Statistics in Medicine 2008, 27, 15-35 1.IntroductionIn a drug development process, the primary goal of Phase II clinical trials is to determine whether there is sufficient evidence of efficacy and safety to make it worth further study in a Phase III clinical trial.For example, suppose a new drug is developed for patients with liver cancer. To investigate whether this new drug extends the life of liver cancer patients, the criterion of drug efficacy is defined to be the presence of tumor shrinkage, which is a binary endpoint with response probability p. In a single-stage design, if the number of patients with tumor shrinkage is large enough, then this new drug may be used as evidence for further testing.Most often, drug experiments employ a Simon's two-stage design [1] in order to avoid giving patients an ineffective drug. Suppose there are n 1 patients participating in the first-stage experiment, and there are n 2 additional patients participating in the second-stage experiment if the new drug design is allowed to continue. The number of responses Y 1 is observed at the first stage, and if Y 1 is less than a specified value a, the design is stopped. Otherwise, this drug design is allowed to continue and the number of responses Y 2 , which is independent of Y 1 , is observed at the second stage.Consequently, the response probability of patients with tumor shrinkage is to be estimated in order to plan a further study.It is natural to use sample proportion to estimate p. However, when the second stage is allowed to continue, this estimator will overestimate the true p, because the number of responses at the first stage is truncated by a and follows a truncated binomial distribution. Therefore, a maximum likelihood estimator based on the truncated binomial distribution is derived to take into account the truncation effect. In addition, to take into account the inherent variability of patients in the measured responses, the confidence interval, that is a range of feasible values within which the true p may lie, is also constructed for p. Two types of interval estimators, the Wald interval without (with) continuity correction [2] and the score interval without (with) continuity correction [3], are constructed in the next section.

In phase II clinical trials, patients are recruited sequentially and consequently the time required to complete the clinical trial will become long if the accrual rate is low. To speed up the drug development process and account for ethical issues, stochastically and non-stochastically curtailed two-stage designs have been proposed in single-arm phase II clinical trials. More recently, randomized phase II clinical trials are being increasingly recommended to avoid biased evaluation of the treatment effect when compared with a historical control. The current patient population and the historical one may be quite heterogeneous. Moreover, it is impossible to randomly assign patients for treatments. Consequently, various two-stage designs have been presented for comparing two arms. Since the sample size required in a randomized phase II trial is usually larger than that required in a single-arm phase II trial, we introduce the concept of curtailed sampling procedure to develop curtailed two-stage design for two-armed, randomized phase II clinical trials. The proposed design does not require pairwise patient response comparison, yet it allows a trial to be stopped early as soon as the difference in therapeutic effect of the experimental therapy and the standard at the end of a trial is foreknown.

When the accrual rate is low and the treatment period is long, a long observational period is required before information concerning the primary end point, such as binary response, becomes available in the study. Simon's two-stage designs are often employed in Phase II clinical trials to avoid giving patient an ineffective drug. Thus, if the new drug is ineffective then this design would certainly accelerate the process of drug discovery and development. However, for a promising new drug this design may still require a long observational period. Therefore, when drug safety is not a primary concern, this paper proposes curtailed two-stage designs to shorten the drug development process as soon as the treatment either shows lack of efficacy or is very effective. The proposed design is superior to Simon's two-stage designs in terms of savings in expected sample size and is much easier to implement in practice than stochastically curtailed Simon's designs.

When phase I clinical trials were found to be unable to precisely estimate the frequency of toxicity, Brayan and Day proposed incorporating toxicity considerations into two-stage designs in phase II clinical trials. Conaway and Petroni further pointed out that it is important to evaluate the clinical activity and safety simultaneously in studying cancer treatments with more toxic chemotherapies in a phase II clinical trial. Therefore, they developed multi-stage designs with two dependent binary endpoints. However, the usual sample sizes in phase II trials make these designs difficult to control the type I error rate at a desired level over the entire null region and still have sufficient power against reasonable alternatives. Therefore, the curtailed sampling procedure summarized by Phatak and Bhatt will be applied to the two-stage designs with two dependent binary endpoints in this paper to reduce sample sizes and speed up the development process for drugs.

In clinical trials, information about certain time points may be of interest in making decisions about treatment effectiveness. Rather than comparing entire survival curves, researchers can focus on the comparison at fixed time points that may have a clinical utility for patients. For two independent samples of right-censored data, Klein et al. (2007) compared survival probabilities at a fixed time point by studying a number of tests based on some transformations of the Kaplan-Meier estimators of the survival function. However, to compare the survival probabilities at a fixed time point for paired right-censored data or clustered right-censored data, their approach would need to be modified. In this paper, we extend the statistics to accommodate the possible within-paired correlation and within-clustered correlation, respectively. We use simulation studies to present comparative results. Finally, we illustrate the implementation of these methods using two real data sets.