Blinded sample size re-estimation accounting for uncertainty in mid-trial estimation

For randomized controlled trials to be conclusive, it is important to set the target sample size accurately at the design stage. Comparing two normal populations, the sample size calculation requires specification of the variance other than the treatment effect and misspecification can lead to underpowered studies. Blinded sample size re-estimation is an approach to minimize the risk of inconclusive studies. Existing methods proposed to use the total (one-sample) variance that is estimable from blinded data without knowledge of the treatment allocation. We demonstrate that, since the expectation of this estimator is greater than or equal to the true variance, the one-sample variance approach can be regarded as providing an upper bound of the variance in blind reviews. This worst-case evaluation can likely reduce a risk of underpowered studies. However, blinded reviews of small sample size may still lead to underpowered studies. We propose a refined method accounting for estimation error in blind reviews using an upper confidence limit of the variance. A similar idea had been proposed in the setting of external pilot studies. Furthermore, we developed a method to select an appropriate confidence level so that the re-estimated sample size attains the target power. Numerical studies showed that our method works well and outperforms existing methods. The proposed procedure is motivated and illustrated by recent randomized clinical trials.

💡 Research Summary

The paper addresses a fundamental problem in the design of randomized controlled trials that compare two normal populations: the need to specify the common variance (σ²) accurately in order to achieve the desired power (1‑β) for detecting a clinically relevant treatment difference (δ). Misspecification of σ² at the planning stage can lead to underpowered studies, especially when external information is scarce or unreliable. Internal pilot studies—mid‑trial data collected before the final analysis—offer a way to re‑estimate the required sample size. However, when the pilot data are reviewed in a blinded fashion (i.e., treatment allocation remains hidden), only the overall (one‑sample) variance can be estimated. Prior work (Gould & Shih 1992; Zucker et al. 1999) showed that this one‑sample variance estimator (σ̂²_OS) has a positive bias: its expectation exceeds the true σ², effectively providing an upper bound on the variance. While this conservatism can protect against under‑power, it is insufficient when the pilot sample is very small, as demonstrated by Kieser & Friede (2003).

To overcome this limitation, the authors propose a new blinded sample‑size re‑estimation method that explicitly accounts for the estimation uncertainty of σ² by using an upper confidence limit (UCL) for the variance. The key steps are:

1. Compute σ̂²_OS from the blinded pilot data.
2. Construct a conservative (1‑γ)×100 % upper confidence bound for σ² based on σ̂²_OS, using asymptotic χ² theory. This bound is larger than σ̂²_OS and therefore more protective.
3. Plug the UCL into the usual sample‑size formula, yielding a re‑estimated total sample size ˆn_f.
4. Choose γ so that the final analysis attains the target power. The authors derive a lower bound for the conditional power given ˆn_f and show that this bound depends only on γ, the pilot size n_int, and the significance level α, but not on the unknown σ². Consequently, γ can be pre‑specified in the protocol, independent of any variance estimate.

The method builds on earlier ideas by Browne (1995) and Kieser & Wassmer (1996), who used upper confidence limits for variance in external pilot studies. The novelty lies in adapting these concepts to the blinded internal‑pilot setting, where the variance estimator is biased and the final analysis data are not independent of the pilot data. By employing a conservative UCL, the authors obtain a tractable expression for power that facilitates the selection of γ.

Simulation studies evaluate four approaches: (i) naïve σ̂²_OS, (ii) bias‑adjusted σ̂²_ADJ, (iii) the inflation‑factor method of Friede & Kieser (2013), and (iv) the proposed UCL method. Across a range of pilot sizes (12–30 subjects) and variance scenarios, the UCL method consistently achieves the nominal power (≥ 0.80) while requiring a smaller increase in total sample size than the inflation‑factor approach. The type‑I error remains at the nominal 0.05 level for all methods, confirming that the blinded nature of the procedure does not inflate false‑positive rates.

Two real clinical trials illustrate practical relevance. In a post‑pancreatectomy acute pancreatitis trial with a 12‑subject pilot, and a Parkinson’s disease deep‑brain‑stimulation timing trial with a 22‑subject pilot, the UCL method required only modest sample‑size expansions (≈ 10–15 %) to reach the target power, whereas the traditional one‑sample variance method would have needed larger expansions or would have risked under‑power.

The authors discuss limitations: the methodology assumes normality, equal variances across groups, and a two‑arm design. Extensions to non‑normal outcomes, multiple arms, or adaptive designs are suggested for future work. They also note that a Bayesian formulation could incorporate prior information about σ², potentially improving efficiency.

In conclusion, the paper provides a theoretically justified and practically implementable blinded sample‑size re‑estimation procedure that uses an upper confidence limit for the variance to control the risk of under‑powered trials, especially when internal pilot studies are small. By allowing the confidence level γ to be predetermined based solely on pilot size, the approach enhances transparency and regulatory acceptability, offering a valuable tool for trials in rare diseases or other settings where only limited pilot data are available.