Response adaptive randomization design for a two-stage study with binary response

Abstract

Response adaptive randomization has the potential to treat more participants in better treatments in a trial to benefit participants. We propose optimal response adaptive randomization designs for a two-stage study with binary response, having the smallest expected sample size or the fewest expected number of failures. Equal randomization is used in the first stage, and data from the first stage is used to determine the adaptive sample size ratio in the second stage. In the proposed optimal designs, the type I error rate and the statistical power are calculated from the asymptotic normal distributions. The new designs that minimize the expected number of failures have the advantage over the existing optimal randomized designs to substantially reduce the number of failures.

1. Introduction

For an early phase trial with binary outcome (e.g., response VS non-response), Simon’s optimal two-stage designs are frequently used among the single-arm designs to evaluate the activity of a new treatment as compared to the historical response rate [1, 2]. When a single-arm clinical trial is not appropriate due to the unreliable response rate from historical data, a randomized clinical trial is recommended for use to compare new treatments with the standard care. As compared to a single-arm study, a randomized clinical trial generally requires more participants to test the same effect size, but it is able to reduce potential biases from a single-arm study [3, 4, 5].

Thall et al. [6] were among the first to develop an optimal two-arm two-stage design for binary response having the smallest expected sample size. The Z test statistic with the pooled variance estimate was used to determine the threshold values for the first stage and the two stages combined.

A two-stage design provides the flexibility allowing early stop due to futility to protect participants being treated by an inferior treatment. A two-stage design is often allowed to be stopped in the first stage due to futility to protect participants being treated by inferior treatments. If a trial proceeds to the second stage with sufficient treatment activity from the first stage, a final decision will be made at the end of the second stage to move the trial to the next phase or not. Thall et al. [6] found that the maximum possible sample size of a two-stage design could be slightly more than that of the conventional randomized design. However, the excepted sample size savings are substantial.

To improve the effectiveness of clinical studies, response adaptive randomization (RAR) approach could be utilized to assign more participants to better treatment(s) in the second stage based on the observed results from the first stage [7]. In a two-stage design, the first stage data are going to be used for two purposes: (1) data to decide the futility stopping; and (2) a burn-in period to estimate the sample size ratio for the second stage. Because of the potential sample size ratio change in the second stage, the RAR designs are more likely to have unbalanced sample sizes across the treatment groups as compared to trials with equal randomization. For that reason, the statistical power of a RAR design may be lower than that of a study with equal randomization given the total sample size. However, a RAR design can reduce the number of failures in a trial to benefit participants. In addition, the use of RAR in trials has the potential to increase recruitment as participants are informed that they have a higher chance to be treated by one of the best treatments [8].

In this article, we propose to develop new optimal randomized two-stage RAR design by using the two optimal sample size allocation formula for binary response [7]. Rosenberger et al. [9] derived the optimal allocation formula for a study to minimize the total number of failures based on a theorem from Melfi et al. [10]. The second optimal allocation rule is from the play-the-winner design given the total number of participants [11, 12]. This optimal allocation is derived from the asymptotic theory of Markov chains. In addition to these two optimal allocation rules, the Neyman allocation [7] may be used to minimize the total sample size when the sample size variances can be considered as fixed. Under this allocation rule, more participants are going to be assigned to an inferior treatment group when the sum of response rates from all groups is more than 100% [9]. For that unethical reason, we are not going to include the Neyman allocation rule in this article for comparison.

The rest of the article is organized as follows. In Section 2, we introduce the optimal RAR designs for a two-stage study with binary response. We then compare the performance of the proposed RAR designs with the existing optimal two-stage design with regards to the expected sample size and the number of failure. We use a clinical trial that was designed by using the existing optimal two-stage design to illustrate the application of the proposed optimal RAR designs. Lastly, we provide some comments and suggestions in Section 4.

2. Methods

In a randomized parallel trial with binary response, suppose p_E and p_C are the response rates for the treatment group and the control group, respectively. The proportion difference between the two groups Δ = p_E − p_C is the parameter of interest. We assume that a treatment with a high response rate is preferable. Then, the statistical hypothesis is presented as

$$H_0:\Delta \le 0\text{against}{H}_a:\Delta >0.$$ (1)

In this article, we are going to develop optimal two-stage designs with the possibility of futility stopping after the first stage due to ineffectiveness observed from the new treatment. Suppose x_Ei and x_Ci are the observed responders from the treatment group and the control group in the i-th stage, i = 1, and 2. In the first stage, participants are equally randomized to one of the two treatments with the sample size of n₁ in each group. Then, the total sample size in the first stage is N₁ = 2n₁. The first stage response rate is estimated as ${\widehat{p}}_{E1}=x_{E1}/n_1$ in the treatment group, and ${\widehat{p}}_{C1}=x_{C1}/n_1$ in the control group. Thus, the proportion difference in the first stage is ${\widehat{\Delta }}_1={\widehat{p}}_{E1}-{\widehat{p}}_{C1}=x_{E1}/n_1-x_{C1}/n_1$. Following Thall et al. [6], we use a two-proportion Z test with the pooled variance estimate to compare the response rate between the treatment group and the control group in the first stage:

$$Z_1=\frac{{\widehat{\Delta }}_1}{\sqrt{2{\widehat{p}}_1(1-{\widehat{p}}_1)/n_1}}\mathrm{,}$$ (2)

where ${\widehat{p}}_1=(x_{E1}+x_{C1})/N_1$ is the overall response rate of the two groups from the first stage. If Z₁ fells in the rejection region as compared to the pre-specified threshold value (r₁), this study is stopped for futility in the first stage. Otherwise, this trial proceeds to the second stage with additional N₂ participants. The sample size allocation in the second stage is determined by using the observed responses from the first stage.

In a two-stage design with equal randomization, the proportion of the sample size for the treatment group remains a constant across the two stages. In this article, we propose to develop optimal RAR designs that allow the sample size ratio in the second stage to be adjusted according to the outcome from the first stage. Suppose ρ is the probability of assigning participants to the treatment group in the second stage. The following two RAR procedures are considered:

$${\text{RAR}}_1:\rho =\frac{\sqrt{p_{E1}}}{\sqrt{p_{E1}}+\sqrt{p_{C1}}}$$

and

$${\text{RAR}}_2:\rho =\frac{q_{C1}}{q_{E1}+q_{C1}}\mathrm{,}$$

where q_G1 = 1 – p_G1 is the non-response rate in the first stage for the group G=E or C. The RAR₁ procedure is the optimal allocation formula to minimize the total number of failures [9, 10], while the RAR₂ procedure is the optimal allocation derived from the play-the-winner design based on the asymptotic theory of Markov chains [11]. It can be seen that ρ is an increasing function of the estimated first stage response rate of the treatment group p_E1 when p_C1 is fixed in each RAR procedure. When a new investigated treatment has a higher response rate than the control, more patients will be assigned to the treatment group in the second stage to benefit more participants in a trial.

The sample size ratio ρ depends on the outcome from the first stage. We divide the Z₁ values into equal sub-ranges. The lower bound of Z₁ is the first stage threshold value r₁, and its upper bound value (say, r_1u) should be large enough such that the probability above r_1u in the standard normal distribution is very small (e.g., r_1u = 6 in the simulation studies). Suppose we equally divide the range (r₁, r_1u) into K complementary sub-ranges:

$$R_k=(r_1+(k-1)\tau \mathrm{,}{r}_1+k\tau )\mathrm{,}$$

where τ = (r_1u – r₁)/K, r_1u r₁ + Kτ, and k = 1, 2, 3, …, K. The value of K is large (e.g., 1000). A large value of K is needed to improve the accuracy of type I error rate and statistical power calculation.

For each sub-range R_k, the corresponding sample size ratio ρ_k is computed as a simple average of the estimated $\widehat{\rho }$ based on all possible outcomes whose test statistics fall in R_k. These possible outcomes are identified through complete enumeration given the sample sizes of n₁ in each group with the sample space size of (n₁ + 1) × (n₁ + 1). In the case with no data points in R_k, we will use ρ_k = ρ_k−1. Once the estimate of ρ_k is calculated, the second stage sample sizes for the treatment group and the control group are n_E2 = ρ_kN₂ and n_C2 (1 − ρ_k)N₂, respectively. Then, the total sample sizes for the treatment group and the control group from both stages are: N_E = n₁ + n_E2 and N_C = n₁ + n_C2.

Let ${\widehat{p}}_{G2}=x_{G2}/n_{G2}$ be the estimated response rate of the G group in the second stage, G = E, C. Their proportion difference in the second stage may be estimated as ${\widehat{\Delta }}_2={\widehat{p}}_{E2}-{\widehat{p}}_{C2}$. Let ${\widehat{p}}_2=({\widehat{p}}_{E2}{n}_{E2}+{\widehat{p}}_{C2}{n}_{C2})/N_2$ be the overall response rate in the second stage. The test statistic for the second stage data is:

$$Z_2=\frac{{\widehat{\Delta }}_2}{\sqrt{{\widehat{p}}_2(1-{\widehat{p}}_2)(\frac{1}{n_{E2}}+\frac{1}{n_{C2}})}}\mathrm{,}$$

When a trial proceeds to the second stage with additional N₂ participants, the final statistic, Z_f, is calculated as a linear combination of Z₁ and Z₂:

$$Z_f=\sqrt{w}{Z}_1+\sqrt{1-w}{Z}_2\mathrm{,}$$ (3)

where w = N₁/(N₁ + N₂) is the ratio of the sample sizes in the first stage. The calculated Z_f is then compared to the pre-specified threshold value r to determine whether the new treatment is effective or not.

2.1. Type I error rate

A two-stage design needs to determine sample sizes for each stage (N₁ and N₂), and threshold values for Z₁ and Z_f (r₁ and r). The second stage sample size ratio ρ will be determined from the first stage outcome. With a one-sided hypothesis in Equation (1), the tail area of the hypothesis testing is

$$\Omega =\left\{(Z_1\mathrm{,}{Z}_f)|Z_1>r_1,\text{and}{Z}_f>r\right\}.$$

Under the null hypothesis, both Z₁ and Z_f follow the standard normal distribution asymptotically. Then, the type I error (TIE) rate is calculated as

$$\begin{array}{ll}\text{TIE}\hfill & =P((Z_1\mathrm{,}{Z}_f)\in \Omega |H_0)\hfill \\ \hfill & =P(Z_1>r_1\mathrm{,}\sqrt{w}{Z}_1+\sqrt{1-w}{Z}_2>r|H_0)\hfill \\ \hfill & ={\int }_{r_1}^{+\infty }\varphi (x)P(Z_2>(r-\sqrt{w}x)/\sqrt{1-w}|H_0)dx\mathrm{,}\hfill \\ \hfill & ={\int }_{r_1}^{+\infty }\varphi (x)[1-\Phi \left(\frac{r-x\sqrt{w}}{\sqrt{1-w}}\right)]dx\mathrm{,}\hfill \end{array}$$ (4)

where ϕ and Φ are the probability density function and the cumulative distribution function of the standard normal distribution.

2.2. Statistical power

In the statistical power calculation, we follow Thall et al. [13] to derive the equivalent threshold values from the normal distribution under the alternative hypothesis. Under the alternative hypothesis, Z₁ follows a normal distribution

$$N({\mu }_{1a}\mathrm{,}{\sigma }_{1a}^2)=N(\frac{p_E-p_C}{{\sigma }_{\mathrm{1,}\mathit{\text{pooled}}}}\mathrm{,}\frac{{\sigma }_{\mathrm{1,}\mathit{\text{unpooled}}}^2}{{\sigma }_{\mathrm{1,}\mathit{\text{pooled}}}^2})\mathrm{,}$$

where ${\sigma }_{\mathrm{1,}\mathit{\text{pooled}}}^2=\frac{p_E+p_C}{2}\left(1-\frac{p_E+p_C}{2}\right)\left(\frac{1}{n_{C1}}+\frac{1}{n_{E1}}\right)$, ${\sigma }_{\mathrm{1,}\mathit{\text{unpooled}}}^2=\frac{p_C(1-p_C)}{n_{C1}}+\frac{p_E(1-p_E)}{n_{E1}}$, n_E1 = n_C1 = n₁ = N₁/2. Similarly, Z₂ follows a normal distribution under the alternative hypothesis as

$$N({\mu }_{2a}\mathrm{,}{\sigma }_{2a}^2)=N\left(\frac{p_E-p_C}{{\sigma }_{\mathrm{2,}\mathit{\text{pooled}}}}\mathrm{,}\frac{{\sigma }_{\mathrm{2,}\mathit{\text{unpooled}}}^2}{{\sigma }_{\mathrm{2,}\mathit{\text{pooled}}}^2}\right)\mathrm{,}$$

where σ_{2, pooled} and σ_{2, unpooled} are calculated by using n_E2 and n_C2 to replace n_E1 and n_C1 in σ_{1, pooled} and σ_{2, unpooled}.

It should be noted that Z₁ and Z₂ may be correlated as the sample size ratio in Z₂ is determined by the first stage outcome (Z₁). A linear combination of Z₁ and Z₂ follows a normal distribution, and the distribution for Z_f under the alternative hypothesis may be approximated by

$$N({\mu }_{fa}\mathrm{,}{\sigma }_{fa}^2)\approx N(\sqrt{w}{\mu }_{1a}+\sqrt{1-w}{\mu }_{2a}\mathrm{,}w{\sigma }_{1a}^2+(1-w){\sigma }_{2a}^2)\mathrm{.}$$

The distributions of Z₁ and Z_f under the alternative hypothesis are used in calculating the statistical power. When Z₁ falls in one of the sub-ranges R_k, the associated threshold value under the alternative hypothesis is r_1a(k):

$$P(Z_1>r_1+(k-1)\tau |H_a)=P(X>r_{1a}(k))\mathrm{,}$$

where $r_{1a}\left(k\right)=\left[r_1+\left(k-1\right)-{\mu }_{1a}\right]/{\sigma }_{1a}$, k = 1, …, K. For Z_f with r as the threshold value in P(Z_f > r|H_a), the associated threshold value is r_a = (r – μ_fa)/σ_fa. Then, the statistical power of a two-stage design may be approximated by

$$\begin{array}{ll}\text{Power}\hfill & =P\left(\left(Z_1\mathrm{,}{Z}_f\right)\in \Omega |H_a\right)\hfill \\ \hfill & =\sum _{k=1}^{K}P\left(Z_1\in R_k\mathrm{,}{Z}_f>r|H_a\right)\hfill \\ \hfill & \approx \sum _{k=1}^K{\int }_{r_{1a}\left(k\right)}^{r_{1a}\left(k+1\right)}\varphi \left(x\right)\left[1-\Phi \left(\frac{r_a-x\sqrt{w}}{\sqrt{1-w}}\right)\right]dx\mathrm{.}\hfill \end{array}$$ (5)

2.3. Optimal design search

Two objective functions are considered here for the optimal designs. The expected sample size (ESS) under the null hypothesis is frequently used to as the criteria for an optimal design:

$$\text{ESS}=N_1+N_2\times P(Z_1>r_1)\mathrm{.}$$

The second objective function is the expected number of non-responders (ENR), which is also known the expected number of failures,

$$\text{ENR}=(q_E+q_C)N_1/2+\sum _{k=1}^{K}P(Z_1\in R_k|H_a)\left[q_E{N}_2{\rho }_k+q_C{N}_2(1-{\rho }_k)\right]\mathrm{.}$$

The two constraints in the design search are the nominal type I error rate and the statistical power:

$$\text{TIE}\le \alpha \mathrm{,}\text{Power}\ge (1-\beta )\mathrm{,}$$

where α and β are the pre-specified type I and II error rates, respectively. We conduct the design search with the first stage and the second stage sample sizes from the design by Thall et al. [6] as the initial value (N₁₀ and N₂₀). For each sample sizes N₁ and N₂ (e.g., N₁ ∈ [N₁₀−15, N₁₀+15], N₂ ∈ [N₂₀−15, N₂₀+15]), we search over all possible designs with r₁ from 0.23 to 0.75 by 0.005, and r from 1.50 to 1.75 by 0.005. Among the designs meet both the TIE and the statistical power, the one with the smallest ESS is the optimal ESS design, and the one with the smallest ENR is the optimal ENR design. We will search for the optimal ESS and ENR designs for each RAR procedure with a total of 4 optimal designs (referred to be as the RAR₁-ESS design, the RAR₂-ESS design, the RAR₁-ENR design, and the RAR₂-ENR design).

3. Results

We compare the performance of the proposed RAR two-stage designs with the existing optimal two-arm two-stage design by Thall et al. [6] (referred to be as the Thall design).

3.1. Simulation study

Suppose a study is designed to achieve 80% or 90% power at the significance level of α = 0.05. The proportion difference between the treatment group and the control group is Δ = p_E − p_C = 15% or 20%. The response rate of the control group p_C is from 20% to 70%. These configurations were studied by Thall et al. [6].

It should be noted that the Thall design was developed under the criteria with the smallest average of ESS under H₀ and ESS under H_a. For that reason, the computed ESS in this article may not be exactly the same as their results, although the difference is often small. From the optimal Thall designs, the identified threshold value r₁ for the first stage is near 0.3 and r for both stages is close to 1.6. For that reason, we search the r₁ value from 0.23 to 0.75 with an increment of 0.005, and r from 1.50 to 1.75 by an increment of 0.005. These ranges and the parameter precision value can be easily adjusted in the software to provide finer designs.

We presented the ESS and ENR comparison between the five designs for a study to achieve 80% statistical power in Table 1 when Δ = 15% and Table 2 when Δ = 20%. The proposed optimal RAR designs with the smallest ESS often have similar ESS as compared to the Thall design, although the new designs have slight more sample sizes. Meanwhile, the proposed RAR₁-ENR and RAR₂-ENR designs can reduce the expected number of failures as compared to the Thall design with the average savings of 9.97% (with the range from 8.85% to 11.68%).

Table 1:

Optimal two-stage response adaptive randomization designs under the criteria of the expected sample size or the expected number of non-responder, as compared to the Thall design when Δ = p_E − p_C = 15%, α = 5%, and 80% power.

Criteria	RAR	N 1	N 2	ESS	ENR	r 1	r
pC = 20%, pE = 35%
The Thall design		86	174	141.2	171.3	0.475	1.520
ESS	RAR1	90	170	141.9	170.2	0.510	1.520
ESS	RAR2	96	170	142.9	172.6	0.595	1.505
ENR	RAR1	132	88	164.5	155.3	0.335	1.630
ENR	RAR2	140	80	170.1	155.6	0.315	1.635
pC = 40%, pE = 55%
The Thall design		114	210	177.3	155.0	0.520	1.520
ESS	RAR1	112	210	178.3	153.0	0.480	1.530
ESS	RAR2	118	206	179.8	153.0	0.525	1.525
ENR	RAR1	160	116	205.2	140.9	0.280	1.630
ENR	RAR2	170	108	206.4	141.0	0.420	1.625
pC = 60%, pE = 75%
The Thall design		98	190	155.6	85.0	0.515	1.515
ESS	RAR1	102	184	156.8	83.3	0.530	1.520
ESS	RAR2	106	184	158.6	83.4	0.565	1.515
ENR	RAR1	136	108	177.9	76.4	0.285	1.625
ENR	RAR2	140	106	179.7	76.4	0.320	1.625

Table 2:

Optimal two-stage response adaptive randomization designs under the criteria of the expected sample size or the expected number of non-responder, as compared to the Thall design when Δ = p_E − p_C = 20%, α = 5%, and 80% power.

Criteria	RAR	N 1	N 2	ESS	ENR	r 1	r
pC = 30%, pE = 50%
The Thall design		60	114	95.0	95.1	0.505	1.520
ESS	RAR1	58	122	95.9	96.0	0.495	1.510
ESS	RAR2	62	120	96.9	96.1	0.550	1.505
ENR	RAR1	102	46	115.5	86.4	0.545	1.630
ENR	RAR2	96	54	112.6	86.6	0.505	1.625
pC = 50%, pE = 70%
The Thall design		60	114	95.0	63.4	0.505	1.520
ESS	RAR1	62	114	96.2	62.6	0.525	1.520
ESS	RAR2	64	120	97.5	63.8	0.585	1.500
ENR	RAR1	96	52	115.0	57.5	0.345	1.635
ENR	RAR2	96	54	114.2	57.5	0.420	1.630
pC = 70%, pE = 90%
The Thall design		38	78	62.6	21.1	0.480	1.520
ESS	RAR1	36	84	64.5	20.3	0.415	1.520
ESS	RAR2	44	78	66.0	20.0	0.575	1.510
ENR	RAR1	48	54	69.5	18.7	0.260	1.610
ENR	RAR2	54	50	71.7	18.6	0.375	1.610

The Thall design was developed under the criteria with the smallest ESS. When that criteria is used, the identified optimal design often has a relatively small first stage sample size. For the studied configurations, the proportion of the first stage sample size in the Thall design ranges from 32.8% to 35.2% with the average proportion of 33.9%. Similar first stage sample size proportion is observed for the proposed RAR designs under the ESS criteria. However, for the proposed optimal RAR designs under the ENR criteria, that trend was reversed with the average proportion of the first stage sample size being 59.5% (between 47.1% and 68.9%). The proposed RAR designs under the ENR criteria allocate slightly more participants in the first stage as compared to the second stage. With more participants in the first stage, more information is collected at this stage to make reliable statistical inference.

In Figure 1, we present the ESS and ENR comparison between the Thall design and the proposed RAR designs when the nominal statistical power is 90%. In general, the estimated ENR is a decreasing function of p_C. The relationship between the estimated ESS and p_C is similar to a bell curve from a normal density function curve. The maximum of ESS occurs when p_C and p_E are close to 50%. The designs under the ESS criteria are often similar to each other with regards to the ESS and the ENR. As compared to the Thall design, the savings in the expected number of failures from the proposed RAR designs under the ENR criteria goes up as p_C is getting smaller, and the savings are substantial when p_C is more than 50%.

Figure 1:

The expected sample size and the expected number of non-responder using the proposed four RAR designs and the Thall design, as a function of the response rate of the control group p_C, when Δ = 15% (top) and Δ = 20% (bottom) for a study to achieve 90% power at the significance level of 0.05.

We also compare the proposed two RAR designs under the ENR criteria. For the configurations in Tables 1 and 2 to achieve 80% power, the RAR₂-ENR design has negligible advantage over the RAR₁-ENR design with regards to the expected number of failures. For the expected sample size, the RAR₁ designs have smaller sample sizes than the RAR₂ designs when p_C is small (e.g., p_C ≤ 30%). But, that trend is reversed when p_C is large and Δ = 20%. As the statistical power is increased to 90% (see, Figure 1), their numbers of failures are very close to each other. The ESS of the RAR₁-ENR design could be slightly less than that of the RAR₂-ENR design when p_C is small. When p_C is large and Δ = 15% (top left figure), the RAR₂-ENR design could save a few sample sizes.

In Table 3, we present the simulated type I error rate and statistical power for the optimal designs with the smallest ESS from Table 1. It can be seen that the type I error rate and the statistical power are very close to the nominal level. In a few cases, the actual statistical power could be slightly below the nominal level. Alternatively, one may use the simulated statistical power instead of using the approximation power in Equation (5). Similar findings are observed for other designs in Table 1, and Table 2. It should be noted that when the estimated ρ based on data from the first stage is near to the boundary (e.g., 0 or 1), all participants in the second stage may be assigned to one group. That would end up with no participant in the other group. For such rare cases, we will assign one participant to the other group in order to calculate the test statistics Z₂ and Z_f.

Table 3:

Simulated type I error rate and statistical power for the optimal designs under the ESS criteria in Table 1, at the nominal level of α = 5%, and 80% statistical power.

p u	p a	Criteria	N 1	N 2	r 1	r	α	Power
0.2	0.35	RAR1	90	170	0.510	1.520	0.0515	0.7926
0.2	0.35	RAR2	96	170	0.595	1.505	0.0487	0.8068
0.4	0.55	RAR1	112	210	0.480	1.530	0.0519	0.8002
0.4	0.55	RAR2	118	206	0.525	1.525	0.0512	0.8137
0.6	0.75	RAR1	102	184	0.530	1.520	0.0502	0.7993
0.6	0.75	RAR2	106	184	0.565	1.515	0.0512	0.8113

3.2. Real trial example

The Thall design was used to design a randomized clinical trial in chemotherapy-naive patients with advanced non-small-cell lung cancer (NSCLC) to compare the activity of a cisplatin-etoposide regimen (CDDP-VP16) and a cisplatin/carboplatin-etoposide-vinorelbine combination, where the former is the standard care [14]. The therapeutic activity rate of the standard CDDP-VP16 regimen was estimated to be p_C = 25%. A 15% higher activity rate was expected from the new experimental treatment, with p_E = 40%.

That NSCLC trial was designed to achieve 80% power at the significance level of 0.05. The Thall two-stage design needs N₁ = 104 in the first stage and N₂ = 150 in the second stage. If the target function was the ENR instead of the ESS, the proposed optimal RAR designs would have more participants in the first stage, with (N₁, N₂) = (144,98) for the proposed RAR₁-ENR design, and (N₁, N₂) = (140,104) for the proposed RAR₂-ENR design. The proposed RAR₁-ENR design has slight smaller ENR=159.29 as compared to the RAR₁-ENR design (ENR=159.62). The savings in the ENR from the proposed RAR₂-ENR and RAR₂-ENR designs are close to 10% as compared to the Thall design (ENR=176.5).

4. Discussion

In optimal randomized two-stage designs to minimize the sample size, the identified design often has a smaller sample size in the first stage than the second stage [13, 15]. With fewer participants in the first stage, the expected sample size could be reduced. However, the variance of the test statistic for the first stage could be larger as compared to that for a study with more participants assigned to the first stage. We developed a Shiny app: https://AdaptiveDesigns.shinyapps.io/RAR2Stage/.

The asymptotic normal distribution of the Z test statistic is used to compute the TIE and the statistical power [13]. In a RAR design, the quantity $\frac{r_a-x\sqrt{w}}{\sqrt{1-w}}$ in the statistical power formula in Equation (5) should be

$$\frac{r_a-x\sqrt{w}\times \frac{{\sigma }_{1a}}{{\sigma }_{fa}}}{\sqrt{1-w}\times \frac{{\sigma }_{2a}}{{\sigma }_{fa}}}\mathrm{,}$$

because n_E2 and n_C2 are often not the same. In the case with equal randomization, both $\frac{{\sigma }_{1a}}{{\sigma }_{fa}}$ and $\frac{{\sigma }_{2a}}{{\sigma }_{fa}}$ are equal to 1. We conducted a simulation study with p_C from 0 to 80%, and Δ = p_E − p_C from 5% to 30%. The ratio of σ_1a and σ_2a is very close to 1 with the mean value of 1.027, and the ratio of σ_1a and σ_fa are even closer to 1. For that reason, we use $\frac{r_a-x\sqrt{w}}{\sqrt{1-w}}$ in the power caluclation to approximate the statstical power. The simulation studies show the statistical power is very close to the nominal level in Table 3.

When the limiting distribution is utilized, the actual error rate functions could be under- or over-estimated. Thall et al. [13] calculated the exact threshold values by using binomial distributions, and found that the results based on normal approximation are close to those based on exact calculation. In the Thall design, participants from the second stage are equally randomized. In the proposed RAR designs, the sample size ratio between the two treatment groups changes as the outcome from the first stage, with a possible of K different ratios depending on the Z₁ value [16, 17, 18]. For each possible value of Z₁, one has to generate the complete sample space. For that reason, the size of the all possible data is huge, and it is a challenge to conduct exact calculation for the proposed RAR designs without efficient integer algorithms [19, 20, 15, 21].

In this article, we use the threshold value r regardless the outcome from the first stage, which reduces the computational intensity. We consider developing a RAR design that allows r to be changed based on the Z₁ value in the future [22, 23]. In addition, a monotonic relationship between the second stage sample size N₂ and Z₁ may be incorporated in the design search. It is an intuitive feature that fewer participants are needed to confirm the activity of a new treatment when a higher effect size is observed from the first stage. In a single-arm two-stage design, we found that intuitive feature would improve the effectiveness of the study design while the improvement is limited in some configurations.

In addition to the propsoed RAR designs for a randomized two-stage design with binary outcome, covariate adaptive randomization (CAR) designs may be considered to balance the importance covariates in a study. For example, in the trials for Alzheimer’s Disease (AD), the apolipoprotein E (ApoE) e4 genotype is found be to be associated with early onset of AD [24, 25, 26, 27]. When the ApoE e4 genotype is included as a covariate in the adaptive randomization, the success rate of AD trials could be increased by reducing the heterogeneity in AD patients [28]. We consider developing CAR designs for a trial with binary outcome as future work.