Improved two‐stage group sequential procedures for testing a secondary endpoint after the primary endpoint achieves significance

In two‐stage group sequential trials with a primary and a secondary endpoint, the overall type I error rate for the primary endpoint is often controlled by an α‐level boundary, such as an O'Brien‐Fleming or Pocock boundary. Following a hierarchical testing sequence, the secondary endpoint is tested only if the primary endpoint achieves statistical significance either at an interim analysis or at the final analysis. To control the type I error rate for the secondary endpoint, this is tested using a Bonferroni procedure or any α‐level group sequential method. In comparison with marginal testing, there is an overall power loss for the test of the secondary endpoint since a claim of a positive result depends on the significance of the primary endpoint in the hierarchical testing sequence. We propose two group sequential testing procedures with improved secondary power: the improved Bonferroni procedure and the improved Pocock procedure. The proposed procedures use the correlation between the interim and final statistics for the secondary endpoint while applying graphical approaches to transfer the significance level from the primary endpoint to the secondary endpoint. The procedures control the familywise error rate (FWER) strongly by construction and this is confirmed via simulation. We also compare the proposed procedures with other commonly used group sequential procedures in terms of control of the FWER and the power of rejecting the secondary hypothesis. An example is provided to illustrate the procedures.     

Interim monitoring in sequential multiple assignment randomized trials

A sequential multiple assignment randomized trial (SMART) facilitates the comparison of multiple adaptive treatment strategies (ATSs) simultaneously. Previous studies have established a framework to test the homogeneity of multiple ATSs by a global Wald test through inverse probability weighting. SMARTs are generally lengthier than classical clinical trials due to the sequential nature of treatment randomization in multiple stages. Thus, it would be beneficial to add interim analyses allowing for an early stop if overwhelming efficacy is observed. We introduce group sequential methods to SMARTs to facilitate interim monitoring based on the multivariate chi-square distribution. Simulation studies demonstrate that the proposed interim monitoring in SMART (IM-SMART) maintains the desired type I error and power with reduced expected sample size compared to the classical SMART. Finally, we illustrate our method by reanalyzing a SMART assessing the effects of cognitive behavioral and physical therapies in patients with knee osteoarthritis and comorbid subsyndromal depressive symptoms.     

Tests for multivariate recurrent events in the presence of a terminal event

In studies involving diseases associated with high rates of mortality, trials are frequently conducted to evaluate the effects of therapeutic interventions on recurrent event processes terminated by death. In this setting, cumulative mean functions form a natural basis for inference for questions of a health economic nature, and Ghosh and Lin (2000) recently proposed a relevant class of test statistics. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications, each of which may be repeatedly experienced by patients over their lifetime. Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have different impacts on patients' quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the specific types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment benefit can then be achieved by carrying out multiplicity adjusted marginal tests, more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences in mean functions between treatment groups which accommodate the fact that each particular event process is ultimately terminated by death. The methods are illustrated by application to a motivating study designed to examine the effect of bisphosphonate therapy on the incidence of skeletal complications among patients with breast cancer metastatic to bone. We find that there is a consistent trend towards a reduction in the cumulative mean for all four types of skeletal complications with bisphosphonate therapy; there is a significant reduction in the need for radiation therapy for the treatment of bone. The global test suggests that bisphosphonate therapy significantly reduces the overall number of skeletal complications.     

Sample size formulae for two-stage randomized trials with survival outcomes

Two-stage randomized trials are growing in importance in developing adaptive treatment strategies, i.e. treatment policies or dynamic treatment regimes. Usually, the first stage involves randomization to one of the several initial treatments. The second stage of treatment begins when an early nonresponse criterion or response criterion is met. In the second-stage, nonresponding subjects are re-randomized among second-stage treatments. Sample size calculations for planning these two-stage randomized trials with failure time outcomes are challenging because the variances of common test statistics depend in a complex manner on the joint distribution of time to the early nonresponse criterion or response criterion and the primary failure time outcome. We produce simple, albeit conservative, sample size formulae by using upper bounds on the variances. The resulting formulae only require the working assumptions needed to size a standard single-stage randomized trial and, in common settings, are only mildly conservative. These sample size formulae are based on either a weighted Kaplan-Meier estimator of survival probabilities at a fixed time-point or a weighted version of the log-rank test.     

Group sequential methods for an ordinal logistic random-effects model under misspecification

Interim analyses in clinical trials are planned for ethical as well as economic reasons. General results have been published in the literature that allow the use of standard group sequential methodology if one uses an efficient test statistic, e.g., when Wald-type statistics are used in random-effects models for ordinal longitudinal data. These models often assume that the random effects are normally distributed. However, this is not always the case. We will show that, when the random-effects distribution is misspecified in ordinal regression models, the joint distribution of the test statistics over the different interim analyses is still a multivariate normal distribution, but a sandwich-type correction to the covariance matrix is needed in order to obtain the correct covariance matrix. The independent increment structure is also investigated. A bias in estimation will occur due to the misspecification. However, we will also show that the treatment effect estimate will be unbiased under the null hypothesis, thus maintaining the type I error. Extensive simulations based on a toenail dermatophyte onychomycosis trial are used to illustrate our results.     

A measure of the proportion of treatment effect explained by a surrogate marker

Randomized clinical trials with rare primary endpoints or long duration times are costly. Because of this, there has been increasing interest in replacing the true endpoint with an earlier measured marker. However, surrogate markers must be appropriately validated. A quantitative measure for the proportion of treatment effect explained by the marker in a specific trial is a useful concept. Freedman, Graubard, and Schatzkin (1992, Statistics in Medicine 11, 167-178) suggested such a measure of surrogacy by the ratio of regression coefficients for the treatment indicator from two separate models with or without adjusting for the surrogate marker. However, it has been shown that this measure is very variable and there is no guarantee that the two models both fit. In this article, we propose alternative measures of the proportion explained that adapts an idea in Tsiatis, DeGruttola, and Wulfsohn (1995, Journal of the American Statistical Association 90, 27-37). The new measures require fewer assumptions in estimation and allow more flexibility in modeling. The estimates of these different measures are compared using data from an ophthalmology clinical trial and a series of simulation studies. The results suggest that the new measures are less variable.     

Exact statistical inference for group sequential trials.

This paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference. The existing group sequential methods for such trials are based on the large-sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses. We demonstrate through extensive numerical studies that, for small and even moderate-sized trials, these approximate procedures may lead to tests with supranominal size (mainly when unpooled estimators of variance are used) and confidence intervals with under-nominal coverage probability. We then study exact methods for group sequential testing, repeated interval estimation, and interval estimation following sequential testing. The new procedures can accommodate any treatment allocation rules. An example using real data is provided.     

Weighting affected sib pairs by marker informativity

For the analysis of affected sib pairs (ASPs), a variety of test statistics is applied in genomewide scans with microsatellite markers. Even in multipoint analyses, these statistics might not fully exploit the power of a given sample, because they do not account for incomplete informativity of an ASP. For meta-analyses of linkage and association studies, it has been shown recently that weighting by informativity increases statistical power. With this idea in mind, the first aim of this article was to introduce a new class of tests for ASPs that are based on the mean test. To take into account how much informativity an ASP contributes, we weighted families inversely proportional to their marker informativity. The weighting scheme is obtained by use of the de Finetti representation of the distribution of identity-by-descent values. We derive the limiting distribution of the weighted mean test and demonstrate the validity of the proposed test. We show that it can be much more powerful than the classical mean test in the case of low marker informativity. In the second part of the article, we propose a Monte Carlo simulation approach for evaluating significance among ASPs. We demonstrate the validity of the simulation approach for both the classical and the weighted mean test. Finally, we illustrate the use of the weighted mean test by reanalyzing two published data sets. In both applications, the maximum LOD score of the weighted mean test is 0.6 higher than that of the classical mean test.     

Approximately sufficient statistics and bayesian computation

The analysis of high-dimensional data sets is often forced to rely upon well-chosen summary statistics. A systematic approach to choosing such statistics, which is based upon a sound theoretical framework, is currently lacking. In this paper we develop a sequential scheme for scoring statistics according to whether their inclusion in the analysis will substantially improve the quality of inference. Our method can be applied to high-dimensional data sets for which exact likelihood equations are not possible. We illustrate the potential of our approach with a series of examples drawn from genetics. In summary, in a context in which well-chosen summary statistics are of high importance, we attempt to put the 'well' into 'chosen.'     

Failure of sequential therapy to eradicate Helicobacter pylori in previously treated subjects

BACKGROUND: Recently, studies in adults and subsequently in children have demonstrated a very high success rate for sequential therapy as a primary therapy when compared to traditional therapy regimens. METHODs: We report our experience with a sequential therapy regimen for the eradication of Helicobacter pylori in five infected children and a young adult, whose conventional therapy regimens had been unsuccessful. RESULTS: Five patients failed the sequential therapy. All of them experienced between two and four failures of traditional therapy prior to the sequential treatment protocol. The only patient who succeeded on the sequential therapy had just one previous failure. All of our patients who had failed sequential therapy achieved eradication of the bacteria with quadruple therapy. CONCLUSIONS: In view of our disappointing results, sequential therapy is unsuccessful as a therapy for children and young adults who have failed previous treatment regimens. At the present time, quadruple therapy is indicated for this group. Well-designed placebo-controlled trials are indicated to further characterize this group of patients.     

Bayesian adaptive trial design for a newly validated surrogate endpoint

The evaluation of surrogate endpoints for primary use in future clinical trials is an increasingly important research area, due to demands for more efficient trials coupled with recent regulatory acceptance of some surrogates as 'valid.' However, little consideration has been given to how a trial that utilizes a newly validated surrogate endpoint as its primary endpoint might be appropriately designed. We propose a novel Bayesian adaptive trial design that allows the new surrogate endpoint to play a dominant role in assessing the effect of an intervention, while remaining realistically cautious about its use. By incorporating multitrial historical information on the validated relationship between the surrogate and clinical endpoints, then subsequently evaluating accumulating data against this relationship as the new trial progresses, we adaptively guard against an erroneous assessment of treatment based upon a truly invalid surrogate. When the joint outcomes in the new trial seem plausible given similar historical trials, we proceed with the surrogate endpoint as the primary endpoint, and do so adaptively-perhaps stopping the trial for early success or inferiority of the experimental treatment, or for futility. Otherwise, we discard the surrogate and switch adaptive determinations to the original primary endpoint. We use simulation to test the operating characteristics of this new design compared to a standard O'Brien-Fleming approach, as well as the ability of our design to discriminate trustworthy from untrustworthy surrogates in hypothetical future trials. Furthermore, we investigate possible benefits using patient-level data from 18 adjuvant therapy trials in colon cancer, where disease-free survival is considered a newly validated surrogate endpoint for overall survival.     

Comparing biomarkers as principal surrogate endpoints

Recently a new definition of surrogate endpoint, the "principal surrogate," was proposed based on causal associations between treatment effects on the biomarker and on the clinical endpoint. Despite its appealing interpretation, limited research has been conducted to evaluate principal surrogates, and existing methods focus on risk models that consider a single biomarker. How to compare principal surrogate value of biomarkers or general risk models that consider multiple biomarkers remains an open research question. We propose to characterize a marker or risk model's principal surrogate value based on the distribution of risk difference between interventions. In addition, we propose a novel summary measure (the standardized total gain) that can be used to compare markers and to assess the incremental value of a new marker. We develop a semiparametric estimated-likelihood method to estimate the joint surrogate value of multiple biomarkers. This method accommodates two-phase sampling of biomarkers and is more widely applicable than existing nonparametric methods by incorporating continuous baseline covariates to predict the biomarker(s), and is more robust than existing parametric methods by leaving the error distribution of markers unspecified. The methodology is illustrated using a simulated example set and a real data set in the context of HIV vaccine trials.     

The analysis of multiple endpoints in clinical trials

Treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall Type I error rate. One option is to select a single primary endpoint for formal statistical inference, but this is not always feasible. Another approach is to apply Bonferroni correction (i.e., multiply each P-value by the total number of endpoints). Its conservatism for correlated endpoints is examined for multivariate normal data. A third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics. Quantitative, binary, and survival endpoints are all considered within this general framework. Two examples are presented and the relative merits of the proposed strategies are discussed.     

Symmetric group sequential test designs

In Phase III clinical trials, ethical considerations often demand interim analyses in order that the better treatment be made available to all patients as soon as possible. Group sequential test designs that do not treat the hypotheses symmetrically may not fully address this concern since early termination of the study may be easier under one of the hypotheses. We present a one-parameter family of symmetric one-sided group sequential designs that are nearly fully efficient in terms of the average sample number. The symmetric tests are then extended to a two-sided hypothesis test. These symmetric two-sided group sequential tests are found to have improved overall efficiency when compared to the tests proposed by Pocock (1977, Biometrika 64, 191-199) and O'Brien and Fleming (1979, Biometrics 35, 549-556). Tables of critical values for both one-sided and two-sided symmetric designs are provided, thus allowing easy determination of sample sizes and stopping boundaries for a group sequential test. Approximate tests based on these designs are proposed for use when the number and timing of analyses are random.     

Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes, or parametric tests

The confirmatory analysis of pre-specified multiple hypotheses has become common in pivotal clinical trials. In the recent past multiple test procedures have been developed that reflect the relative importance of different study objectives, such as fixed sequence, fallback, and gatekeeping procedures. In addition, graphical approaches have been proposed that facilitate the visualization and communication of Bonferroni-based closed test procedures for common multiple test problems, such as comparing several treatments with a control, assessing the benefit of a new drug for more than one endpoint, combined non-inferiority and superiority testing, or testing a treatment at different dose levels in an overall and a subpopulation. In this paper, we focus on extended graphical approaches by dissociating the underlying weighting strategy from the employed test procedure. This allows one to first derive suitable weighting strategies that reflect the given study objectives and subsequently apply appropriate test procedures, such as weighted Bonferroni tests, weighted parametric tests accounting for the correlation between the test statistics, or weighted Simes tests. We illustrate the extended graphical approaches with several examples. In addition, we describe briefly the gMCP package in R, which implements some of the methods described in this paper.     

Motor commands for fast point-to-point arm movements are customized for small changes in inertial load

For repeated point-to-point arm movements it is often assumed that motor commands are customized in a trial-to-trial manner, based on previous endpoint error. To test this assumption, we perturbed movement execution without affecting the endpoint error by using a modest manipulation of inertia. Participants made point-to-point elbow flexion and extension movements in the horizontal plane, under the instruction to move as fast as possible from one target area to another. In selected trials the moment of inertia of the lower arm was increased or decreased by 25%. First, we found that an unexpected increase or decrease of inertia did not affect the open loop controlled part of the movement path (and thus endpoint error was not affected). Second, we found that when the increased or decreased inertia was presented repeatedly, after 5-11 trials motor commands were customized: the first 100ms of agonistic muscle activity in the smoothed and rectified electromyographic signal of agonistic muscles was higher for the high inertia compared to the low inertia. We conclude that endpoint error is not the only parameter that is used to evaluate if motor commands lead to movements as planned.     

Perspectives from the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial--Lipid Lowering Trial and the Anglo-Scandinavian Cardiac Outcomes Trial--Lipid Lowering Arm

PURPOSE OF REVIEW: The design, process and outcomes are compared between two large clinical trials of LDL cholesterol reduction with statin treatment in patients with known high blood pressure. This new information is placed in the context of previous clinical trials of cholesterol reduction, which have provided analyses of sub-groups with high blood pressure. RECENT FINDINGS: The Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial failed to find a significant reduction of total mortality (primary endpoint), cardiovascular mortality or major cardiovascular events. This differed sharply from the Anglo-Scandinavian Cardiac Outcomes Trial, Lipid Lowering Arm, which was stopped before the planned termination due to a marked reduction (36%) in coronary death or myocardial infarction (primary endpoint). This trial also found significant reductions in stroke (27%) and separately, all major vascular events (21%). The two studies were similar in that they each contained over 10 000 participants with documented high blood pressure requiring drug therapy and they both used a fixed dose of a single statin. Pravastatin (40 mg/day) was used in the former and atorvastatin 10 mg/day in the latter. The major difference was that the control group in the Anglo-Scandinavian trial was treated with placebo with a double blind design whereas antihypertensive and lipid-lowering trial was open label with controls receiving usual care. SUMMARY: The benefit of achieving and maintaining significant LDL cholesterol reduction in patients with high blood pressure was convincingly demonstrated in the Anglo-Scandinavian trial. The Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial failed to achieve similar success due to use of a less effective drug and loss of the differential effect with increasing statin treatment in the usual care control group.     

Flexible designs by adaptive plans of generalized Pocock- and O'Brien-Fleming-type and by self-designing clinical trials

Flexible designs are provided by adaptive planning of sample sizes as well as by introducing the weighted inverse normal combining method and the generalized inverse chi-square combining method in the context of conducting trials consecutively step by step. These general combining methods allow quite different weighting of sequential study parts, also in a completely adaptive way, based on full information from unblinded data in previously performed stages. So, in reviewing some basic developments of flexible designing, we consider a generalizing approach to group sequentially performed clinical trials of Pocock-type, of O'Brien-Fleming-type, and of Self-designing-type. A clinical trial may be originally planned either to show non-inferiority or superiority. The proposed flexible designs, however, allow in each interim analysis to change the planning from showing non-inferiority to showing superiority and vice versa. Several examples of clinical trials with normal and binary outcomes are worked out in detail. We demonstrate the practicable performance of the discussed approaches, confirmed in an extensive simulation study. Our flexible designing is a useful tool, provided that a priori information about parameters involved in the trial is not available or subject to uncertainty.     

Efficient and robust approaches for analysis of sequential multiple assignment randomized trials: Illustration using the ADAPT-R trial

Personalized intervention strategies, in particular those that modify treatment based on a participant's own response, are a core component of precision medicine approaches. Sequential multiple assignment randomized trials (SMARTs) are growing in popularity and are specifically designed to facilitate the evaluation of sequential adaptive strategies, in particular those embedded within the SMART. Advances in efficient estimation approaches that are able to incorporate machine learning while retaining valid inference can allow for more precise estimates of the effectiveness of these embedded regimes. However, to the best of our knowledge, such approaches have not yet been applied as the primary analysis in SMART trials. In this paper, we present a robust and efficient approach using targeted maximum likelihood estimation (TMLE) for estimating and contrasting expected outcomes under the dynamic regimes embedded in a SMART, together with generating simultaneous confidence intervals for the resulting estimates. We contrast this method with two alternatives (G-computation and inverse probability weighting estimators). The precision gains and robust inference achievable through the use of TMLE to evaluate the effects of embedded regimes are illustrated using both outcome-blind simulations and a real-data analysis from the Adaptive Strategies for Preventing and Treating Lapses of Retention in Human Immunodeficiency Virus (HIV) Care (ADAPT-R) trial (NCT02338739), a SMART with a primary aim of identifying strategies to improve retention in HIV care among people living with HIV in sub-Saharan Africa.