The complex nature of rare diseases and limited data present unique challenges in the design of clinical trials [1, 2, 3, 4]. To better capture the intended effects of an investigational treatment, multiple endpoints, rather than a single endpoint, are commonly used in rare disease trials. For example, in the ENDEAR trial [5] that evaluated the efficacy and safety of nusinersen in infants with spinal muscular atrophy (SMA), both the motor milestone response and event-free survival were used as primary endpoints. Multiple endpoints allow for a more comprehensive assessment of treatment effects, particularly when there are no historical data available to prioritize one aspect of the disease over another. Furthermore, when the drug has an effect on each of the multiple endpoints, using multiple endpoints may improve statistical power and reduce the required sample size by aggregating information in different clinically significant outcomes [6, 7]. In the setting of dual primary endpoints, where statistical significance can be claimed if any endpoint is statistically significant, there is a multiplicity issue, and the family-wise type I error rate (FWER) must be strongly controlled in order to make valid endpoint-level claims [7, 8, 9]. In this setting, multiple multiplicity adjustment methods can be used to strongly control the FWER [7, 10, 11, 12, 13].

In the setting of rare disease trials, when a single endpoint cannot be a complete representation of the treatment effect, two common methods are used to control the type I error. One is using the composite endpoint, which combines multiple clinical results with the same type of data into a single variable [7, 14]. Another is the co-primary endpoint approach [15], which requires statistical significance of all co-primary endpoints [7, 16]. In this paper, we consider the global testing method, which aggregates multivariate summary statistics [17, 18, 19] such as z-scores [3, 20, 21] and p-values [22, 23] into a single univariate test framework to assess the totality of evidence. Sun et al. [20] proposed a method using the average z-scores as a univariate test statistic, coupled with the Wilcoxon rank-sum test [20]. Li et al. [21] introduced a nonparametric approach through a permutation test based on the average z-scores. Although this approach was originally applied to combine the same data type of endpoints, it can also be generalized to apply to mixed data types of endpoints. A modification of Sun et al.’s method was developed by replacing the Wilcoxon rank-sum test with an exact small sample ordinary least squares (OLS) test, called the z-score OLS, proposed by Zhang et al. [3]. They further extended this approach to the hybrid OLS, which combines individual endpoint’s test statistics (e.g., t-test) into a univariate OLS test statistic, with correlations among test statistics estimated via permutation.

Simulations comparing these methods with non-prioritized mixed type endpoints by Zhang et al. [3] suggest that the hybrid OLS provides type I error control and tends to be conservative, particularly in smaller sample sizes. Regarding power, both the z-score OLS and the hybrid OLS perform well across various scenarios for moderate to no correlations between endpoints.

A common challenge in the setting of rare diseases is that clinical data and scientific knowledge are limited. As a result, it may not be straightforward to pre-specify the endpoints and their magnitude of treatment effect to support regulatory approval when designing a clinical trial for development of a new therapy. Conventional fixed size designs can lead to the risk of designing an under-powered study or unnecessary resource waste in cost and timelines [3]. Adaptive sample size designs allowing modifications based on interim data that are not available at the time of the trial design stage are particularly appealing in rare disease settings. These designs can potentially increase the likelihood of trial success and optimize resource utilization, which are desirable for small patient populations [24, 25]. Among these, the promising zone design based on conditional power is a widely adopted approach [26, 27, 28, 29, 30]. The core concept is partitioning the interim conditional power into three zones: favorable, promising, and unfavorable, with sample size increases occurring only in the promising zone. Originally developed by Chen, DeMets, and Lan [27] and extended by Mehta and Pocock [26], this design defines the promising zone as having conditional power greater than or equal to a specified threshold (e.g., 50%) and utilizes the conventional test statistic to control type I error inflation associated with unblinded SSR, as an alternative to the CHW weighted test statistic [31]. A recent study [30] has shown that the CHW weighted test is uniformly more powerful than the method of Mehta and Pocock within the promising zone. Despite this, the promising zone concept, as a general framework, remains highly valuable and practical in adaptive sample size designs.

In this paper, we propose a study design that allows sample size adjustment at interim based on global testing of multiple endpoints when considering the totality of evidence in the context of rare diseases. Two global tests are considered including the OLS test and a nonparametric permutation test. Sample size re-estimation is conducted within a generalized promising zone framework, where the zones are determined based on the interim conditional power. In the promising zone, two distinct SSR approaches are considered: one is based on power (SSR-Power) and the other on conditional power (SSR-CP). For SSR-Power, the sample size is recalculated using the original power formula that is used for initial planning of the trial, but with updated estimates for the effect size and other nuisance parameters from interim data. In contrast, SSR-CP re-estimates the sample size based on the conditional power calculated using the interim data. We derive the power and conditional power formulas in the setting of multiple endpoints. The SSR-Power approach achieves power directly, irrespective of the timing of the interim analysis; while the SSR-CP approach achieves conditional power, i.e., the chance of statistical success, which incorporates the timing of the interim analysis. Additionally, we integrate the weighted combination test proposed by Cui et al. [31] and Lehmacher and Wassmer [32] to ensure rigorous type I error control. This integration allows greater flexibility in defining conditional power thresholds, which need not be identical to those proposed by Chen et al. (2004) [27] or Mehta and Pocock (2011) [26]. It is also straightforward to implement using traditional group sequential design boundaries and offers notable flexibility by allowing data from two stages to be combined without relying on interim decision criteria. We evaluate the performance of the proposed design through the simulation study under various scenarios, including settings with small sample sizes.

The remainder of the paper is organized as follows. Section 2 introduces the proposed study design, including two types of totality-of-evidence tests and two SSR procedures. Section 3 details the setup, scenarios and implementation of the simulation study. Section 4 reports the empirical results of the simulation study. Section 5 provides an example illustrating how to use the proposed design with our open-source R package SSRTE. Finally, Section 6 discusses limitations and practical considerations in implementing the method.

To investigate the impact of the proposed adaptive design on type I error control and power in various scenarios, we performed an extensive series of simulations with a two-stage setting and a target power of 80%. The trial considered a total of six clinically important endpoints, generated from a multivariate normal distribution with arm-specific means μ T , μ C and a common covariance matrix Σ. The control mean was set to μ C = μ = ( 4 , 6 , 4 , 5 , 7 , 6 ) T , and the treatment mean was defined by the overall treatment effect and the standard deviation of each endpoint as μ T , k = μ C , k + θ ¯ σ k , ∀ k = 1 , … , 6. For Σ, two sets of values were considered: Σ 0 and Σ 1 . Σ 0 assumes equal variances and equal correlations across the six endpoints, having the diagonal values of 1 and the off-diagonal values of 0.3. That is, standard deviation σ k = 1, ∀ k = 1 , … , 6 and correlation coefficient ρ p q = 0.3, ∀ 1 ≤ p < q ≤ 6. In contrast, Σ 1 assumes varying variances and correlations across endpoints, with σ = ( 0.5 , 0.5 , 1 , 1 , 2 , 2 ) T and ρ = 1 0.1 0.3 0.7 0.1 0.3 0.1 1 0.7 0.1 0.3 0.7 0.3 0.7 1 0.1 0.3 0.7 0.7 0.1 0.1 1 0.1 0.3 0.1 0.3 0.3 0.1 1 0.7 0.3 0.7 0.7 0.3 0.7 1 .

Thus, Σ 1 = 0.25 0.025 0.15 0.35 0.1 0.3 0.025 0.25 0.35 0.05 0.3 0.7 0.15 0.35 1 0.1 0.6 1.4 0.35 0.05 0.1 1 0.2 0.6 0.1 0.3 0.6 0.2 4 2.8 0.3 0.7 1.4 0.6 2.8 4 .

The simulation can be reproduced using the SSRTE_simstudy() function from our R package SSRTE, which is available on GitHub. Installation instructions are provided in Section 5.

Because the simulation results under Σ 0 and Σ 1 are highly similar, we present only the results under Σ 0 in the main text and include the results under Σ 1 in Appendix B.

We present an illustrative example demonstrating how users can implement the proposed design using our user-friendly R package SSRTE, available on https://github.com/Slan1997/SSRTE. The package can be easily installed and loaded with the following R commands:

Suppose we aim to design a two-stage clinical trial with six continuous endpoints, an allocation ratio of r = 1, a one-sided type I error rate of α = 2.5 %, a target power of 1 − β = 80 %, and an interim analysis planned at 50% information fraction.

At the initial design stage, based on limited prior information, we assume that the best estimate of the mean Cohen’s d ( θ ¯) is 0.4, and the pairwise correlation among endpoints is ρ p q = 0.5 for all 1 ≤ p < q ≤ 6. The initially planned total and stage 1 sample sizes can be calculated using the function get_initial_tot_ss() as follows:

The planned total sample size is 116, with 58 participants in the interim stage, equally divided between treatment and control groups.

Next, stage 1 data is collected. For illustration purposes, a simulated data set is used as the true data, with a true control arm mean vector μ C = ( 4 , 6 , 4 , 5 , 7 , 6 ) T , a true mean Cohen’s d of 0.3, shared standard deviations across arms σ = ( 1.5 , 0.5 , 1.6 , 1 , 0.6 , 1.2 ) T and the true correlation matrix ρ = 1 0.1 0.3 0.5 0.1 0.3 0.1 1 0.5 0.1 0.3 0.5 0.3 0.5 1 0.1 0.3 0.5 0.5 0.1 0.1 1 0.1 0.3 0.1 0.3 0.3 0.1 1 0.5 0.3 0.5 0.5 0.3 0.5 1 . From a reproducible research perspective, using a simulated dataset that can be publicly shared is preferable to relying on real data that cannot be made available.

The control and treatment data for stage 1 are generated using the following code:

The interim analysis is then performed using the function interim_analysis(). Suppose the exact OLS test and SSR-CP are used, we can create an interim analysis report by following commands (if the permutation test is used, a random seed is needed to ensure reproducibility):

The output of print(ia) is presented in Figure 5, which indicates that the null hypothesis is not rejected at interim ( z 1 ∗ = 2.07 < Z α 1 = 2.96). The conditional power is estimated as 41.3%, which falls within the promising zone ( 0.2 , 0.8 ), leading to a re-estimation of the sample size: 61 participants per arm for stage 2 (122 total) and 90 per arm for the overall final (180 total).

A new set of stage 2 data is simulated to represent the true data collected in this stage:

The same underlying distributional parameters as those used in sim1 are applied here and are therefore omitted.

Finally, the final analysis is conducted using the function final_analysis(), which takes the interim analysis results and stage 2 data as inputs:

The final analysis report (Figure 6) shows that z f i n a l ∗ = 3.45 > Z α 2 = 1.97, and therefore the null hypothesis is rejected.

We have proposed a sample size re-estimation design based on the promising zone framework that incorporates two distinct totality of evidence methods to perform a global test of mean effect size across multiple endpoints. Specifically, we begin by considering an exact OLS test tailored for small sample sizes, as the asymptotic theory in the approximate z-test becomes unreliable in rare populations where sample sizes are typically very small [3]. Additionally, we adopt a nonparametric test due to its robustness and independence from assumptions about the underlying distribution of the test statistics. For the SSR approach, we explore two distinct methods: one is based on power (SSR-Power) and the other on conditional power (SSR-CP). To facilitate these approaches, we have derived the initial sample size formula as well as a conditional power formula capable of handling global tests for multiple endpoints.

To assess the impact of our approach on type I error control and power, we conducted a simulation study under various scenarios, including different timings of interim analyses and initial sample size settings, within a two-stage SSR design framework involving six continuous endpoints.

The results of our simulation study demonstrate that the proposed design, which incorporates both the totality of evidence tests and SSR methods, effectively controls type I error rates in trials of rare diseases with multiple continuous endpoints. In particular, the OLS test offers stricter type I error control in smaller sample settings, while the permutation test experiences a large variability in all scenarios due to its random resampling nature. Regarding power assessment, in general, the SSR design maintains adequate power even when the initially planned sample size is underestimated or the treatment effect size is overly optimistic. Specifically, the permutation test achieves slightly higher power and requires fewer samples in most scenarios. Consequently, we recommend the permutation test for situations where strict type I error control is not required, and when users prefer a less assumption-dependent approach with a moderately timed interim analysis. Furthermore, SSR-CP consistently achieves higher power than SSR-Power but tends to require larger sample sizes and more frequently reaches the maximum allowable sample size. Lastly, conducting the interim analysis at a later stage consistently helps reduce the variability in the type I error of the permutation test and enhances the trial’s power for both tests.

Our study has limitations. We consider situations where there is no clear hierarchical structure among endpoints. For prioritized endpoints, more specialized methods may be considered [42, 43]. Moreover, our current simulation assumes that all endpoints are continuous, whereas some trials can include other data types. For example, when dealing with binary endpoints, using Cohen’s d as the global effect size measure may not be appropriate, since the assumption of common variance is violated by nature. In Chapter 4.3.8 of Zhang et al. (2023) [3], the authors proposed using z-scores to handle mixed types of endpoints and conducted simulations comparing different tests, such as the z-score OLS, the permutation test and hybrid OLS. However, their z-scores for all continuous, binary and ordinal endpoints are obtained by “subtracting the pooled groups’ mean and dividing the pooled groups’ standard deviation.” Thus, these z-scores also assume a common variance, which suffers from the same limitation as Cohen’s d when applied to binary endpoints. More research is needed to develop methods that can appropriately accommodate different types of endpoints.

Furthermore, to test the totality of evidence we use the unweighted average of Cohen’s d across endpoints, which implicitly assumes that all endpoints have equal clinical importance. When endpoints have heterogeneous clinical relevance to the treatment, a weighted average of Cohen’s d can be used instead to reflect their relative importance. Note that the use of Cohen’s d in this work does not replace endpoint-specific estimands or model-based analyses. Rather, it is used as a standardized parameter to facilitate a global assessment of treatment effect across multiple endpoints measured on different scales. This approach is intended for settings in which multiple clinically relevant endpoints exist, a single primary endpoint may be difficult to prespecify, and a global assessment is scientifically justified. Endpoint-specific analyses and clinically interpretable estimands remain essential and complementary components of the overall analysis strategy.

In addition, we only implement efficacy boundaries in our sequential design. However, it is possible to add futility boundaries if there is sufficient insight on the minimum clinically important difference. In such cases, during interim analysis, the critical values for efficacy and futility, denoted as Z e , 1 and Z f , 1 , respectively, could be used to refine the rejection rule at the interim stage as follows:

Finally, the current design may lead to an excessive boost in power when initial estimates are accurate, as it allows for retaining or increasing the sample size but not decreasing it. This could potentially result in an inefficient use of resources for some studies. Therefore, decisions on whether or not to increase the sample size must be made carefully and tailored to the specific requirements of each study. The study with adaptive sample size is more complex in conduct; therefore, it is essential to perform extensive simulations to assess the operating characteristics. In addition, for trial integrity considerations, information access and the process of performing SSR should be well planned in advance.

At the time the research was conducted, Yong Lin, Philip He, and Di Shu were employees of Daiichi Sankyo, Inc. Please note that the views and opinions expressed in this paper are those of the authors and are not intended to reflect the views and/or opinions of their employer(s).