Adaptive extensions of a two-stage group sequential procedure for testing primary and secondary endpoints (II): sample size re-estimation

In this part II of the paper on adaptive extensions of a two‐stage group sequential procedure (GSP) for testing primary and secondary endpoints, we focus on the second stage sample size re‐estimation based on the first stage data. First, we show that if we use the Cui–Huang–Wang statistics at the second stage, then we can use the same primary and secondary boundaries as for the original procedure (without sample size re‐estimation) and still control the type I familywise error rate. This extends their result for the single endpoint case. We further show that the secondary boundary can be sharpened in this case by taking the unknown correlation coefficient ρ between the primary and secondary endpoints into account through the use of the confidence limit method proposed in part I of this paper. If we use the sufficient statistics instead of the CHW statistics, then we need to modify both the primary and secondary boundaries; otherwise, the error rate can get inflated. We show how to modify the boundaries of the original group sequential procedure to control the familywise error rate. We provide power comparisons between competing procedures. We illustrate the procedures with a clinical trial example. Copyright © 2012 John Wiley & Sons, Ltd.     

Fallback tests for co‐primary endpoints

When efficacy of a treatment is measured by co‐primary endpoints, efficacy is claimed only if for each endpoint an individual statistical test is significant at level α. While such a strategy controls the family‐wise type I error rate (FWER), it is often strictly conservative and allows for no inference if not all null hypotheses can be rejected. In this paper, we investigate fallback tests, which are defined as uniform improvements of the classical test for co‐primary endpoints. They reject whenever the classical test rejects but allow for inference also in settings where only a subset of endpoints show a significant effect. Similarly to the fallback tests for hierarchical testing procedures, these fallback tests for co‐primary endpoints allow one to continue testing even if the primary objective of the trial was not met. We propose examples of fallback tests for two and three co‐primary endpoints that control the FWER in the strong sense under the assumption of multivariate normal test statistics with arbitrary correlation matrix and investigate their power in a simulation study. The fallback procedures for co‐primary endpoints are illustrated with a clinical trial in a rare disease and a diagnostic trial. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

An efficient method for accommodating potentially underpowered primary endpoints

If a trial is adequately powered for two clinically important endpoints, A and B, each of which can fully characterize a treatment benefit to support a regulatory claim by itself, then both endpoints are usually labeled primary, and the trial is deemed positive if either endpoint is statistically significant after a multiplicity adjustment. However, if only A is adequately powered, then should B be designated a secondary endpoint, or should it be retained in the primary family despite being (potentially) underpowered? The former option can lead to a negative trial if A is not statistically significant, no matter how positive the results are for B, since no familywise type I error rate (FWER) is allocated to B, while the latter can reduce the likelihood of a positive trial if an inefficient multiplicity adjustment is used. We underscore this contemporary problem with real examples and offer a novel and intuitively appealing solution for accommodating clinically important but potentially underpowered endpoint(s) in the primary family. In our proposal, for the above scenario with two endpoints, A is tested at a prespecified level α₁=α?ε (e.g. ε=0.01 when α=0.05), and B at an ‘adaptive’ level α₂ (?α) calculated using a prespecified non‐increasing function of the p‐value for A. Our method controls the FWER at level α and can notably increase the probability of achieving a positive trial compared with a fixed prospective alpha allocation scheme (Control. Clin. Trials 2000; 20 :40–49), and with Hochberg's method applied to the family of primary endpoints. Importantly, our proposal enables strong results for potentially underpowered primary endpoint(s) to be interpreted in a conclusive rather than exploratory light. Copyright © 2008 John Wiley & Sons, Ltd.     

A new partition testing strategy for multiple endpoints

To evaluate efficacy in multiple endpoints in confirmatory clinical trials is a challenging problem in multiple hypotheses testing. The difficulty comes from the different importance of each endpoint and their underlying correlation. Current approaches to this problem, which test the efficacy in certain dose–endpoint combinations and collate the results, are based on closed testing or partition testing. Despite their different formulations, all current approaches test their dose–endpoint combinations as intersection hypotheses and apply various union‐intersection tests. Likelihood ratio test is seldom used owing to the extensive computation and lack of consistent inferences. In this article, we first generalize the formulation of multiple endpoints problem to include the cases of alternative primary endpoints and co‐primary endpoints. Then we propose a new partition testing approach that is based on consonance‐adjusted likelihood ratio test. The new procedure provides consistent inferences, and yet, it is still conservative and does not rely on the estimation of endpoint correlation or independence assumptions that might be challenged by regulatory agencies. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive group-sequential design with population enrichment in phase 3 randomized controlled trials with two binary co-primary endpoints

The use of co-primary endpoints in drug development allows investigators to capture an experimental intervention's multidimensional effect more comprehensively than a single primary endpoint. We propose the theoretical basis and development of an adaptive population enrichment design with co-primary endpoints, provide stage-wise boundary values for futility and efficacy, and discuss power under different efficacy configurations, subgroup prevalence, and analysis times using a pre-specified decision criterion. We considered a two-arm, two-stage, parallel group design where population enrichment occurs at the interim analysis by dropping any non-responsive subgroups. A test for efficacy is conducted only in the enriched population. Two binary endpoints are evaluated as co-primary endpoints. Our trial objective is to determine whether the experimental intervention is superior to the control intervention, with superiority required in both endpoints. We define the stopping boundary using alpha spending functions. Using a 0.025 significance level for each endpoint, we obtain the stage I threshold boundary values for futility and efficacy as −0.1040 and 2.2761, respectively, and the stage II boundary value for futility and efficacy is 2.2419. We show that in the presence of substantial heterogeneity of treatment effect, we gain more power to observe an effect in the subgroup where the benefits are greater. By allowing the dropping of non-responsive subgroups at an early stage, our design reduces the likelihood of obtaining false-negative results due to inclusion of the heterogeneous treatment effects of both subgroups, which would dilute the responsive subgroup's results.     

Validity of the Hochberg procedure revisited for clinical trial applications

There is much interest in using the Hochberg procedure (HP) for statistical tests on primary endpoints of confirmatory clinical trials. The procedure is simple to use and enjoys more power than the Bonferroni and the Holm procedures. However, the HP is not assumption free like the other two procedures. It controls the familywise type I error rate when test statistics (used for statistical tests) are independent or if dependent satisfy a conditionally independent formulation. Otherwise, its properties for dependent tests at present are not fully understood. Consequently, its use for confirmatory trials, especially for their primary endpoints, remains worrisome. Confirmatory trials are typically designed with 1–2 primary endpoints. Therefore, a question was raised at the Food and Drug Administration as to whether the HP is a valid test for the simple case of performing treatment‐to‐control comparisons on two primary endpoints when their test statistics are not independent. Confirmatory trials for statistical tests normally use simple test statistics, such as the normal Z, student's t, and chi‐square. The literature does include some work on the HP for dependent cases covering these test statistics, but concerns remain regarding its use for confirmatory trials for which endpoint tests are mostly of the dependent kind. The purpose of this paper is therefore to revisit this procedure and provide sufficient details for better understanding of its performance for dependent cases related to the aforementioned question. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.     

Weighted analysis of composite endpoints with simultaneous inference for flexible weight constraints

Composite endpoints are widely used as primary endpoints of randomized controlled trials across clinical disciplines. A common critique of the conventional analysis of composite endpoints is that all disease events are weighted equally, whereas their clinical relevance may differ substantially. We address this by introducing a framework for the weighted analysis of composite endpoints and interpretable test statistics, which are applicable to both binary and time‐to‐event data. To cope with the difficulty of selecting an exact set of weights, we propose a method for constructing simultaneous confidence intervals and tests that asymptotically preserve the family‐wise type I error in the strong sense across families of weights satisfying flexible inequality or order constraints based on the theory of ‐distributions. We show that the method achieves the nominal simultaneous coverage rate with substantial efficiency gains over Scheffé's procedure in a simulation study and apply it to trials in cardiovascular disease and enteric fever. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Multi‐arm trials with multiple primary endpoints and missing values

We present an extension of multiple contrast tests for multiple endpoints to the case of missing values. The endpoints are assumed to be normally distributed and correlated and to have equal covariance matrices for the different treatments. Different multivariate t distributions will be applied, differing in endpoint‐specific degrees of freedom. In contrast to competing methods, the familywise error type I is maintained in the strong sense in an admissible range, and the problem of different marginal errors type I is avoided. The information of all observations is exploited, thereby enabling a gain in power compared with a complete case analysis.     

Maximum type I error rate inflation from sample size reassessment when investigators are blind to treatment labels

Consider a parallel group trial for the comparison of an experimental treatment to a control, where the second‐stage sample size may depend on the blinded primary endpoint data as well as on additional blinded data from a secondary endpoint. For the setting of normally distributed endpoints, we demonstrate that this may lead to an inflation of the type I error rate if the null hypothesis holds for the primary but not the secondary endpoint. We derive upper bounds for the inflation of the type I error rate, both for trials that employ random allocation and for those that use block randomization. We illustrate the worst‐case sample size reassessment rule in a case study. For both randomization strategies, the maximum type I error rate increases with the effect size in the secondary endpoint and the correlation between endpoints. The maximum inflation increases with smaller block sizes if information on the block size is used in the reassessment rule. Based on our findings, we do not question the well‐established use of blinded sample size reassessment methods with nuisance parameter estimates computed from the blinded interim data of the primary endpoint. However, we demonstrate that the type I error rate control of these methods relies on the application of specific, binding, pre‐planned and fully algorithmic sample size reassessment rules and does not extend to general or unplanned sample size adjustments based on blinded data. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Sample size determination in group‐sequential clinical trials with two co‐primary endpoints

We discuss sample size determination in group‐sequential designs with two endpoints as co‐primary. We derive the power and sample size within two decision‐making frameworks. One is to claim the test intervention's benefit relative to control when superiority is achieved for the two endpoints at the same interim timepoint of the trial. The other is when superiority is achieved for the two endpoints at any interim timepoint, not necessarily simultaneously. We evaluate the behaviors of sample size and power with varying design elements and provide a real example to illustrate the proposed sample size methods. In addition, we discuss sample size recalculation based on observed data and evaluate the impact on the power and Type I error rate. Copyright © 2014 John Wiley & Sons, Ltd.     

Clinical endpoints for efficacy studies

Well-established, validated and clinically meaningful primary and secondary endpoints are critical in advancing vaccines through proof of principal studies, licensure and pre-qualification. To that end, the field of vaccine development for Shigella, enterotoxigenic Escherichia coli (ETEC) as well as other enteric pathogens would benefit greatly from a focused review of clinical endpoints and the use of common endpoints across the field to enable study-to-study comparisons as well as comparative assessments between vaccine candidates. A workshop was conducted to review clinical endpoints from controlled human challenge studies, field studies in naïve adult travelers and pediatric studies in low-middle income countries and to develop a consensus on clinical endpoints for future vaccine trials.Following sequential presentations on different study designs (CHIM, travelers’ efficacy and pediatric efficacy), workshop participants broke into three simultaneous workgroups focused on those study designs to discuss a number of topics key to clinical endpoints specific to each study design. Previously utilized endpoints were reviewed with an eye towards potentially novel endpoints for future studies and consideration of the disease parameters and spectrum of disease targeted for prevention. The strength of support among workshop participants for the use of various endpoints is summarized as are recommendations for additional endpoints to be considered in future studies. It is anticipated that this report will facilitate endpoint determination in future efficacy trials of vaccine candidates.     

Surrogate endpoints in clinical trials: cancer

Investigators use a surrogate endpoint when the endpoint of interest is too difficult and/or expensive to measure routinely and when they can define some other, more readily measurable, endpoint, that is sufficiently well correlated with the first to justify its use as a substitute. A surrogate endpoint is usually proposed on the basis of a biologic rationale. In cancer studies with survival time as the primary endpoint, surrogate endpoints frequently employed are tumour response, time to progression, or time to reappearance of disease, since these events occur earlier and are unaffected by use of secondary therapies. In early drug development studies, tumour response is often the true primary endpoint. We discuss the investigation of the validity of carcinoembryonic antigen (a tumour marker present in the blood) as a surrogate for tumour response. In considering the validity of surrogate endpoints, one must distinguish between study endpoints that provide a basis for reliable comparisons of therapeutic effect, and clinical endpoints that are useful for patient management but have insufficient sensitivity and/or specificity to provide reproducible assessments of the effects of particular therapies.     

Changes are still needed on multiple co‐primary endpoints

The Food and Drug Administration in the United States issued a much‐awaited draft guidance on ‘Multiple Endpoints in Clinical Trials’ in January 2017. The draft guidance is well written and contains consistent message on the technical implementation of the principles laid out in the guidance. In this commentary, we raise a question on applying the principles to studies designed from a safety perspective. We then direct our attention to issues related to multiple co‐primary endpoints. In a paper published in the Drug Information Journal in 2007, Offen et al. give examples of disorders where multiple co‐primary endpoints are required by regulators. The standard test for multiple co‐primary endpoints is the min test which tests each endpoint individually, at the one‐sided 2.5% level, for a confirmatory trial. This approach leads to a substantial loss of power when the number of co‐primary endpoints exceeds 2, a fact acknowledged in the draft guidance. We review approaches that have been proposed to tackle the problem of power loss and propose a new one. Using recommendations by Chen et al. for the assessment of drugs for vulvar and vaginal atrophy published in the Drug Information Journal in 2010, we argue the need for more changes and urge a path forward that uses different levels of claims to reflect the effectiveness of a product on multiple endpoints that are equally important and mostly unrelated. Copyright © 2017 John Wiley & Sons, Ltd.     

Statistical power of multiplicity adjustment strategies for correlated binary endpoints

There are numerous alternatives to the so-called Bonferroni adjustment to control for familywise Type I error among multiple tests. Yet, for the most part, these approaches disregard the correlation among endpoints. This can prove to be a conservative hypothesis testing strategy if the null hypothesis is false. The James procedure was proposed to account for the correlation structure among multiple continuous endpoints. Here, a simulation study evaluates the statistical power of the Hochberg and James adjustment strategies relative to that of the Bonferroni approach when used for multiple correlated binary variables. The simulations demonstrate that relative to the Bonferroni approach, neither alternative sacrifices power. The Hochberg approach has more statistical power for rho相似文献   

Sizing clinical trials when comparing bivariate time‐to‐event outcomes

Clinical trials with multiple primary time‐to‐event outcomes are common. Use of multiple endpoints creates challenges in the evaluation of power and the calculation of sample size during trial design particularly for time‐to‐event outcomes. We present methods for calculating the power and sample size for randomized superiority clinical trials with two correlated time‐to‐event outcomes. We do this for independent and dependent censoring for three censoring scenarios: (i) the two events are non‐fatal; (ii) one event is fatal (semi‐competing risk); and (iii) both are fatal (competing risk). We derive the bivariate log‐rank test in all three censoring scenarios and investigate the behavior of power and the required sample sizes. Separate evaluations are conducted for two inferential goals, evaluation of whether the test intervention is superior to the control on: (1) all of the endpoints (multiple co‐primary) or (2) at least one endpoint (multiple primary). Copyright © 2017 John Wiley & Sons, Ltd.     

Assessment of equivalence on multiple endpoints.

Some clinical trials aim to demonstrate therapeutic equivalence on multiple primary endpoints. For example, therapeutic equivalence studies of agents for the treatment of osteoarthritis use several primary endpoints including investigator's global assessment of disease activity, patient's global assessment of response to therapy, and pain. In this paper, thoughts on simultaneous equivalence assessment on three endpoints are presented. As pointed out by Berger and Hsu (1996), the conventional intersection-union test can be conservative. Simulation and computation are conducted to provide an insight on the conservativeness. We also provide a method to lower the confidence level and at the same time maintain the type I error when endpoints have normal distributions and are independent. If, in a particular analysis, the goal is to demonstrate equivalence on as many endpoints as possible, a step-up procedure can be used for selecting those endpoints for which equivalence may be demonstrated. This step-up procedure at the same time controls experimentwise error rate. The techniques are illustrated by a data example.     

Resampling-based methods for the analysis of multiple endpoints in clinical trials

We consider multivariate tests for comparing two treatments with multiple endpoints. The test decision is drawn from the simultaneous consideration of the univariate tests for the single endpoints. A general class of these tests, called cut-off tests, can be given, which, however, can lead to highly conservative procedures because the dependencies among the endpoints are not taken into account. In applying resampling-based methods considerable improvements for these tests can be achieved. Resampling-based cut-off tests are proposed which are sensitive against a treatment difference in a single endpoint, in a subgroup of endpoints, or in all endpoints. The results of Monte Carlo simulations demonstrate that a remarkable gain in statistical power as compared to the crude simultaneous consideration can be reached. In particular, for the multivariate one-sided test situation the proposed tests can be recommended. As an example the application of the tests is demonstrated by data from a clinical trial.     

Optimizing the order of hypotheses in serial testing of multiple endpoints in clinical trials

Clinical trials usually collect information on a large number of variables or endpoints, including one or more primary endpoints as well as a number of secondary endpoints representing different aspects of treatment effectiveness and safety. In this article, we focus on serial testing procedures that test multiple endpoints in a pre‐specified order, and consider how to optimize the order of endpoints subject to any clinical constraints, with respect to the expected number of successes (i.e., endpoints that reach statistical significance) or the expected gain (if endpoints are associated with numerical utilities). We consider some common approaches to this problem and propose two new approaches: a greedy algorithm based on conditional power and a simulated annealing algorithm that attempts to improve a given sequence in a random and iterative fashion. Simulation results indicate that the proposed algorithms are useful for finding a high‐performing sequence, and that optimized fixed sequence procedures can be competitive against traditional multiple testing procedures such as Holm's. The methods and findings are illustrated with two examples concerning migraine and asthma. Copyright © 2015 John Wiley & Sons, Ltd.     

Some comments on frequently used multiple endpoint adjustment methods in clinical trials

Confirmatory clinical trials often classify clinical response variables into primary and secondary endpoints. The presence of two or more primary endpoints in a clinical trial usually means that some adjustments of the observed p-values for multiplicity of tests may be required for the control of the type I error rate. In this paper, we discuss statistical concerns associated with some commonly used multiple endpoint adjustment procedures. We also present limited Monte Carlo simulation results to demonstrate the performance of selected p-value-based methods in protecting the type I error rate. © 1997 by John Wiley & Sons, Ltd.     

Validation of surrogate endpoints in cancer clinical trials via principal stratification with an application to a prostate cancer trial

Increasing attention has been focused on the use and validation of surrogate endpoints in cancer clinical trials. Previous literature on validation of surrogate endpoints are classified into four approaches: the proportion explained approach; the indirect effects approach; the meta‐analytic approach; and the principal stratification approach. The mainstream in cancer research has seen the application of a meta‐analytic approach. However, VanderWeele (2013) showed that all four of these approaches potentially suffer from the surrogate paradox. It was also shown that, if a principal surrogate satisfies additional criteria called one‐sided average causal sufficiency, the surrogate cannot exhibit a surrogate paradox. Here, we propose a method for estimating principal effects under a monotonicity assumption. Specifically, we consider cancer clinical trials which compare a binary surrogate endpoint and a time‐to‐event clinical endpoint under two naturally ordered treatments (e.g. combined therapy vs. monotherapy). Estimation based on a mean score estimating equation will be implemented by the expectation‐maximization algorithm. We will also apply the proposed method as well as other surrogacy criteria to evaluate the surrogacy of prostate‐specific antigen using data from a phase III advanced prostate cancer trial, clarifying the complementary roles of both the principal stratification and meta‐analytic approaches in the evaluation of surrogate endpoints in cancer. Copyright © 2017 John Wiley & Sons, Ltd.