Sample Size Re-estimation Methods Research Articles

Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design

ABSTRACT Single-arm phase II trials are very common in oncology. A fixed sample trial may lack sufficient power if the true efficacy is less than the assumed one. Adaptive designs have been proposed in the literature. We propose a Simon’s design based, adaptive sequential design. Simon’s design is the most used fixed sample design for single-arm phase II oncology trials. A prominent feature of Simon’s design is that it minimizes the sample size when there is no clinically meaningful efficacy. We identify Simon’s trial as a special group sequential design. Established methods for sample size re-estimation (SSR) can be readily applied to Simon’s design. Simulations show that simply adding SSR to Simon’s design may still not provide desirable power. We propose some expansions to Simon’s design. The expanded design with SSR can provide even more power.

Sample size re-estimation for confirmatory two-stage flexible multi-arm trial with normal outcomes

ABSTRACTThe need for efficient clinical trial designs has led to the development of flexible multi-arm designs. In this paper, we investigated sample size re-estimation (SSR) methods for fully flexible designs that allow for any unplanned treatment selection, early stopping for efficacy or futility. Even though SSR methods for two-arm trials are well developed, they are not directly applicable to certain fully flexible multi-arm trials due to the application of the closed test procedure. We derive SSR procedures based on the conditional power approach in the context of confirmatory multi-arm trials for three designs: inverse normal combination test, Fisher’s combination test and flexible group sequential design. We conduct extensive simulation studies to evaluate the performance of the three designs with and without SSR. Results show that the relative performance of the three designs depends on many factors. In practice, simulation studies are usually necessary to help determine the most appropriate design.

Sample size re-estimation for clinical trials with longitudinal negative binomial counts including time trends.

In some diseases, such as multiple sclerosis, lesion counts obtained from magnetic resonance imaging (MRI) are used as markers of disease progression. This leads to longitudinal, and typically overdispersed, count data outcomes in clinical trials. Models for such data invariably include a number of nuisance parameters, which can be difficult to specify at the planning stage, leading to considerable uncertainty in sample size specification. Consequently, blinded sample size re-estimation procedures are used, allowing for an adjustment of the sample size within an ongoing trial by estimating relevant nuisance parameters at an interim point, without compromising trial integrity. To date, the methods available for re-estimation have required an assumption that the mean count is time-constant within patients. We propose a new modeling approach that maintains the advantages of established procedures but allows for general underlying and treatment-specific time trends in the mean response. A simulation study is conducted to assess the effectiveness of blinded sample size re-estimation methods over fixed designs. Sample sizes attained through blinded sample size re-estimation procedures are shown to maintain the desired study power without inflating the Type I error rate and the procedure is demonstrated on MRI data from a recent study in multiple sclerosis.

Modelling and sample size reestimation for longitudinal count data with incomplete follow up.

We consider modelling and inference as well as sample size estimation and reestimation for clinical trials with longitudinal count data as outcomes. Our approach is general but is rooted in design and analysis of multiple sclerosis trials where lesion counts obtained by magnetic resonance imaging are important endpoints. We adopt a binomial thinning model that allows for correlated counts with marginal Poisson or negative binomial distributions. Methods for sample size planning and blinded sample size reestimation for randomised controlled clinical trials with such outcomes are developed. The models and approaches are applicable to data with incomplete observations. A simulation study is conducted to assess the effectiveness of sample size estimation and blinded sample size reestimation methods. Sample sizes attained through these procedures are shown to maintain the desired study power without inflating the type I error. Data from a recent trial in patients with secondary progressive multiple sclerosis illustrate the modelling approach.

Comparison of Operating Characteristics of Commonly Used Sample Size Re-Estimation Procedures in a Two-Stage Design

In group sequential clinical trials, there are several sample size re-estimation methods proposed in the literature that allow for change of sample size at the interim analysis. Most of these methods are based on either the conditional error function or the interim effect size. Our simulation studies compared the operating characteristics of three commonly used sample size re-estimation methods, Chen et al. (2004), Cui et al. (1999), and Muller and Schafer (2001). Gao et al. (2008) extended the CDL method and provided an analytical expression of lower and upper threshold of conditional power where the type I error is preserved. Recently, Mehta and Pocock (2010) extensively discussed that the real benefit of the adaptive approach is to invest the sample size resources in stages and increasing the sample size only if the interim results are in the so called “promising zone” which they define in their article. We incorporated this concept in our simulations while comparing the three methods. To test the robustness of these methods, we explored the impact of incorrect variance assumption on the operating characteristics. We found that the operating characteristics of the three methods are very comparable. In addition, the concept of promising zone, as suggested by MP, gives the desired power and smaller average sample size, and thus increases the efficiency of the trial design.

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.

Innovative approaches to clinical development and trial design.

Pharmaceutical innovation is increasingly risky, costly and at times inefficient, which has led to a decline in industry productivity. Despite the increased investment in R&D by the industry, the number of new molecular entities achieving marketing authorization is not increasing. Novel approaches to clinical development and trial design could have a key role in overcoming some of these challenges by improving efficiency and reducing attrition rates. The effectiveness of clinical development can be improved by adopting a more integrated model that increases flexibility and maximizes the use of accumulated knowledge. Central to this model of drug development are novel tools, including modelling and simulation, Bayesian methodologies, and adaptive designs, such as seamless adaptive designs and sample-size re-estimation methods. Applications of these methodologies to early- and late-stage drug development are described with some specific examples, along with advantages, challenges, and barriers to implementation. Because they are so flexible, these new trial designs require significant statistical analyses, simulations and logistical considerations to verify their operating characteristics, and therefore tend to require more time for the planning and protocol development phase. Greater awareness of the distinct advantages of innovative designs by regulators and sponsors are crucial to increasing the adoption of these modern tools.

Sample size re‐estimation in clinical trials

Adaptive clinical trials are becoming very popular because of their flexibility in allowing mid-stream changes of sample size, endpoints, populations, etc. At the same time, they have been regarded with mistrust because they can produce bizarre results in very extreme settings. Understanding the advantages and disadvantages of these rapidly developing methods is a must. This paper reviews flexible methods for sample size re-estimation when the outcome is continuous.

Adaptive Clinical Trials

Adaptive designs promise the flexibility to redesign clinical trials at interim stages. This flexibility would provide greater efficiency in drug development. However, despite this promise, many hesitate to implement such designs. Here we explore three possible reasons for the hesitation: (i) confusion with respect to the definition of an 'adaptive design'; (ii) controversy surrounding the use of sample size re-estimation methods; and (iii) logistical barriers that must be overcome in order to use adaptive designs within existing trial frameworks.The large volume of recent work has created confusion with respect to the definition of an 'adaptive design'. Unfortunately, this has resulted in reduced usage of many acceptable methods because of guilt by association with the more controversial methods. This review attempts to clarify the differences among many common types of proposed adaptive designs. Once the differences are noted, it becomes apparent that some adaptive designs are well accepted while others remain very controversial. In fact, much of the controversy and criticism surrounding adaptive designs has focused on their use for sample size re-estimation. Hence, this review also examines the different types of adaptive designs for sample size re-estimation in order to clarify the controversy surrounding the use of these methods. Specifically, separating the controversial from good practice requires clarifying differences between adaptive designs with sample size re-estimation based on a revised treatment effect and re-estimation based only on nuisance parameters (internal pilot designs). Finally, many logistical barriers must be overcome in order to use adaptive designs within existing trial frameworks.If the promise of adaptive designs is to be achieved, it will be important to bring together large groups of individuals from funding sources and regulatory agencies to address these limitations. Very few discussions of these issues have appeared in journals that are targeted to clinical audiences. In fact, current use of adaptive designs is not really hindered by the lack of statistical methods to accommodate the adaptations. Rather, there is a need for education as to which adaptive designs are acceptable and which are not acceptable. These discussions will require the involvement of many individuals outside the statistical community. In this review, we summarize the existing methods and current controversies with the intent of providing a clarification that will enable these individuals to participate in these much-needed discussions.

Operating characteristics of sample size re-estimation with futility stopping based on conditional power

Various methods have been described for re-estimating the final sample size in a clinical trial based on an interim assessment of the treatment effect. Many re-weight the observations after re-sizing so as to control the pursuant inflation in the type I error probability alpha. Lan and Trost (Estimation of parameters and sample size re-estimation. Proceedings of the American Statistical Association Biopharmaceutical Section 1997; 48-51) proposed a simple procedure based on conditional power calculated under the current trend in the data (CPT). The study is terminated for futility if CPT < or = CL, continued unchanged if CPT > or = CU, or re-sized by a factor m to yield CPT = CU if CL < CPT < CU, where CL and CU are pre-specified probability levels. The overall level alpha can be preserved since the reduction due to stopping for futility can balance the inflation due to sample size re-estimation, thus permitting any form of final analysis with no re-weighting. Herein the statistical properties of this approach are described including an evaluation of the probabilities of stopping for futility or re-sizing, the distribution of the re-sizing factor m, and the unconditional type I and II error probabilities alpha and beta. Since futility stopping does not allow a type I error but commits a type II error, then as the probability of stopping for futility increases, alpha decreases and beta increases. An iterative procedure is described for choice of the critical test value and the futility stopping boundary so as to ensure that specified alpha and beta are obtained. However, inflation in beta is controlled by reducing the probability of futility stopping, that in turn dramatically increases the possible re-sizing factor m. The procedure is also generalized to limit the maximum sample size inflation factor, such as at m max = 4. However, doing so then allows for a non-trivial fraction of studies to be re-sized at this level that still have low conditional power. These properties also apply to other methods for sample size re-estimation with a provision for stopping for futility. Sample size re-estimation procedures should be used with caution and the impact on the overall type II error probability should be assessed.

Adaptive, group sequential and decision theoretic approaches to sample size determination

This paper presents two adaptive methods for sample size re-estimation within a unified group sequential framework. The conceptual and practical distinction between these adaptive modifications and more traditional sample size changes due to revised estimates of nuisance parameters is highlighted. The motivation for the adaptive designs is discussed. Having established that adaptive sample size modifications can be made without inflating the type 1 error, the paper concludes with a novel decision theoretic approach for determining the magnitude of the sample size modification.

Sample size re-estimation in group-sequential response-adaptive clinical trials.

In clinical trials where the variances of the response variables are unknown, in accurate estimates of these can affect the type II error rate considerably. More accurate estimates of the variances may be obtained by taking a look at the data available part way through the trial and re-calculating the required sample size based on these new estimates. The main impetus for sample size re-estimation came from a two-stage procedure developed by Stein in 1945 and the literature is now replete with variations on this approach. In this paper, existing sample size re-estimation methods for both fixed sample and sequential clinical trial models will be reviewed. These will then be extended for use in group-sequential response-adaptive designs. In particular, a test for a recently developed group-sequential response-adaptive design, which compares two treatments with immediate normally distributed responses and unknown variances, is presented based on a modified version of Stein's test. The principal modifications involve updating the required sample size at each interim analysis and calculating the test statistic based on the current estimates of the variances. Hence, all the available information is used at each stage. Simulation is used to assess to what extent the updating of the required sample size at each interim analysis in the new test helps to attain the nominal error rates. The test is compared to modified versions of a simple test and a Stein-type group sequential t-test studied in the recent literature. These tests calculate the required sample sizes based on less accurate estimates of the variances. The type I error rate is close to the nominal value and the power is more accurately maintained in the new test.

An Example of an Unblinded, Third-Party Interim Analysis for Sample Size Re-estimation

An interim analysis for sample size re-estimation is often carried out in a clinical trial when there is concern about the uncertainty of the estimate of variability used for the prestudy sample size estimation. The use of an unblinded third party to do the interim analysis instead of conventional blinded methods may be possible and may lead to the desired power for the study being more likely to be achieved. An example of a clinical trial where the unblinded, third-party approach was used is described. For the interim results obtained, the decision not to increase the sample size would also have been reached had the project team used either of two commonly adopted blinded sample size re-estimation methods. However, simulation results from some plausible alternative scenarios based on the example show that an unblinded, third-party interim analysis for sample size re-estimation can provide more accurate results than the two blinded methods.

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.

Sample Size Reestimation in Clinical Trials

This paper describes an approach to performing interim sample size reestimation based on the observed treatment difference in clinical trials. The approach combines the advantages of the group sequential and sample size reestimation methods and provides an efficient design for clinical trials. It provides flexibility but still maintains the integrity of the trial. To control the overall type I error level, a method is proposed to adjust the group sequential stopping boundaries adjusted for sample size reestimation and negative stops. The adjusted stopping boundaries are flexible to different rules of sample size reestimation and reuse the alpha values saved by negative stops. The adjustment is based on the exact type I error change and, therefore, the penalty for the type I error inflation due to such an interim reestimation is kept to a minimum. The efficiency of sample size reestimation without positive stops is compared with the group sequential method using unconditional power and expected sample size. All results are based on sufficient mathematical justifications.