, n, c) can be found in Appendix. The following search algorithm is used to identify the optimal design given the design parameters (α, β, t_c, Λ₀(t_c), Λ₁(t_c), θ).

Step 1: Sample size for the one-stage design is calculated using the approach by Wu [10], n_single.

Step 2: We search the optimal design with the maximum possible sample size, n, with a range from 0.5n_single to 1.5n_single.

Step 3: For each given n from Step 2, the total accrual time is t_a = n/θ. We assume that all the patients are uniformly accrued between 0 and t_a. The possible stopping times for the first stage are t₁ = 1/ θ, 2/ θ, ⋯, (n − 1) / θ, which is equivalent to having n₁ = 1, 2, ⋯, n − 1 in the first stage.

Step 4: For given (n₁, n) or (t₁, n), the first stage critical value c₁ has the possible values from −1.65 to 1.65 by an increment of 0.005.

Step 5: For each combination of (n₁, c₁, n), we compute the largest c such that the TIE from Equation (5) in Appendix is less than or equal to α.

Step 6: Power is then computed by using Equation (6) in Appendix after the critical value of c is determined from Step 5. If the computed power is above the nominal level (1 − β), this set of sample sizes and critical values (n₁, c₁, n, c) is saved as a candidate for the optimal design.

Step 5 and Step 6 are used iteratively to compute power of all the possible designs (n₁, c₁, n, c) and identify all the designs who have the power above the nominal level. When n is large, we would suggest reducing the number of possible first stopping in Step 3 to every half month, instead of a total of n − 1 possible first stage stopping times to reduce the computational boundary.

Among the designs that meet the power requirement, the one with the smallest expected sample size under the null hypothesis (ESS₀) is the optimal two-stage design. Following Simon’s design, we also introduce the minimax design that has the smallest ESS₀ among the ones having the smallest n. In the following section, we compare the performance of the proposed two-stage designs for a study with time-to-event endpoint with other existing one-stage and two-stage designs.

3. Simulation study

We compare the proposed two-stage optimal designs based on exact variance, with the existing one-stage and two-stage optimal designs with regards to expected sample size and maximum possible sample size.

Kwak and Jung [9] (referred to be as the KJ design) compute the variances of and under the alternative by using the average of Λ₀(t) and Λ₁(t) to estimate Λ(t) under the alternative, which could lead to an reduced estimate of Λ(t) under H_a, and an over-estimated sample size. This approach may be appropriate for use in practice when Λ₀(t) and Λ₁(t) are close to each other. Under a one-stage design setting, Wu [10] showed that the power of a study designed by the KJ approach is generally below the nominal level, and the design by using exact variance derived by Wu [10] has the power that is always above the nominal level.

We compare the proposed two-stage minimax design and the two-stage minimax KJ design in Table 1, for studies with design parameters: the accrual rate of θ = 30 patients per year, the clinically meaningful follow-up time t_c = 1. Only the exponential distribution is considered in the KJ design, with k = 1 in the Weibull distribution. The CHF Λ(t_c) under the null hypothesis and that under the alternative are log(2) and log(2)/δ, respectively, where δ is the HR with the values of 1.4, 1.5, 1.6, and 1.7. The new two-stage design is similar to the KJ design with regards to the maximum possible sample size n and the simulated power. The simulated power is close to be 0.02 below the nominal level. For this reason, we increase the nominal level by 0.02, and search for the optimal two-stage design again with the design parameters (α, β −0.02, t_c, Λ₀(t_c), Λ₁(t_c), θ). For these new designs at the nominal power level of 92%, their simulated power values are all above the nominal level. Although the ESS₀ and n for the designs with 0.02 higher in power (with β = 0.08) are larger than these from the original designs (with β = 0.10), we suggest to utilize a study design having the guaranteed power to detect the clinically meaningful difference between the new treatment to historical data in a single-arm trial. For this reason, we increase the power nominal level by 0.02 for the proposed two-stage optimal designs in the following comparisons.

Table 1:

Comparison between the proposed two-stage minimax designs with the existing two-stage minimax design by Kwak and Jung (2014), when α = 0.05, β = 0.1, the accrual rate θ = 30 patients per year, the follow-up time t_c = 1. The cumulative hazard function is assumed to be log(2) under the null hypothesis, and log(2)/δ under the alternative hypothesis, where δ is the hazard ratio (HR).

HR	KJ design (β = 0.10)			New design (β = 0.10)			New design (β = 0.08)
δ	n	ESS0	power	n	ESS0	power	n	ESS0	power	TIE
α = 5%
1.4	98	82.8	0.876	99	91.1	0.885	107	95.6	0.908	0.043
1.5	74	65.6	0.879	74	68.9	0.881	80	72.0	0.907	0.043
1.6	59	54.9	0.889	59	55.7	0.881	64	57.7	0.905	0.042
1.7	51	45.1	0.895	49	47.8	0.878	53	48.7	0.903	0.041
α = 10%
1.4	78	72.3	0.882	81	75.6	0.890	88	81.6	0.913	0.091
1.5	59	55.7	0.881	61	56.4	0.888	66	61.0	0.910	0.090
1.6	48	44.0	0.887	48	46.2	0.886	52	49.8	0.909	0.089
1.7	40	38.5	0.886	40	38.1	0.884	43	42.3	0.908	0.089

We further compare the performance of the proposed two-stage designs with the one-stage design by Wu [10]. The proposed two-stage designs can be considered as an extension of Wu’s one-stage design based on exact variance estimate of W under the alternative. We use the survival function of S(t) = exp ^{−log(2)(t / m)k} in the comparison, where m is the median survival time, and k is the shape parameter of the Weibull distribution. When the shape parameter k is assumed to be the same under the null and the alternative, the HR between the null and the alternative is calculated as: δ = (m₁ / m₀)^k, which is a proportional hazard model.

Table 2 shows the sample size comparison between the proposed two-stage designs and the one-stage design with design parameters: (α, β, t_c) = (0.05, 0.1, 1), δ = 1.2 and 1.8, under the Weibull distribution with the shape parameter k = 0.1, 0.25, 0.5, 1, and 2. It should be noted that the power nominal level is 92% for the two-stage designs, while it is still 90% in the one-stage design. As compared to the one-stage design, the proposed two-stage minimax designs save sample sizes with an average of 14.4% (ranges from 2% to 31%) for the cases studied in Table 2, and the average sample size saving is increased to 21.7% for the proposed two-stage optimal designs. The empirical TIE and power of these two-stage designs are presented in Table 3. The size of all studies is well controlled at the nominal level, and the simulated power is often above 90% except the cases with a small shape parameter, e.g., k = 0.1. For a Weibull distribution with a very small shape parameter (e.g., 0.1), the survival time for the majority of patients is very short. For the cases with the power being less than the nominal level, we would suggest searching for two-stage optimal designs with an increased power level adjustment, such as, 4%. In general, the proposed two-stage designs have a satisfactory performance with regards to TIE rate and power.

Table 2:

Proposed two-stage optimal and minimax designs with the design parameters, the shape parameter in the Weibull distribution k from 0.1 to 2, the accrual rate θ = n_single / 3 with t_c = 3 for the one-stage design by Wu (2015), HR δ = 0.05 and 1.8, α = 0.05 and β = 0.10. The optimal two-stage designs presented in the table are the ones searched at the nominal power of 92%. ESS₀: expected sample size under the null hypothesis; PET: probability of early termination.

	Two-stage minimax design						Two-stage optimal design
k	n1	n	c1	c	ESS0	PET	n1	n	c1	c	ESS0	PET	nsingle
HR: δ =1.2
0.1	282	562	−0.045	−1.645	417.0	0.52	230	599	−0.295	−1.645	371.7	0.62	534
0.25	307	515	0.145	−1.645	423.0	0.44	239	572	−0.255	−1.644	372.0	0.60	491
0.5	306	450	0.450	−1.645	403.0	0.33	240	504	−0.165	−1.641	354.7	0.57	432
1	282	365	0.865	−1.645	348.9	0.19	228	403	−0.025	−1.638	313.8	0.51	356
2	251	308	1.000	−1.645	299.0	0.16	204	332	0.165	−1.639	276.4	0.43	306
HR: δ =1.8
0.1	29	62	−0.135	−1.645	43.7	0.55	24	67	−0.290	−1.645	40.6	0.61	63
0.25	33	56	0.240	−1.645	46.7	0.41	26	62	−0.245	−1.644	40.5	0.60	57
0.5	28	49	0.220	−1.644	40.3	0.41	25	54	−0.080	−1.641	38.6	0.53	50
1	26	38	0.785	−1.645	35.4	0.22	23	41	0.235	−1.640	33.7	0.41	41
2	21	31	1.460	−1.645	30.3	0.07	21	33	0.600	−1.642	29.7	0.27	36

Table 3:

The empirical TIE and power of the two-stage optimal and minimax designs in Table (2) at the significance level of 0.05 and the nominal power level of 92%, based on 10,000 simulations. TIE: actual type I error rate.

	Two-stage minimax design			Two-stage optimal design
k	TIE	Power	n	TIE	Power	n
HR: δ =1.2
0.1	0.046	0.908	562	0.041	0.892	599
0.25	0.047	0.913	515	0.041	0.901	572
0.5	0.047	0.915	450	0.042	0.906	504
1	0.047	0.913	365	0.044	0.912	403
2	0.045	0.913	308	0.044	0.912	332
HR: δ =1.8
0.1	0.040	0.894	62	0.037	0.884	67
0.25	0.042	0.903	56	0.037	0.893	62
0.5	0.039	0.899	49	0.038	0.900	54
1	0.040	0.901	38	0.038	0.902	41
2	0.038	0.901	31	0.038	0.908	33

We also provide sample size comparison between the proposed two-stage designs with the one-stage design based on exact variance [10] in Figure 1 with HR: δ = 1.3, and 1.6. The two-stage optimal designs have the smallest ESS₀, followed by the two-stage minimax designs, and the one-stage design. The maximum possible sample size of the two-stage minimax designs is close to that of the one-stage design, and the two-stage optimal designs often need more patients than the other two designs.

Figure 1:

Comparison the expected sample size ESS₀ and the maximum possible sample size n between the proposed two-stage designs and the one-stage design by Wu (2015) when HR δ = 1.3, and 1.6.

3.1. An example

We consider an example from a clinical trial to compare a new treatment to the standard care of the drug D-penicillamine (DPCA) for patients with primary biliary cirrhosis of the liver [14]. The historical data with the DPCA treatment can be fit well by the Weibull distribution with the shape parameter k = 1.22, and the median survival time m₀ = 9 years. Suppose the survival distribution of the new treatment has the same shape parameter k = 1.22 as the standard care, and the HR is δ = 1.75. The total accrual time is t_a = 5 years, and the clinically meaningful follow-up time is t_c = 3 years. The one-stage design developed by Wu [10] requires the sample size of n_single = 88 to attain 80% power at the significance level of 0.05.

To compute the proposed two-stage designs, we use the same accrual rate θ = n_single / t_a = 88 / 5 as in the one-stage design. The total study time of the one-stage design is 5+3=8 years. We provide the detailed two-stage optimal designs in Table 4 with no power adjustment, 0.02 more, and 0.04 more power with the nominal power level of 80%, 82%, and 84%. When the power nominal level is either 80% or 82%, the simulated power is slightly below the nominal level. The study designs found by using the power level at 84% have the simulated power above 80%, and the type I error rate is well controlled for all the cases. For this example, we would recommend the two-stage designs computed at the power nominal level of 84% for use in practice to make sure that a study has sufficient enough patients to detect the clinically meaningful activity of the new treatment. Let the expected total study length under the null hypothesis be

$$ETS{L}_0=t_1+\left(t_a+t_c-t_1\right)\left(1-PET\right),$$

where PET is the probability of early termination that is the probability that a trial is being stopped at the end of the first stage. The ETSL₀ of the optimal design and the minimax design is 1 year and 1.5 years shorter than that of the one-stage design (ETSL₀ = 5 + 3 = 8). The sample size savings are within the range of the findings from the aforementioned numerical studies.

Table 4:

The two-stage optimal and minimax designs under the nominal power levels of 80%, 82%, and 84% for a trial to investigate the activity of a new treatment for patients with primary biliary cirrhosis of the liver, as compared to the one-stage design with n_single = 88, and the expected total study length ETSL₀ = 5 + 3 = 8 years, where t_a = 5 is the accrual time and t_c = 3 is the follow-up time. ESS₀ : expected sample size under the null hypothesis; ETSL₀ : expected total study length under the null hypothesis; PET : probability of early termination.

Nominal Power	n1	c1	n	c	DA1	ESS0	ETSL0	TIE	Power
Two-stage minimax design
80%	47	0.800	85	−1.632	2.667	76.9	6.7	0.042	0.787
82%	54	0.710	88	−1.633	3.068	79.9	6.8	0.042	0.797
84%	61	0.710	91	−1.636	3.466	83.8	7.0	0.042	0.817
Two-stage optimal design
80%	44	0.310	92	−1.597	2.500	73.8	6.1	0.044	0.795
82%	45	0.345	95	−1.601	2.557	76.7	6.3	0.043	0.796
84%	50	0.350	97	−1.608	2.841	79.9	6.5	0.043	0.820

Given the design parameters, the null and alternative survival rates at t_c = 3 years are estimated as 83.41% and 90.15%, respectively. As suggested by the reviewers, we include Simon’s two-stage minimax and optimal designs when the outcomes are considered as binary. In this case, the binary outcome is defined as the 3-year survival. The expected sample sizes under the null hypothesis are 100.5 for the optimal design and 156.2 for the minimax design, which are larger than those from the proposed two-stage designs with survival endpoints as seen in Table 4. The proposed two-stage designs are preferable when the outcome is survival.

4. Conclusion

Wu [10] derived the exact variance estimate of W under the alternative to calculate sample size for a single-arm one-stage design. The empirical power of these designs is always above the nominal level while that of other designs based on asymptotic variance estimates is often underestimated. For this reason, we extend the one-stage design to two-stage designs based on exact variance [10]. From our numerical studies, it can be seen that the empirical power of the proposed two-stage designs is not guaranteed. Although the variances of W₁ and W under the alternative can be estimated by using the exact method, the correlation between W₁ and W is still based on the limiting distribution of the joint function of W₁ and W. The exact correlation coefficient may be estimated by using non-parametric approaches, which could be then used to improve the performance of the proposed two-stage designs [15, 16].

The computed type I error rate could be conservative for the proposed two-stage minimax and optimal designs with survival outcome. In this article, we search for both designs at the same time until power of both designs is above the nominal level [17, 18]. When one type of designs is chosen in the study design, the type I error rate could be closer to the nominal level. Alternative, one could add an additional constraint for the computed type I error (e.g., at least 4.5% when α=5%). We provide statistical software program in R for the design search. Researchers can modify the code to add a constraint on type I error rate.

For the existing two-stage KJ designs, the power of these optimal designs is calculated at the average of Λ₀ and Λ₁. This approach increases the required sample size, however, the empirical power of these two-stage designs is still underestimated. We would suggest computing the power of the study at the cumulative hazard function of Λ₁. In practice, we recommend adjusting the nominal power level to guarantee the power of the two-stage designs.

Appendix

Suppose L₁ and L are the one-sample log-rank test at first stage and the two stages combined. The two-dimensional variable, (L₁, L), follows a bivariate normal distribution asymptotically. Then, the conditional distribution of L₁ given L = z follows a normal distribution. Let ϕ and Φ be the probability density function and the cumulative distribution function of the standard normal distribution. The TIE of a two-stage design is computed as:

$$TIE={\int }_{-\infty }^c\varphi \left(t\right)\Phi \left(\frac{c_1-{\rho }_{0}t}{\sqrt{1-{\rho }_0^2}}\right)dt,$$ (5)

where ρ₀ is the correlation coefficient between W₁ and W.

We assume that the censoring distribution at the end of study, G(t), is a uniform distribution U(t_c, t_a + t_c), and the censoring distribution at the end of the first stage, G₁(t), is a uniform distribution, U(0, t₁) [9]. Under the null hypothesis, the variances of W₁ and W are estimated as

$${\sigma }_{01}^2=Var\left(W_1\right)=-{\int }_0^{t_1}{G}_1\left(t\right)d{S}_0\left(t\right)\text{and}{\sigma }_{02}^2=Var\left(W\right)=-{\int }_0^{t_a+t_c}G\left(t\right)d{S}_0\left(t\right).$$

It follows that the correlation between W₁ and W under H₀ is ρ₀ = σ₀₁ / σ₀₂.

Under the alternative hypothesis, the mean values of W₁ and W are

$$E\left(W_1\right)=\sqrt{n_1}{\omega }_1\text{and}E\left(W\right)=\sqrt{n}\omega$$

where ω = p₁ − p₀, $p_1={\int }_0^{t_a+t_c}G(t)S_1(t)d{\Lambda }_1(t)$, $p_0={\int }_0^{t_a+t_c}G(t)S_1(t)d{\Lambda }_0(t)$, and ω₁ = p_1f − p_0f, $p_{1f}={\int }_0^{t_1}{G}_1(t)S_1(t)d{\Lambda }_1(t)$, $p_{0f}={\int }_0^{t_1}{G}_1(t)S_1(t)d{\Lambda }_0(t)$. Recently, Wu [10] derived the exact variance of W under H_a as

$${\sigma }_{12}^2=Var\left(W\right)=p_1-p_1^2-p_0^2+2P_0{p}_1-2P_{00}+2p_{01},$$

where $p_{00}={\int }_0^{t_a+t_c}G(t)S_1(t){\Lambda }_0(t)d{\Lambda }_0(t)$ and $p_{01}={\int }_0^{t_a+t_c}G(t)S_1(t){\Lambda }_0(t)d{\Lambda }_1(t)$. The exact variance of W₁, ${\sigma }_{11}^2=\mathit{Var}(W_1)$, can be derived by a similar approach. Then, the correlation between W₁ and W under H_a is ρ₀ = σ₁₁ / σ₁₂, and the power of a two-stage design is computed as

$$Power={\int }_{-\infty }^{\tilde{c}}\varphi \left(t\right)\Phi \left(\frac{{\tilde{c}}_1-{\rho }_{a}t}{\sqrt{1-{\rho }_a^2}}\right)dt,$$ (6)

where ${\tilde{c}}_1=\frac{{\sigma }_{01}}{{\sigma }_{11}}\left(c_1-\frac{{\omega }_1\sqrt{n_1}}{{\sigma }_{01}}\right),$ and $\tilde{c}=\frac{{\sigma }_{02}}{{\sigma }_{12}}\left(c-\frac{{\omega }_2\sqrt{n_2}}{{\sigma }_{02}}\right).$