Sample Size Reassessment in Adaptive Clinical Trials Using a Bias Corrected Estimate

Point estimation in group sequential and adaptive trials is an important issue in analysing a clinical trial. Most literature in this area is only concerned with estimation after completion of a trial. Since adaptive designs allow reassessment of sample size during the trial, reliable point estimation of the true effect when continuing the trial is additionally needed. We present a bias adjusted estimator which allows a more exact sample size determination based on the conditional power principle than the naive sample mean does.     

Conditional Maximum Likelihood Estimation Following a Group Sequential Test

We consider estimation after a group sequential test. An estimator that is unbiased or has small bias may have substantial conditional bias (Troendle and Yu, 1999, Coburger and Wassmer, 2001). In this paper we derive the conditional maximum likelihood estimators of both the primary parameter and a secondary parameter, and investigate their properties within a conditional inference framework. The method applies to both the usual and adaptive group sequential test designs. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Bootstrap Procedure for Adaptive Selection of the Test Statistic in Flexible Two‐Stage Designs

Adaptive two‐stage designs allow a data‐driven change of design characteristics during the ongoing trial. One of the available options is an adaptive choice of the test statistic for the second stage of the trial based on the results of the interim analysis. Since there is often only a vague knowledge of the distribution shape of the primary endpoint in the planning phase of a study, a change of the test statistic may then be considered if the data indicate that the assumptions underlying the initial choice of the test are not correct. Collings and Hamilton proposed a bootstrap method for the estimation of the power of the two‐sample Wilcoxon test for shift alternatives. We use this approach for the selection of the test statistic. By means of a simulation study, we show that the gain in terms of power may be considerable when the initial assumption about the underlying distribution was wrong, whereas the loss is relatively small when in the first instance the optimal test statistic was chosen. The results also hold true for comparison with a one‐stage design. Application of the method is illustrated by a clinical trial example.     

Estimation of a Secondary Parameter in a Group Sequential Clinical Trial

In this article, we investigate estimation of a secondary parameter in group sequential tests. We study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects. The bias of the naive secondary parameter estimator is studied. It is shown that the sampling proportions of the subgroup have a crucial effect on the bias: As the sampling proportion of the subgroup at or just before the stopping time increases, the bias of the naive subgroup parameter estimator increases as well. An unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived. Using simulations, we compare the mean squared error of the unbiased estimator to that of the naive estimator, and we show that the differences are negligible. As an example, the methods of estimation are applied to an actual group sequential clinical trial, The Beta-Blocker Heart Attack Trial.     

Adaptive sample size calculations in group sequential trials

A method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed. Without exaggerating the Type I error rate, this method enables data-driven sample size reassessments during the course of the study. It uses the stopping boundaries of the classical group sequential tests. Furthermore, exact test procedures may be derived for a wide range of applications. The procedure is compared with the classical designs in terms of power and expected sample size.     

Adaptive Two Stage Designs and the Conditional Error Function

Proschan and Hunsberger (1995) suggest the use of a conditional error function to construct a two stage test that meets the α level and allows a very flexible reassessment of the sample size after the interim analysis. In this note we show that several adaptive designs can be formulated in terms of such an error function. The conditional power function defined similarly provides a simple method for sample size reassessment in adaptive two stage designs.     

Interim analysis and sample size reassessment

This article deals with sample size reassessment for adaptive two-stage designs based on conditional power arguments utilizing the variability observed at the first stage. Fisher's product test for the p-values from the disjoint samples at the two stages is considered in detail for the comparison of the means of two normal populations. We show that stopping rules allowing for the early acceptance of the null hypothesis that are optimal with respect to the average sample size may lead to a severe decrease of the overall power if the sample size is a priori underestimated. This problem can be overcome by choosing designs with low probabilities of early acceptance or by midtrial adaptations of the early acceptance boundary using the variability observed in the first stage. This modified procedure is negligibly anticonservative and preserves the power.     

Interim analysis incorporating short‐ and long‐term binary endpoints

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.     

BAYESian Point Estimation of the Unknown Upper Limit of a Uniform Distribution

BAYESian fixed sample size and sequential procedures are considered for the estimation of the unknown parameter v of a uniform distribution over (0, v), under several loss functions.     

Internal pilots for observational studies

Study planning often involves selecting an appropriate sample size. Power calculations require specifying an effect size and estimating “nuisance” parameters, e.g. the overall incidence of the outcome. For observational studies, an additional source of randomness must be estimated: the rate of the exposure. A poor estimate of any of these parameters will produce an erroneous sample size. Internal pilot (IP) designs reduce the risk of this error ‐ leading to better resource utilization ‐ by using revised estimates of the nuisance parameters at an interim stage to adjust the final sample size. In the clinical trials setting, where allocation to treatment groups is pre‐determined, IP designs have been shown to achieve the targeted power without introducing substantial inflation of the type I error rate. It has not been demonstrated whether the same general conclusions hold in observational studies, where exposure‐group membership cannot be controlled by the investigator. We extend the IP to observational settings. We demonstrate through simulations that implementing an IP, in which prevalence of the exposure can be re‐estimated at an interim stage, helps ensure optimal power for observational research with little inflation of the type I error associated with the final data analysis.     

Continual reassessment designs with early termination

The continual reassessment method (CRM) is an increasingly popular approach for estimating the maximum tolerated dose (MTD) in phase I dose finding studies. In its original formulation, the scheme is based on a fixed sample size. Many experimenters feel that, whenever possible, it may be advantageous to bring these trials to an early halt and thus reduce average sample size required to complete the study. To address this issue a stopping rule has been proposed (O'Quigley and Reiner, 1998) based on the idea that continuing the study would not lead to a change in recommendation with high probability. The rule, based on precise probabilistic calculation, is quite involved and not straightforward to implement. A much simpler rule can be constructed based on the idea of having settled at some level. In this work we investigate more deeply the essential ingredients behind these rules and consider more closely their operating characteristics.     

Adaptive group sequential designs for clinical trials: combining the advantages of adaptive and of classical group sequential approaches

A general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing. This allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs. The concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans.     

Sequential methods for comparing years of life saved in the two-sample censored data problem

This research develops nonparametric strategies for sequentially monitoring clinical trial data where detecting years of life saved is of interest. The recommended test statistic looks at integrated differences in survival estimates during the time frame of interest. In many practical situations, the test statistic presented has an independent increments covariance structure. Hence, with little additional work, we may apply these testing procedures using available methodology. In the case where an independent increments covariance structure is present, we suggest how clinical trial data might be monitored using these statistics in an information-based design. The resulting study design maintains the desired stochastic operating characteristics regardless of the shapes of the survival curves being compared. This offers an advantage over the popular log-rank-based design strategy since more restrictive assumptions relating to the behavior of the hazards are required to guarantee the planned power of the test. Recommendations for how to sequentially monitor clinical trial progress in the nonindependent increments case are also provided along with an example.     

A simple algorithm for designing group sequential clinical trials

This article describes a simple algorithm for calculating probabilities associated with group sequential trials. This allows the choice of boundaries that may not be among those implemented in available software.