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.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

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.     

Trendtests with adaptive scoring

The concept of adaptive two‐stage designs is applied to the problem of testing the equality of several normal means against an ordered (monotone) alternative. The likelihood‐ratio‐test proposed by Bartholomew is known to have favorable power properties when testing against a monotonic trend. Tests based on contrasts provide a flexible way to incorporate available information regarding the pattern of the unknown true means through appropriate specification of the scores. The basic idea of the presented concept is the combination of Bartholomew 's test (first stage) with an “adaptive score test” (second stage) which utilizes the information resulting from isotonic regression estimation at the first stage. In a Monte Carlo simulation study the adaptive scoring procedure is compared to the non‐adaptive two‐stage procedure using the Bartholomew test at both stages. We found that adaptive scoring may improve the power of the two stage design, in particular if the sample size at the first stage is considerably larger than at the second stage.     

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.     

Optimal promising zone designs

Clinical trials with adaptive sample size reassessment based on an unblinded analysis of interim results are perhaps the most popular class of adaptive designs (see Elsäßer et al., 2007). Such trials are typically designed by prespecifying a zone for the interim test statistic, termed the promising zone, along with a decision rule for increasing the sample size within that zone. Mehta and Pocock (2011) provided some examples of promising zone designs and discussed several procedures for controlling their type‐1 error. They did not, however, address how to choose the promising zone or the corresponding sample size reassessment rule, and proposed instead that the operating characteristics of alternative promising zone designs could be compared by simulation. Jennison and Turnbull (2015) developed an approach based on maximizing expected utility whereby one could evaluate alternative promising zone designs relative to a gold‐standard optimal design. In this paper, we show how, by eliciting a few preferences from the trial sponsor, one can construct promising zone designs that are both intuitive and achieve the Jennison and Turnbull (2015) gold‐standard for optimality.     

Adaptive dose-finding: Proof of concept with type I error control

We consider an adaptive dose‐finding study with two stages. The doses for the second stage will be chosen based on the first stage results. Instead of considering pairwise comparisons with placebo, we apply one test to show an upward trend across doses. This is a possibility according to the ICH‐guideline for dose‐finding studies (ICH‐E4). In this article, we are interested in trend tests based on a single contrast or on the maximum of multiple contrasts. We are interested in flexibly choosing the Stage 2 doses including the possibility to add doses. If certain requirements for the interim decision rules are fulfilled, the final trend test that ignores the adaptive nature of the trial (naïve test) can control the type I error. However, for the more common case that these requirements are not fulfilled, we need to take the adaptivity into account and discuss a method for type I error control. We apply the general conditional error approach to adaptive dose‐finding and discuss special issues appearing in this application. We call the test based on this approach Adaptive Multiple Contrast Test. For an example, we illustrate the theory discussed before and compare the performance of several tests for the adaptive design in a simulation study.     

A Booststrap Adaptive Test for Two-way Analysis of Variance

We propose a method to construct adaptive tests based on a bootstrap technique. The procedure leads to a nearly exact adaptive test depending on the size of the sample. With the use of the estimated Pitman's relative efficacy as selector statistic, we show that the adaptive test has a power that is asymptotically equal to the power of it's better component. We apply the idea to construct an adaptive test for two-way analysis of variance model. Finally, we use simulations to observe the behaviour of the method for small sample sizes.     

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.     

Optimal conditional error functions for the control of conditional power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.     

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

False discovery rate control in two-stage designs

ABSTRACT: BACKGROUND: For gene expression or gene association studies with a large number of hypotheses the number of measurements per marker in a conventional single-stage design is often low due to limited resources. Two-stage designs have been proposed where in a first stage promising hypotheses are identified and further investigated in the second stage with larger sample sizes. For two types of two-stage designs proposed in the literature we derive multiple testing procedures controlling the False Discovery Rate (FDR) demonstrating FDR control by simulations: designs where a fixed number of top-ranked hypotheses are selected and designs where the selection in the interim analysis is based on an FDR threshold. In contrast to earlier approaches which use only the second-stage data in the hypothesis tests (pilot approach), the proposed testing procedures are based on the pooled data from both stages (integrated approach). Results: For both selection rules the multiple testing procedures control the FDR in the considered simulation scenarios. This holds for the case of independent observations across hypotheses as well as for certain correlation structures. Additionally, we show that in scenarios with small effect sizes the testing procedures based on the pooled data from both stages can give a considerable improvement in power compared to tests based on the second-stage data only. Conclusion: The proposed hypothesis tests provide a tool for FDR control for the considered two-stage designs. Comparing the integrated approaches for both selection rules with the corresponding pilot approaches showed an advantage of the integrated approach in many simulation scenarios.     

A comparison of methods for constructing confidence intervals after phase II/III clinical trials

Recently, in order to accelerate drug development, trials that use adaptive seamless designs such as phase II/III clinical trials have been proposed. Phase II/III clinical trials combine traditional phases II and III into a single trial that is conducted in two stages. Using stage 1 data, an interim analysis is performed to answer phase II objectives and after collection of stage 2 data, a final confirmatory analysis is performed to answer phase III objectives. In this paper we consider phase II/III clinical trials in which, at stage 1, several experimental treatments are compared to a control and the apparently most effective experimental treatment is selected to continue to stage 2. Although these trials are attractive because the confirmatory analysis includes phase II data from stage 1, the inference methods used for trials that compare a single experimental treatment to a control and do not have an interim analysis are no longer appropriate. Several methods for analysing phase II/III clinical trials have been developed. These methods are recent and so there is little literature on extensive comparisons of their characteristics. In this paper we review and compare the various methods available for constructing confidence intervals after phase II/III clinical trials.     

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.     

Adaptive web sampling

A flexible class of adaptive sampling designs is introduced for sampling in network and spatial settings. In the designs, selections are made sequentially with a mixture distribution based on an active set that changes as the sampling progresses, using network or spatial relationships as well as sample values. The new designs have certain advantages compared with previously existing adaptive and link-tracing designs, including control over sample sizes and of the proportion of effort allocated to adaptive selections. Efficient inference involves averaging over sample paths consistent with the minimal sufficient statistic. A Markov chain resampling method makes the inference computationally feasible. The designs are evaluated in network and spatial settings using two empirical populations: a hidden human population at high risk for HIV/AIDS and an unevenly distributed bird population.     

A Shift Algorithm for Exact Computation in Sequential Tests with Binary Outcome

A simple shift algorithm is described enabling the exact determination of power functions and sample size distributions for a large variety of closed sequential two‐sample designs with a binary outcome variable. The test statistics are assumed to be based on relative frequencies of successes or failures, but the number of interim analyses, the monitoring times, and the continuation regions may be specified as desired. To give examples, exact properties of designs proposed by the program package EaSt (Cytel , 1992) are determined, and plans with interim analyses are considered where decisions are based on the conditional power given the observations obtained so far.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.     

Simultaneous Non‐inferiority Test of Sensitivity and Specificity for Two Diagnostic Procedures in the Presence of a Gold Standard

Sensitivity and specificity have traditionally been used to assess the performance of a diagnostic procedure. Diagnostic procedures with both high sensitivity and high specificity are desirable, but these procedures are frequently too expensive, hazardous, and/or difficult to operate. A less sophisticated procedure may be preferred, if the loss of the sensitivity or specificity is determined to be clinically acceptable. This paper addresses the problem of simultaneous testing of sensitivity and specificity for an alternative test procedure with a reference test procedure when a gold standard is present. The hypothesis is formulated as a compound hypothesis of two non‐inferiority (one‐sided equivalence) tests. We present an asymptotic test statistic based on the restricted maximum likelihood estimate in the framework of comparing two correlated proportions under the prospective and retrospective sampling designs. The sample size and power of an asymptotic test statistic are derived. The actual type I error and power are calculated by enumerating the exact probabilities in the rejection region. For applications that require high sensitivity as well as high specificity, a large number of positive subjects and a large number of negative subjects are needed. We also propose a weighted sum statistic as an alternative test by comparing a combined measure of sensitivity and specificity of the two procedures. The sample size determination is independent of the sampling plan for the two tests.     

Optimization of two-stage genetic designs where data are combined using an accurate and efficient approximation for Pearson's statistic

There is an increasing interest in the use of two-stage case-control studies to reduce genotyping costs in the search for genes underlying common disorders. Instead of analyzing the data from the second stage separately, a more powerful test can be performed by combining the data from both stages. However, standard tests cannot be used because only the markers that are significant in the first stage are selected for the second stage and the test statistics at both stages are dependent because they partly involve the same data. Theoretical approximations are not available for commonly used test statistics and in this specific context simulations can be problematic because of the computational burden. We therefore derived a cost-effective, that is, accurate but fast in terms of central processing unit (CPU) time, approximation for the distribution of Pearson's statistic on 2 xm contingency tables in two-stage design with combined data. We included this approximation in an iterative method for designing optimal two-stage studies. Simulations supported the accuracy of our approximation. Numerical results confirmed that the use of two-stage designs reduces the genotyping burden substantially. Compared to not combining data, combining the data decreases the required sample sizes on average by 15% and the genotyping burden by 5%.