Gatekeeping procedures in dose-response clinical trials based on the Dunnett test

This paper discusses multiple testing procedures in dose-response clinical trials with primary and secondary endpoints. A general gatekeeping framework for constructing multiple tests is proposed, which extends the Dunnett test [Journal of the American Statistical Association 1955; 50: 1096-1121] and Bonferroni-based gatekeeping tests developed by Dmitrienko et al. [Statistics in Medicine 2003; 22:2387-2400]. The proposed procedure accounts for the hierarchical structure of the testing problem; for example, it restricts testing of secondary endpoints to the doses for which the primary endpoint is significant. The multiple testing approach is illustrated using a dose-response clinical trial in patients with diabetes. Monte-Carlo simulations demonstrate that the proposed procedure provides a power advantage over the Bonferroni gatekeeping procedure. The power gain generally increases with increasing correlation among the endpoints, especially when all primary dose-control comparisons are significant.     

A temporal correlation function for the association between patient‐reported outcome and clinical endpoints

For clinical trials with multiple endpoints, the primary interest is usually to evaluate the relationship of these endpoints and treatment interventions. Studying the correlation of two clinical trial endpoints can also be of interests. For example, the association between patient‐reported outcome and clinically assessed endpoint could answer important research questions and also generate interesting hypothesis for future research. However, it is not straightforward to quantify such association. In this article, we proposed a multiple event approach to profile such association with a temporal correlation function, visualized by a correlation function plot over time with a confidence band. We developed this approach by extending the existing methodology in recurrent event literature. This approach was shown to be generally unbiased and could be a useful tool for data visualization and inference. We demonstrated the use of this method with data from a real clinical trial. Although this approach was developed to evaluate the association between patient‐reported outcome and adverse events, it can also be used to evaluate the association of any two endpoints that can be translated to time‐to‐event endpoints. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Analyzing multiple endpoints in a confirmatory randomized clinical trial—an approach that addresses stratification,missing values,baseline imbalance and multiplicity for strictly ordinal outcomes

Confirmatory randomized clinical trials with a stratified design may have ordinal response outcomes, ie, either ordered categories or continuous determinations that are not compatible with an interval scale. Also, multiple endpoints are often collected when 1 single endpoint does not represent the overall efficacy of the treatment. In addition, random baseline imbalances and missing values can add another layer of difficulty in the analysis plan. Therefore, the development of an approach that provides a consolidated strategy to all issues collectively is essential. For a real case example that is from a clinical trial comparing a test treatment and a control for the pain management for patients with osteoarthritis, which has all aforementioned issues, multivariate Mann‐Whitney estimators with stratification adjustment are applicable to the strictly ordinal responses with stratified design. Randomization based nonparametric analysis of covariance is applied to account for the possible baseline imbalances. Several approaches that handle missing values are provided. A global test followed by a closed testing procedure controls the family wise error rate in the strong sense for the analysis of multiple endpoints. Four outcomes indicating joint pain, stiffness, and functional status were analyzed collectively and also individually through the procedures. Treatment efficacy was observed in the combined endpoint as well as in the individual endpoints. The proposed approach is effective in addressing the aforementioned problems simultaneously and straightforward to implement.     

An investigation into the two‐stage meta‐analytic copula modelling approach for evaluating time‐to‐event surrogate endpoints which comprise of one or more events of interest

Clinical trials of experimental treatments must be designed with primary endpoints that directly measure clinical benefit for patients. In many disease areas, the recognised gold standard primary endpoint can take many years to mature, leading to challenges in the conduct and quality of clinical studies. There is increasing interest in using shorter‐term surrogate endpoints as substitutes for costly long‐term clinical trial endpoints; such surrogates need to be selected according to biological plausibility, as well as the ability to reliably predict the unobserved treatment effect on the long‐term endpoint. A number of statistical methods to evaluate this prediction have been proposed; this paper uses a simulation study to explore one such method in the context of time‐to‐event surrogates for a time‐to‐event true endpoint. This two‐stage meta‐analytic copula method has been extensively studied for time‐to‐event surrogate endpoints with one event of interest, but thus far has not been explored for the assessment of surrogates which have multiple events of interest, such as those incorporating information directly from the true clinical endpoint. We assess the sensitivity of the method to various factors including strength of association between endpoints, the quantity of data available, and the effect of censoring. In particular, we consider scenarios where there exist very little data on which to assess surrogacy. Results show that the two‐stage meta‐analytic copula method performs well under certain circumstances and could be considered useful in practice, but demonstrates limitations that may prevent universal use.     

Testing multiple primary endpoints in clinical trials with sample size adaptation

In this paper, we propose a design that uses a short‐term endpoint for accelerated approval at interim analysis and a long‐term endpoint for full approval at final analysis with sample size adaptation based on the long‐term endpoint. Two sample size adaptation rules are compared: an adaptation rule to maintain the conditional power at a prespecified level and a step function type adaptation rule to better address the bias issue. Three testing procedures are proposed: alpha splitting between the two endpoints; alpha exhaustive between the endpoints; and alpha exhaustive with improved critical value based on correlation. Family‐wise error rate is proved to be strongly controlled for the two endpoints, sample size adaptation, and two analysis time points with the proposed designs. We show that using alpha exhaustive designs greatly improve the power when both endpoints are effective, and the power difference between the two adaptation rules is minimal. The proposed design can be extended to more general settings. Copyright © 2015 John Wiley & Sons, Ltd.     

Nested combination tests with a time‐to‐event endpoint using a short‐term endpoint for design adaptations

Adaptive trial methodology for multiarmed trials and enrichment designs has been extensively discussed in the past. A general principle to construct test procedures that control the family‐wise Type I error rate in the strong sense is based on combination tests within a closed test. Using survival data, a problem arises when using information of patients for adaptive decision making, which are under risk at interim. With the currently available testing procedures, either no testing of hypotheses in interim analyses is possible or there are restrictions on the interim data that can be used in the adaptation decisions as, essentially, only the interim test statistics of the primary endpoint may be used. We propose a general adaptive testing procedure, covering multiarmed and enrichment designs, which does not have these restrictions. An important application are clinical trials, where short‐term surrogate endpoints are used as basis for trial adaptations, and we illustrate how such trials can be designed. We propose statistical models to assess the impact of effect sizes, the correlation structure between the short‐term and the primary endpoint, the sample size, the timing of interim analyses, and the selection rule on the operating characteristics.     

How to avoid concerns with the interpretation of two primary endpoints if significant superiority in one is sufficient for formal proof of efficacy

Formal proof of efficacy of a drug requires that in a prospective experiment, superiority over placebo, or either superiority or at least non-inferiority to an established standard, is demonstrated. Traditionally one primary endpoint is specified, but various diseases exist where treatment success needs to be based on the assessment of two primary endpoints. With co-primary endpoints, both need to be “significant” as a prerequisite to claim study success. Here, no adjustment of the study-wise type-1-error is needed, but sample size is often increased to maintain the pre-defined power. Studies that use an at-least-one concept have been proposed where study success is claimed if superiority for at least one of the endpoints is demonstrated. This is sometimes also called the dual primary endpoint concept, and an appropriate adjustment of the study-wise type-1-error is required. This concept is not covered in the European Guideline on multiplicity because study success can be claimed if one endpoint shows significant superiority, despite a possible deterioration in the other. In line with Röhmel's strategy, we discuss an alternative approach including non-inferiority hypotheses testing that avoids obvious contradictions to proper decision-making. This approach leads back to the co-primary endpoint assessment, and has the advantage that minimum requirements for endpoints can be modeled flexibly for several practical needs. Our simulations show that, if planning assumptions are correct, the proposed additional requirements improve interpretation with only a limited impact on power, that is, on sample size.     

A generalized analytic solution to the win ratio to analyze a composite endpoint considering the clinical importance order among components

A composite endpoint consists of multiple endpoints combined in one outcome. It is frequently used as the primary endpoint in randomized clinical trials. There are two main disadvantages associated with the use of composite endpoints: a) in conventional analyses, all components are treated equally important; and b) in time‐to‐event analyses, the first event considered may not be the most important component. Recently Pocock et al. (2012) introduced the win ratio method to address these disadvantages. This method has two alternative approaches: the matched pair approach and the unmatched pair approach. In the unmatched pair approach, the confidence interval is constructed based on bootstrap resampling, and the hypothesis testing is based on the non‐parametric method by Finkelstein and Schoenfeld (1999). Luo et al. (2015) developed a close‐form variance estimator of the win ratio for the unmatched pair approach, based on a composite endpoint with two components and a specific algorithm determining winners, losers and ties. We extend the unmatched pair approach to provide a generalized analytical solution to both hypothesis testing and confidence interval construction for the win ratio, based on its logarithmic asymptotic distribution. This asymptotic distribution is derived via U‐statistics following Wei and Johnson (1985). We perform simulations assessing the confidence intervals constructed based on our approach versus those per the bootstrap resampling and per Luo et al. We have also applied our approach to a liver transplant Phase III study. This application and the simulation studies show that the win ratio can be a better statistical measure than the odds ratio when the importance order among components matters; and the method per our approach and that by Luo et al., although derived based on large sample theory, are not limited to a large sample, but are also good for relatively small sample sizes. Different from Pocock et al. and Luo et al., our approach is a generalized analytical method, which is valid for any algorithm determining winners, losers and ties. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian single-to-double arm transition design using both short-term and long-term endpoints

The choice between single-arm designs versus randomized double-arm designs has been contentiously debated in the literature of phase II oncology trials. Recently, as a compromise, the single-to-double arm transition design was proposed, combining the two designs into one trial over two stages. Successful implementation of the two-stage transition design requires a suspension period at the end of the first stage to collect the response data of the already enrolled patients. When the evaluation of the primary efficacy endpoint is overly long, the between-stage suspension period may unfavorably prolong the trial duration and cause a delay in treating future eligible patients. To accelerate the trial, we propose a Bayesian single-to-double arm design with short-term endpoints (BSDS), where an intermediate short-term endpoint is used for making early termination decisions at the end of the single-arm stage, followed by an evaluation of the long-term endpoint at the end of the subsequent double-arm stage. Bayesian posterior probabilities are used as the primary decision-making tool at the end of the trial. Design calibration steps are proposed for this Bayesian monitoring process to control the frequentist operating characteristics and minimize the expected sample size. Extensive simulation studies have demonstrated that our design has comparable power and average sample size but a much shorter trial duration than conventional single-to-double arm design. Applications of the design are illustrated using two phase II oncology trials with binary endpoints.     

Closure properties of classes of multiple testing procedures

Statistical discoveries are often obtained through multiple hypothesis testing. A variety of procedures exists to evaluate multiple hypotheses, for instance the ones of Benjamini–Hochberg, Bonferroni, Holm or Sidak. We are particularly interested in multiple testing procedures with two desired properties: (solely) monotonic and well-behaved procedures. This article investigates to which extent the classes of (monotonic or well-behaved) multiple testing procedures, in particular the subclasses of so-called step-up and step-down procedures, are closed under basic set operations, specifically the union, intersection, difference and the complement of sets of rejected or non-rejected hypotheses. The present article proves two main results: First, taking the union or intersection of arbitrary (monotonic or well-behaved) multiple testing procedures results in new procedures which are monotonic but not well-behaved, whereas the complement or difference generally preserves neither property. Second, the two classes of (solely monotonic or well-behaved) step-up and step-down procedures are closed under taking the union or intersection, but not the complement or difference.     

Utilization of treatment effect on a surrogate endpoint for planning a study to evaluate treatment effect on a final endpoint

To design a phase III study with a final endpoint and calculate the required sample size for the desired probability of success, we need a good estimate of the treatment effect on the endpoint. It is prudent to fully utilize all available information including the historical and phase II information of the treatment as well as external data of the other treatments. It is not uncommon that a phase II study may use a surrogate endpoint as the primary endpoint and has no or limited data for the final endpoint. On the other hand, external information from the other studies for the other treatments on the surrogate and final endpoints may be available to establish a relationship between the treatment effects on the two endpoints. Through this relationship, making full use of the surrogate information may enhance the estimate of the treatment effect on the final endpoint. In this research, we propose a bivariate Bayesian analysis approach to comprehensively deal with the problem. A dynamic borrowing approach is considered to regulate the amount of historical data and surrogate information borrowing based on the level of consistency. A much simpler frequentist method is also discussed. Simulations are conducted to compare the performances of different approaches. An example is used to illustrate the applications of the methods.     

A flexible fixed-sequence testing method for hierarchically ordered correlated multiple endpoints in clinical trials

Statistical approaches for addressing multiplicity in clinical trials range from the very conservative (the Bonferroni method) to the least conservative the fixed sequence approach. Recently, several authors proposed methods that combine merits of the two extreme approaches. Wiens [2003. A fixed sequence Bonferroni procedure for testing multiple endpoints. Pharmaceutical Statist. 2003, 2, 211–215], for example, considered an extension of the Bonferroni approach where the type I error rate (α)$(\alpha )$ is allocated among the endpoints, however, testing proceeds in a pre-determined order allowing the type I error rate to be saved for later use as long as the null hypotheses are rejected. This leads to a higher power of the test in testing later null hypotheses. In this paper, we consider an extension of Wiens’ approach by taking into account correlations among endpoints for achieving higher flexibility in testing. We show strong control of the family-wise type I error rate for this extension and provide critical values and significance levels for testing up to three endpoints with equal correlations and show how to calculate them for other correlation structures. We also present results of a simulation experiment for comparing the power of the proposed method with those of Wiens’ and others. The results of this experiment show that the magnitude of the gain in power of the proposed method depends on the prospective ordering of testing of the endpoints, the magnitude of the treatment effects of the endpoints and the magnitude of correlation between endpoints. Finally, we consider applications of the proposed method for clinical trials with multiple time points and multiple doses, where correlations among endpoints frequently arise.     

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.     

A generalized step-up-down multiple test procedure

A generalization of step-up and step-down multiple test procedures is proposed. This step-up-down procedure is useful when the objective is to reject a specified minimum number, q, out of a family of k hypotheses. If this basic objective is met at the first step, then it proceeds in a step-down manner to see if more than q hypotheses can be rejected. Otherwise it proceeds in a step-up manner to see if some number less than q hypotheses can be rejected. The usual step-down procedure is the special case where q = 1, and the usual step-up procedure is the special case where q = k. Analytical and numerical comparisons between the powers of the step-up-down procedures with different choices of q are made to see how these powers depend on the actual number of false hypotheses. Examples of application include comparing the efficacy of a treatment to a control for multiple endpoints and testing the sensitivity of a clinical trial for comparing the efficacy of a new treatment with a set of standard treatments.     

A fixed sequence Bonferroni procedure for testing multiple endpoints

A method for controlling the familywise error rate combining the Bonferroni adjustment and fixed testing sequence procedures is proposed. This procedure allots Type I error like the Bonferroni adjustment, but allows the Type I error to accumulate whenever a null hypothesis is rejected. In this manner, power for hypotheses tested later in a prespecified order will be increased. The order of the hypothesis tests needs to be prespecified as in a fixed sequence testing procedure, but unlike the fixed sequence testing procedure all hypotheses can always be tested, allowing for an a priori method of concluding a difference in the various endpoints. An application will be in clinical trials in which mortality is a concern, but it is expected that power to distinguish a difference in mortality will be low. If the effect on mortality is larger than anticipated, this method allows a test with a prespecified method of controlling the Type I error rate. Copyright © 2003 John Wiley & Sons, Ltd.     

Treatment effect quantification for time‐to‐event endpoints–Estimands,analysis strategies,and beyond

A draft addendum to ICH E9 has been released for public consultation in August 2017. The addendum focuses on two topics particularly relevant for randomized confirmatory clinical trials: estimands and sensitivity analyses. The need to amend ICH E9 grew out of the realization of a lack of alignment between the objectives of a clinical trial stated in the protocol and the accompanying quantification of the “treatment effect” reported in a regulatory submission. We embed time‐to‐event endpoints in the estimand framework and discuss how the four estimand attributes described in the addendum apply to time‐to‐event endpoints. We point out that if the proportional hazards assumption is not met, the estimand targeted by the most prevalent methods used to analyze time‐to‐event endpoints, logrank test, and Cox regression depends on the censoring distribution. We discuss for a large randomized clinical trial how the analyses for the primary and secondary endpoints as well as the sensitivity analyses actually performed in the trial can be seen in the context of the addendum. To the best of our knowledge, this is the first attempt to do so for a trial with a time‐to‐event endpoint. Questions that remain open with the addendum for time‐to‐event endpoints and beyond are formulated, and recommendations for planning of future trials are given. We hope that this will provide a contribution to developing a common framework based on the final version of the addendum that can be applied to design, protocols, statistical analysis plans, and clinical study reports in the future.     

P-value adjustment for multiple binary endpoints

The p-value-based adjustment of individual endpoints and the global test for an overall inference are the two general approaches for the analysis of multiple endpoints. Statistical procedures developed for testing multivariate outcomes often assume that the multivariate endpoints are either independent or normally distributed. This paper presents a general approach for the analysis of multivariate binary data under the framework of generalized linear models. The generalized estimating equations (GEE) approach is applied to estimate the correlation matrix of the test statistics using the identity and exchangeable working correlation matrices with the model-based as well as robust estimators. The objectives of the approaches are the adjustment of p-values of individual endpoints to identify the affected endpoints as well as the global test of an overall effect. A Monte Carlo simulation was conducted to evaluate the overall family wise error (FWE) rates of the single-step down p-value adjustment approach from two adjustment methods to three global test statistics. The p-value adjustment approach seems to control the FWE better than the global approach Applications of the proposed methods are illustrated by analyzing a carcinogenicity experiment designed to study the dose response trend for 10 tumor sites, and a developmental toxicity experiment with three malformation types: external, visceral, and skeletal.     

Modifications of the Bonferroni-Holm procedure for a multi-way ANOVA

We present step-wise test procedures based on the Bonferroni-Holm principle for multi-way ANOVA-type models. It is shown for two plausible modifications that the multiple level α is preserved. These theoretical results are supplemented by a simulation study, in a two-way ANOVA setting, to compare the multiple procedures with respect to their simultaneous power and the relative frequency of correctly rejected false hypotheses. Financial support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.     

Blinded assessment of treatment effects for survival endpoint in an ongoing trial

Many assumptions, including assumptions regarding treatment effects, are made at the design stage of a clinical trial for power and sample size calculations. It is desirable to check these assumptions during the trial by using blinded data. Methods for sample size re‐estimation based on blinded data analyses have been proposed for normal and binary endpoints. However, there is a debate that no reliable estimate of the treatment effect can be obtained in a typical clinical trial situation. In this paper, we consider the case of a survival endpoint and investigate the feasibility of estimating the treatment effect in an ongoing trial without unblinding. We incorporate information of a surrogate endpoint and investigate three estimation procedures, including a classification method and two expectation–maximization (EM) algorithms. Simulations and a clinical trial example are used to assess the performance of the procedures. Our studies show that the expectation–maximization algorithms highly depend on the initial estimates of the model parameters. Despite utilization of a surrogate endpoint, all three methods have large variations in the treatment effect estimates and hence fail to provide a precise conclusion about the treatment effect. Copyright © 2012 John Wiley & Sons, Ltd.