Modification of sample size in group sequential clinical trials

Adaptive design and estimation in randomized clinical trials with correlated observations

Clinical trial designs involving correlated data often arise in biomedical research. The intracluster correlation needs to be taken into account to ensure the validity of sample size and power calculations. In contrast to the fixed-sample designs, we propose a flexible trial design with adaptive monitoring and inference procedures. The total sample size is not predetermined, but adaptively re-estimated using observed data via a systematic mechanism. The final inference is based on a weighted average of the block-wise test statistics using generalized estimating equations, where the weight for each block depends on cumulated data from the ongoing trial. When there are no significant treatment effects, the devised stopping rule allows for early termination of the trial and acceptance of the null hypothesis. The proposed design updates information regarding both the effect size and within-cluster correlation based on the cumulated data in order to achieve a desired power. Estimation of the parameter of interest and its confidence interval are proposed. We conduct simulation studies to examine the operating characteristics and illustrate the proposed method with an example.

A practical comparison of group-sequential and adaptive designs

Sequential methods provide a formal framework by which clinical trial data can be monitored as they accumulate. The results from interim analyses can be used either to modify the design of the remainder of the trial or to stop the trial as soon as sufficient evidence of either the presence or absence of a treatment effect is available. The circumstances under which the trial will be stopped with a claim of superiority for the experimental treatment, must, however, be determined in advance so as to control the overall type I error rate. One approach to calculating the stopping rule is the group-sequential method. A relatively recent alternative to group-sequential approaches is the adaptive design method. This latter approach provides considerable flexibility in changes to the design of a clinical trial at an interim point. However, a criticism is that the method by which evidence from different parts of the trial is combined means that a final comparison of treatments is not based on a sufficient statistic for the treatment difference, suggesting that the method may lack power. The aim of this paper is to compare two adaptive design approaches with the group-sequential approach. We first compare the form of the stopping boundaries obtained using the different methods. We then focus on a comparison of the power of the different trials when they are designed so as to be as similar as possible. We conclude that all methods acceptably control type I error rate and power when the sample size is modified based on a variance estimate, provided no interim analysis is so small that the asymptotic properties of the test statistic no longer hold. In the latter case, the group-sequential approach is to be preferred. Provided that asymptotic assumptions hold, the adaptive design approaches control the type I error rate even if the sample size is adjusted on the basis of an estimate of the treatment effect, showing that the adaptive designs allow more modifications than the group-sequential method.

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.

Sample size calculations in randomised trials: mandatory and mystical

Investigators should properly calculate sample sizes before the start of their randomised trials and adequately describe the details in their published report. In these a-priori calculations, determining the effect size to detect--eg, event rates in treatment and control groups--reflects inherently subjective clinical judgments. Furthermore, these judgments greatly affect sample size calculations. We question the branding of trials as unethical on the basis of an imprecise sample size calculation process. So-called underpowered trials might be acceptable if investigators use methodological rigor to eliminate bias, properly report to avoid misinterpretation, and always publish results to avert publication bias. Some shift of emphasis from a fixation on sample size to a focus on methodological quality would yield more trials with less bias. Unbiased trials with imprecise results trump no results at all. Clinicians and patients deserve guidance now.

Testing and estimation in flexible group sequential designs with adaptive treatment selection

Integrating selection and confirmation phases into a single trial can expedite the development of new treatments and allows to use all accumulated data in the decision process. In this paper we review adaptive treatment selection based on combination tests and propose overall adjusted p-values and simultaneous confidence intervals. Also point estimation in adaptive trials is considered. The methodology is illustrated in a detailed example based on an actual planned study.

Group sequential design for comparative diagnostic accuracy studies: implications and guidelines for practitioners

PURPOSE

Comparative diagnostic accuracy (CDA) studies are typically small retrospective studies supporting a higher accuracy for one modality over another for either staging a particular disease or assessing response to therapy, and they are used to generate hypotheses for larger prospective trials. The purpose of this article is to introduce the group sequential design (GSD) approach in planning these larger trials.

METHODS

Methodology needed for using GSD in the CDA studies is recently developed. In this article, GSD with the O'Brien and Fleming (OBF) stopping rule is described and guidelines for sample size calculation are provided. Simulated data is used to demonstrate the application of GSD in the design/analysis of a clinical trial in the CDA study setting.

RESULTS

The expected sample size needed for planning a trial with GSD (under the OBF stopping rule) is slightly inflated but may ultimately result in greater savings of patient resources.

CONCLUSION

GSD is a specialized statistical method that is helpful in balancing the ethical and financial advantages of stopping a study early against the risk of an incorrect conclusion and should be adopted for planning CDA studies.

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.

Data adaptive interim modification of sample sizes for candidate-gene association studies

OBJECTIVES

The use of conventional Transmission/Disequilibrium tests in the analysis of candidate-gene association studies requires the precise and complete pre-specification of the total number of trios to be sampled to obtain sufficient power at a certain significance level (type I error risk). In most of these studies, very little information about the genetic effect size will be available beforehand and thus it will be difficult to calculate a reasonable sample size. One would therefore wish to reassess the sample size during the course of a study.

METHOD

We propose an adaptive group sequential procedure which allows for both early stopping of the study with rejection of the null hypothesis (H0) and for recalculation of the sample size based on interim effect size estimates when H0 cannot be rejected. The applicability of the method which was developed by Müller and Schäfer [Biometrics 2001;57:886-891] in a clinical context is demonstrated by a numerical example. Monte Carlo simulations are performed comparing the adaptive procedure with a fixed sample and a conventional group sequential design.

RESULTS

The main advantage of the adaptive procedure is its flexibility to allow for design changes in order to achieve a stabilized power characteristic while controlling the overall type I error and using the information already collected.

CONCLUSIONS

Given these advantages, the procedure is a promising alternative to traditional designs.

A group sequential approach to evaluation of bridging studies

The International Conference on Harmonization (ICH) E5 guideline defines a bridging study as a supplementary study conducted in the new region to provide pharmacodynamic or clinical data on efficacy, safety, dosage, and dose regimen to allow extrapolation of the foreign clinical data to the population of the new region. Therefore, a bridging study is usually conducted in the new region only after the test product has been approved for commercial marketing in the original region due to its proven efficacy and safety. The issue of analysis of clinical data generated by the bridging study conducted in the new region to evaluate the similarity for extrapolation of the foreign clinical data to the population of the new region is the information on efficacy, safety, dosage, and dose regimen of the original region that cannot be concurrently obtained from the local bridging studies but is available in the trials conducted in the original region. A group sequential approach is therefore proposed to overcome the issue of internal validity. In particular, we use the region as a group sequence to enroll the patients from the original region first and then to enroll patients from the new region subsequently. Methods for sample size determination for the bridging study in the new region are also proposed.

The advantages and disadvantages of adaptive designs for clinical trials

Flexible designs for clinical trials permit mid-trial design modifications, which are based on interim information from inside or outside the trial while meeting (regulatory) requirements for the control of the type I error rate. The basic principle is to combine stage standardized test statistics such as p-values or z-scores in a pre-specified way. The flexibility covers changes of sample sizes, treatment allocation ratios and the number of interim analyses, as well as the selection of treatments, doses and end points. The price to be paid is that non-standard test statistics must be used after an adaptation.

Mid-course sample size modification in clinical trials based on the observed treatment effect

It is not uncommon to set the sample size in a clinical trial to attain specified power at a value for the treatment effect deemed likely by the experimenters, even though a smaller treatment effect would still be clinically important. Recent papers have addressed the situation where such a study produces only weak evidence of a positive treatment effect at an interim stage and the organizers wish to modify the design in order to increase the power to detect a smaller treatment effect than originally expected. Raising the power at a small treatment effect usually leads to considerably higher power than was first specified at the original alternative. Several authors have proposed methods which are not based on sufficient statistics of the data after the adaptive redesign of the trial. We discuss these proposals and show in an example how the same objectives can be met while maintaining the sufficiency principle, as long as the eventuality that the treatment effect may be small is considered at the design stage. The group sequential designs we suggest are quite standard in many ways but unusual in that they place emphasis on reducing the expected sample size at a parameter value under which extremely high power is to be achieved. Comparisons of power and expected sample size show that our proposed methods can out-perform L. Fisher's 'variance spending' procedure. Although the flexibility to redesign an experiment in mid-course may be appealing, the cost in terms of the number of observations needed to correct an initial design may be substantial.

Sequential tests for noninferiority and superiority

The problem of simultaneous sequential tests for noninferiority and superiority of a treatment, as compared to an active control, is considered in terms of continuous hierarchical families of one-sided null hypotheses, in the framework of group sequential and adaptive two-stage designs. The crucial point is that the decision boundaries for the individual null hypotheses may vary over the parameter space. This allows one to construct designs where, e.g., a rigid stopping criterion is chosen, rejecting or accepting all individual null hypotheses simultaneously. Another possibility is to use monitoring type stopping boundaries, which leave some flexibility to the experimenter: he can decide, at the interim analysis, whether he is satisfied with the noninferiority margin achieved at this stage, or wants to go for more at the second stage. In the case where he proceeds to the second stage, he may perform midtrial design modifications (e.g., reassess the sample size). The proposed approach allows one to "spend," e.g., less of alpha for an early proof of noninferiority than for an early proof of superiority, and is illustrated by typical examples.

Practical midcourse sample size modification in clinical trials

Power calculations are very important in the planning of a well-designed clinical trial. Sometimes there is limited information available before the trial, making it highly desirable to adjust the sample size after seeing actual trial data. Indeed, there has been a recent proliferation of papers promising great flexibility in midcourse correction of sample size and other design features, such as choice of primary endpoint. We point out the difficulty in accurately estimating the treatment effect midway through a trial, and we encourage the use of a simple, conservative approach whereby sample size can be increased but not decreased from what was originally planned. We show how to compute the p value and confidence interval for this two-stage procedure. If the original sample size is maintained, analysis of the data is the same as for a fixed sample procedure.

TACT method for non-inferiority testing in active controlled trials

In active controlled trials without a placebo arm, non-inferiority testing is often considered but has different objectives. For the objective of demonstrating the efficacy of an experimental treatment or retention of a fraction of the control effect by the treatment, there are two types of statistical methods for testing - the synthesis method and the confidence interval method. According to the study of Wang, Hung and Tsong, the former is efficient under the so-called constancy condition but may have the alpha error rate inflate rapidly if the condition does not hold. In contrast, the latter method with careful selection of the non-inferiority margin tends to be conservative if the condition holds and may still have a valid alpha error otherwise unless the effect of the active control is less to a large extent in the active controlled trial than in the historical trials. We developed the TACT method, Two-stage Active Control Testing, as a viable compromise between the two methods. Through the TACT method, the uninterpretable non-inferiority testing may be avoided prior to the end of the trial. The TACT method carefully constructed can have a valid alpha error rate and the power close to the synthesis method if the constancy condition holds. In addition, the TACT method is more powerful than the confidence interval method for testing for the efficacy of the new treatment relative to the putative placebo and for showing that the new treatment is not inferior to the active control comparator.

Methods for mid-course corrections in clinical trials with survival outcomes

Interim monitoring of randomized controlled clinical trials often requires that the assumptions under which the trial was designed be evaluated and that appropriate design modifications be made. Furthermore, if there is little hope of ultimately demonstrating benefit, the study may be terminated on either ethical or economic grounds. Design modifications may be made based on either blinded or unblinded analyses. Since observed patterns of recruitment and patient outcomes may differ from those required by standard methods, specialized tools may be required to perform the necessary computations. In this paper we demonstrate how the Markov chain model of Lakatos can accommodate arbitrary observed patterns of recruitment, hazard rates, and other design parameters for performing mid-course corrections in trials with survival outcomes.

Stopping boundaries adjusted for sample size reestimation and negative stop

We propose an approach to specify group sequential stopping boundaries adjusted for sample size reestimation and negative stop in interim analyses of a clinical trial. Sample size can be adjusted based on the observed delta at each interim to maintain the targeted power. The calculation of stopping boundaries incorporates possible changes in the type-I error due to sample size reestimation and/or negative stops; hence the overall type-I error is well controlled. This approach combines the advantages of the group sequential and sample size reestimation methods and is more efficient than either one alone. It provides flexibility in clinical trials and still maintains the integrity of these trials. When no early stop is planned, the stopping boundaries will be adjusted only for sample size reestimation. All calculations are given in closed mathematical forms and adjustments in stopping boundaries are based on the exact type-I error change. Therefore, the penalty for the type-I error inflation due to such interim conductions is kept to a minimum.

On changing a long-term clinical trial midstream

Clinical triallists are often reluctant to alter the protocol or the design of an ongoing study in part because of concern that changes may affect the integrity of the study. This paper encourages considering changes in long-term clinical trials when the medical environment changes or when the accruing data from the trial lead to questioning the assumptions underlying the original design. One must make such changes in a way that will not cast doubt on the integrity of the trial. An important aspect of the design is choice of sample size. Methodology for sample size recalculation has matured over the decade. Several techniques are now available for the usual types of endpoints - continuous, discrete and time-to-failure - as well as for longitudinal analysis. Published papers describe the choices of variance at the time of recalculation, approaches to estimation and testing at the end of the study, and the time of recalculation. In a study with a sponsor, a Data Safety Monitoring Board (DSMB) and an Executive Committee, one or more of these three groups is responsible for recommending increases in sample size when the accruing data indicate that the assumed variance underestimated the true variance. Sometimes a statistician independent of all three bodies performs the relevant calculations and sends the recommendation to the sponsor. This paper argues that the DSMB should not generally be the responsible body because knowledge of treatment effect can place it in an uncomfortable position.

African-American Heart Failure Trial (A-HeFT): rationale, design, and methodology

BACKGROUND

Hydralazine and isosorbide dinitrate combination (H+ISDN), angiotensin-converting enzyme inhibitors, and beta-blockers have improved outcomes in heart failure (HF). Analysis of previous trials has shown that H+ISDN appears especially beneficial in African American patients.

METHODS AND RESULTS

The African-American Heart Failure Trial (A-HeFT) is double-blind, placebo-controlled, and includes African American patients with stable New York Heart Association Class III-IV HF on standard therapy. Patients must have prior HF-related events and left ventricular ejection fraction (LVEF) < or = 35% or LVEF <45% with left ventricular internal diastolic dimension >2.9 cm/m(2). Randomization to addition of placebo or BiDil (Nitro Med, Inc., Bedford, MA), a fixed combination of H+ISDN, is stratified for beta-blocker usage. All patients are treated and followed until the last patient entered completes 6 months of follow-up. The primary efficacy endpoint is a composite score including quality of life, death, and hospitalization for HF. At least 600 patients will be randomized; the first was randomized in June 2001.

CONCLUSIONS

In addition to providing additional information on BiDil efficacy in HF, A-HeFT is the first HF trial aimed at a selected subgroup of patients and the first to use a new composite HF score as its primary efficacy endpoint.

Self-designing two-stage trials to minimize expected costs

In the design of clinical trials, the sample size for the trial is traditionally calculated from estimates of parameters of interest, such as the mean treatment effect, which can often be inaccurate. However, recalculation of the sample size based on an estimate of the parameter of interest that uses accumulating data from the trial can lead to inflation of the overall Type I error rate of the trial. The self-designing method of Fisher, also known as the variance-spending method, allows the use of all accumulating data in a sequential trial (including the estimated treatment effect) in determining the sample size for the next stage of the trial without inflating the Type I error rate. We propose a self-designing group sequential procedure to minimize the expected total cost of a trial. Cost is an important parameter to consider in the statistical design of clinical trials due to limited financial resources. Using Bayesian decision theory on the accumulating data, the design specifies sequentially the optimal sample size and proportion of the test statistic's variance needed for each stage of a trial to minimize the expected cost of the trial. The optimality is with respect to a prior distribution on the parameter of interest. Results are presented for a simple two-stage trial. This method can extend to nonmonetary costs, such as ethical costs or quality-adjusted life years.

Strategies for changing the test statistic during a clinical trial

This article discusses the design of a clinical trial where a new treatment will be compared to a control. For a specific type of endpoint, there are a wide variety of test statistics that can be used. Also, the investigator must decide how many patients to accrue in each arm as well as the duration of the study. After an interim look at the data, the investigator may decide that a different test statistic would be more powerful or that more patients or longer follow-up is needed. In this article, we discuss a strategy for making these types of changes. This strategy controls the probability of making a type I error and can result in a procedure that has higher power than the test without adaptation.

Sample size re-estimation: recent developments and practical considerations

Interim findings of a clinical trial often will be useful for increasing the sample size if necessary to provide the required power against the null hypothesis when the alternative hypothesis is true. Strategies for carrying out the interim examination that have been described over the past several years include "internal pilot studies", blinded interim sample size adjustment and conditional power. Simulation studies show that the alternative methods generally control the type I error rate satisfactorily, although the power properties are more variable. The important issues associated with sample size re-estimation are strategic, not numeric. Clearly expressed regulatory preferences suggest that methods not requiring unblinding the data before completion of the trial would be most appropriate. Extending a trial has its risks. The investigators/patients enrolled later in the course of a trial are not necessarily the same as those recruited/entered early. Re-activating the enrollment process may be sufficiently complicated and expensive to justify enrolling more investigators/patients at the outset. Since sample size re-estimation adjusts the sample size on the basis of variability while efficacy interim analysis adjusts the sample size based on the basis of estimated effect size, both principles can be used in the same trial. Sample size re-estimation may not be advisable for trials involving extended follow-up of individual patients or, more generally, when the follow-up time is long relative to the recruitment time. In such cases, it may be better to estimate the sample size conservatively and introduce an interim efficacy evaluation.

Adaptive group sequential designs for clinical trials: combining the advantages of adaptive and of classical group sequential approaches

A general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing. This allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs. The concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans.

Group sequential test strategies for superiority and non-inferiority hypotheses in active controlled clinical trials

In a group sequential active controlled clinical trial, the study hypothesis may be a superiority hypothesis that an experimental treatment is more effective than the active control therapy or a non-inferiority hypothesis that the treatment is no worse than the active control within some non-inferiority range. When it is necessary to plan for testing the superiority and the non-inferiority hypotheses, we propose an adaptive group sequential closed test strategy by which the sample size is planned for testing superiority and is to be increased for showing non-inferiority given that it is deemed more plausible than superiority based on the observed sample path during the course of the trial. The proposed adaptive test strategy is valid in terms of having the type I error probability maintained at the targeted alpha level for both superiority and non-inferiority. It has power advantage or sample size saving over the traditional group sequential test designed for testing either superiority only or non-inferiority only.

Type I error in sample size re-estimations based on observed treatment difference

Sample size re-estimation based on an observed difference can ensure an adequate power and potentially save a large amount of time and resources in clinical trials. One of the concerns for such an approach is that it may inflate the type I error. However, such a possible inflation has not been mathematically quantified. In this paper the mathematical mechanism of this inflation is explored for two-sample normal tests. A (conditional) type I error function based on normal data is derived. This function not only provides the quantification but also gives mathematical mechanisms of possible inflation in the type I error due to the sample size re-estimation. Theoretically, based on their decision rules (certain upper and lower bounds), people can calculate this function and exactly visualize the changes in type I error. Computer simulations are performed to ensure the results. If there are no bounds for the adjustment, the inflation is evident. If proper adjusting rules are used, the inflation can be well controlled. In some cases the type I error can even be reduced. The trade-off is to give up some 'unrealistic power'. We investigated several scenarios in which the mechanisms to change the type I error are different. Our simulations show that similar results may apply to other distributions.

Interim analysis and sample size reassessment

This article deals with sample size reassessment for adaptive two-stage designs based on conditional power arguments utilizing the variability observed at the first stage. Fisher's product test for the p-values from the disjoint samples at the two stages is considered in detail for the comparison of the means of two normal populations. We show that stopping rules allowing for the early acceptance of the null hypothesis that are optimal with respect to the average sample size may lead to a severe decrease of the overall power if the sample size is a priori underestimated. This problem can be overcome by choosing designs with low probabilities of early acceptance or by midtrial adaptations of the early acceptance boundary using the variability observed in the first stage. This modified procedure is negligibly anticonservative and preserves the power.