Three-stage designs for monitoring clinical trials

Optimal three-stage designs with equal sample sizes at each stage are presented and compared to fixed sample designs, fully sequential designs, designs restricted to use the fixed sample critical value at the final stage, and to modifications of other group sequential designs previously proposed in the literature. Typically, the greatest savings realized with interim analyses are obtained by the first interim look. More than 50% of the savings possible with a fully sequential design can be realized with a simple two-stage design. Three-stage designs can realize as much as 75% of the possible savings. Without much loss in efficiency, the designs can be modified so that the critical value at the final stage equals the usual fixed sample value while maintaining the overall level of significance, alleviating some potential confusion should a final stage be necessary. Some common group sequential designs, modified to allow early acceptance of the null hypothesis, are shown to be nearly optimal in some settings while performing poorly in others. An example is given to illustrate the use of several three-stage plans in the design of clinical trials.     

Outcome-adaptive allocation with natural lead-in for three-group trials with binary outcomes

ABSTRACT

Just as Bayes extensions of the frequentist optimal allocation design have been developed for the two-group case, we provide a Bayes extension of optimal allocation in the three-group case. We use the optimal allocations derived by Jeon and Hu [Optimal adaptive designs for binary response trials with three treatments. Statist Biopharm Res. 2010;2(3):310–318] and estimate success probabilities for each treatment arm using a Bayes estimator. We also introduce a natural lead-in design that allows adaptation to begin as early in the trial as possible. Simulation studies show that the Bayesian adaptive designs simultaneously increase the power and expected number of successfully treated patients compared to the balanced design. And compared to the standard adaptive design, the natural lead-in design introduced in this study produces a higher expected number of successes whilst preserving power.     

Blinded continuous monitoring in clinical trials with recurrent event endpoints

In studies with recurrent event endpoints, misspecified assumptions of event rates or dispersion can lead to underpowered trials or overexposure of patients. Specification of overdispersion is often a particular problem as it is usually not reported in clinical trial publications. Changing event rates over the years have been described for some diseases, adding to the uncertainty in planning. To mitigate the risks of inadequate sample sizes, internal pilot study designs have been proposed with a preference for blinded sample size reestimation procedures, as they generally do not affect the type I error rate and maintain trial integrity. Blinded sample size reestimation procedures are available for trials with recurrent events as endpoints. However, the variance in the reestimated sample size can be considerable in particular with early sample size reviews. Motivated by a randomized controlled trial in paediatric multiple sclerosis, a rare neurological condition in children, we apply the concept of blinded continuous monitoring of information, which is known to reduce the variance in the resulting sample size. Assuming negative binomial distributions for the counts of recurrent relapses, we derive information criteria and propose blinded continuous monitoring procedures. The operating characteristics of these are assessed in Monte Carlo trial simulations demonstrating favourable properties with regard to type I error rate, power, and stopping time, ie, sample size.     

Sample size estimation for comparing rates of change in K-group repeated count outcomes

Sample size estimation for comparing the rates of change in two-arm repeated measurements has been investigated by many investigators. In contrast, the literature has paid relatively less attention to sample size estimation for studies with multi-arm repeated measurements where the design and data analysis can be more complex than two-arm trials. For continuous outcomes, Jung and Ahn (2004 Jung, S., Ahn, C. (2004). K-sample test and sample size calculation for comparing slopes in data with repeated measurements. Biometrical J. 46(5):554–564.[Crossref], [Web of Science ®] , [Google Scholar]) and Zhang and Ahn (2013 Zhang, S., Ahn, C. (2013). Sample size calculation for comparing time-averaged responses in k-group repeated measurement studies. Comput. Stat. Data Anal. 58:283–291.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) have presented sample size formulas to compare the rates of change and time-averaged responses in multi-arm trials, using the generalized estimating equation (GEE) approach. To our knowledge, there has been no corresponding development for multi-arm trials with count outcomes. We present a sample size formula for comparing the rates of change in multi-arm repeated count outcomes using the GEE approach that accommodates various correlation structures, missing data patterns, and unbalanced designs. We conduct simulation studies to assess the performance of the proposed sample size formula under a wide range of designing configurations. Simulation results suggest that empirical type I error and power are maintained close to their nominal levels. The proposed method is illustrated using an epileptic clinical trial example.     

Statistical testing strategies for assessing treatment efficacy and marker accuracy in phase III trials

When a candidate predictive marker is available, but evidence on its predictive ability is not sufficiently reliable, all‐comers trials with marker stratification are frequently conducted. We propose a framework for planning and evaluating prospective testing strategies in confirmatory, phase III marker‐stratified clinical trials based on a natural assumption on heterogeneity of treatment effects across marker‐defined subpopulations, where weak rather than strong control is permitted for multiple population tests. For phase III marker‐stratified trials, it is expected that treatment efficacy is established in a particular patient population, possibly in a marker‐defined subpopulation, and that the marker accuracy is assessed when the marker is used to restrict the indication or labelling of the treatment to a marker‐based subpopulation, ie, assessment of the clinical validity of the marker. In this paper, we develop statistical testing strategies based on criteria that are explicitly designated to the marker assessment, including those examining treatment effects in marker‐negative patients. As existing and developed statistical testing strategies can assert treatment efficacy for either the overall patient population or the marker‐positive subpopulation, we also develop criteria for evaluating the operating characteristics of the statistical testing strategies based on the probabilities of asserting treatment efficacy across marker subpopulations. Numerical evaluations to compare the statistical testing strategies based on the developed criteria are provided.     

Blinded sample size re‐estimation in superiority and noninferiority trials: bias versus variance in variance estimation

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. Copyright © 2013 John Wiley & Sons, Ltd.     

Sample size recalculation for binary data in internal pilot study designs

For binary endpoints, the required sample size depends not only on the known values of significance level, power and clinically relevant difference but also on the overall event rate. However, the overall event rate may vary considerably between studies and, as a consequence, the assumptions made in the planning phase on this nuisance parameter are to a great extent uncertain. The internal pilot study design is an appealing strategy to deal with this problem. Here, the overall event probability is estimated during the ongoing trial based on the pooled data of both treatment groups and, if necessary, the sample size is adjusted accordingly. From a regulatory viewpoint, besides preserving blindness it is required that eventual consequences for the Type I error rate should be explained. We present analytical computations of the actual Type I error rate for the internal pilot study design with binary endpoints and compare them with the actual level of the chi‐square test for the fixed sample size design. A method is given that permits control of the specified significance level for the chi‐square test under blinded sample size recalculation. Furthermore, the properties of the procedure with respect to power and expected sample size are assessed. Throughout the paper, both the situation of equal sample size per group and unequal allocation ratio are considered. The method is illustrated with application to a clinical trial in depression. Copyright © 2004 John Wiley & Sons Ltd.     

Duration of Accrual and Follow-up for Two-Stage Clinical Trials

Group sequential trialswith time to event end points can be complicated to design. Notonly are there unlimited choices for the number of events requiredat each stage, but for each of these choices, there are unlimitedcombinations of accrual and follow-up at each stage that providethe required events. Methods are presented for determining optimalcombinations of accrual and follow-up for two-stage clinicaltrials with time to event end points. Optimization is based onminimizing the expected total study length as a function of theexpected accrual duration or sample size while providing an appropriateoverall size and power. Optimal values of expected accrual durationand minimum expected total study length are given assuming anexponential proportional hazards model comparing two treatmentgroups. The expected total study length can be substantiallydecreased by including a follow-up period during which accrualis suspended. Conditions that warrant an interim follow-up periodare considered, and the gain in efficiency achieved by includingan interim follow-up period is quantified. The gain in efficiencyshould be weighed against the practical difficulties in implementingsuch designs. An example is given to illustrate the use of thesetechniques in designing a clinical trial to compare two chemotherapyregimens for lung cancer. Practical considerations of includingan interim follow-up period are discussed.     

Accounting for uncertainty in the historical response rate of the standard treatment in single‐arm two‐stage designs based on Bayesian power functions

In phase II single‐arm studies, the response rate of the experimental treatment is typically compared with a fixed target value that should ideally represent the true response rate for the standard of care therapy. Generally, this target value is estimated through previous data, but the inherent variability in the historical response rate is not taken into account. In this paper, we present a Bayesian procedure to construct single‐arm two‐stage designs that allows to incorporate uncertainty in the response rate of the standard treatment. In both stages, the sample size determination criterion is based on the concepts of conditional and predictive Bayesian power functions. Different kinds of prior distributions, which play different roles in the designs, are introduced, and some guidelines for their elicitation are described. Finally, some numerical results about the performance of the designs are provided and a real data example is illustrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Accounting for informative non‐compliance with a bivariate exponential model in the design of endpoint trials

Failure to adjust for informative non‐compliance, a common phenomenon in endpoint trials, can lead to a considerably underpowered study. However, standard methods for sample size calculation assume that non‐compliance is non‐informative. One existing method to account for informative non‐compliance, based on a two‐subpopulation model, is limited with respect to the degree of association between the risk of non‐compliance and the risk of a study endpoint that can be modelled, and with respect to the maximum allowable rates of non‐compliance and endpoints. In this paper, we introduce a new method that largely overcomes these limitations. This method is based on a model in which time to non‐compliance and time to endpoint are assumed to follow a bivariate exponential distribution. Parameters of the distribution are obtained by equating them with the study design parameters. The impact of informative non‐compliance is investigated across a wide range of conditions, and the method is illustrated by recalculating the sample size of a published clinical trial. Copyright © 2005 John Wiley & Sons, Ltd.     

A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.