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.     

Robustness of an odds-ratio test in a stratified group sequential trial with a binary outcome measure

Many clinical trials compare two or more treatment groups by using a binary outcome measure. For example, the goal could be to determine whether the frequency of pain episodes is significantly reduced in the treatment group (arm A) as compared to the control group (arm B). However, for ethical or regulatory reasons, group sequential designs are commonly employed. Then, based on a binomial distribution, the stopping boundaries for the interim analyses are constructed for assessing the difference in the response probabilities between the two groups. This is easily accomplished by using any of the standard procedures, e.g., those discussed by Jennison and Turnbull (2000), and using one of the most commonly used software packages, East (2000). Several factors are known to often affect the primary outcome of interest, but their true distributions are not known in advance. In addition, these factors may cause heterogeneous treatment responses among individuals in a group, and their exact effect size may be unknown. To limit the effect of such factors on the comparison of the two arms, stratified randomization is used in the actual conduct of the trial. Then, a stratified analysis based on the odds ratio proposed in Jennison and Turnbull (2000, pages 251-252) and consistent with the stratified design is undertaken. However, the stopping rules used for the interim analyses are those obtained for determining the differences in response rates in a design that was not stratified. The purpose of this paper is to assess the robustness of such an approach on the performance of the odds ratio test when the underlying distribution and effect size of the factors that influence the outcome may vary. The simulation studies indicate that, in general, the stratified approach offers consistently better results than does the unstratified approach, as long as the difference in the weighted average of the response probabilities across strata between the two groups remains closer to the hypothesized values, irrespective of the differences in the (allocation) distributions and heterogeneous response rate. However, if the response probabilities deviate significantly from the hypothesized values so that the difference in the weighted average is less than the hypothesized value, then the proposed study could be significantly underpowered.     

Adaptive treatment allocation for comparative clinical studies with recurrent events data

In long-term clinical studies, recurrent event data are sometimes collected and used to contrast the efficacies of two different treatments. The event reoccurrence rates can be compared using the popular negative binomial model, which incorporates information related to patient heterogeneity into a data analysis. For treatment allocation, a balanced approach in which equal sample sizes are obtained for both treatments is predominately adopted. However, if one treatment is superior, then it may be desirable to allocate fewer subjects to the less-effective treatment. To accommodate this objective, a sequential response-adaptive treatment allocation procedure is derived based on the doubly adaptive biased coin design. Our proposed treatment allocation schemes have been shown to be capable of reducing the number of subjects receiving the inferior treatment while simultaneously retaining a test power level that is comparable to that of a balanced design. The redesign of a clinical study illustrates the advantages of using our procedure.     

Testing a Primary and a Secondary Endpoint in a Group Sequential Design

Summary We consider a clinical trial with a primary and a secondary endpoint where the secondary endpoint is tested only if the primary endpoint is significant. The trial uses a group sequential procedure with two stages. The familywise error rate (FWER) of falsely concluding significance on either endpoint is to be controlled at a nominal level α. The type I error rate for the primary endpoint is controlled by choosing any α‐level stopping boundary, e.g., the standard O'Brien–Fleming or the Pocock boundary. Given any particular α‐level boundary for the primary endpoint, we study the problem of determining the boundary for the secondary endpoint to control the FWER. We study this FWER analytically and numerically and find that it is maximized when the correlation coefficient ρ between the two endpoints equals 1. For the four combinations consisting of O'Brien–Fleming and Pocock boundaries for the primary and secondary endpoints, the critical constants required to control the FWER are computed for different values of ρ. An ad hoc boundary is proposed for the secondary endpoint to address a practical concern that may be at issue in some applications. Numerical studies indicate that the O'Brien–Fleming boundary for the primary endpoint and the Pocock boundary for the secondary endpoint generally gives the best primary as well as secondary power performance. The Pocock boundary may be replaced by the ad hoc boundary for the secondary endpoint with a very little loss of secondary power if the practical concern is at issue. A clinical trial example is given to illustrate the methods.     

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.     

Obtaining Critical Values for Simultaneous Confidence Intervals and Multiple Testing

There are many situations where it is desired to make simultaneous tests or give simultaneous confidence intervals for linear combinations (contrasts) of population or treatment means. Somerville (1997, 1999) developed algorithms for calculating the critical values for a large class of simultaneous tests and simultaneous confidence intervals. Fortran 90 and SAS‐IML batch programs and interactive programs were developed. These programs calculate the critical values for 15 different simultaneous confidence interval procedures (and the corresponding simultaneous tests) and for arbitrary procedures where the user specifies a combination of one and two sided contrasts. The programs can also be used to obtain the constants for “step‐down” testing of multiple hypotheses. This paper gives examples of the use of the algorithms and programs and illustrates their versatility and generality. The designs need not be balanced, multiple covariates may be present and there may be many missing values. The use of multiple regression and dummy variables to obtain the required variance covariance matrix is illustrated. Under weak normality assumptions the methods are “exact” and make the use of approximate methods or “simulation” unnecessary.     

Truncation of a Two Sample Sequential Wilcoxon Test

To reduce the number of patients needed in a clinical trial, a sequential plan is sometimes used. This paper deals with the truncation of a sequential two sample Wilcoxon test which was originally an open plan (SKOVLUND and WALLØE , 1988). The truncated boundaries have been developed by stochastic simulation. These boundaries form a triangular continuation region. The simulated constants of the boundaries are presented in a table which may be used directly. Boundaries for truncation at 50, 100, 150 and 200 patients are developed. The chosen significance level may be 0.01, 0.025 or 0.05, and the power 0.95. To extend the applicability to other treatment differences than those tabulated, smooth curves are fitted to the simulated points. These make the test applicable to any treatment difference within the simulated range.     

Ethical Considerations for Outcome‐adaptive Trial Designs: A Clinical Researcher's Perspective

In a typical comparative clinical trial the randomization scheme is fixed at the beginning of the study, and maintained throughout the course of the trial. A number of researchers have championed a randomized trial design referred to as ‘outcome‐adaptive randomization.’ In this type of trial, the likelihood of a patient being enrolled to a particular arm of the study increases or decreases as preliminary information becomes available suggesting that treatment may be superior or inferior. While the design merits of outcome‐adaptive trials have been debated, little attention has been paid to significant ethical concerns that arise in the conduct of such studies. These include loss of equipoise, lack of processes for adequate informed consent, and inequalities inherent in the research design which could lead to perceptions of injustice that may have negative implications for patients and the research enterprise. This article examines the ethical difficulties inherent in outcome‐adaptive trials.     

A regulatory perspective on choice of margin and statistical inference issue in non-inferiority trials

Without a placebo arm, any non-inferiority inference involving assessment of the placebo effect under the active control trial setting is difficult. The statistical risk for falsely concluding non-inferiority cannot be evaluated unless the constancy assumption approximately holds that the effect of the active control under the historical trial setting where the control effect can be assessed carries to the noninferiority trial setting. The constancy assumption cannot be checked because of missing the placebo arm in the non-inferiority trial. Depending on how serious the violation of the assumption is thought to be, one may need to seek an alternative design strategy that includes a cushion for a very conservative non-inferiority analysis or shows superiority of the experimental treatment over the control. Determination of the non-inferiority margin depends on what objective the non-inferiority analysis is intended to achieve. The margin can be a fixed margin or a margin functionally defined. Between-trial differences always exist and need to be properly considered.     

Exploitations and their complications: the necessity of identifying the multiple forms of exploitation in pharmaceutical trials

Human subject trials of pharmaceuticals in low and middle income countries (LMICs) have been associated with the moral wrong of exploitation on two grounds. First, these trials may include a placebo control arm even when proven treatments for a condition are in use in other (usually wealthier) parts of the world. Second, the trial researchers or sponsors may fail to make a successful treatment developed through the trial available to either the trial participants or the host community following the trial. Many commentators have argued that a single form of exploitation takes place during human subject research in LMICs. These commentators do not, however, agree as to what kind of moral wrong exploitation is or when exploitation is morally impermissible. In this paper, I have two primary goals. First, I will argue for a taxonomy of exploitation that identifies three distinct forms of exploitation. While each of these forms of exploitation has its critics, I will argue that they can each be developed into plausible accounts of exploitation tied to different vulnerabilities and different forms of wrongdoing. Second, I will argue that each of these forms of exploitation can coexist in single situations, including human subject trials of pharmaceuticals. This lesson is important, since different forms of exploitation in a single relationship can influence, among other things, whether the relationship is morally permissible.     

Optimal conditional error functions for the control of conditional power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.     

A note on P-values under group sequential testing and nonproportional hazards

Summary .   Group sequential designs are often used for periodically assessing treatment efficacy during the course of a clinical trial. Following a group sequential test, P -values computed under the assumption that the data were gathered according to a fixed sample design are no longer uniformly distributed under the null hypothesis of no treatment effect. Various sample space orderings have been proposed for computing proper P -values following a group sequential test. Although many of the proposed orderings have been compared in the setting of time-invariant treatment effects, little attention has been given to their performance when the effect of treatment within an individual varies over time. Our interest here is to compare two of the most commonly used methods for computing proper P -values following a group sequential test, based upon the analysis time (AT) and Z -statistic orderings, with respect to resulting power functions when treatment effects on survival are delayed. Power under the AT ordering is shown to be heavily influenced by the presence of a delayed treatment effect, while power functions corresponding to the Z -statistic ordering remain robust under time-varying treatment effects.     

Treatment allocation for nonlinear models in clinical trials: the logistic model

Many clinical trials have a binary outcome variable. If covariate adjustment is necessary in the analysis, the logistic-regression model is frequently used. Optimal designs for allocating treatments for this model, or for any nonlinear or heteroscedastic model, are generally unbalanced with regard to overall treatment totals and totals within strata. However, all treatment-allocation methods that have been recommended for clinical trials in the literature are designed to balance treatments within strata, either directly or asymptotically. In this paper, the efficiencies of balanced sequential allocation schemes are measured relative to sequential Ds-optimal designs for the logistic model, using as examples completed trials conducted by the Eastern Cooperative Oncology Group and systematic simulations. The results demonstrate that stratified, balanced designs are quite efficient, in general. However, complete randomization is frequently inefficient, and will occasionally result in a trial that is very inefficient.     

Periodontal Treatment for Preventing Adverse Pregnancy Outcomes: A Meta- and Trial Sequential Analysis

Objectives

Periodontal treatment might reduce adverse pregnancy outcomes. The efficacy of periodontal treatment to prevent preterm birth, low birth weight, and perinatal mortality was evaluated using meta-analysis and trial sequential analysis.

Methods

An existing systematic review was updated and meta-analyses performed. Risk of bias, heterogeneity, and publication bias were evaluated, and meta-regression performed. Subgroup analysis was used to compare different studies with low and high risk of bias and different populations, i.e., risk groups. Trial sequential analysis was used to assess risk of random errors.

Results

Thirteen randomized clinical trials evaluating 6283 pregnant women were meta-analyzed. Four and nine trials had low and high risk of bias, respectively. Overall, periodontal treatment had no significant effect on preterm birth (odds ratio [95% confidence interval] 0.79 [0.57-1.10]) or low birth weight (0.69 [0.43-1.13]). Trial sequential analysis demonstrated that futility was not reached for any of the outcomes. For populations with moderate occurrence (<20%) of preterm birth or low birth weight, periodontal treatment was not efficacious for any of the outcomes, and trial sequential analyses indicated that further trials might be futile. For populations with high occurrence (≥20%) of preterm birth and low birth weight, periodontal treatment seemed to reduce the risk of preterm birth (0.42 [0.24-0.73]) and low birth weight (0.32 [0.15-0.67]), but trial sequential analyses showed that firm evidence was not reached. Periodontal treatment did not significantly affect perinatal mortality, and firm evidence was not reached. Risk of bias, but not publication bias or patients’ age modified the effect estimates.

Conclusions

Providing periodontal treatment to pregnant women could potentially reduce the risks of perinatal outcomes, especially in mothers with high risks. Conclusive evidence could not be reached due to risks of bias, risks of random errors, and unclear effects of confounding. Further randomized clinical trials are required.     

An initial top-down proteomic analysis of the standard cuprizone mouse model of multiple sclerosis

An initial proteomic analysis of the cuprizone mouse model to characterise the breadth of toxicity by assessing cortex, skeletal muscle, spleen and peripheral blood mononuclear cells. Cuprizone treated vs. control mice for an initial characterisation. Select tissues from each group were pooled, analysed in triplicate using two-dimensional gel electrophoresis (2DE) and deep imaging and altered protein species identified using liquid chromatography tandem mass spectrometry (LC/MS/MS). Forty-three proteins were found to be uniquely detectable or undetectable in the cuprizone treatment group across the tissues analysed. Protein species identified in the cortex may potentially be linked to axonal damage in this model, and those in the spleen and peripheral blood mononuclear cells to the minimal peripheral immune cell infiltration into the central nervous system during cuprizone mediated demyelination. Primary oligodendrocytosis has been observed in type III lesions in multiple sclerosis. However, the underlying mechanisms are poorly understood. Cuprizone treatment results in oligodendrocyte apoptosis and secondary demyelination. This initial analysis identified proteins likely related to axonal damage; these may link primary oligodendrocytosis and secondary axonal damage. Furthermore, this appears to be the first study of the cuprizone model to also identify alterations in the proteomes of skeletal muscle, spleen and peripheral blood mononuclear cells. Notably, protein disulphide isomerase was not detected in the cuprizone cohort; its absence has been linked to reduced major histocompatibility class I assembly and reduced antigen presentation. Overall, the results suggest that, like experimental autoimmune encephalomyelitis, results from the standard cuprizone model should be carefully considered relative to clinical multiple sclerosis.     

Two Stage Sampling for Simultaneously Testing Main and Side Effects in Clinical Trials

In clinical trials for the comparison of two treatments it seems reasonable to stop the study if either one treatment has worked out to be markedly superior in the main effect, or one to be severely inferior with respect to an adverse side effect. Two stage sampling plans are considered for simultaneously testing a main and side effect, assumed to follow a bivariate normal distribution with known variances, but unknown correlation. The test procedure keeps the global significance level under the null hypothesis of no differences in main and side effects. The critical values are chosen under the side condition, that the probability for ending at the first or second stage with a rejection of the elementary null hypothesis for the main effect is controlled, when a particular constellation of differences in mean holds; analogously the probability of ending with a rejection of the null hypotheses for the side effect, given certain treatment differences, is controlled too. Plans “optimal” with respect to sample size are given.     

A group sequential type design for three‐arm non‐inferiority trials with binary endpoints

The three‐arm design with a test treatment, an active control and a placebo group is the gold standard design for non‐inferiority trials if it is ethically justifiable to expose patients to placebo. In this paper, we first use the closed testing principle to establish the hierarchical testing procedure for the multiple comparisons involved in the three‐arm design. For the effect preservation test we derive the explicit formula for the optimal allocation ratios. We propose a group sequential type design, which naturally accommodates the hierarchical testing procedure. Under this proposed design, Monte Carlo simulations are conducted to evaluate the performance of the sequential effect preservation test when the variance of the test statistic is estimated based on the restricted maximum likelihood estimators of the response rates under the null hypothesis. When there are uncertainties for the placebo response rate, the proposed design demonstrates better operating characteristics than the fixed sample design.     

Simple sequential boundaries for treatment selection in multi-armed randomized clinical trials with a control

Summary .   In situations when many regimens are possible candidates for a large phase III study, but too few resources are available to evaluate each relative to the standard, conducting a multi-armed randomized selection trial is a useful strategy to remove inferior treatments from further consideration. When the study has a relatively quick endpoint such as an imaging-based lesion volume change in acute stroke patients, frequent interim monitoring of the trial is ethically and practically appealing to clinicians. In this article, I propose a class of sequential selection boundaries for multi-armed clinical trials, in which the objective is to select a treatment with a clinically significant improvement upon the control group, or to declare futility if no such treatment exists. The proposed boundaries are easy to implement in a blinded fashion, and can be applied on a flexible monitoring schedule in terms of calendar time. Design calibration with respect to prespecified levels of confidence is simple, and can be accomplished when the response rate of the control group is known only up to an interval. One of the proposed methods is applied to redesign a selection trial with an imaging endpoint in acute stroke patients, and is compared to an optimal two-stage design via simulations: The proposed method imposes smaller sample size on average than the two-stage design; this advantage is substantial when there is in fact a superior treatment to the control group.     

Study duration for clinical trials with survival response and early stopping rule

A comparative clinical trial with built-in sequential stopping rules allows earlier-than-scheduled stopping, should there be a significant indication of treatment difference. In a clinical trial where the major outcome is time (survival time or response) to a certain event such as failure, the design of the study should determine how long one needs to accrue patients and follow through until there is a sufficient number of events observed during the entire study duration. This paper proposes a unified design procedure for group sequential clinical trials with survival response. The time to event is assumed to be exponentially distributed, but the arguments extend naturally to the proportional hazards model after suitable transformation on the time scale. An example from the Eastern Cooperative Oncology Group (ECOG) is given to illustrate how this procedure can be implemented. The same example is used to explore the overall operating characteristics and the robustness of the proposed group sequential design.     

Methods for comparison of parameters from longitudinal rhythmometric models with multiple components

Multiple components linear least-squares methods have been proposed for the detection of periodic components in nonsinusoidal longitudinal time series. However, a proper test for comparison of parameters obtained from this method for two or more time series is not yet available. Accordingly, we propose two methods, one parametric and one nonparametric, to compare parameters from rhythmometric models with multiple components. The parametric method is based on techniques commonly and generally employed in linear regression analysis. The comparison of parameters among two or more time series is accomplished by the use of so-called dummy variables. The nonparametric method is based on bootstrap techniques. This approach basically tests if the difference in any given parameter obtained by fitting a model with the same periods to two different longitudinal time series differs from zero. This method calculates a confidence interval for the difference in the tested parameter. If this interval does not contain zero, it can be concluded that the parameters obtained from the two time series are different with high probability. An estimation of the p-value for the corresponding test can also be calculated. By the use of similar bootstrap techniques, confidence intervals can also be obtained for any parameter derived from the multiple component fit of several periods to nonsinusoidal longitudinal time series, including the orthophase (peak time), bathyphase (trough time), and global amplitude (difference between the maximum and the minimum) of the fitted model waveform. These methods represent a valuable tool for the comparison of rhythm parameters obtained by multiple component analysis, and they render this approach as a generally applicable one for waveform representation and detection of periodicities in nonsinusoidal, sparse, and noisy longitudinal time series sampled with either equidistant or unequidistant observations.