Conditional power (CP) serves as a widely utilized approach for futility monitoring in group sequential designs. However, adopting the CP methods may lead to inadequate control of the type II error rate at the desired level. In this study, we introduce a flexible beta spending function tailored to regulate the type II error rate while employing CP based on a predetermined standardized effect size for futility monitoring (a so-called CP-beta spending function). This function delineates the expenditure of type II error rate across the entirety of the trial. Unlike other existing beta spending functions, the CP-beta spending function seamlessly incorporates beta spending concept into the CP framework, facilitating precise stagewise control of the type II error rate during futility monitoring. In addition, the stopping boundaries derived from the CP-beta spending function can be calculated via integration akin to other traditional beta spending function methods. Furthermore, the proposed CP-beta spending function accommodates various thresholds on the CP-scale at different stages of the trial, ensuring its adaptability across different information time scenarios. These attributes render the CP-beta spending function competitive among other forms of beta spending functions, making it applicable to any trials in group sequential designs with straightforward implementation. Both simulation study and example from an acute ischemic stroke trial demonstrate that the proposed method accurately captures expected power, even when the initially determined sample size does not consider futility stopping, and exhibits a good performance in maintaining overall type I error rates for evident futility.

A multi-stage randomized trial design can significantly improve efficiency by allowing early termination of the trial when the experimental arm exhibits either low or high efficacy compared to the control arm during the study. However, proper inference methods are necessary because the underlying distribution of the target statistic changes due to the multi-stage structure. This article focuses on multi-stage randomized phase II trials with a dichotomous outcome, such as treatment response, and proposes exact conditional confidence intervals for the odds ratio. The usual single-stage confidence intervals are invalid when used in multi-stage trials. To address this issue, we propose a linear ordering of all possible outcomes. This ordering is conditioned on the total number of responders in each stage and utilizes the exact conditional distribution function of the outcomes. This approach enables the estimation of an exact confidence interval accounting for the multi-stage designs.

A multi-stage design for a randomized trial is to allow early termination of the study when the experimental arm is found to have low or high efficacy compared to the control during the study. In such a trial, an early stopping rule results in bias in the maximum likelihood estimator of the treatment effect. We consider multi-stage randomized trials on a dichotomous outcome, such as treatment response, and investigate the estimation of the odds ratio. Typically, randomized phase II cancer clinical trials have two-stage designs with small sample sizes, which makes the estimation of odds ratio more challenging. In this paper, we evaluate several existing estimation methods of odds ratio and propose bias-corrected estimators for randomized multi-stage trials, including randomized phase II cancer clinical trials. Through numerical studies, the proposed estimators are shown to have a smaller bias and a smaller mean squared error overall.

We explore frequentist operating characteristics of a Bayesian adaptive design that allows continuous early stopping for futility. In particular, we focus on the power versus sample size relationship when more patients are accrued than originally planned.
We consider the case of a phase II single-arm study and a Bayesian phase II outcome-adaptive randomization design. For the former, analytical calculations are possible; for the latter, simulations are conducted.
Results for both cases show a decrease in power with an increasing sample size. It appears that this effect is due to the increasing cumulative probability of incorrectly stopping for futility.
The increase in cumulative probability of incorrectly stopping for futility is related to the continuous nature of the early stopping, which increases the number of interim analyses with accrual. The issue can be addressed by, for instance, delaying the start of testing for futility, reducing the number of futility tests to be performed or by setting stricter criteria for concluding futility.

A phase II trial is conducted to investigate if an experimental therapy is efficacious enough to proceed to a large-scale phase III trial or not. In spite of the fast progress in design and analysis methods, single-arm two-stage design is still the most popular for phase II cancer clinical trials. In this review article, we discuss two design and analysis methods that are popularly used for phase II clinical trials, but that can cause serious bias. One is about using the sample proportion as the estimator of the true response rate from single-arm two-stage trials. For a two-stage design with a futility interim test, the sample proportion is negatively biased by ignoring the two-stage design. The other is about the design and analysis of single-arm phase II trials for patient populations consisting of multiple sub-populations with different response rates. In this case, a standard design method is to project the prevalence of each subpopulation and select a standard two-stage design based on the expected response rate for the whole population. By using an unstratified statistical testing in this case, the standard analysis method can be seriously biased if the observed prevalence is very different from the projected one. In this paper, we review appropriate design and analysis methods that are proposed to avoid these sources of bias.

In this paper, we propose a randomized Bayesian optimal phase II (RBOP2) design with a binary endpoint (e.g., response rate). A beta-binomial distribution is used to model the binary endpoint for a two-arm phase II trial. Posterior probabilities of the endpoint of interest are evaluated at each interim look and used in the decision to stop the trial due to futility. Compared with other Bayesian designs, the proposed RBOP2 design has the following merits: (i) strongly controls the type I error rate at a pre-defined level; (ii) optimizes the stopping boundaries, thus maximizing the power to detect treatment effects and minimizing the expected sample size for futile treatment; (iii) does not limit the number of interim looks, thus enabling frequent trial monitoring; and (iv) allows the stopping boundaries to be pre-defined in the protocol and is easy to implement. We conduct simulation studies to compare the proposed design with a group sequential design and other Bayesian randomized designs and evaluate its operating characteristics under different scenarios.

Stopping for futility is a useful tool in a clinical trial. It is widely used in single-arm trials in oncology and in many two-arm trials. We review three stopping rules for futility. We give recommendations for the optimal timing of futility looks in two-stage trials in terms of the information fraction and the probability of stopping under the alternative hypothesis. We discuss futility stopping in trials with substantial uncertainty about the variability of the outcome and in crossover trials.

Designs incorporating more than one endpoint have become popular in drug development. One of such designs allows for incorporation of short-term information in an interim analysis if the long-term primary endpoint has not been yet observed for some of the patients. At first we consider a two-stage design with binary endpoints allowing for futility stopping only based on conditional power under both fixed and observed effects. Design characteristics of three estimators: using primary long-term endpoint only, short-term endpoint only, and combining data from both are compared. For each approach, equivalent cut-off point values for fixed and observed effect conditional power calculations can be derived resulting in the same overall power. While in trials stopping for futility the type I error rate cannot get inflated (it usually decreases), there is loss of power. In this study, we consider different scenarios, including different thresholds for conditional power, different amount of information available at the interim, different correlations and probabilities of success. We further extend the methods to adaptive designs with unblinded sample size reassessments based on conditional power with inverse normal method as the combination function. Two different futility stopping rules are considered: one based on the conditional power, and one from P-values based on Z-statistics of the estimators. Average sample size, probability to stop for futility and overall power of the trial are compared and the influence of the choice of weights is investigated.

Two-arm group sequential designs have been widely used for over 40 years, especially for studies with mortality endpoints. The natural generalization of such designs to trials with multiple treatment arms and a common control (MAMS designs) has, however, been implemented rarely. While the statistical methodology for this extension is clear, the main limitation has been an efficient way to perform the computations. Past efforts were hampered by algorithms that were computationally explosive. With the increasing interest in adaptive designs, platform designs, and other innovative designs that involve multiple comparisons over multiple stages, the importance of MAMS designs is growing rapidly. This article provides break-through algorithms that can compute MAMS boundaries rapidly thereby making such designs practical. For designs with efficacy-only boundaries the computational effort increases linearly with number of arms and number of stages. For designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages.