Sample size determination for a binary response in a superiority clinical trial using a hybrid classical and Bayesian procedure
 Maria M. Ciarleglio^{1, 2}Email authorView ORCID ID profile and
 Christopher D. Arendt^{3}
DOI: 10.1186/s1306301717910
© The Author(s) 2017
Received: 28 January 2016
Accepted: 9 January 2017
Published: 23 February 2017
Abstract
Background
When designing studies that have a binary outcome as the primary endpoint, the hypothesized proportion of patients in each population experiencing the endpoint of interest (i.e., π _{1},π _{2}) plays an important role in sample size and power calculations. Point estimates for π _{1} and π _{2} are often calculated using historical data. However, the uncertainty in these estimates is rarely addressed.
Methods
This paper presents a hybrid classical and Bayesian procedure that formally integrates prior information on the distributions of π _{1} and π _{2} into the study’s power calculation. Conditional expected power (CEP), which averages the traditional power curve using the prior distributions of π _{1} and π _{2} as the averaging weight conditional on the presence of a positive treatment effect (i.e., π _{2}>π _{1}), is used, and the sample size is found that equates the prespecified frequentist power (1−β) and the conditional expected power of the trial.
Results
Notional scenarios are evaluated to compare the probability of achieving a target value of power with a trial design based on traditional power and a design based on CEP. We show that if there is uncertainty in the study parameters and a distribution of plausible values for π _{1} and π _{2}, the performance of the CEP design is more consistent and robust than traditional designs based on point estimates for the study parameters. Traditional sample size calculations based on point estimates for the hypothesized study parameters tend to underestimate the required sample size needed to account for the uncertainty in the parameters. The greatest marginal benefit of the proposed method is achieved when the uncertainty in the parameters is not large.
Conclusions
Through this procedure, we are able to formally integrate prior information on the uncertainty and variability of the study parameters into the design of the study while maintaining a frequentist framework for the final analysis. Solving for the sample size that is necessary to achieve a high level of CEP given the available prior information helps protect against misspecification of hypothesized treatment effect and provides a substantiated estimate that forms the basis for discussion about the study’s feasibility during the design phase.
Keywords
Sample size Clinical trial Proportions Binary endpoint Conditional expected power Hybrid classicalBayesianBackground
When designing a study that has a binary outcome as the primary endpoint, the hypothesized proportion of patients in each population experiencing the endpoint of interest (i.e., π _{1},π _{2}) plays an important role in sample size determination. In a twoarm study comparing two independent proportions, π _{2}−π _{1} represents the true hypothesized difference between groups, sometimes also known as the minimal relevant difference [1]. While the treatment effect may also be parameterized equivalently using an odds ratio or relative risk, when appropriate, the most frequently used sample size formula expresses the treatment effect using the difference between groups [2, 3]. In the case of proportions, the variance of the difference depends on the individual hypothesized values for the population parameters π _{1} and π _{2} under the alternative hypothesis. Thus, the sample size required to detect a particular difference of interest is affected by both the magnitude of the difference and the individual hypothesized values.
Traditional sample size formulas incorporate beliefs about π _{1} and π _{2} through single point estimates [1]. However, there is often uncertainty in these hypothesized proportions and, thus, a distribution of plausible values that should be considered when determining sample size. Misspecification of these hypothesized proportions in the sample size calculation may lead to an underpowered study, or one that has a low probability of detecting a smaller and potentially clinically relevant difference when such a difference exists [4]. Alternatively, if there is strong evidence in favor of a large difference, a study may be overpowered to detect a small hypothesized difference. Thus, a method for determining sample size that formally uses prior information on the distribution of study design parameters can mitigate the risk that the power calculation will be overly optimistic or overly conservative.
Similar difficulty surrounding the choice of study parameters for a continuous endpoint with known variance [5] and for a continuous endpoint with unknown variance [6] has been discussed previously. We have presented methods that formally incorporate the distribution of prior information on both the treatment effect and the variability of the endpoint into sample size determination. In this paper, we extend these methods to a binary endpoint by using a “hybrid classical and Bayesian” [7] technique based on conditional expected power (CEP) [8] to account for the uncertainty in study parameters π _{1} and π _{2} when determining the sample size of a superiority clinical trial. Unlike traditional power, which is calculated assuming the truth of a point alternative hypothesis (π _{2}−π _{1}=Δ _{ A }) for given values of π _{1} and π _{2}, CEP conditions on the truth of a composite alternative of superiority (e.g., π _{2}−π _{1}>0 or π _{2}>π _{1}). CEP formally incorporates available prior information on both π _{1} and π _{2} into the power calculations by averaging the traditional power curve using the product of the prior distribution of π _{1} and the conditional prior distribution of π _{2},p(π _{2}  π _{2}>π _{1}), as the averaging weight. Based on the available prior information, the sample size that yields the desired level of CEP can be used when estimating the required sample size of the study.
While there has been much research in the area of Bayesian sample size determination [9–12], the hybrid classical and Bayesian method presented here aligns more with the ideas found in traditional frequentist sample size determination. Unlike traditional frequentist methods, however, we do not assume that the true parameters under the alternative hypothesis are known. This assumption rarely holds; typically, parameter values are estimated from early phase or pilot studies, studies of the intervention in different populations, or studies of similar agents in the current population [13, 14]. Thus, there is uncertainty surrounding the estimation of these population parameters and natural prior distributions of plausible values of these parameters that should be incorporated into the assessment of a trial’s power. Our method incorporates knowledge on the magnitude and uncertainty in the parameters into the traditional frequentist notion of power through explicit prior distributions on these unknown parameters to give CEP. As discussed in the “Methods” Section, CEP is not only well behaved, but it allows us to maintain a definition of power that intuitively converges to the traditional definition. Bayesian methodology is used only during the study design to allow prior information, through the prior distributions, to inform a choice for the sample size. Traditional type I and type II error rates, which have been accepted in practice, are maintained, and inferences are based on the likelihood of the data. The probability of achieving a target value of power using this method is compared to the performance of a traditional design. It is our hope that this formal method for incorporating prior knowledge into the study design will form the basis of thoughtful discussion about the feasibility of the study in order to reduce the number of poorly designed, underpowered studies that are conducted.
Methods
CEP for dichotomous outcome
Suppose that the study endpoint is dichotomous so that the probability (risk) of experiencing the event of interest in group 2 (the experimental treatment group), π _{2}, is compared to that in group 1 (the control group), π _{1}. The responses (i.e., the number of successes) in each group follow a binomial distribution. Assume that after n observations in each independent group or N=2n total observations, the twosample Ztest of proportions is performed to test the null hypothesis H _{0}:π _{2}=π _{1} (i.e., π _{2}−π _{1}=Δ=0) versus the twosided alternative hypothesis H _{1}:π _{2}≠π _{1} (i.e., π _{2}−π _{1}=Δ≠0), where π _{2}>π _{1} indicates benefit of the experimental treatment over the control. The test is based on the test statistic T=p _{2}−p _{1}, or the difference in the proportion of successes in each sample. Under H _{0}:π _{2}=π _{1}=π,T · ∼ N(0,σ _{0}) in large samples, where σ _{0} is the standard deviation of the normal distribution. Assuming equal sample sizes n in each group gives \(\sigma _{0} = \sqrt {2 \pi (1\pi)/n }\), where π=(π _{1}+π _{2})/2. In this setting, H _{0} is rejected at the αlevel of significance if \(T \geq z_{{}_{1\alpha /2}} \, \hat {\sigma }_{0}\), where \(\phantom {\dot {i}\!}z_{{}_{1\alpha /2}}\) is the critical value for lower tail area 1−α/2 of the standard normal distribution and π is estimated by p=(p _{1}+p _{2})/2 in \(\hat {\sigma }_{0}\). A positive conclusion, D _{1}, occurs if \(Z = T/\hat {\sigma }_{0} \geq z_{{}_{1\alpha /2}}\).
The unconditional prior distributions p(π _{1}) and p(π _{2}) are defined such that π _{1}∉ [0,1]⇒p(π _{1})=0 and π _{2}∉ [ 0,1]⇒p(π _{2})=0 (e.g., beta or uniform(0,1) distributions).
Note, any appropriate sample size and power formulas may be used to evaluate CEP in (5). For example, continuitycorrected versions of (2) or the arcsine approximation [25, 26] may alternatively be utilized instead of (2) to determine sample size, while related power formulas may be used instead of (1) for CEP calculations.
To evaluate CEP under a proposed design, find N in (2) for the hypothesized values of π _{1} and π _{2}, significance level α, and traditional power level 1−β. Numerical integration may then be used to evaluate CEP (5) for the assumed prior distributions p(π _{1}) and p(π _{2}). If CEP for the proposed design is less than 1−β, the study is expected to be underpowered under the treatment superiority assumption, and if the CEP is greater than 1−β, the study is expected to be overpowered. To ensure that the study is expected to be appropriately powered under the treatment superiority assumption, an iterative search procedure can be used to find the value of the sample size N in (5) that gives CEP equal to the threshold of traditional power 1−β. The value of N that achieves this desired level is denoted N ^{∗}. As in traditional power, we would like the probability of detecting a difference when a positive difference exists to be high (i.e., 0.80≤1−β≤0.90). Pseudocode 1 outlines the steps for this process.
Previous work in this area almost exclusively uses expected power P(D _{1}) to account for uncertainty in study design parameters [8, 9, 15–24], and finds the sample size that gives the desired level of P(D _{1}). Our preference for using CEP as opposed to EP to inform the design of a study is twofold. EP gives the predictive probability of a positive conclusion, regardless of the truth of the alternative hypothesis. CEP, however, is conceptually analogous to traditional power in that it is conditional on the truth of the benefit of the experimental treatment, which provides a more familiar framework for setting the desired level of CEP for a study. Secondly, if P(π _{1}>π _{2})>0, then EP will not approach 1 as the sample size goes to infinity because \({\lim }_{N\to \infty } P(D_{1})=1P(\pi _{1}>\pi _{2})\). CEP, however, is conditioned on π _{2}>π _{1}, so it approaches 1 as the sample size increases since \({\lim }_{N\to \infty } P(D_{1} \,\, \pi _{2} > \pi _{1}) = \frac {1P(\pi _{1}>\pi _{2})}{P(\pi _{2}>\pi _{1})}=1\). Thus, CEP is also more mathematically analogous to traditional power in that the probability of correctly reaching a positive conclusion is assured as the sample size goes to infinity.
Prior distributions
The prior distributions p(π _{1}) and p(π _{2}) reflect the current knowledge about the response rate in each treatment group before the trial is conducted. In the design phase of a clinical trial, a review of the literature is often performed. This collection of prior evidence forms a natural foundation for specifying the prior distributions. Historical data are commonly pooled using traditional metaanalysis techniques to calculate an overall point estimate [27, 28]; however, a Bayesian randomeffects metaanalysis [29–31] may be more appropriate when the goal is to hypothesize a prior distribution. The priors can also incorporate the pretrial consensus of experts in the field [9] or Phase II trial data [22]. Translating and combining prior evidence and opinions to form a prior distribution is often hailed as the most challenging part of using a Bayesian framework [7], and several works [32–35] describe techniques for eliciting a prior distribution.
A uniform prior distribution may also be assumed for π _{ j } with limits within the interval [ 0,1]. The uniform prior has lower bound a and upper bound b, or π _{ j }∼U(a,b), and is constant over the range [ a,b]. The prior is centered at μ=(a+b)/2 with variance τ ^{2}=(b−a)^{2}/12. The noninformative prior distribution that assumes no values of π _{ j } are more probable than any others is U(0,1)≡Beta(1,1). One may also restrict the range of the uniform distribution to focus on smaller ranges for π _{1} and π _{2}. Rather than setting the lower and upper bounds of the uniform, one may set the mean μ<1 and variance \(\tau ^{2} < \frac {\min (\mu ^{2}, (1\mu)^{2})}{3}\) of the prior distribution, which gives lower bound \(a = \mu  \sqrt {3 \, \tau ^{2}} \) and upper bound \(b = \mu + \sqrt {3 \, \tau ^{2}}\). Again, under a traditional design, the difference in means μ _{2}−μ _{1} is a natural estimate for the hypothesized difference in proportions when presented with uniform prior evidence. (Table 3 in the Appendix reports the values of a and b for given μ and τ ^{2}.) Notice that restrictions exist for the variances assumed for certain means to maintain bounds between [ 0,1].
Results
The procedures described in the “Methods” Section were applied to a set of notional scenarios to compare traditionally designed studies to those designed using CEP. The integration step of Pseudocode 1 was approximated using Riemann sums with step size 0.0001.
For the same scenario, a CEPdesigned study would select a sample size of N ^{∗}=80 based on Pseudocode 1 to achieve 80% CEP with a twosided type I error of 5%. Figure 2 b displays the joint distribution of π _{1} and π _{2}, conditional on π _{2}>π _{1}, and highlights the region where power would be less than 80% under a CEP design when the sample size is N ^{∗}=80. For this scenario, the study with sample size N ^{∗}=80 would achieve power less than the target value in approximately 33% of instances when π _{2}>π _{1}. Note that the intersection of the two regions corresponds to values of π _{1} and π _{2} that give power from (1) equal to 80% with sample size N=80.
Thus, the traditionally designed study from the example scenario produced a performance of (100−56)%=44%, while the CEP design, which explicitly accounts for uncertainty, produced a more robust performance of (100−33)%=67%. However, this increase in performance required an increase in sample size from N=48 to N ^{∗}=80. The increase in performance divided by the increase in sample size is here termed the marginal benefit for the scenario due to CEP. The marginal benefit for the example scenario due to CEP is given by (67−44)%/(80−48)=0.71%. If there is no uncertainty in the design parameters, then there would be no marginal benefit due to CEP, since the probability of achieving less than the target power would be assumed 0 for a traditionally designed study and the CEPdesigned study would give N ^{∗}=N. On the other hand, if the uncertainty in the design parameters is very large, the marginal benefit may approach 0, since the CEPdesigned study could give N ^{∗}>>N with limited increase in performance. This is important to consider, since a very small marginal benefit could make it impractical to achieve a desired value for CEP or a desired threshold of performance.
 1.
The modes, m _{1} and m _{2}, and variances, \(\tau ^{2}_{1}=\tau ^{2}_{2}\), were used to hypothesize a beta prior distribution for π _{1} and π _{2}, respectively.
 2.For each pair of prior distributions (p(π _{1}),p(π _{2})) considered:
 (a)
Traditional sample size is found using (2) by setting the hypothesized values of π _{1} and π _{2} equal to the mode of each prior, m _{1} and m _{2}, respectively. Twosided type I error α=0.05 and traditional power 1−β=0.80 are assumed. Traditional sample size is denoted \(\hat {N}\). If \(\hat {N}\) is odd, the sample size is increased by 1 to provide equal sample size for both groups.
 (b)
The CEP of the traditional design is found using (5), with \(N=\hat {N}\), twosided α=0.05, and 1−β=0.80.
 (c)
The performance of the traditional design is found using (6), with \(N=\hat {N}\), twosided α=0.05, and 1−β=0.80.
 (d)
The smallest sample size for which CEP evaluates to ≥1−β is found using PseudoCode 1 and is denoted N ^{∗}. If N ^{∗} is odd, the sample size is increased by 1 to provide equal sample size for both groups.
 (e)
The probability of a positive treatment effect, P(π _{2}>π _{1}), is found using (4) with Riemann sum integral approximations.
 (f)The conditional expected difference, E(π _{2}−π _{1}π _{2}>π _{1}), is found using Riemann sum integral approximations of$$\begin{aligned} {}E(\pi_{2} \! \!\pi_{1}\pi_{2} \!\!>\!\! \pi_{1})\,=\,\frac{1}{P(\pi_{2}\!\!>\!\!\pi_{1})}\!\! \int \limits_{\pi_{1}=0}^{1} \int \limits_{\pi_{2}=\pi_{1}}^{1} \!\!(\pi_{2}\,\,\pi_{1}) p\!\left(\pi_{1}\right) \!p\!\left(\pi_{2} \right) \!d\pi_{2} d\pi_{1}. \end{aligned} $$
 (g)
The performance of the CEP design is found using (6), with N=N ^{∗}, twosided α=0.05, and 1−β=0.80.
 (h)
The marginal benefit due to CEP for the scenario is found by dividing the difference between the CEP design performance and the traditional design performance by the difference between the CEP sample size and the traditional sample size, \(N^{*}\hat {N}\).
 (a)
Table 4 in the Appendix shows that when m _{2}−m _{1}>1/3, the performance of the traditional design decreases as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases. This is explained by the fact that the conditional expected difference is less than the hypothesized difference that was used in the traditional design sample size calculation. This occurs for m _{2}−m _{1}>1/3 since both prior distributions are approaching U(0,1) as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases, and E(π _{2}−π _{1}π _{2}>π _{1})=1/3 for π _{1},π _{2}∼U(0,1). Thus, when m _{2}−m _{1}<1/3, the performance of the traditional design increases as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases since the hypothesized difference is less than the limit of the conditional expected difference. When m _{2}−m _{1} is smaller than E(π _{2}−π _{1}π _{2}>π _{1}), CEP will be high for a traditional design with hypothesized difference m _{2}−m _{1}, since it is designed to detect a difference smaller than the expected difference.
The procedure was also applied to scenarios where \(\tau _{1}^{2} = 0.001\) and \(\tau _{2}^{2} > 0.001\) (Table 5 in the Appendix) and scenarios where \(\tau _{1}^{2} = 0.08\) and \(\tau _{2}^{2} < 0.08\) (Table 6 in the Appendix), corresponding to small and large uncertainty, respectively, in the proportion experiencing the outcome in the control group. Table 5 in the Appendix shows that the performance of the traditional design is similar to the performance seen in Table 4 in the Appendix. However, when \(\tau _{1}^{2}\) is fixed at 0.001,E(π _{2}−π _{1}π _{2}>π _{1}) begins near m _{2}−m _{1} and approaches (1−m _{1})/2 as \(\tau _{2}^{2}\) increases because p(π _{2}π _{2}>π _{1}) is approaching U(m _{1},1). Thus, when m _{2}−m _{1}>(1−m _{1})/2, the performance of the traditional design decreases as \(\tau _{2}^{2}\) increases, and when m _{2}−m _{1}<(1−m _{1})/2, the performance of the traditional design increases as \(\tau _{2}^{2}\) increases.
Analysis of traditional design performance
Uncertainty  m _{2}−m _{1}  Performance 

\(\tau _{1}^{2}=\tau _{2}^{2}\)  <1/3  Increases as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases 
>1/3  Decreases as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases  
\(\tau _{1}^{2}\) small  <(1−m _{1})/2  Increases as \(\tau _{2}^{2}\) increases 
>(1−m _{1})/2  Decreases as \(\tau _{2}^{2}\) increases  
\(\tau _{1}^{2}\) large  >2/3−m _{1}  Increases as \(\tau _{2}^{2}\) increases 
<2/3−m _{1}  Decreases as \(\tau _{2}^{2}\) increases 
Excursions with uniform priors were performed. Table 7 in the Appendix shows that the performance of a traditional design under a uniform prior is similar to the performance observed in Table 4 in the Appendix. However, fewer trends are visible because the parameters of the uniform distribution are more restricted than the parameters of the beta distribution.
As expected, the performance of the CEP design changes minimally as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases, since N ^{∗} is chosen to explicitly account for changes in \(\tau _{1}^{2}=\tau _{2}^{2}\). Note, N ^{∗} is directly tied to E(π _{2}−π _{1}π _{2}>π _{1}): N ^{∗} increases as the conditional expected difference decreases, and N ^{∗} decreases as the conditional expected difference increases. This occurs because increasing the variability can increase the conditional expected difference if the resulting conditional priors give more relative weight to larger differences and less relative weight to smaller differences compared to the unconditional priors. This is more likely to occur when m _{1} is large, since increasing the variability when m _{1} is large will make smaller values of π _{1} more likely due to the condition that π _{2}>π _{1}. Similarly, when m _{2} is small, larger values of π _{2} are more likely under the assumption that π _{2}>π _{1}.
The marginal benefit due to CEP is greatest for small values of \(\tau _{1}^{2}=\tau _{2}^{2}\). This is so because the relative difference between \(\hat {N}\) and N ^{∗} is smallest when the uncertainty is low (i.e., when the traditional assumptions closely approximate the CEP assumptions). However, the marginal benefit due to CEP decreases minimally or remains constant as the uncertainty increases because the difference in performance is always less than 1, while the difference in sample size, \(N^{*}\hat {N}\), can be greater than 200 in some cases. Furthermore, as \(\tau _{1}^{2}=\tau _{2}^{2}\) increases, the performance of the traditional design can improve even though \(\hat {N}\) remains constant, while N ^{∗} may have to increase to maintain the performance of the CEP design.
When \(\tau _{1}^{2}\) is fixed at 0.001, the performance of the CEP design remains stable at approximately 0.7. However, the marginal benefit is greater with fixed, low uncertainty in π _{1} compared with the changing uncertainty in Table 4 in the Appendix. The sample size required to achieve CEP of 1−β with fixed \(\tau _{1}^{2}\) is reduced compared to scenarios with changing \(\tau _{1}^{2}\). This is because uncertainty in the control group is small, which indicates that reducing the uncertainty in the control parameter can increase the benefit of CEP to the study.
When \(\tau _{1}^{2}\) is fixed at 0.08, the performance of the CEP design remains stable at approximately 0.71. However, the marginal benefit is very small because N ^{∗} is always greater than that in Table 4 or Table 5 in the Appendix due to the larger uncertainty in π _{1}. Again, this demonstrates that it is beneficial to minimize the uncertainty in π _{1} to increase the marginal benefit.
Note that for small differences in m _{2}−m _{1} and any large variance, the CEP design can reduce the sample size from the value determined from a traditional design. The reason is that increased uncertainty under the treatment superiority assumption increases the likelihood of differences greater than m _{2}−m _{1}.
Discussion
Many underpowered clinical trials are conducted with limited justification for the chosen study parameters used to determine the required sample size [36, 37] with scientific, economic, and ethical implications [36, 38]. While sample size calculations based on traditional power assume no uncertainty in the study parameters, the hybrid classical and Bayesian procedure presented here formally accounts for the uncertainty in the study parameters by incorporating the prior distributions for π _{1} and π _{2} into the calculation of conditional expected power (CEP). This method allows available evidence on both the magnitude and the variability surrounding the parameters to play a formal role in determining study power and sample size.
In this paper, we explored several notional scenarios to compare the performance of the CEP design to that of a design based on traditional power. We show that if there is uncertainty in the study parameters and a distribution of plausible values for π _{1} and π _{2}, the performance of the CEP design is more consistent and robust than that of traditional designs based on point estimates for the study parameters. Traditional sample size calculations based on point estimates for the hypothesized study parameters tend to underestimate the required sample size needed to account for the uncertainty in the parameters.
The scenarios demonstrate that reducing uncertainty in the control parameter π _{1} can lead to greater benefit from the CEPdesigned study, because the relative difference between \(\hat {N}\) and N ^{∗} is smallest when uncertainty is low. Therefore, it is worthwhile to use historical information to reduce the variability in the control group proportion rather than focusing only on the prior for the experimental treatment group. Nonetheless, when there is significant overlap between the prior distributions and a small hypothesized difference m _{2}−m _{1}, traditional study designs can be overpowered under the treatment superiority assumption compared to the CEP design, and the CEP design would result in a smaller sample size. This happens because increased uncertainty under the treatment superiority assumption increases the relative likelihood of differences greater than m _{2}−m _{1}.
In the scenarios we evaluated, the performance of the traditional design was highly dependent on the prior distributions but exhibited predictable behavior. The CEP design, however, consistently generated performance near 70% across all scenarios. This indicates that power greater than the target 1−β would not be uncommon for a CEP design. This begs the question of whether or not 1−β is an appropriate target for CEP, since it could apparently lead to overpowered studies. To avoid this issue, one may use a lower target for CEP or instead design the study using a target value of performance and use our iterative N ^{∗} search to find the design that achieves acceptable performance.
Additionally, when comparing the method based on CEP to similar methods based on expected power, the sample size from a CEP design will always be less than or equal to the sample size required to achieve equivalent EP. While pure Bayesian methods of sample size determination that compute prior effective sample size to count the information contained in the prior towards the current study will generally yield a smaller sample size than traditional frequentist methods [10], the method presented here does not assume that prior information will be incorporated into the final analysis.
Conclusions
The hybrid classical and Bayesian procedure presented here integrates available prior information about the study design parameters into the calculation of study sample size for a binary endpoint. This method allows prior information on both the magnitude and uncertainty surrounding the parameters π _{1} and π _{2} to inform the design of the current study through the use of conditional expected power. When there is a distribution of plausible study parameters, the design based on conditional expected power tends to outperform the traditional design. Note that if the determined sample size N ^{∗} is greater than what can be feasibly recruited in the proposed trial, this may indicate excessive uncertainty about the study parameters and should encourage serious discussion concerning the advisability of the study. Thus, we do not recommend that N ^{∗} be blindly used as the final study sample size, but we hope that this method encourages a careful synthesis of the prior information and motivates thoughtful discussion about the feasibility of the study in order to reduce the number of poorly designed, underpowered studies that are conducted.
Appendix
Table 2 presents the values of shape paramaters [a, b] for given m and τ ^{2} for the beta distribution. Table 3 reports the values of minimum and maximum parameters [ a,b] for given μ and τ ^{2} for the uniform distribution.
Shape parameters [ a,b] of beta distribution given mode m and variance τ ^{2}
m/ τ ^{2}  0.001  0.005  0.01  0.015  0.02  0.025  0.03  0.035  0.04  0.045  0.05  0.055  0.06  0.065  0.07  0.075  0.08 

0.01  [1.33,33.99]  [1.12,12.73]  [1.07,8.18]  [1.05,6.21]  [1.04,5.05]  [1.03,4.26]  [1.03,3.67]  [1.02,3.21]  [1.02,2.84]  [1.02,2.53]  [1.01,2.26]  [1.01,2.03]  [1.01,1.82]  [1.01,1.63]  [1.01,1.45]  [1,1.28]  [1,1.12] 
0.05  [3.89,55.92]  [1.77,15.55]  [1.43,9.21]  [1.3,6.74]  [1.23,5.36]  [1.18,4.45]  [1.15,3.79]  [1.12,3.29]  [1.1,2.89]  [1.08,2.56]  [1.07,2.28]  [1.05,2.03]  [1.04,1.82]  [1.03,1.62]  [1.02,1.44]  [1.02,1.28]  [1.01,1.11] 
0.1  [10.36,85.26]  [3.05,19.4]  [2.07,10.59]  [1.72,7.44]  [1.53,5.76]  [1.41,4.7]  [1.33,3.95]  [1.27,3.39]  [1.22,2.95]  [1.18,2.59]  [1.14,2.3]  [1.12,2.04]  [1.09,1.82]  [1.07,1.62]  [1.05,1.44]  [1.03,1.27]  [1.01,1.11] 
0.15  [20.24,110]  [4.88,23.01]  [2.93,11.91]  [2.25,8.1]  [1.91,6.14]  [1.69,4.92]  [1.55,4.09]  [1.44,3.48]  [1.35,3]  [1.29,2.62]  [1.23,2.31]  [1.18,2.04]  [1.14,1.81]  [1.11,1.61]  [1.08,1.43]  [1.05,1.26]  [1.02,1.1] 
0.2  [32.84,128.38]  [7.22,25.89]  [4,13]  [2.91,8.65]  [2.36,6.45]  [2.03,5.11]  [1.8,4.2]  [1.64,3.54]  [1.51,3.04]  [1.41,2.64]  [1.33,2.31]  [1.26,2.04]  [1.2,1.8]  [1.15,1.6]  [1.1,1.41]  [1.06,1.25]  [1.02,1.1] 
0.25  [47.46,140.37]  [9.95,27.84]  [5.25,13.75]  [3.68,9.03]  [2.89,6.66]  [2.41,5.23]  [2.09,4.27]  [1.86,3.57]  [1.68,3.05]  [1.55,2.63]  [1.43,2.3]  [1.34,2.02]  [1.26,1.78]  [1.19,1.58]  [1.13,1.4]  [1.08,1.24]  [1.03,1.09] 
0.3  [63.33,146.43]  [12.92,28.82]  [6.62,14.11]  [4.52,9.21]  [3.46,6.75]  [2.83,5.27]  [2.4,4.28]  [2.1,3.57]  [1.87,3.03]  [1.69,2.61]  [1.55,2.27]  [1.43,1.99]  [1.33,1.76]  [1.24,1.56]  [1.16,1.38]  [1.1,1.23]  [1.04,1.09] 
0.35  [79.71,147.18]  [16.01,28.88]  [8.05,14.09]  [5.39,9.16]  [4.06,6.69]  [3.27,5.21]  [2.73,4.22]  [2.35,3.51]  [2.07,2.98]  [1.84,2.56]  [1.66,2.23]  [1.52,1.96]  [1.39,1.73]  [1.29,1.53]  [1.2,1.36]  [1.12,1.21]  [1.04,1.08] 
0.4  [95.87,143.3]  [19.07,28.1]  [9.47,13.7]  [6.27,8.9]  [4.67,6.5]  [3.71,5.06]  [3.06,4.1]  [2.61,3.41]  [2.26,2.89]  [2,2.49]  [1.78,2.17]  [1.61,1.91]  [1.46,1.69]  [1.33,1.5]  [1.23,1.34]  [1.13,1.2]  [1.05,1.08] 
0.45  [111.04,135.5]  [21.94,26.6]  [10.81,12.99]  [7.09,8.45]  [5.24,6.18]  [4.12,4.82]  [3.38,3.91]  [2.85,3.26]  [2.45,2.77]  [2.14,2.4]  [1.89,2.09]  [1.69,1.85]  [1.52,1.64]  [1.38,1.46]  [1.26,1.31]  [1.15,1.18]  [1.06,1.07] 
0.5  [124.5,124.5]  [24.5,24.5]  [12,12]  [7.83,7.83]  [5.75,5.75]  [4.5,4.5]  [3.67,3.67]  [3.07,3.07]  [2.63,2.63]  [2.28,2.28]  [2,2]  [1.77,1.77]  [1.58,1.58]  [1.42,1.42]  [1.29,1.29]  [1.17,1.17]  [1.06,1.06] 
0.55  [135.5,111.04]  [26.6,21.94]  [12.99,10.81]  [8.45,7.09]  [6.18,5.24]  [4.82,4.12]  [3.91,3.38]  [3.26,2.85]  [2.77,2.45]  [2.4,2.14]  [2.09,1.89]  [1.85,1.69]  [1.64,1.52]  [1.46,1.38]  [1.31,1.26]  [1.18,1.15]  [1.07,1.06] 
0.6  [143.3,95.87]  [28.1,19.07]  [13.7,9.47]  [8.9,6.27]  [6.5,4.67]  [5.06,3.71]  [4.1,3.06]  [3.41,2.61]  [2.89,2.26]  [2.49,2]  [2.17,1.78]  [1.91,1.61]  [1.69,1.46]  [1.5,1.33]  [1.34,1.23]  [1.2,1.13]  [1.08,1.05] 
0.65  [147.18,79.71]  [28.88,16.01]  [14.09,8.05]  [9.16,5.39]  [6.69,4.06]  [5.21,3.27]  [4.22,2.73]  [3.51,2.35]  [2.98,2.07]  [2.56,1.84]  [2.23,1.66]  [1.96,1.52]  [1.73,1.39]  [1.53,1.29]  [1.36,1.2]  [1.21,1.12]  [1.08,1.04] 
0.7  [146.43,63.33]  [28.82,12.92]  [14.11,6.62]  [9.21,4.52]  [6.75,3.46]  [5.27,2.83]  [4.28,2.4]  [3.57,2.1]  [3.03,1.87]  [2.61,1.69]  [2.27,1.55]  [1.99,1.43]  [1.76,1.33]  [1.56,1.24]  [1.38,1.16]  [1.23,1.1]  [1.09,1.04] 
0.75  [140.37,47.46]  [27.84,9.95]  [13.75,5.25]  [9.03,3.68]  [6.66,2.89]  [5.23,2.41]  [4.27,2.09]  [3.57,1.86]  [3.05,1.68]  [2.63,1.55]  [2.3,1.43]  [2.02,1.34]  [1.78,1.26]  [1.58,1.19]  [1.4,1.13]  [1.24,1.08]  [1.09,1.03] 
0.8  [128.38,32.84]  [25.89,7.22]  [13,4]  [8.65,2.91]  [6.45,2.36]  [5.11,2.03]  [4.2,1.8]  [3.54,1.64]  [3.04,1.51]  [2.64,1.41]  [2.31,1.33]  [2.04,1.26]  [1.8,1.2]  [1.6,1.15]  [1.41,1.1]  [1.25,1.06]  [1.1,1.02] 
0.85  [110,20.24]  [23.01,4.88]  [11.91,2.93]  [8.1,2.25]  [6.14,1.91]  [4.92,1.69]  [4.09,1.55]  [3.48,1.44]  [3,1.35]  [2.62,1.29]  [2.31,1.23]  [2.04,1.18]  [1.81,1.14]  [1.61,1.11]  [1.43,1.08]  [1.26,1.05]  [1.1,1.02] 
0.9  [85.26,10.36]  [19.4,3.05]  [10.59,2.07]  [7.44,1.72]  [5.76,1.53]  [4.7,1.41]  [3.95,1.33]  [3.39,1.27]  [2.95,1.22]  [2.59,1.18]  [2.3,1.14]  [2.04,1.12]  [1.82,1.09]  [1.62,1.07]  [1.44,1.05]  [1.27,1.03]  [1.11,1.01] 
0.95  [55.93,3.89]  [15.55,1.77]  [9.21,1.43]  [6.74,1.3]  [5.36,1.23]  [4.45,1.18]  [3.79,1.15]  [3.29,1.12]  [2.89,1.1]  [2.56,1.08]  [2.28,1.07]  [2.03,1.05]  [1.82,1.04]  [1.62,1.03]  [1.44,1.02]  [1.28,1.02]  [1.11,1.01] 
0.99  [33.99,1.33]  [12.73,1.12]  [8.18,1.07]  [6.21,1.05]  [5.05,1.04]  [4.26,1.03]  [3.67,1.03]  [3.21,1.02]  [2.84,1.02]  [2.53,1.02]  [2.26,1.01]  [2.03,1.01]  [1.82,1.01]  [1.63,1.01]  [1.45,1.01]  [1.28,1.003]  [1.12,1.001] 
Minimum and maximum parameters [ a,b] of uniform distribution given mean μ and variance τ ^{2}
μ/ τ ^{2}  0.001  0.005  0.01  0.015  0.02  0.025  0.03  0.035  0.04  0.045  0.05  0.055  0.06  0.065  0.07  0.075  0.08  0.0833 

0.01  
0.05  
0.1  [0.045,0.155]  
0.15  [0.095,0.205]  [0.028,0.272]  
0.2  [0.145,0.255]  [0.078,0.322]  [0.027,0.373]  
0.25  [0.195,0.305]  [0.128,0.372]  [0.077,0.423]  [0.038,0.462]  [0.005,0.495]  
0.3  [0.245,0.355]  [0.178,0.422]  [0.127,0.473]  [0.088,0.512]  [0.055,0.545]  [0.026,0.574]  [0,0.6]  
0.35  [0.295,0.405]  [0.228,0.472]  [0.177,0.523]  [0.138,0.562]  [0.105,0.595]  [0.076,0.624]  [0.05,0.65]  [0.026,0.674]  [0.004,0.696]  
0.4  [0.345,0.455]  [0.278,0.522]  [0.227,0.573]  [0.188,0.612]  [0.155,0.645]  [0.126,0.674]  [0.1,0.7]  [0.076,0.724]  [0.054,0.746]  [0.033,0.767]  [0.013,0.787]  
0.45  [0.395,0.505]  [0.328,0.572]  [0.277,0.623]  [0.238,0.662]  [0.205,0.695]  [0.176,0.724]  [0.15,0.75]  [0.126,0.774]  [0.104,0.796]  [0.083,0.817]  [0.063,0.837]  [0.044,0.856]  [0.026,0.874]  [0.008,0.892]  
0.5  [0.445,0.555]  [0.378,0.622]  [0.327,0.673]  [0.288,0.712]  [0.255,0.745]  [0.226,0.774]  [0.2,0.8]  [0.176,0.824]  [0.154,0.846]  [0.133,0.867]  [0.113,0.887]  [0.094,0.906]  [0.076,0.924]  [0.058,0.942]  [0.042,0.958]  [0.026,0.974]  [0.01,0.99]  [0,1] 
0.55  [0.495,0.605]  [0.428,0.672]  [0.377,0.723]  [0.338,0.762]  [0.305,0.795]  [0.276,0.824]  [0.25,0.85]  [0.226,0.874]  [0.204,0.896]  [0.183,0.917]  [0.163,0.937]  [0.144,0.956]  [0.126,0.974]  [0.108,0.992]  
0.6  [0.545,0.655]  [0.478,0.722]  [0.427,0.773]  [0.388,0.812]  [0.355,0.845]  [0.326,0.874]  [0.3,0.9]  [0.276,0.924]  [0.254,0.946]  [0.233,0.967]  [0.213,0.987]  
0.65  [0.595,0.705]  [0.528,0.772]  [0.477,0.823]  [0.438,0.862]  [0.405,0.895]  [0.376,0.924]  [0.35,0.95]  [0.326,0.974]  [0.304,0.996]  
0.7  [0.645,0.755]  [0.578,0.822]  [0.527,0.873]  [0.488,0.912]  [0.455,0.945]  [0.426,0.974]  [0.4,1]  
0.75  [0.695,0.805]  [0.628,0.872]  [0.577,0.923]  [0.538,0.962]  [0.505,0.995]  
0.8  [0.745,0.855]  [0.678,0.922]  [0.627,0.973]  
0.85  [0.795,0.905]  [0.728,0.972]  
0.9  [0.845,0.955]  
0.95  
0.99 
Sample scenarios assuming beta priors p(π _{1}) and p(π _{2}) where \(\tau _{1}^{2}=\tau _{2}^{2}\). Hypothesized values of π _{1} and π _{2} set equal to m _{1} and m _{2}, respectively, under the traditional design. Twosided α=0.05 and 1−β=0.80 assumed
Traditional design  CEP design  

(m _{1},m _{2})  \(\tau _{1}^{2}=\tau _{2}^{2}\)  \(\hat {N}\)  Performance  CEP  N ^{∗}  Performance  E(π _{2}−π _{1}π _{2}>π _{1})  P(π _{2}>π _{1})  Marginal benefit 
(0.1,0.9)  0.001  10  0.797  0.518  12  0.742  0.783  1  0.1120 
0.01  10  0.622  0.242  16  0.627  0.674  1  0.0642  
0.02  10  0.508  0.164  26  0.673  0.585  0.993  0.0318  
0.03  10  0.433  0.123  40  0.687  0.519  0.965  0.0188  
0.04  10  0.381  0.099  60  0.698  0.469  0.915  0.0120  
0.05  10  0.342  0.082  86  0.705  0.429  0.846  0.0082  
0.06  10  0.311  0.070  120  0.710  0.397  0.761  0.0058  
0.07  10  0.285  0.060  160  0.713  0.369  0.662  0.0044  
0.08  10  0.262  0.052  214  0.716  0.342  0.545  0.0033  
(0.1,0.8) or (0.2,0.9)  0.001  14  0.804  0.559  14  0.559  0.688  1  0 
0.01  14  0.656  0.314  22  0.623  0.602  1  0.0386  
0.02  14  0.553  0.237  34  0.674  0.529  0.989  0.0218  
0.03  14  0.486  0.195  52  0.692  0.476  0.953  0.0131  
0.04  14  0.439  0.168  76  0.702  0.438  0.896  0.0086  
0.05  14  0.405  0.149  102  0.706  0.408  0.823  0.0063  
0.06  14  0.377  0.134  134  0.711  0.383  0.740  0.0048  
0.07  14  0.353  0.121  172  0.713  0.361  0.646  0.0037  
0.08  14  0.331  0.110  218  0.716  0.340  0.540  0.0030  
(0.1,0.7) or (0.3,0.9)  0.001  20  0.814  0.616  20  0.616  0.590  1  0 
0.01  20  0.670  0.364  30  0.636  0.518  0.999  0.0272  
0.02  20  0.578  0.295  48  0.677  0.463  0.979  0.0136  
0.03  20  0.523  0.260  72  0.697  0.426  0.931  0.0084  
0.04  20  0.488  0.238  98  0.705  0.401  0.866  0.0060  
0.05  20  0.462  0.223  126  0.708  0.382  0.792  0.0046  
0.06  20  0.442  0.211  154  0.712  0.367  0.712  0.0037  
0.07  20  0.424  0.200  186  0.714  0.353  0.627  0.0031  
0.08  20  0.407  0.189  222  0.716  0.338  0.534  0.0026  
(0.2,0.8)  0.001  20  0.800  0.537  22  0.644  0.593  1  0.0537 
0.01  20  0.676  0.372  30  0.643  0.530  0.999  0.0271  
0.02  20  0.588  0.308  48  0.680  0.474  0.982  0.0133  
0.03  20  0.532  0.272  70  0.695  0.435  0.936  0.0085  
0.04  20  0.495  0.247  94  0.703  0.408  0.872  0.0062  
0.05  20  0.467  0.229  122  0.709  0.387  0.798  0.0047  
0.06  20  0.445  0.215  152  0.711  0.370  0.717  0.0038  
0.07  20  0.425  0.202  184  0.714  0.354  0.630  0.0031  
0.08  20  0.407  0.189  222  0.716  0.339  0.534  0.0026  
(0.1,0.6) or (0.4,0.9)  0.001  28  0.804  0.572  28  0.572  0.491  1  0 
0.01  28  0.655  0.368  48  0.660  0.430  0.996  0.0146  
0.02  28  0.578  0.319  76  0.689  0.393  0.958  0.0077  
0.03  28  0.540  0.299  106  0.701  0.374  0.895  0.0052  
0.04  28  0.518  0.290  132  0.706  0.362  0.824  0.0040  
0.05  28  0.504  0.285  158  0.710  0.355  0.752  0.0033  
0.06  28  0.494  0.281  180  0.713  0.349  0.680  0.0028  
0.07  28  0.486  0.278  202  0.714  0.343  0.606  0.0025  
0.08  28  0.476  0.272  226  0.715  0.336  0.528  0.0022  
(0.2,0.7) or (0.3,0.8)  0.001  30  0.803  0.558  30  0.558  0.494  1  0 
0.01  30  0.682  0.412  46  0.653  0.446  0.998  0.0151  
0.02  30  0.609  0.365  72  0.686  0.410  0.966  0.0076  
0.03  30  0.570  0.342  100  0.700  0.388  0.907  0.0051  
0.04  30  0.546  0.328  126  0.706  0.374  0.837  0.0039  
0.05  30  0.529  0.317  150  0.710  0.363  0.763  0.0033  
0.06  30  0.515  0.309  174  0.713  0.354  0.688  0.0028  
0.07  30  0.503  0.300  198  0.714  0.346  0.611  0.0025  
0.08  30  0.491  0.291  226  0.716  0.337  0.529  0.0022  
(0.1,0.5) or (0.5,0.9)  0.001  40  0.781  0.488  42  0.566  0.392  1  0.0393 
0.01  40  0.627  0.358  80  0.675  0.343  0.985  0.0079  
0.02  40  0.573  0.335  124  0.697  0.326  0.918  0.0043  
0.03  40  0.553  0.334  160  0.706  0.322  0.841  0.0031  
0.04  40  0.546  0.339  182  0.709  0.324  0.769  0.0026  
0.05  40  0.545  0.346  200  0.712  0.327  0.703  0.0023  
0.06  40  0.545  0.353  210  0.713  0.331  0.642  0.0021  
0.07  40  0.546  0.359  220  0.715  0.334  0.584  0.0020  
0.08  40  0.545  0.362  230  0.716  0.334  0.522  0.0019  
(0.2,0.6) or (0.4,0.8)  0.001  46  0.791  0.518  48  0.587  0.395  1  0.0345 
0.01  46  0.668  0.422  80  0.668  0.360  0.991  0.0072  
0.02  46  0.617  0.401  118  0.694  0.343  0.933  0.0041  
0.03  46  0.597  0.397  150  0.704  0.339  0.860  0.0030  
0.04  46  0.588  0.397  172  0.709  0.338  0.787  0.0025  
0.05  46  0.584  0.399  190  0.712  0.338  0.719  0.0022  
0.06  46  0.580  0.400  202  0.713  0.337  0.654  0.0020  
0.07  46  0.577  0.400  216  0.714  0.337  0.589  0.0018  
0.08  46  0.572  0.398  230  0.715  0.335  0.523  0.0017  
(0.3,0.7)  0.001  48  0.793  0.524  50  0.558  0.396  1  0.0167 
0.01  48  0.678  0.438  80  0.665  0.365  0.992  0.0071  
0.02  48  0.629  0.419  118  0.695  0.349  0.938  0.0039  
0.03  48  0.610  0.415  146  0.704  0.344  0.866  0.0029  
0.04  48  0.601  0.415  168  0.709  0.342  0.794  0.0024  
0.05  48  0.595  0.415  186  0.711  0.341  0.724  0.0021  
0.06  48  0.591  0.415  200  0.714  0.340  0.658  0.0020  
0.07  48  0.586  0.413  214  0.715  0.338  0.591  0.0018  
0.08  48  0.580  0.409  230  0.715  0.335  0.523  0.0017  
(0.1,0.4) or (0.6,0.9)  0.001  64  0.763  0.461  72  0.588  0.292  1  0.0159 
0.01  64  0.612  0.377  156  0.689  0.262  0.951  0.0034  
0.02  64  0.588  0.381  216  0.705  0.265  0.850  0.0021  
0.03  64  0.588  0.397  242  0.709  0.276  0.768  0.0018  
0.04  64  0.594  0.414  250  0.712  0.289  0.703  0.0016  
0.05  64  0.603  0.432  250  0.714  0.302  0.649  0.0015  
0.06  64  0.613  0.449  246  0.715  0.314  0.603  0.0015  
0.07  64  0.622  0.464  238  0.715  0.325  0.560  0.0014  
0.08  64  0.628  0.474  234  0.715  0.332  0.516  0.0014  
(0.2,0.5) or (0.5,0.8)  0.001  78  0.776  0.491  84  0.584  0.296  1  0.0154 
0.01  78  0.654  0.441  158  0.686  0.277  0.964  0.0031  
0.02  78  0.635  0.450  210  0.703  0.282  0.872  0.0019  
0.03  78  0.635  0.465  232  0.709  0.292  0.791  0.0016  
0.04  78  0.640  0.480  238  0.711  0.302  0.724  0.0014  
0.05  78  0.647  0.493  240  0.713  0.312  0.666  0.0014  
0.06  78  0.653  0.504  238  0.714  0.321  0.615  0.0013  
0.07  78  0.658  0.514  234  0.715  0.328  0.566  0.0013  
0.08  78  0.661  0.520  234  0.716  0.332  0.517  0.0013  
(0.3,0.6) or (0.4,0.7)  0.001  84  0.775  0.483  92  0.584  0.297  1  0.0126 
0.01  84  0.666  0.457  162  0.683  0.283  0.968  0.0029  
0.02  84  0.649  0.471  210  0.702  0.289  0.881  0.0018  
0.03  84  0.652  0.489  228  0.708  0.300  0.802  0.0015  
0.04  84  0.658  0.504  234  0.712  0.309  0.735  0.0014  
0.05  84  0.663  0.516  234  0.713  0.318  0.676  0.0013  
0.06  84  0.668  0.525  232  0.714  0.324  0.621  0.0013  
0.07  84  0.671  0.532  232  0.715  0.329  0.569  0.0012  
0.08  84  0.672  0.536  234  0.716  0.333  0.517  0.0012  
(0.1,0.3) or (0.7,0.9)  0.001  124  0.736  0.453  152  0.614  0.194  1  0.0057 
0.01  124  0.625  0.434  346  0.703  0.193  0.866  0.0012  
0.02  124  0.636  0.470  374  0.710  0.216  0.750  0.0010  
0.03  124  0.654  0.501  358  0.712  0.239  0.680  0.0009  
0.04  124  0.671  0.528  334  0.714  0.261  0.631  0.0009  
0.05  124  0.687  0.553  306  0.714  0.281  0.594  0.0009  
0.06  124  0.701  0.575  280  0.715  0.300  0.565  0.0009  
0.07  124  0.714  0.594  258  0.715  0.317  0.538  0.0009  
0.08  124  0.725  0.610  240  0.716  0.330  0.510  0.0009  
(0.2,0.4) or (0.6,0.8)  0.001  164  0.754  0.487  190  0.605  0.197  1  0.0045 
0.01  164  0.667  0.496  372  0.702  0.205  0.887  0.0010  
0.02  164  0.683  0.537  380  0.709  0.229  0.774  0.0008  
0.03  164  0.701  0.568  356  0.712  0.251  0.702  0.0007  
0.04  164  0.716  0.593  326  0.714  0.271  0.650  0.0007  
0.05  164  0.730  0.614  300  0.714  0.289  0.609  0.0007  
0.06  164  0.742  0.632  276  0.715  0.305  0.574  0.0007  
0.07  164  0.752  0.647  254  0.715  0.319  0.543  0.0008  
0.08  164  0.760  0.659  240  0.716  0.330  0.511  0.0008  
(0.3,0.5) or (0.5,0.7)  0.001  186  0.755  0.487  214  0.605  0.198  1  0.0042 
0.01  186  0.679  0.513  394  0.700  0.210  0.896  0.0009  
0.02  186  0.699  0.560  390  0.709  0.236  0.787  0.0007  
0.03  186  0.719  0.594  356  0.712  0.259  0.715  0.0007  
0.04  186  0.735  0.620  324  0.713  0.278  0.662  0.0007  
0.05  186  0.748  0.640  294  0.714  0.294  0.619  0.0007  
0.06  186  0.759  0.656  272  0.715  0.308  0.581  0.0007  
0.07  186  0.768  0.669  252  0.715  0.321  0.546  0.0007  
0.08  186  0.774  0.679  238  0.716  0.331  0.511  0.0007  
(0.4,0.6)  0.001  194  0.757  0.491  222  0.603  0.198  1  0.0040 
0.01  194  0.683  0.518  402  0.700  0.211  0.898  0.0009  
0.02  194  0.704  0.567  396  0.709  0.238  0.791  0.0007  
0.03  194  0.724  0.602  358  0.712  0.261  0.720  0.0007  
0.04  194  0.741  0.628  322  0.713  0.280  0.666  0.0007  
0.05  194  0.754  0.648  294  0.714  0.296  0.622  0.0007  
0.06  194  0.765  0.664  270  0.715  0.310  0.583  0.0007  
0.07  194  0.773  0.677  252  0.715  0.321  0.547  0.0007  
0.08  194  0.779  0.686  238  0.716  0.331  0.511  0.0007  
(0.1,0.2) or (0.8,0.9)  0.001  398  0.683  0.470  676  0.675  0.098  0.981  0.0007 
0.01  398  0.712  0.584  794  0.712  0.141  0.708  0.0003  
0.02  398  0.750  0.642  618  0.714  0.179  0.626  0.0003  
0.03  398  0.775  0.679  504  0.715  0.211  0.586  0.0003  
0.04  398  0.794  0.706  424  0.715  0.239  0.562  0.0003  
0.05  398  0.809  0.728  364  0.715  0.265  0.544  0.0004  
0.06  398  0.822  0.747  316  0.716  0.288  0.530  0.0004  
0.07  398  0.833  0.763  276  0.716  0.310  0.518  0.0004  
0.08  398  0.843  0.777  244  0.716  0.328  0.505  0.0004  
(0.2,0.3) or (0.7,0.8)  0.001  588  0.701  0.495  924  0.672  0.100  0.985  0.0005 
0.01  588  0.746  0.633  922  0.711  0.149  0.727  0.0002  
0.02  588  0.787  0.695  666  0.714  0.188  0.644  0.0002  
0.03  588  0.812  0.731  524  0.715  0.218  0.601  0.0003  
0.04  588  0.829  0.756  432  0.715  0.245  0.573  0.0003  
0.05  588  0.842  0.775  366  0.716  0.269  0.552  0.0003  
0.06  588  0.853  0.791  316  0.716  0.291  0.535  0.0003  
0.07  588  0.863  0.805  276  0.716  0.311  0.520  0.0003  
0.08  588  0.871  0.816  244  0.716  0.328  0.506  0.0003  
(0.3,0.4) or (0.6,0.7)  0.001  712  0.704  0.500  1100  0.671  0.101  0.986  0.0004 
0.01  712  0.756  0.647  1034  0.711  0.153  0.736  0.0002  
0.02  712  0.800  0.713  714  0.714  0.193  0.654  0.0005  
0.03  712  0.825  0.751  544  0.714  0.224  0.611  0.0002  
0.04  712  0.843  0.776  440  0.715  0.250  0.581  0.0002  
0.05  712  0.856  0.795  368  0.715  0.273  0.559  0.0002  
0.06  712  0.867  0.810  314  0.716  0.294  0.540  0.0002  
0.07  712  0.875  0.823  274  0.716  0.312  0.523  0.0002  
0.08  712  0.882  0.833  244  0.716  0.329  0.506  0.0003  
(0.4,0.5) or (0.5,0.6)  0.001  776  0.706  0.502  1190  0.670  0.101  0.986  0.0004 
0.01  776  0.760  0.652  1096  0.711  0.154  0.740  0.0002  
0.02  776  0.804  0.720  744  0.713  0.195  0.659  0.0002  
0.03  776  0.831  0.758  558  0.714  0.226  0.615  0.0002  
0.04  776  0.849  0.784  446  0.715  0.253  0.586  0.0002  
0.05  776  0.862  0.803  368  0.715  0.275  0.562  0.0002  
0.06  776  0.872  0.818  314  0.716  0.295  0.542  0.0002  
0.07  776  0.881  0.830  274  0.716  0.313  0.524  0.0002  
0.08  776  0.887  0.840  244  0.716  0.329  0.506  0.0002 
Sample scenarios assuming beta priors p(π _{1}) and p(π _{2}) where \(\tau _{1}^{2}=0.001\). Hypothesized values of π _{1} and π _{2} set equal to m _{1} and m _{2}, respectively, under the traditional design. Twosided α=0.05 and 1−β=0.80 assumed
(\(\tau _{1}^{2}=0.001\))  Traditional design  CEP design  

(m _{1},m _{2})  \(\tau _{2}^{2}\)  \(\hat {N}\)  Performance  CEP  N ^{∗}  Performance  E(π _{2}−π _{1}π _{2}>π _{1})  P(π _{2}>π _{1})  Marginal benefit 
(0.1,0.9)  0.01  10  0.344  0.706  14  0.640  0.728  1  0.0740 
0.02  10  0.280  0.639  16  0.615  0.682  1  0.0559  
0.03  10  0.239  0.586  20  0.655  0.640  0.999  0.0416  
0.04  10  0.208  0.539  24  0.659  0.601  0.997  0.0322  
0.05  10  0.184  0.497  30  0.671  0.564  0.991  0.0244  
0.06  10  0.163  0.459  38  0.687  0.528  0.979  0.0187  
0.07  10  0.144  0.424  50  0.693  0.494  0.955  0.0137  
0.08  10  0.127  0.389  70  0.703  0.459  0.913  0.0096  
(0.1,0.8)  0.01  14  0.439  0.748  16  0.586  0.656  1  0.0735 
0.02  14  0.389  0.697  20  0.623  0.624  1  0.0390  
0.03  14  0.353  0.652  24  0.651  0.592  0.999  0.0298  
0.04  14  0.324  0.611  28  0.667  0.561  0.997  0.0245  
0.05  14  0.299  0.574  34  0.674  0.533  0.990  0.0187  
0.06  14  0.278  0.540  44  0.687  0.505  0.975  0.0136  
0.07  14  0.259  0.509  56  0.695  0.480  0.950  0.0104  
0.08  14  0.241  0.480  72  0.703  0.455  0.910  0.0080  
(0.1,0.7)  0.01  20  0.507  0.773  22  0.603  0.572  1  0.0481 
0.02  20  0.466  0.732  26  0.623  0.552  1  0.0262  
0.03  20  0.438  0.695  30  0.643  0.532  0.999  0.0205  
0.04  20  0.416  0.662  36  0.667  0.513  0.995  0.0157  
0.05  20  0.397  0.633  42  0.680  0.494  0.986  0.0129  
0.06  20  0.382  0.607  52  0.691  0.478  0.969  0.0097  
0.07  20  0.369  0.585  62  0.696  0.464  0.943  0.0078  
0.08  20  0.358  0.566  76  0.705  0.450  0.906  0.0062  
(0.2,0.8)  0.01  20  0.430  0.734  24  0.601  0.561  1  0.0428 
0.02  20  0.382  0.676  30  0.641  0.528  0.999  0.0259  
0.03  20  0.349  0.628  38  0.666  0.499  0.995  0.0176  
0.04  20  0.324  0.589  48  0.681  0.473  0.982  0.0127  
0.05  20  0.305  0.557  62  0.693  0.452  0.960  0.0092  
0.06  20  0.289  0.530  76  0.699  0.434  0.927  0.0073  
0.07  20  0.276  0.508  96  0.706  0.418  0.881  0.0057  
0.08  20  0.264  0.488  118  0.710  0.403  0.821  0.0046  
(0.1,0.6)  0.01  28  0.511  0.770  32  0.604  0.483  1  0.0231 
0.02  28  0.490  0.737  36  0.631  0.474  1  0.0175  
0.03  28  0.477  0.708  40  0.651  0.465  0.998  0.0145  
0.04  28  0.467  0.685  46  0.667  0.457  0.991  0.0111  
0.05  28  0.461  0.667  54  0.685  0.452  0.978  0.0086  
0.06  28  0.458  0.653  62  0.691  0.448  0.959  0.0069  
0.07  28  0.457  0.643  70  0.698  0.446  0.934  0.0057  
0.08  28  0.458  0.636  78  0.704  0.446  0.903  0.0049  
(0.2,0.7)  0.01  30  0.479  0.750  36  0.608  0.477  1  0.0214 
0.02  30  0.447  0.702  44  0.649  0.458  0.999  0.0144  
0.03  30  0.426  0.664  54  0.671  0.441  0.991  0.0102  
0.04  30  0.411  0.636  66  0.686  0.427  0.974  0.0076  
0.05  30  0.401  0.615  78  0.695  0.417  0.947  0.0061  
0.06  30  0.395  0.599  92  0.702  0.410  0.912  0.0050  
0.07  30  0.391  0.587  106  0.706  0.404  0.868  0.0042  
0.08  30  0.388  0.578  122  0.711  0.400  0.816  0.0035  
(0.1,0.5)  0.01  40  0.490  0.751  48  0.614  0.392  1  0.0155 
0.02  40  0.493  0.725  54  0.643  0.392  0.999  0.0107  
0.03  40  0.498  0.708  60  0.666  0.395  0.993  0.0084  
0.04  40  0.504  0.697  66  0.678  0.400  0.981  0.0067  
0.05  40  0.514  0.692  72  0.689  0.407  0.965  0.0055  
0.06  40  0.525  0.692  76  0.695  0.417  0.945  0.0047  
0.07  40  0.538  0.694  80  0.701  0.429  0.923  0.0041  
0.08  40  0.552  0.699  80  0.705  0.442  0.899  0.0038  
(0.2,0.6)  0.01  46  0.487  0.742  56  0.624  0.388  1  0.0137 
0.02  46  0.475  0.704  68  0.659  0.380  0.996  0.0083  
0.03  46  0.471  0.679  82  0.678  0.376  0.981  0.0057  
0.04  46  0.472  0.665  94  0.691  0.376  0.956  0.0046  
0.05  46  0.477  0.658  106  0.698  0.379  0.925  0.0037  
0.06  46  0.484  0.655  114  0.704  0.384  0.890  0.0032  
0.07  46  0.493  0.655  120  0.707  0.390  0.851  0.0029  
0.08  46  0.502  0.658  126  0.711  0.396  0.810  0.0026  
(0.3,0.7)  0.01  48  0.465  0.724  62  0.630  0.379  1  0.0118 
0.02  48  0.440  0.673  82  0.670  0.363  0.990  0.0068  
0.03  48  0.429  0.643  104  0.689  0.354  0.963  0.0046  
0.04  48  0.426  0.626  124  0.698  0.349  0.924  0.0036  
0.05  48  0.425  0.616  144  0.705  0.347  0.879  0.0029  
0.06  48  0.427  0.610  158  0.708  0.347  0.829  0.0026  
0.07  48  0.431  0.607  172  0.711  0.348  0.776  0.0023  
0.08  48  0.435  0.605  182  0.713  0.349  0.718  0.0021  
(0.1,0.4)  0.01  64  0.497  0.735  82  0.635  0.300  0.999  0.0077 
0.02  64  0.519  0.720  92  0.667  0.312  0.992  0.0053  
0.03  64  0.540  0.717  98  0.680  0.328  0.977  0.0041  
0.04  64  0.563  0.722  102  0.691  0.346  0.960  0.0034  
0.05  64  0.585  0.730  100  0.696  0.366  0.942  0.0031  
0.06  64  0.608  0.741  96  0.700  0.388  0.925  0.0029  
0.07  64  0.632  0.754  90  0.703  0.412  0.909  0.0027  
0.08  64  0.656  0.768  84  0.706  0.437  0.895  0.0025  
(0.2,0.5)  0.01  78  0.496  0.726  104  0.644  0.297  0.998  0.0057 
0.02  78  0.505  0.701  126  0.677  0.302  0.982  0.0036  
0.03  78  0.521  0.695  140  0.691  0.313  0.954  0.0027  
0.04  78  0.540  0.698  148  0.699  0.327  0.922  0.0023  
0.05  78  0.559  0.705  148  0.704  0.342  0.890  0.0021  
0.06  78  0.579  0.715  144  0.707  0.358  0.860  0.0019  
0.07  78  0.599  0.726  138  0.709  0.375  0.831  0.0018  
0.08  78  0.620  0.738  130  0.711  0.393  0.804  0.0018  
(0.3,0.6)  0.01  84  0.474  0.708  120  0.652  0.291  0.997  0.0049 
0.02  84  0.477  0.677  156  0.684  0.290  0.970  0.0029  
0.03  84  0.489  0.668  180  0.697  0.297  0.928  0.0022  
0.04  84  0.504  0.669  194  0.704  0.305  0.882  0.0018  
0.05  84  0.520  0.674  198  0.707  0.315  0.838  0.0016  
0.06  84  0.536  0.681  198  0.710  0.325  0.795  0.0015  
0.07  84  0.551  0.689  194  0.712  0.336  0.753  0.0015  
0.08  84  0.566  0.698  188  0.713  0.346  0.712  0.0014  
(0.1,0.3)  0.01  124  0.525  0.718  182  0.671  0.214  0.987  0.0025 
0.02  124  0.574  0.730  186  0.688  0.242  0.960  0.0018  
0.03  124  0.614  0.749  174  0.695  0.271  0.937  0.0016  
0.04  124  0.650  0.768  156  0.699  0.301  0.920  0.0016  
0.05  124  0.681  0.787  138  0.702  0.332  0.908  0.0015  
0.06  124  0.710  0.805  120  0.704  0.363  0.899  0.0015  
0.07  124  0.738  0.822  102  0.704  0.397  0.894  0.0015  
0.08  124  0.765  0.840  86  0.705  0.433  0.891  0.0016  
(0.2,0.4)  0.01  164  0.524  0.711  254  0.678  0.210  0.979  0.0017 
0.02  164  0.565  0.718  272  0.695  0.233  0.934  0.0012  
0.03  164  0.602  0.735  260  0.702  0.258  0.895  0.0010  
0.04  164  0.635  0.753  236  0.706  0.283  0.864  0.0010  
0.05  164  0.665  0.771  208  0.708  0.309  0.841  0.0010  
0.06  164  0.692  0.789  182  0.710  0.335  0.822  0.0010  
0.07  164  0.718  0.805  156  0.710  0.362  0.809  0.0010  
0.08  164  0.742  0.822  134  0.711  0.389  0.798  0.0010  
(0.3,0.5)  0.01  186  0.509  0.697  314  0.683  0.205  0.971  0.0014 
0.02  186  0.545  0.701  348  0.699  0.224  0.909  0.0010  
0.03  186  0.580  0.717  336  0.705  0.244  0.857  0.0008  
0.04  186  0.610  0.734  310  0.708  0.265  0.815  0.0008  
0.05  186  0.638  0.751  280  0.711  0.285  0.780  0.0008  
0.06  186  0.663  0.767  250  0.711  0.304  0.751  0.0008  
0.07  186  0.686  0.782  222  0.713  0.324  0.726  0.0008  
0.08  186  0.707  0.796  194  0.714  0.343  0.704  0.0008  
(0.4,0.6)  0.01  194  0.495  0.684  354  0.686  0.200  0.962  0.0012 
0.02  194  0.526  0.685  406  0.702  0.214  0.885  0.0008  
0.03  194  0.556  0.698  402  0.707  0.229  0.820  0.0007  
0.04  194  0.582  0.712  382  0.710  0.244  0.767  0.0007  
0.05  194  0.605  0.726  354  0.712  0.258  0.722  0.0007  
0.06  194  0.626  0.739  326  0.713  0.272  0.682  0.0007  
0.07  194  0.645  0.751  300  0.714  0.284  0.646  0.0007  
0.08  194  0.662  0.763  274  0.715  0.296  0.610  0.0007  
(0.1,0.2)  0.01  398  0.626  0.752  562  0.701  0.145  0.897  0.0005 
0.02  398  0.703  0.799  402  0.705  0.190  0.870  0.0005  
0.03  398  0.751  0.830  302  0.706  0.230  0.861  0.0005  
0.04  398  0.786  0.853  234  0.706  0.268  0.861  0.0005  
0.05  398  0.814  0.872  184  0.707  0.305  0.864  0.0005  
0.06  398  0.837  0.888  146  0.707  0.343  0.870  0.0005  
0.07  398  0.859  0.903  116  0.707  0.384  0.878  0.0005  
0.08  398  0.879  0.917  88  0.706  0.429  0.887  0.0006  
(0.2,0.3)  0.01  588  0.623  0.747  874  0.704  0.140  0.864  0.0003 
0.02  588  0.697  0.792  630  0.708  0.180  0.813  0.0003  
0.03  588  0.743  0.823  470  0.710  0.216  0.791  0.0003  
0.04  588  0.778  0.846  362  0.711  0.250  0.782  0.0003  
0.05  588  0.806  0.865  284  0.711  0.283  0.779  0.0003  
0.06  588  0.829  0.881  224  0.711  0.316  0.780  0.0003  
0.07  588  0.849  0.895  176  0.711  0.350  0.785  0.0003  
0.08  588  0.868  0.909  138  0.711  0.386  0.793  0.0004  
(0.3,0.4)  0.01  712  0.614  0.739  1126  0.706  0.136  0.841  0.0002 
0.02  712  0.684  0.783  830  0.710  0.171  0.774  0.0002  
0.03  712  0.730  0.813  628  0.711  0.202  0.740  0.0002  
0.04  712  0.764  0.836  488  0.712  0.231  0.720  0.0002  
0.05  712  0.791  0.854  388  0.713  0.259  0.708  0.0002  
0.06  712  0.814  0.870  310  0.713  0.286  0.701  0.0003  
0.07  712  0.834  0.884  250  0.713  0.313  0.697  0.0003  
0.08  712  0.852  0.896  200  0.714  0.341  0.697  0.0003  
(0.4,0.5)  0.01  776  0.605  0.732  1298  0.707  0.132  0.823  0.0002 
0.02  776  0.672  0.773  984  0.711  0.163  0.742  0.0002  
0.03  776  0.715  0.802  762  0.712  0.189  0.698  0.0002  
0.04  776  0.747  0.824  608  0.713  0.213  0.668  0.0002  
0.05  776  0.773  0.841  494  0.714  0.234  0.646  0.0002  
0.06  776  0.794  0.856  406  0.714  0.255  0.629  0.0002  
0.07  776  0.813  0.869  338  0.715  0.275  0.614  0.0002  
0.08  776  0.829  0.880  282  0.715  0.294  0.602  0.0002 
Sample scenarios assuming beta priors p(π _{1}) and p(π _{2}) where \(\tau _{1}^{2}=0.08\). Hypothesized values of π _{1} and π _{2} set equal to m _{1} and m _{2}, respectively, under the traditional design. Twosided α=0.05 and 1−β=0.80 assumed
(\(\tau _{1}^{2}=0.08\))  Traditional design  CEP design  

(m _{1},m _{2})  \(\tau _{2}^{2}\)  \(\hat {N}\)  Performance  CEP  N ^{∗}  Performance  E(π _{2}−π _{1}π _{2}>π _{1})  P(π _{2}>π _{1})  Marginal benefit 
(0.1,0.9)  0.001  10  0.127  0.389  70  0.703  0.459  0.913  0.0096 
0.01  10  0.097  0.356  88  0.707  0.435  0.861  0.0078  
0.02  10  0.083  0.335  102  0.709  0.417  0.815  0.0068  
0.03  10  0.075  0.319  118  0.711  0.402  0.774  0.0059  
0.04  10  0.069  0.306  132  0.712  0.389  0.734  0.0053  
0.05  10  0.064  0.294  148  0.713  0.378  0.693  0.0047  
0.06  10  0.060  0.283  166  0.713  0.366  0.650  0.0042  
0.07  10  0.056  0.273  186  0.714  0.355  0.602  0.0037  
(0.1,0.8)  0.001  14  0.147  0.399  114  0.710  0.407  0.825  0.0056 
0.01  14  0.134  0.387  124  0.710  0.396  0.794  0.0052  
0.02  14  0.128  0.377  134  0.711  0.386  0.760  0.0049  
0.03  14  0.124  0.368  146  0.712  0.378  0.728  0.0045  
0.04  14  0.121  0.360  158  0.713  0.370  0.695  0.0041  
0.05  14  0.118  0.353  170  0.713  0.362  0.661  0.0038  
0.06  14  0.116  0.346  184  0.714  0.355  0.625  0.0035  
0.07  14  0.113  0.339  198  0.715  0.348  0.586  0.0033  
(0.1,0.7)  0.001  20  0.174  0.415  172  0.713  0.355  0.731  0.0035 
0.01  20  0.172  0.414  176  0.713  0.353  0.712  0.0035  
0.02  20  0.174  0.413  182  0.713  0.350  0.691  0.0033  
0.03  20  0.177  0.412  188  0.714  0.348  0.669  0.0032  
0.04  20  0.180  0.411  192  0.714  0.346  0.646  0.0031  
0.05  20  0.183  0.410  200  0.715  0.344  0.622  0.0030  
0.06  20  0.185  0.409  206  0.714  0.342  0.596  0.0028  
0.07  20  0.187  0.408  214  0.715  0.340  0.567  0.0027  
(0.2,0.8)  0.001  20  0.264  0.488  118  0.710  0.403  0.821  0.0046 
0.01  20  0.247  0.475  128  0.711  0.393  0.789  0.0043  
0.02  20  0.233  0.462  138  0.712  0.384  0.755  0.0041  
0.03  20  0.223  0.451  150  0.712  0.375  0.723  0.0038  
0.04  20  0.215  0.442  162  0.713  0.367  0.690  0.0035  
0.05  20  0.208  0.433  174  0.713  0.360  0.656  0.0033  
0.06  20  0.201  0.424  188  0.714  0.353  0.620  0.0031  
0.07  20  0.195  0.416  202  0.715  0.346  0.580  0.0029  
(0.1,0.6)  0.001  28  0.193  0.426  252  0.715  0.303  0.632  0.0023 
0.01  28  0.199  0.431  250  0.715  0.307  0.623  0.0023  
0.02  28  0.209  0.437  246  0.715  0.311  0.613  0.0023  
0.03  28  0.220  0.443  244  0.715  0.316  0.601  0.0023  
0.04  28  0.231  0.449  240  0.715  0.320  0.589  0.0023  
0.05  28  0.241  0.456  238  0.715  0.324  0.576  0.0023  
0.06  28  0.252  0.463  234  0.715  0.328  0.562  0.0022  
0.07  28  0.262  0.469  230  0.715  0.332  0.546  0.0022  
(0.2,0.7)  0.001  30  0.302  0.510  178  0.713  0.352  0.725  0.0028 
0.01  30  0.298  0.507  182  0.713  0.350  0.706  0.0027  
0.02  30  0.295  0.504  186  0.714  0.348  0.685  0.0027  
0.03  30  0.293  0.502  192  0.714  0.346  0.663  0.0026  
0.04  30  0.292  0.499  198  0.714  0.344  0.640  0.0025  
0.05  30  0.291  0.497  204  0.714  0.342  0.616  0.0024  
0.06  30  0.291  0.495  210  0.715  0.340  0.590  0.0024  
0.07  30  0.291  0.493  218  0.715  0.338  0.561  0.0023  
(0.1,0.5)  0.001  40  0.213  0.437  364  0.716  0.253  0.531  0.0016 
0.01  40  0.226  0.448  350  0.716  0.262  0.530  0.0016  
0.02  40  0.244  0.461  334  0.716  0.272  0.529  0.0016  
0.03  40  0.264  0.474  318  0.716  0.282  0.528  0.0016  
0.04  40  0.283  0.488  300  0.715  0.293  0.527  0.0017  
0.05  40  0.303  0.502  284  0.716  0.303  0.526  0.0017  
0.06  40  0.323  0.516  266  0.716  0.313  0.525  0.0017  
0.07  40  0.342  0.530  248  0.716  0.324  0.523  0.0018  
(0.2,0.6)  0.001  46  0.343  0.536  258  0.715  0.301  0.625  0.0018 
0.01  46  0.347  0.539  256  0.715  0.305  0.617  0.0018  
0.02  46  0.352  0.543  252  0.715  0.309  0.606  0.0018  
0.03  46  0.359  0.547  250  0.715  0.314  0.595  0.0017  
0.04  46  0.366  0.552  246  0.715  0.318  0.583  0.0017  
0.05  46  0.373  0.556  242  0.715  0.322  0.570  0.0017  
0.06  46  0.382  0.562  238  0.715  0.326  0.556  0.0017  
0.07  46  0.390  0.567  234  0.716  0.330  0.540  0.0017  
(0.3,0.7)  0.001  48  0.435  0.605  182  0.713  0.349  0.718  0.0021 
0.01  48  0.431  0.602  186  0.713  0.347  0.700  0.0020  
0.02  48  0.427  0.599  192  0.714  0.345  0.679  0.0020  
0.03  48  0.423  0.596  196  0.714  0.343  0.657  0.0020  
0.04  48  0.419  0.592  202  0.714  0.342  0.634  0.0019  
0.05  48  0.416  0.589  208  0.715  0.340  0.610  0.0019  
0.06  48  0.414  0.586  214  0.715  0.338  0.584  0.0018  
0.07  48  0.411  0.583  222  0.715  0.336  0.555  0.0017  
(0.1,0.4)  0.001  64  0.258  0.468  532  0.716  0.202  0.428  0.0010 
0.01  64  0.281  0.486  494  0.716  0.217  0.435  0.0010  
0.02  64  0.308  0.506  452  0.716  0.234  0.444  0.0011  
0.03  64  0.336  0.526  412  0.716  0.250  0.454  0.0011  
0.04  64  0.363  0.546  374  0.716  0.267  0.464  0.0011  
0.05  64  0.391  0.566  338  0.716  0.283  0.475  0.0012  
0.06  64  0.418  0.586  302  0.716  0.299  0.487  0.0013  
0.07  64  0.446  0.607  268  0.716  0.316  0.501  0.0013  
(0.2,0.5)  0.001  78  0.404  0.577  374  0.716  0.251  0.524  0.0011 
0.01  78  0.414  0.585  358  0.716  0.260  0.523  0.0011  
0.02  78  0.427  0.594  342  0.716  0.270  0.522  0.0011  
0.03  78  0.441  0.604  326  0.716  0.281  0.521  0.0011  
0.04  78  0.455  0.615  308  0.716  0.291  0.521  0.0011  
0.05  78  0.470  0.625  290  0.716  0.301  0.520  0.0012  
0.06  78  0.486  0.637  272  0.716  0.312  0.519  0.0012  
0.07  78  0.503  0.649  254  0.716  0.322  0.518  0.0012  
(0.3,0.6)  0.001  84  0.497  0.647  266  0.715  0.299  0.618  0.0012 
0.01  84  0.501  0.649  264  0.715  0.303  0.609  0.0012  
0.02  84  0.505  0.652  260  0.715  0.307  0.599  0.0012  
0.03  84  0.509  0.655  256  0.715  0.311  0.588  0.0012  
0.04  84  0.514  0.658  252  0.715  0.316  0.576  0.0012  
0.05  84  0.519  0.661  248  0.715  0.320  0.564  0.0012  
0.06  84  0.524  0.665  244  0.715  0.324  0.550  0.0012  
0.07  84  0.530  0.668  240  0.716  0.328  0.534  0.0012  
(0.1,0.3)  0.001  124  0.339  0.526  808  0.717  0.153  0.323  0.0006 
0.01  124  0.376  0.554  708  0.717  0.176  0.341  0.0006  
0.02  124  0.414  0.582  612  0.717  0.200  0.361  0.0006  
0.03  124  0.450  0.608  530  0.717  0.223  0.382  0.0007  
0.04  124  0.483  0.632  458  0.717  0.245  0.404  0.0007  
0.05  124  0.515  0.656  394  0.717  0.266  0.428  0.0007  
0.06  124  0.546  0.679  338  0.717  0.287  0.452  0.0008  
0.07  124  0.578  0.702  286  0.716  0.308  0.480  0.0009  
(0.2,0.4)  0.001  164  0.502  0.647  546  0.717  0.201  0.420  0.0006 
0.01  164  0.519  0.659  506  0.716  0.216  0.428  0.0006  
0.02  164  0.537  0.672  464  0.717  0.233  0.437  0.0006  
0.03  164  0.557  0.686  422  0.717  0.249  0.447  0.0006  
0.04  164  0.576  0.700  382  0.716  0.265  0.457  0.0006  
0.05  164  0.596  0.715  344  0.716  0.282  0.469  0.0007  
0.06  164  0.617  0.730  308  0.716  0.298  0.482  0.0007  
0.07  164  0.638  0.745  272  0.716  0.314  0.495  0.0007  
(0.3,0.5)  0.001  186  0.597  0.716  384  0.716  0.249  0.516  0.0006 
0.01  186  0.605  0.721  368  0.716  0.258  0.515  0.0006  
0.02  186  0.614  0.728  352  0.716  0.269  0.515  0.0006  
0.03  186  0.623  0.735  334  0.716  0.279  0.514  0.0006  
0.04  186  0.633  0.742  314  0.716  0.289  0.514  0.0006  
0.05  186  0.644  0.749  296  0.716  0.299  0.513  0.0007  
0.06  186  0.655  0.757  278  0.716  0.310  0.513  0.0007  
0.07  186  0.667  0.766  258  0.716  0.320  0.512  0.0007  
(0.4,0.6)  0.001  194  0.662  0.763  274  0.715  0.296  0.610  0.0007 
0.01  194  0.664  0.764  270  0.715  0.301  0.602  0.0007  
0.02  194  0.667  0.766  266  0.715  0.305  0.592  0.0007  
0.03  194  0.669  0.768  262  0.715  0.309  0.581  0.0007  
0.04  194  0.672  0.770  258  0.715  0.314  0.569  0.0007  
0.05  194  0.675  0.772  254  0.715  0.318  0.557  0.0007  
0.06  194  0.679  0.774  250  0.716  0.322  0.543  0.0007  
0.07  194  0.682  0.777  244  0.716  0.326  0.528  0.0007  
(0.1,0.2)  0.001  398  0.506  0.648  1346  0.717  0.104  0.218  0.0002 
0.01  398  0.563  0.691  978  0.714  0.139  0.247  0.0003  
0.02  398  0.606  0.720  814  0.717  0.172  0.286  0.0003  
0.03  398  0.642  0.746  660  0.717  0.201  0.319  0.0003  
0.04  398  0.673  0.768  542  0.717  0.227  0.352  0.0003  
0.05  398  0.700  0.788  450  0.717  0.253  0.386  0.0003  
0.06  398  0.726  0.807  372  0.717  0.277  0.422  0.0004  
0.07  398  0.751  0.825  304  0.716  0.302  0.460  0.0004  
(0.2,0.3)  0.001  588  0.667  0.765  828  0.717  0.152  0.316  0.0002 
0.01  588  0.691  0.783  694  0.714  0.175  0.330  0.0002  
0.02  588  0.709  0.794  624  0.717  0.199  0.355  0.0002  
0.03  588  0.728  0.808  540  0.717  0.222  0.376  0.0002  
0.04  588  0.747  0.821  466  0.717  0.243  0.398  0.0002  
0.05  588  0.764  0.834  402  0.717  0.265  0.422  0.0003  
0.06  588  0.781  0.846  344  0.717  0.286  0.447  0.0003  
0.07  588  0.799  0.858  292  0.716  0.307  0.474  0.0003  
(0.3,0.4)  0.001  712  0.748  0.822  560  0.717  0.200  0.413  0.0002 
0.01  712  0.761  0.832  502  0.715  0.215  0.416  0.0002  
0.02  712  0.767  0.836  474  0.717  0.231  0.430  0.0002  
0.03  712  0.778  0.843  432  0.716  0.248  0.440  0.0002  
0.04  712  0.788  0.851  390  0.716  0.264  0.451  0.0002  
0.05  712  0.799  0.859  352  0.716  0.280  0.462  0.0002  
0.06  712  0.810  0.866  314  0.716  0.296  0.475  0.0002  
0.07  712  0.821  0.874  278  0.716  0.312  0.490  0.0002  
(0.4,0.5)  0.001  776  0.797  0.857  394  0.716  0.247  0.508  0.0002 
0.01  776  0.803  0.863  368  0.714  0.257  0.503  0.0002  
0.02  776  0.806  0.863  360  0.716  0.267  0.507  0.0002  
0.03  776  0.811  0.867  342  0.716  0.277  0.507  0.0002  
0.04  776  0.816  0.871  322  0.716  0.287  0.507  0.0002  
0.05  776  0.822  0.875  302  0.716  0.298  0.506  0.0002  
0.06  776  0.827  0.879  284  0.716  0.308  0.506  0.0002  
0.07  776  0.834  0.883  264  0.716  0.318  0.506  0.0002 
Sample scenarios assuming uniform priors p(π _{1}) and p(π _{2}) where \(\tau _{1}^{2}=\tau _{2}^{2}\). Hypothesized values of π _{1} and π _{2} set equal to μ _{1} and μ _{2}, respectively, under the traditional design. Twosided α=0.05 and 1−β=0.80 assumed
Traditional design  CEP design  

(μ _{1},μ _{2})  \(\tau _{1}^{2}=\tau _{2}^{2}\)  \(\hat {N}\)  Performance  CEP  N ^{∗}  Performance  E(π _{2}−π _{1}π _{2}>π _{1})  P(π _{2}>π _{1})  Marginal benefit 
(0.1,0.9)  0.001  10  0.638  0.826  10  0.638  0.800  1  0 
(0.1,0.8) or (0.2,0.9)  0.001  14  0.643  0.824  14  0.643  0.700  1  0 
(0.1,0.7) or (0.3,0.9)  0.001  20  0.680  0.831  20  0.680  0.600  1  0 
(0.2,0.8)  0.001  20  0.593  0.811  20  0.593  0.600  1  0 
0.01  20  0.556  0.775  22  0.615  0.600  1  0.030  
(0.1,0.6)or (0.4,0.9)  0.001  28  0.633  0.822  28  0.633  0.500  1  0 
(0.2,0.7)or (0.3,0.8)  0.001  30  0.599  0.812  30  0.599  0.500  1  0 
0.01  30  0.554  0.763  36  0.650  0.500  1  0.016  
(0.1,0.5) or (0.5,0.9)  0.001  40  0.551  0.801  40  0.551  0.400  1  0 
(0.2,0.6) or (0.4,0.8)  0.001  46  0.553  0.800  46  0.553  0.400  1  0 
0.01  46  0.535  0.735  62  0.663  0.400  1  0.008  
(0.3,0.7)  0.001  48  0.555  0.800  48  0.555  0.400  1  0 
0.01  48  0.536  0.735  64  0.660  0.400  1  0.008  
0.02  48  0.549  0.711  80  0.694  0.407  0.983  0.005  
0.03  48  0.575  0.718  82  0.702  0.427  0.945  0.004  
(0.1,0.4) or (0.6,0.9)  0.001  64  0.523  0.787  68  0.589  0.300  1  0.017 
(0.2,0.5) or (0.5,0.8)  0.001  78  0.521  0.786  82  0.581  0.300  1  0.015 
0.01  78  0.526  0.704  130  0.689  0.303  0.991  0.003  
(0.3,0.6) or (0.4,0.7)  0.001  84  0.506  0.781  90  0.591  0.300  1  0.014 
0.01  84  0.521  0.701  142  0.690  0.303  0.991  0.003  
0.02  84  0.566  0.712  150  0.704  0.330  0.925  0.002  
0.03  84  0.603  0.731  140  0.709  0.357  0.875  0.002  
(0.1,0.3) or (0.7,0.9)  0.001  124  0.510  0.764  140  0.600  0.200  1  0.006 
(0.2,0.4) or (0.6,0.8)  0.001  164  0.512  0.764  184  0.603  0.200  1  0.005 
0.01  164  0.566  0.713  292  0.703  0.224  0.911  0.001  
(0.3,0.5) or (0.5,0.7)  0.001  186  0.504  0.761  210  0.603  0.200  1  0.004 
0.01  186  0.562  0.710  336  0.704  0.224  0.911  0.001  
0.02  186  0.626  0.745  284  0.709  0.263  0.825  0.001  
0.03  186  0.669  0.772  236  0.710  0.295  0.778  0.001  
(0.4,0.6)  0.001  194  0.504  0.761  220  0.607  0.200  1  0.004 
0.01  194  0.562  0.709  350  0.704  0.224  0.911  0.001  
0.02  194  0.625  0.745  296  0.708  0.263  0.825  0.001  
0.03  194  0.668  0.771  248  0.711  0.295  0.778  0.001  
0.04  194  0.700  0.792  210  0.712  0.323  0.747  0.001  
0.05  194  0.725  0.809  194  0.725  0.349  0.725  0 
Abbreviations
 CEP:

Conditional expected power
 EP:

Expected power
Declarations
Acknowledgements
The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government.
Authors’ contributions
MMC developed the concept and performed the literature search, simulations, data analysis, interpretation of results, and manuscript writing. CDA performed simulations and data analysis, interpretation of results, and manuscript writing. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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