Multiple Co-primary Endpoints Research Articles

Seamless phase 2/3 design for trials with multiple co-primary endpoints using Bayesian predictive power

Seamless phase 2/3 design has become increasingly popular in clinical trials with a single endpoint. Trials that define success based on the achievement of all co-primary endpoints (CPEs) encounter the challenge of inflated type 2 error rates, often leading to an overly large sample size. To tackle this challenge, we introduced a seamless phase 2/3 design strategy that employs Bayesian predictive power (BPP) for futility monitoring and sample size re-estimation at interim analysis. The correlations among multiple CPEs are incorporated using a Dirichlet-multinomial distribution. An alternative approach based on conditional power (CP) was also discussed for comparison. A seamless phase 2/3 vaccine trial employing four binary endpoints under the non-inferior hypothesis serves as an example. Our results spotlight that, in scenarios with relatively small phase 2 sample sizes (e.g., 50 or 100 subjects), the BPP approach either outperforms or matches the CP approach in terms of overall power. Particularly, with n1 = 50 and ρ = 0, BPP showcases an overall power advantage over CP by as much as 8.54%. Furthermore, when the phase 2 stage enrolled more subjects (e.g., 150 or 200), especially with a phase 2 sample size of 200 and ρ = 0, the BPP approach evidences a peak difference of 5.76% in early stop probability over the CP approach, emphasizing its better efficiency in terminating futile trials. It’s noteworthy that both BPP and CP methodologies maintained type 1 error rates under 2.5%. In conclusion, the integration of the Dirichlet-Multinominal model with the BPP approach offers improvement in certain scenarios over the CP approach for seamless phase 2/3 trials with multiple CPEs.

Simplification or simulation: Power calculation in clinical trials

Background and objectivesA justifiable sample size is essential at trial design stage. Generally this task is completed by forming the main research question into a statistical procedure and then implementing the published formulae or software packages. When these standard statistical formulae/software packages become unavailable for studies with complex statistical procedures, some statisticians choose to fill this gap by assuming an alternative simplified sample size calculation. Monte Carlo simulations can also be deployed, particularly for complex trials. However, it is still unclear on how to determine the appropriate approach under certain practical scenarios. MethodsWe adopted real clinical trials as examples and investigated on simplification and simulation-based sample size calculation approaches. ResultsCompared to simplified sample size calculation, the simulation approach can better address the non-ignorable impact of baseline/follow-up outcome correlation on study power. For studies with multiple endpoints and multiple co-primary endpoints, the sample sizes calculated by simplification approach should be scrutinized. ConclusionsDirectly using the simplification approach for sample size calculation should be restricted. We recommend to utilize the simulation approach, particularly for complex trials, at least as a sensitivity checking and a useful triangulation to the simplification approach outlined.

Power analysis for cluster randomized trials with multiple binary co-primary endpoints.

Cluster randomized trials (CRTs) are widely used in different areas of medicine and public health. Recently, with increasing complexity of medical therapies and technological advances in monitoring multiple outcomes, many clinical trials attempt to evaluate multiple co-primary endpoints. In this study, we present a power analysis method for CRTs with binary co-primary endpoints. It is developed based on the GEE (generalized estimating equation) approach, and three types of correlations are considered: inter-subject correlation within each endpoint, intra-subject correlation across endpoints, and inter-subject correlation across endpoints. A closed-form joint distribution of the K test statistics is derived, which facilitates the evaluation of power and type I error for arbitrarily constructed hypotheses. We further present a theorem that characterizes the relationship between various correlations and testing power. We assess the performance of the proposed power analysis method based on extensive simulation studies. An application example to a real clinical trial ispresented.

Sequential parallel comparison design with two coprimary endpoints.

A placebo-controlled randomized clinical trial is required to demonstrate that an experimental treatment is superior to its corresponding placebo on multiple coprimary endpoints. This is particularly true in the field of neurology. In fact, clinical trials for neurological disorders need to show the superiority of an experimental treatment over a placebo in two coprimary endpoints. Unfortunately, these trials often fail to detect a true treatment effect for the experimental treatment versus the placebo owing to an unexpectedly high placebo response rate. Sequential parallel comparison design (SPCD) can be used to address this problem. However, the SPCD has not yet been discussed in relation to clinical trials with coprimary endpoints. In this article, our aim was to develop a hypothesis-testing method and a method for calculating the corresponding sample size for the SPCD with two coprimary endpoints. In a simulation, we show that the proposed hypothesis-testing method achieves the nominal type I error rate and power and that the proposed sample size calculation method has adequate power accuracy. In addition, the usefulness of our methods is confirmed by returning to an SPCD trial with a single primary endpoint of Alzheimer disease-related agitation.

Bayesian Uncertainty Directed Trial Designs

ABSTRACTMost Bayesian response-adaptive designs unbalance randomization rates toward the most promising arms with the goal of increasing the number of positive treatment outcomes during the study, even though the primary aim of the trial is different. We discuss Bayesian uncertainty directed designs (BUD), a class of Bayesian designs in which the investigator specifies an information measure tailored to the experiment. All decisions during the trial are selected to optimize the available information at the end of the study. The approach can be applied to several designs, ranging from early stage multi-arm trials to biomarker-driven and multi-endpoint studies. We discuss the asymptotic limit of the patient allocation proportion to treatments, and illustrate the finite-sample operating characteristics of BUD designs through examples, including multi-arm trials, biomarker-stratified trials, and trials with multiple co-primary endpoints. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

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.

Sample size determination for a specific region in multiregional clinical trials with multiple co-primary endpoints.

Recently, multi-regional clinical trials (MRCTs), which incorporate subjects from many countries/regions around the world under the same protocol, have been widely conducted by many global pharmaceutical companies. The objective of such trials is to accelerate the development process for a drug and shorten the drug’s approval time in key markets. Several statistical methods have been purposed for the design and evaluation of MRCTs, as well as for assessing the consistency of treatment effects across all regions with one primary endpoint. However, in some therapeutic areas (e.g., Alzheimer’s disease), the clinical efficacy of a new treatment may be characterized by a set of possibly correlated endpoints, known as multiple co-primary endpoints. In this paper, we focus on a specific region and establish three statistical criteria for evaluating consistency between the specific region and overall results in MRCTs with multiple co-primary endpoints. More specifically, two of those criteria are used to assess whether the treatment effect in the region of interest is as large as that of the other regions or of the regions overall, while the other criterion is used to assess the consistency of the treatment effect of the specific region achieving a pre-specified threshold. The sample size required for the region of interest can also be evaluated based on these three criteria.

Interim evaluation of efficacy or futility in group-sequential trials with multiple co-primary endpoints.

We discuss group-sequential designs in superiority clinical trials with multiple co-primary endpoints, that is, when trials are designed to evaluate if the test intervention is superior to the control on all primary endpoints. We consider several decision-making frameworks for evaluating efficacy or futility, based on boundaries using group-sequential methodology. We incorporate the correlations among the endpoints into the calculations for futility boundaries and sample sizes as a function of other design parameters, including mean differences, the number of analyses, and efficacy boundaries. We investigate the operating characteristics of the proposed decision-making frameworks in terms of efficacy/futility boundaries, power, the Type I error rate, and sample sizes, while varying the number of analyses, the correlations among the endpoints, and the mean differences. We provide an example to illustrate the methods and discuss practical considerations when designing efficient group-sequential designs in clinical trials with co-primary endpoints.

Reducing unnecessary measurements in clinical trials with multiple primary endpoints

Clinical trials often involve two or more primary endpoints. However, observing or measuring high-cost endpoints often reduces the efficiency of the study because of high medical costs, highly invasive measurements, or long-term follow-up. Further, the individual powers to demonstrate the overall efficacy of a new intervention for the multiple endpoints often differ under a given sample size. We propose an efficient clinical trial design in which the sample size for each of the endpoints is individually determined, taking into consideration both the cost and the individual power for each endpoint. We compared the efficiency of the proposed design with that of the conventional design using three variables: (1) the number of participants in the study, (2) the total number of measurements for all endpoints, and (3) the cost of enrolling the participants and obtaining the measurements for all endpoints. We extended the proposed design to a group-sequential design. Numerical examples show that the proposed design can reduce unnecessary measurements and adjust the individual powers for the endpoints, especially when the individual power for one endpoint is relatively higher than that for other endpoints in a study with multiple co-primary endpoints.

Group-Sequential Strategies in Clinical Trials with Multiple Co-Primary Outcomes

We discuss the decision-making frameworks for clinical trials with multiple co-primary endpoints in a group-sequential setting. The decision-making frameworks can account for flexibilities, such as a varying number of analyses, equally or unequally spaced increments of information, and fixed or adaptive Type I error allocation among endpoints. The frameworks can provide efficiency, that is, potentially fewer trial participants, than the fixed sample size designs. We investigate the operating characteristics of the decision-making frameworks and provide guidance on constructing efficient group-sequential strategies in clinical trials with multiple co-primary endpoints.

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.

Power estimation for multiple co-primary endpoints: a comparison among conservative solutions

The problem of estimating the power of the multivariate Intersection Union test (IUT) is studied. Four classical parametric solutions and a bootstrap non-parametric one, providing statistical lower bounds (i.e. one directional confidence intervals) for the power, are considered. The performances of these techniques in several bivariate IUT settings are compared through a simulation study. All solutions are biased, since their actual coverage probabilities are higher than the nominal one. The bootstrap solution shows the smallest bias, and the variability of its estimates is the lowest. Moreover, the bias of the bootstrap solution reduces faster than those of the other techniques when the pilot sample size, or the correlation, or the rate between the two non-centrality parameters increases. Also, the non-parametric bootstrap solution can be improved by calibration, with a considerable bias reduction.

Sample size determination in clinical trials with multiple co‐primary endpoints including mixed continuous and binary variables

In the field of pharmaceutical drug development, there have been extensive discussions on the establishment of statistically significant results that demonstrate the efficacy of a new treatment with multiple co-primary endpoints. When designing a clinical trial with such multiple co-primary endpoints, it is critical to determine the appropriate sample size for indicating the statistical significance of all the co-primary endpoints with preserving the desired overall power because the type II error rate increases with the number of co-primary endpoints. We consider overall power functions and sample size determinations with multiple co-primary endpoints that consist of mixed continuous and binary variables, and provide numerical examples to illustrate the behavior of the overall power functions and sample sizes. In formulating the problem, we assume that response variables follow a multivariate normal distribution, where binary variables are observed in a dichotomized normal distribution with a certain point of dichotomy. Numerical examples show that the sample size decreases as the correlation increases when the individual powers of each endpoint are approximately and mutually equal.

A convenient formula for sample size calculations in clinical trials with multiple co‐primary continuous endpoints

The clinical efficacy of a new treatment may often be better evaluated by two or more co-primary endpoints. Recently, in pharmaceutical drug development, there has been increasing discussion regarding establishing statistically significant favorable results on more than one endpoint in comparisons between treatments, which is referred to as a problem of multiple co-primary endpoints. Several methods have been proposed for calculating the sample size required to design a trial with multiple co-primary correlated endpoints. However, because these methods require users to have considerable mathematical sophistication and knowledge of programming techniques, their application and spread may be restricted in practice. To improve the convenience of these methods, in this paper, we provide a useful formula with accompanying numerical tables for sample size calculations to design clinical trials with two treatments, where the efficacy of a new treatment is demonstrated on continuous co-primary endpoints. In addition, we provide some examples to illustrate the sample size calculations made using the formula. Using the formula and the tables, which can be read according to the patterns of correlations and effect size ratios expected in multiple co-primary endpoints, makes it convenient to evaluate the required sample size promptly.

Sample Size Determination in Superiority Clinical Trials with Multiple Co-Primary Correlated Endpoints

In pharmaceutical drug development, for regulatory purposes, there are increasing discussions on the establishment of statistically significant results demonstrating the efficacy of a new treatment on multiple co-primary endpoints. At the design stage with multiple co-primary endpoints, it is critical to determine the appropriate sample size for indicating statistical significance for all co-primary endpoints with preserving the intended power set, since the type II error increases as the number of co-primary endpoints increases. We provide fundamental formulae for power and sample size calculation with multiple co-primary endpoints and illustrate the aspect of the provided methods through numerical tables and examples.

Sample size determination in clinical trials with multiple co‐primary binary endpoints

Clinical trials often employ two or more primary efficacy endpoints. One of the major problems in such trials is how to determine a sample size suitable for multiple co-primary correlated endpoints. We provide fundamental formulae for the calculation of power and sample size in order to achieve statistical significance for all the multiple primary endpoints given as binary variables. On the basis of three association measures among primary endpoints, we discuss five methods of power and sample size calculation: the asymptotic normal method with and without continuity correction, the arcsine method with and without continuity correction, and Fisher's exact method. For all five methods, the achieved sample size decreases as the value of association measure increases when the effect sizes among endpoints are approximately equal. In particular, a high positive association has a greater effect on the decrease in the sample size. On the other hand, such a relationship is not very strong when the effect sizes are different.

Sample size for simultaneous testing of rate differences in non-inferiority trials with multiple endpoints

This paper discusses sample size estimation for several simultaneous tests of rate differences in non-inferiority trials. Formulae corresponding to the commonly used methods in a single endpoint study, which were proposed by [Blackwelder, W.C., 1982. Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3, 345–353] and [Farrington, C.P., Manning, G., 1990. Test statistics and sample size formulae for comparative binomial trials with null hypothesis on non-zero risk difference or non-unity relative risk. Statistics in Medicine 9, 1447–1454] are presented. The focus of this study is a non-inferiority trial with multiple co-primary endpoints. The sample size for simultaneous testing is not only determined by Type I and II error rates, but also by several other factors including the effect sizes of each endpoint, the number of endpoints and the correlation among test statistics in each endpoint. To demonstrate the sample sizes required to achieve various levels of statistical significance under different correlations, this manuscript presents plots of the distribution of P-values as a function of sample size. These plots are obtained simultaneously from each endpoint under the alternative hypothesis. The results are applied to the design of a non-inferiority trial.

Multiple Co-primary Endpoints: Medical and Statistical Solutions: A Report from the Multiple Endpoints Expert Team of the Pharmaceutical Research and Manufacturers of America

There are quite a few disorders for which regulatory agencies have required a treatment to demonstrate a statistically significant effect on multiple endpoints, each at the one-sided 2.5% level, before accepting the treatment’s efficacy for the disorders. Depending on the correlation among the endpoints, this requirement could lead to a substantial reduction in the study’s power to conclude the efficacy of a treatment. To investigate the prevalence of this requirement and propose possible solutions, a multiple-disciplinary Multiple Endpoints Expert Team sponsored by Pharmaceutical Research and Manufacturers of America was formed in November 2003. The team recognized early that many researchers were not fully aware of the implications of requiring multiple co-primary endpoints. The team proposes possible solutions from both the medical and the statistical perspectives. The optimal solution is to reduce the number of multiple co-primary endpoints. If after careful considerations, multiple co-primary endpoints remain a scientific requirement, the team proposes statistical solutions and encourages that regulatory agencies be receptive to approaches that adopt modest upward adjustments of the nominal significance levels for testing individual endpoints. Finally, the team hopes that this report will draw more attention to the problem of multiple co-primary endpoints and stimulate further research.

Challenge of multiple co‐primary endpoints: a new approach

There are many disorders where regulatory agencies have required a new treatment to demonstrate efficacy on multiple co-primary endpoints, all significant at the one-sided 2.5 per cent level, before accepting the treatment's effect for the disorder. This requirement, rooted in the intersection-union (IU) test, has led many researchers to increase the study sample size to make up for the reduction in the statistical power at the study level. Unfortunately, the increase in sample size could be substantial when the endpoints are minimally correlated and the treatment effects on the multiple endpoints are comparable. In this paper, we demonstrate that the frequentist concept of controlling the maximum false positive rate, even when applied to a restricted null space, has only limited success in keeping the sample size increase at a reasonable level. We therefore propose an approach that is based on the notion of controlling an average type I error rate. By employing an upper bound for the average type I error rate, the new approach provides an adjustment to the significance level that depends only on the correlation among the endpoints. For the most common case of two or three co-primary endpoints, the adjusted significance level is at most 5 per cent (one-sided) when the endpoints are moderately correlated. We show how sample size could be calculated under the proposed approach and contrast the needed sample size with that required under the IU test. We provide additional comments and discuss why the new approach is consistent with the principle requiring evidence of significance in the drug development and approval process.