STOPPING BOUNDARIES ADJUSTED FOR SAMPLE SIZE REESTIMATION AND NEGATIVE STOP

We propose an approach to specify group sequential stopping boundaries adjusted for sample size reestimation and negative stop in interim analyses of a clinical trial. Sample size can be adjusted based on the observed delta at each interim to maintain the targeted power. The calculation of stopping boundaries incorporates possible changes in the type-I error due to sample size reestimation and/or negative stops; hence the overall type-I error is well controlled. This approach combines the advantages of the group sequential and sample size reestimation methods and is more efficient than either one alone. It provides flexibility in clinical trials and still maintains the integrity of these trials. When no early stop is planned, the stopping boundaries will be adjusted only for sample size reestimation. All calculations are given in closed mathematical forms and adjustments in stopping boundaries are based on the exact type-I error change. Therefore, the penalty for the type-I error inflation due to such interim conductions is kept to a minimum.     

Stopping boundaries adjusted for sample size reestimation and negative stop

We propose an approach to specify group sequential stopping boundaries adjusted for sample size reestimation and negative stop in interim analyses of a clinical trial. Sample size can be adjusted based on the observed delta at each interim to maintain the targeted power. The calculation of stopping boundaries incorporates possible changes in the type-I error due to sample size reestimation and/or negative stops; hence the overall type-I error is well controlled. This approach combines the advantages of the group sequential and sample size reestimation methods and is more efficient than either one alone. It provides flexibility in clinical trials and still maintains the integrity of these trials. When no early stop is planned, the stopping boundaries will be adjusted only for sample size reestimation. All calculations are given in closed mathematical forms and adjustments in stopping boundaries are based on the exact type-I error change. Therefore, the penalty for the type-I error inflation due to such interim conductions is kept to a minimum.     

Futility Rules in Bioequivalence Trials with Sequential Designs

Health Canada, the US Food and Drug Administration, as well as the European Medicines Agency consider sequential designs acceptable for bioequivalence studies as long as the type I error is controlled at 5%. The EU guideline explicitly asks for specification of stopping rules, so the goal of this work is to investigate how stopping rules may affect type I errors and power for recently published sequential bioequivalence trial designs. Using extensive trial simulations, five different futility rules were evaluated for their effect on type I error rates and power in two-stage scenarios. Under some circumstances, notably low sample size in stage 1 and/or high variability power may be very severely affected by the stopping rules, whereas type I error rates appear less affected. Because applicants may initiate sequential studies when the variability is not known in advance, achieving sufficient power and thereby complying with certain guideline requirements may be challenging and application of optimistic futility rules could possibly be unethical. This is the first work to investigate how futility rules affect type I errors and power in sequential bioequivalence trials.     

Statistical Monitoring of Clinical Trials With Multiple Co-Primary Endpoints Using Multivariate B-value

This article develops methods of statistical monitoring of clinical trials with multiple co-primary endpoints, where success is defined as meeting both endpoints simultaneously. In practice, a group sequential design (GSD) method is used to stop trials early for promising efficacy, and conditional power (CP) is used for futility stopping rules. In this article, we show that stopping boundaries for the GSD with multiple co-primary endpoints should be the same as those for studies with single endpoints. Lan and Wittes proposed the B-value tool to calculate the CP of single endpoint trials and we extend this tool to calculate the CP for studies with multiple co-primary endpoints. We consider the cases of two-arm studies with co-primary normal and provide an example of implementation with simulated trial. A fixed-weight sample size reestimation approach based on CP is introduced.     

肿瘤药物临床试验中成组序贯设计的统计学考虑

成组序贯设计因其拥有较少的病例样本数和较早终止试验的可能性成为肿瘤药物临床试验设计方法的较好选择。如何科学有效地设计和应用成组序贯设计,本文通过Monte Carlo试验模拟,探讨肿瘤药物临床试验中成组序贯设计的期中分析次数、实施时间以及α消耗函数选取等问题,为读者系统指明如何去规划一次成组序贯试验以及如何确定其最优的试验参数。模拟结果表明,成组序贯设计以时间点2∶1∶1折半划分的三次期中分析为好,其期望样本含量仅为420.53。Lan-Demets的五种α消耗函数中,1.5次幂和2次幂的α消耗函数拥有最小期望样本含量约393例,相对于O＇Brien-Fleming设计和Po-cock设计在整体上更显优势。     

The Type II Error Probability of a Group Sequential Test of Efficacy and Futility,and Considerations for Power and Sample Size

For a clinical trial incorporating a group sequential test that allows early stopping for efficacy or futility (GSTEF), the primary hypothesis concerns efficacy. However, the type II error probability of the tests of efficacy is neither specified nor known. The type II error probability of a GSTEF is partitioned into the sum of its component type II error probabilities of futility and efficacy. This partitioning provides transparency, allowing researchers flexibility to set these component error probabilities directly and to know the impact on the total type II error probability and vice versa. This transparency and flexibility should improve the application of GSTEF to clinical trials.     

Assessing the impact of safety monitoring on the efficacy analysis in large Phase III group sequential trials with non-trivial safety event rate

In Phase III clinical trials for life-threatening conditions, some serious but expected adverse events, such as early deaths or congestive heart failure, are often treated as the secondary or co-primary endpoint, and are closely monitored by the Data and Safety Monitoring Committee (DSMC). A naïve group sequential design (GSD) for such a study is to specify univariate statistical boundaries for the efficacy and safety endpoints separately, and then implement the two boundaries during the study, even though the two endpoints are typically correlated. One problem with this naïve design, which has been noted in the statistical literature, is the potential loss of power. In this article, we develop an analytical tool to evaluate this negative impact for trials with non-trivial safety event rates, particularly when the safety monitoring is informal. Using a bivariate binary power function for the GSD with a random-effect component to account for subjective decision-making in safety monitoring, we demonstrate how, under common conditions, the power loss in the naïve design can be substantial. This tool may be helpful to entities such as the DSMCs when they wish to deviate from the prespecified stopping boundaries based on safety measures.     

Group Sequential Design With Change of Endpoint

ABSTRACT

In Rheumatoid Arthritis studies, scientists usually need to focus on multiple endpoints simultaneously. Regulatory agencies require ACR20 as the primary endpoint for approval and making decisions. However, the proportional measurement for a binary variable is less reliable with a small sample size. DAS28 is an increasingly popular co-primary or key secondary endpoint for decision making in interim analyses because it is a continuous variable as well as a linear combination of multiple measurements that are also contained in ACR20. In a group sequential (GS) design, one of the most important parts is to compute the correlation among test statistics in interim and final analyses. We perform a logistic regression to link DAS28 and ACR20 and compute the correlation numerically. We also prove that the covariance matrix between binary and continuous variables with underlying logistic regression is a consistent estimator. Three methods including “GS design with univariate variable (ACR20),” “GS design with change of endpoints (DAS28 and ACR20),” and “Bonferroni tests with change of endpoints” are compared in three cases—“early stopping for efficacy,” “early stopping for futility,” and “early stopping for both” through simulation. The results show that in the first and third cases, the power of tests with a change of endpoint is much higher than tests that only use ACR20. In the case of “early stopping for futility,” the methods with a change of endpoint also provide a higher probability of correctly stopping early for futility.     

Group sequential extensions of a standard bioequivalence testing procedure

Bioequivalence trials compare the relative bioavailability of different formulations of a drug. Regulatory requirements for demonstrating average bioequivalence of two formulations generally include showing that a (say) 90% confidence interval for the ratio of expected pharmacologic end point values of the formulations lies between specified end points, e.g., 0.8–1.25. The likelihood of demonstrating bioequivalence when the formulations truly are equivalent depends on the sample size and on the variability of the pharmacologic end point. Group sequential bioequivalence testing provides a statistically valid way to accommodate misspecification of the variability in designing the trial by allowing for additional observations if a clear decision to accept or reject bioequivalence cannot be reached with the initial set of observations. This paper describes group sequential bioequivalence designs applicable in most practical situations that allow a decision to be reached with fewer observations than fixed-sample designs about 60% of the time at approximately the same average cost. The designs can be used in trials where the formulations are expected to have equal bioavailability and in trials where the formulations are expected to differ slightly. Data analyses are carried out exactly as for fixed-sample designs. Providing the capability of sequential decisions modestly affects the nominal significance levels, e.g., the required confidence level may be 93–94% instead of 90%.     

Pilot and Repeat Trials as Development Tools Associated with Demonstration of Bioequivalence

The purpose of this work is to use simulated trials to study how pilot trials can be implemented in relation to bioequivalence testing, and how the use of the information obtained at the pilot stage can influence the overall chance of showing bioequivalence (power) or the chance of approving a truly bioinequivalent product (type I error). The work also covers the use of repeat pivotal trials since the difference between a pilot trial followed by a pivotal trial and a pivotal trial followed by a repeat trial is mainly a question of whether a conclusion of bioequivalence can be allowed after the first trial. Repeating a pivotal trial after a failed trial involves dual or serial testing of the bioequivalence null hypothesis, and the paper illustrates how this may inflate the type I error up to almost 10%. Hence, it is questioned if such practice is in the interest of patients. Tables for power, type I error, and sample sizes are provided for a total of six different decision trees which allow the developer to use either the observed geometric mean ratio (GMR) from the first or trial or to assume that the GMR is 0.95. In cases when the true GMR can be controlled so as not to deviate more from unity than 0.95, sequential design methods ad modum Potvin may be superior to pilot trials. The tables provide a quantitative basis for choosing between sequential designs and pivotal trials preceded by pilot trials.

Electronic supplementary material

The online version of this article (doi:10.1208/s12248-015-9744-6) contains supplementary material, which is available to authorized users.KEY WORDS: bioequivalence, pilot trials, power, type I error     

On the Optimal Timing of Futility Interim Analyses

Futility analyses provide a mechanism to stop a trial early because of low likelihood to achieve its efficacy objective. They are usually motivated by ethical and economic purposes, so that stopping a trial with poor efficacy could save patients and resources for other promising trials. There are various methods to address futility analyses in the literature but most focus on equally spaced interim looks. We consider a constrained optimization framework where the timing and the futility boundary are decided jointly to balance the risks between stopping trials which should continue, and continuing trials which should stop. The average sample size is used as a key parameter, which is evaluated under different degrees of power loss. Alternative objective functions and constraints are compared to assess the operating characteristics of the optimal futility scheme. Numerical results for single and multiple futility looks are provided. Supplementary materials for this article are available online.     

Alternative Views On Setting Clinical Trial Futility Criteria

A feature increasingly utilized in clinical trial practice is to allow a study to stop early when it seems unlikely to achieve its primary efficacy objectives. This is commonly referred to as stopping for futility, and can be motivated by ethical and financial considerations. A number of methods for addressing futility have been described in the literature, including rules based upon conditional power, predictive probability, beta spending functions, and others. We consider futility stopping from the point of view of quantifying and providing an objective sensible balance between risks of incorrect decisions (e.g., stopping trials which should continue, and continuing trials which should stop), and discuss how specific considerations within a trial can lead to choice of a sensible scheme. This approach is not specific to any particular scales in the literature such as those just mentioned, and we describe interrelationships among criteria expressed on different scales. As futility may be evaluated multiple times in a long-term trial and the amount of information available at scheduled interim analyses may be difficult to predict in advance, we present a specific optimality criterion and discuss which of the familiar scales tend to produce schemes simple to describe and implement, and with better behavior across different timepoints at which futility might be evaluated.     

Conditional and unconditional confidence intervals following a group sequential test

After a group sequential test, the naive confidence interval (CI) is usually biased in the sense that it does not cover the true parameter at the correct nominal level. Furthermore, when the stopping time is taken into account, the actual conditional confidence coverage probability can be much less accurate. In this article, we study the conditional coverage probability and other related properties of the naive CI and different versions of exact CI's. It is demonstrated that only correcting the overall confidence level does not necessarily improve the confidence level at any given stopping stage. Conditional inference can be applied to construct an exact conditional CI but it is not without serious undesirable properties. We propose a two-step restricted conditional confidence interval (RCCI) which considerably improves the conditional confidence level while minimizing the undesirable properties. Numerical comparisons are made between the proposed method and existing methods. The results show that the RCCI not only improves the conditional coverage probability considerably from the exact CI's but also is free of the major undesirable properties displayed by the pure conditional CI. Differences between the conditional and unconditional CI's and their respective strengths are also discussed.     

Single-Arm Phase II Group Sequential Trial Design With Survival Endpoint at a Fixed Time Point

In this article, three nonparametric test statistics are proposed to design single-arm phase II group sequential trials for monitoring survival probability. The small-sample properties of these test statistics are studied through simulations. Sample size formulas are derived for the fixed sample test. The Brownian motion property of the test statistics allowed us to develop a flexible group sequential design using a sequential conditional probability ratio test procedure. An example is given to illustrate the trial design by using the proposed method.     

Controlling alpha for mixed effects models for repeated measures

Mixed Effects Models for Repeated Measures (MMRM) is often used in clinical trials with longitudinal data. However, there has not been an in-depth examination available on how investigators can implement interim analysis while also controlling the overall alpha for clinical trials under an MMRM analysis framework. Statistical independence among measurements, which is often assumed in group sequential testing (GST), is not valid under an MMRM framework due to the correlations induced by longitudinal within-subject measurements. Therefore, methods associated with GST derived under independence need to be adjusted accordingly. While these correlations can be estimated from the study data, regulatory agencies may not accept results based on these estimated correlations since there is no guarantee that the overall alpha is strongly controlled. In this article, we propose a new AC-Hybrid-approach for controlling the overall alpha. The AC-Hybrid-approach has two key attributes. First, we apply the MMRM analysis framework on all available data at every analysis timepoint. Second, we use complete-case information fractions to derive the group sequential stopping boundaries. We prove that the overall alpha is controlled regardless of the correlations among within-subject measurements. We also show the impact of this approach on the alpha and the power through examples.     

Conditional and Unconditional Confidence Intervals Following a Group Sequential Test

After a group sequential test, the naïive confidence interval (CI) is usually biased in the sense that it does not cover the true parameter at the correct nominal level. Furthermore, when the stopping time is taken into account, the actual conditional confidence coverage probability can be much less accurate. In this article, we study the conditional coverage probability and other related properties of the naïve CI and different versions of exact CI's. It is demonstrated that only correcting the overall confidence level does not necessarily improve the confidence level at any given stopping stage. Conditional inference can be applied to construct an exact conditional CI but it is not without serious undesirable properties. We propose a two-step restricted conditional confidence interval (RCCI) which considerably improves the conditional confidence level while minimizing the undesirable properties. Numerical comparisons are made between the proposed method and existing methods. The results show that the RCCI not only improves the conditional coverage probability considerably from the exact CI's but also is free of the major undesirable properties displayed by the pure conditional CI. Differences between the conditional and unconditional CI's and their respective strengths are also discussed.     

On efficient two-stage adaptive designs for clinical trials with sample size adjustment

Group sequential designs are rarely used for clinical trials with substantial over running due to fast enrollment or long duration of treatment and follow-up. Traditionally, such trials rely on fixed sample size designs. Recently, various two-stage adaptive designs have been introduced to allow sample size adjustment to increase statistical power or avoid unnecessarily large trials. However, these adaptive designs can be seriously inefficient. To address this infamous problem, we propose a likelihood-based two-stage adaptive design where sample size adjustment is derived from a pseudo group sequential design using cumulative conditional power. We show through numerical examples that this design cannot be improved by group sequential designs. In addition, the approach may uniformly improve any existing two-stage adaptive designs with sample size adjustment. For statistical inference, we provide methods for sequential p-values and confidence intervals, as well as median unbiased and minimum variance unbiased estimates. We show that the claim of inefficiency of adaptive designs by Tsiatis and Mehta ( 2003 ) is logically flawed, and thereby provide a strong defense of Cui et al. ( 1999 ).     

A Novel Approach to Testing for Average Bioequivalence Based on Modeling the Within-Period Dependence Structure

Bioequivalence trials are commonly conducted to assess therapeutic equivalence between a generic and an innovator brand formulations. In such trials, drug concentrations are obtained repeatedly over time and are summarized using a metric such as the area under the concentration vs. time curve (AUC) for each subject. The usual practice is to then conduct two one-sided tests using these areas to evaluate for average bioequivalence. A major disadvantage of this approach is the loss of information encountered when ignoring the correlation structure between repeated measurements in the computation of areas. In this article, we propose a general linear model approach that incorporates the within-subject covariance structure for making inferences on mean areas. The model-based method can be seen to arise naturally from the reparameterization of the AUC as a linear combination of outcome means. We investigate and compare the inferential properties of our proposed method with the traditional two one-sided tests approach using Monte Carlo simulation studies. We also examine the properties of the method in the event of missing data. Simulations show that the proposed approach is a cost-effective, viable alternative to the traditional method with superior inferential properties. Inferential advantages are particularly apparent in the presence of missing data. To illustrate our approach, a real working example from an asthma study is utilized.     

Hierarchical Testing of a Primary and a Secondary Endpoint in a Group Sequential Design With Different Information Times

ABSTRACT

We consider a confirmatory clinical trial where a primary and a secondary endpoints are tested hierarchically to control the family-wise error rate at level α. The trial uses a group sequential design with an interim and a final analysis. When the information times at the interim analysis are the same for the primary and the secondary hypotheses, it has been shown in the literature that the secondary hypothesis has to be tested using a group sequential boundary at a level no more than α. In many event-driven trials, however, the information times are usually different because of different event rates for the two endpoints. The information times may also be different for a noninferiority hypothesis and a superiority hypothesis due to different analysis sets. We consider this general setup and derive a sharp upper bound on the probability of rejecting the secondary hypothesis at an interim and a final analysis. This bound suggests that the secondary boundary can be refined so that the group sequential design can be tested at a significance level greater than α while still controlling the family-wise error rate at level α. We carry out a simulation study to illustrate the power gain by using the refined boundary for different choices of boundaries for the two hypotheses. The proposed approach is illustrated in two oncology clinical trials with more than two analyses.     

Noninferiority testing beyond simple two-sample comparison

In order to fulfill the requirement of a new drug application, a sponsor often need to conduct multiple clinical trials. Often these trials are of designs more complicated than a randomized two-sample single-factor study. For example, these trials could be designed with multiple centers, multiple factors, covariates, group sequential and/or adaptive scheme, etc. When an active standard treatment used as the control treatment in a two-arm clinical trial, the efficacy of the test treatment is often established by performing a noninferiority test through comparison of the test treatment and the active standard treatment. Typically, the noninferiority trials are designed with either a generalized historical control approach (i.e., noninferiority margin approach or delta-margin approach) or a cross-trial comparison approach (i.e., synthesis approach or lambda-margin approach). Many of the statistical properties of the approaches discussed in the literature were focused on testing in a simple two sample comparison form. We studied the limitations of the two approaches for the consideration of switching between superiority and noninferiority testing, feasibility to be applied with group sequential design, constancy assumption requirements, test dependency in multiple trials, analysis of homogeneity of efficacy among centers in a multi-center trial, data transformation and changing analysis method from the historical studies. Our evaluation shows that the cross-trial comparison approach is more restricted to simple two sample comparison with normal approximation test because of its poor properties with more complicated design and analysis. On the other hand, the generalized historical control comparison approach may have more flexible properties when the variability of the margin delta is indeed negligibly small.