Event Size Re-estimation in Randomized Clinical Trials

Adaptive designs for clinical trials in cardiovascular diseases

In many clinical experiments, particularly in randomized clinical trials, the sample size required needs to be assessed and justified. For calculating a clinical trial’s sample size, assumptions have to be made regarding the clinical trial’s outcome data. These assumptions are based on prior clinical trials or merely on expert knowledge and always subject to some degree of uncertainty. To cope with this uncertainty in sample size estimation, adaptive designs were developed to re-estimate the sample size within a running trial. Especially adaptive designs for blinded sample size re-estimation, also referred to as non-comparative adaptive designs, have gained popularity, as these generally do not require an adjustment of the significance level to maintain type I error rates. In the first part of this thesis, we will consider developing sample size re-estimation methods for longitudinal overdispersed count data. As a first step, such data is modeled by a negative binomial counting process, and techniques for inference, sample size estimation and sample size re-estimation are provided. In a second step, presented methods are extended to handle time trends, which may occur during the course of a clinical trial. These trends are modeled by a gamma frailty model, for which inference, sample size estimation and sample size re-estimation techniques are also described in detail. As an application, we consider lesion counts measured by magnetic resonance imaging (MRI), which play an important role in phase II multiple sclerosis (MS) trials for measuring disease progression. These lesion counts are generally overdispersed and often measured multiple times per patient during a running trial, therefore resembling the statistical model. Methods are kept general to allow for applications outside of MS, whenever similar data arise, and shown to preserve type I error rates while correcting the sample size, such that a desired power level is reached, in extensive simulation runs. The second part of this thesis will consider univariate negative binomial data with baseline covariates. For example, such data arise in MS when the total number of lesions at the end of a clinical trial, corrected for the number of lesions at baseline or other baseline variables, is taken as an endpoint. Developed sample size re-estimation techniques are also shown to preserve type I error rates while correcting the sample size such that a desired power level is reached. The summarized results are made available as R-functions and extend current methodology in the field of non-comparative adaptive designs.

ABSTRACTSample size estimation (SSE) is an important issue in the planning of clinical studies. While larger studies are likely to have sufficient power, it may be unethical to expose more patients than necessary to answer a scientific question. Budget considerations may also cause one to limit the study to an adequate size to answer the question at hand. Typically at the planning stage, a statistically based justification for sample size is provided. An effective sample size is usually planned under a pre-specified type I error rate, a desired power under a particular alternative and variability associated with the observations recorded. The nuisance parameter such as the variance is unknown in practice. Thus, information from a preliminary pilot study is often used to estimate the variance. However, calculating the sample size based on the estimated nuisance parameter may not be stable. Sample size re-estimation (SSR) at the interim analysis may provide an opportunity to re-evaluate the uncertainties using accrued data and continue the trial with an updated sample size. This article evaluates a proposed SSR method based on controlling the variability of nuisance parameter. A numerical study is used to assess the performance of proposed method with respect to the control of type I error. The proposed method and concepts could be extended to SSR approaches with respect to other criteria, such as maintaining effect size, achieving conditional power, and reaching a desired reproducibility probability.

BackgroundWhile the clinical trials and statistical methodology literature on sample size re-estimation (SSRE) is robust, evaluation of SSRE procedures following the completion of a clinical trial has been sparsely reported. In blinded sample size re-estimation, only nuisance parameters are re-estimated, and the blinding of the current trial treatment effect is preserved. Blinded re-estimation procedures are well-accepted by regulatory agencies and funders. We review our experience of sample size re-estimation in a large international, National Institutes of Health funded clinical trial for adjuvant breast cancer treatment, and evaluate our blinded sample size re-estimation procedure for this time-to-event trial. We evaluated the SSRE procedure by examining assumptions made during the re-estimation process, estimates resulting from re-estimation, and the impact on final trial results with and without the addition of participants, following sample size re-estimation.MethodsWe compared the control group failure probabilities estimated at the time of SSRE to estimates used in the original planning, to the final un-blinded control group failure probability estimates for those included in the SSRE procedure (SSRE cohort), and to the final total control group failure probability estimates. The impact of re-estimation on the final comparison between randomized treatment groups is evaluated for those in the originally planned cohort (n = 340) and for the combination of those recruited in the originally planned cohort and those added after re-estimation (n = 509).ResultsVery little difference is observed between the originally planned cohort and all randomized patients in the control group failure probabilities over time or in the overall hazard ratio estimating treatment effect (originally planned cohort HR 1.25 (0.86, 1.79); all randomized cohort HR 1.24 95% CI (0.91, 1.68)). At the time of blinded SSRE, the estimated control group failure probabilities at 3 years (0.24) and 5 years (0.40) were similar to those for the SSRE cohort once un-blinded (3 years, 0.22 (0.16, 0.30); 5 years, 0.33 (0.26, 0.41)).ConclusionsWe found that our re-estimation procedure performed reasonably well in estimating the control group failure probabilities at the time of re-estimation. Particularly for time-to-event outcomes, pre-planned blinded SSRE procedures may be the best option to aid in maintaining power.Trial registrationClinicalTrials.gov, NCT00201851. Registered on 9 September 2005. Retrospectively registered.

A sample size re-estimation (SSR) design is a flexible, adaptive design with the primary purpose of allowing sample size of a study to be reassessed in the mid-course of the study to ensure adequate power. In real world drug product, biologic, and device development, there may be large uncertainty in key factors that drive the sample size estimation for a confirmatory clinical trial. For example, early phase studies may have encouraging results but could be of shorter duration, or use a different endpoint than what is required for confirmatory phase clinical trials. The negative impact of high uncertainty at design stage for a confirmatory trial can be mitigated by an SSR design. Recent surveys have reported an encouraging upward trend in the use of SSR designs in clinical trials since the release of the draft guidance for adaptive design clinical trials for drugs and biologics by the U.S. Food and Drug Administration in 2010 (U.S. Food and Drug Administration (FDA) (February, 2010), Draft Guidance for Industry: Adaptive Design Clinical Trials for Drugs and Biologics). To support broad understanding and acceptance of SSR designs in confirmatory settings, especially unblinded SSR designs, we summarize statistical methods pertaining to SSR designs, including recent development in this field, and discuss design alternatives among blinded SSR, unblinded SSR, and conventional group sequential designs. To support appropriate implementation of SSR designs, we make recommendations on operational logistics for trial conduct based on accumulated experience in recent years, and provide points to consider for final data analysis and reporting for studies where the sample size has been increased following either a blinded or an unblinded SSR algorithm.

Sequential monitoring of response-adaptive randomized clinical trials with sample size re-estimation

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.

Covariate-adaptive randomization (CAR) procedures have been developed in clinical trials to mitigate the imbalance of treatments among covariates. In recent years, an increasing number of trials have started to use CAR for the advantages in statistical efficiency and enhancing credibility. At the same time, sample size re-estimation (SSR) has become a common technique in industry to reduce time and cost while maintaining a good probability of success. Despite the widespread popularity of combining CAR designs with SSR, few researchers have investigated this combination theoretically. More importantly, the existing statistical inference must be adjusted to protect the desired type I error rate when a model that omits some covariates is used. In this article, we give a framework for the application of SSR in CAR trials and study the underlying theoretical properties. We give the adjusted test statistic and derive the sample size calculation formula under the CAR setting. We can tackle the difficulties caused by the adaptive features in CAR and prove the asymptotic independence between stages. Numerical studies are conducted under multiple parameter settings and scenarios that are commonly encountered in practice. The results show that all advantages of CAR and SSR can be preserved and further improved in terms of power and sample size.

BackgroundThe sample size required to power a study to a nominal level in a paired comparative diagnostic accuracy study, i.e. studies in which the diagnostic accuracy of two testing procedures is compared relative to a gold standard, depends on the conditional dependence between the two tests - the lower the dependence the greater the sample size required. A priori, we usually do not know the dependence between the two tests and thus cannot determine the exact sample size required. One option is to use the implied sample size for the maximal negative dependence, giving the largest possible sample size. However, this is potentially wasteful of resources and unnecessarily burdensome on study participants as the study is likely to be overpowered. A more accurate estimate of the sample size can be determined at a planned interim analysis point where the sample size is re-estimated.MethodsThis paper discusses a sample size estimation and re-estimation method based on the maximum likelihood estimates, under an implied multinomial model, of the observed values of conditional dependence between the two tests and, if required, prevalence, at a planned interim. The method is illustrated by comparing the accuracy of two procedures for the detection of pancreatic cancer, one procedure using the standard battery of tests, and the other using the standard battery with the addition of a PET/CT scan all relative to the gold standard of a cell biopsy. Simulation of the proposed method illustrates its robustness under various conditions.ResultsThe results show that the type I error rate of the overall experiment is stable using our suggested method and that the type II error rate is close to or above nominal. Furthermore, the instances in which the type II error rate is above nominal are in the situations where the lowest sample size is required, meaning a lower impact on the actual number of participants recruited.ConclusionWe recommend multinomial model maximum likelihood estimation of the conditional dependence between paired diagnostic accuracy tests at an interim to reduce the number of participants required to power the study to at least the nominal level.Trial registrationISRCTN ISRCTN73852054. Registered 9th of January 2015. Retrospectively registered.

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter-based sample size re-estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta-analytic-predictive approach. To incorporate external information into the sample size re-estimation, we propose to update the meta-analytic-predictive prior based on the results of the internal pilot study and to re-estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re-estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior-data conflict is present, incorporating external information into the sample size re-estimation improves the operating characteristics compared to the traditional approach. In the case of a prior-data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re-estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re-estimation, the potential gains should be balanced against the risks.

In clinical trials, sample size reestimation is a useful strategy for mitigating the risk of uncertainty in design assumptions and ensuring sufficient power for the final analysis. In particular, sample size reestimation based on unblinded interim effect size can often lead to sample size increase, and statistical adjustment is usually needed for the final analysis to ensure that type I error rate is appropriately controlled. In current literature, sample size reestimation and corresponding type I error control are discussed in the context of maintaining the original randomization ratio across treatment groups, which we refer to as "proportional increase." In practice, not all studies are designed based on an optimal randomization ratio due to practical reasons. In such cases, when sample size is to be increased, it is more efficient to allocate the additional subjects such that the randomization ratio is brought closer to an optimal ratio. In this research, we propose an adaptive randomization ratio change when sample size increase is warranted. We refer to this strategy as "nonproportional increase," as the number of subjects increased in each treatment group is no longer proportional to the original randomization ratio. The proposed method boosts power not only through the increase of the sample size, but also via efficient allocation of the additional subjects. The control of type I error rate is shown analytically. Simulations are performed to illustrate the theoreticalresults.

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.

Sequential monitoring of covariate adaptive randomized clinical trials with sample size re-estimation

In the planning of clinical trials with count outcomes such as the number of exacerbations in chronic obstructive pulmonary disease (COPD) often considerable uncertainty exists with regard to the overall event rate and the level of overdispersion which are both crucial for sample size calculations. To develop a sample size reestimation strategy that maintains the blinding of the trial, controls the type I error rate and is robust against misspecification of the nuisance parameters in the planning phase in that the actual power is close to the target. The operation characteristics of the developed sample size reestimation procedure are investigated in a Monte Carlo simulation study. Estimators of the overall event rate and the overdispersion parameter that do not require unblinding can be used to effectively adjust the sample size without inflating the type I error rate while providing power values close to the target. If only little information is available regarding the size of the overall event rate and the overdispersion parameter in the design phase of a trial, we recommend the use of a design with sample size reestimation as the one suggested here. Trials in COPD are expected to benefit from the proposed sample size reestimation strategy.

The internal pilot study design allows for modifying the sample size during an ongoing study based on a blinded estimate of the variance thus maintaining the trial integrity. Various blinded sample size re-estimation procedures have been proposed in the literature. We compare the blinded sample size re-estimation procedures based on the one-sample variance of the pooled data with a blinded procedure using the randomization block information with respect to bias and variance of the variance estimators, and the distribution of the resulting sample sizes, power, and actual type I error rate. For reference, sample size re-estimation based on the unblinded variance is also included in the comparison. It is shown that using an unbiased variance estimator (such as the one using the randomization block information) for sample size re-estimation does not guarantee that the desired power is achieved. Moreover, in situations that are common in clinical trials, the variance estimator that employs the randomization block length shows a higher variability than the simple one-sample estimator and in turn the sample size resulting from the related re-estimation procedure. This higher variability can lead to a lower power as was demonstrated in the setting of noninferiority trials. In summary, the one-sample estimator obtained from the pooled data is extremely simple to apply, shows good performance, and is therefore recommended for application.