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Existing multi-outcome designs focus almost entirely on evaluating whether all outcomes show evidence of efficacy or whether at least one outcome shows evidence of efficacy. While a small number of authors have provided multi-outcome designs that evaluate when a general number of outcomes show promise, these designs have been single-stage in nature only. We therefore propose two designs, of group-sequential and drop the loser form, that provide this design characteristic in a multi-stage setting. Previous such multi-outcome multi-stage designs have allowed only for a maximum of two outcomes; our designs thus also extend previous related proposals by permitting any number of outcomes., Comment: 38 pages, 11 figures, 4 tables. Submitted for the degree of Doctor of Philosophy at University of Cambridge

Purpose: Two-stage single-arm trial designs are commonly used in phase II oncology to infer treatment effects for a binary primary outcome (e.g., tumour response). It is imperative that such studies be designed, analysed, and reported effectively. However, there is little available evidence on whether this is the case, particularly for key statistical considerations. We therefore comprehensively review such trials, examining in particular quality of reporting. Methods: Published oncology trials that utilised "Simon's two-stage design" over a 5 year period were identified and reviewed. Articles were evaluated on whether they reported sufficient design details, such as the required sample size, and analysis details, such as a confidence interval (CI). The articles that did not adjust their inference for the incorporation of an interim analysis were re-analysed to evaluate the impact on their reported point estimate and CI. Results: Four hundred and twenty five articles that reported the results of a single treatment arm were included. Of these, 47.5% provided the five components that ensure design reproducibility. Only 1.2% and 2.1% reported an adjusted point estimate or CI, respectively. Just 55.3% of trials provided the final stage rejection bound, indicating many trials did not test the hypothesis the design is constructed to assess. Re-analysis of the trials suggests that reported point estimates underestimated treatment effects and that reported CIs were too narrow. Conclusion: Key design details of two-stage single-arm trials are often unreported. Whilst inference is regularly performed, it is rarely done so in a way that removes the bias introduced by the interim analysis. In order to maximise their value, future studies must improve the way that they are analysed and reported.

In single-arm phase II oncology trials, the most popular choice of design is Simon's two-stage design, which allows early stopping at one interim analysis. However, the expected trial sample size can be reduced further by allowing curtailment. Curtailment is stopping when the final go or no-go decision is certain, so-called non-stochastic curtailment, or very likely, known as stochastic curtailment. In the context of single-arm phase II oncology trials, stochastic curtailment has previously been restricted to stopping in the second stage and/or stopping for a no-go decision only. We introduce two designs that incorporate stochastic curtailment and allow stopping after every observation, for either a go or no-go decision. We obtain optimal stopping boundaries by searching over a range of potential conditional powers, beyond which the trial will stop for a go or no-go decision. This search is novel: firstly, the search is undertaken over a range of values unique to each possible design realisation. Secondly, these values are evaluated taking into account the possibility of early stopping. Finally, each design realisation's operating characteristics are obtained exactly. The proposed designs are compared to existing designs in a real data example. They are also compared under three scenarios, both with respect to four single optimality criteria and using a loss function. The proposed designs are superior in almost all cases. Optimising for the expected sample size under either the null or alternative hypothesis, the saving compared to the popular Simon's design ranges from 22% to 55%.

The determination of the sample size required by a crossover trial typically depends on the specification of one or more variance components. Uncertainty about the value of these parameters at the design stage means that there is often a risk a trial may be under- or over-powered. For many study designs, this problem has been addressed by considering adaptive design methodology that allows for the re-estimation of the required sample size during a trial. Here, we propose and compare several approaches for this in multi-treatment crossover trials. Specifically, regulators favour re-estimation procedures to maintain the blinding of the treatment allocations. We therefore develop blinded estimators for the within and between person variances, following simple or block randomisation. We demonstrate that, provided an equal number of patients are allocated to sequences that are balanced for period, the proposed estimators following block randomisation are unbiased. We further provide a formula for the bias of the estimators following simple randomisation. The performance of these procedures, along with that of an unblinded approach, is then examined utilising three motivating examples, including one based on a recently completed four-treatment four-period crossover trial. Simulation results show that the performance of the proposed blinded procedures is in many cases similar to that of the unblinded approach, and thus they are an attractive alternative.

Recently, methodology was presented to facilitate the incorporation of interim analyses in stepped-wedge (SW) cluster randomised trials (CRTs). Here, we extend this previous discussion. We detail how the stopping boundaries, allocation sequences, and per-cluster per-period sample size of a group sequential SW-CRT can be optimised. We then describe methods by which point estimates, p-values, and confidence intervals, which account for the sequential nature of the design, can be calculated. We demonstrate that optimal sequential designs can reduce the expected required number of measurements under the null hypothesis, compared to the classical design, by up to 30%, with no cost to the maximal possible required number of measurements. Furthermore, the adjusted analysis procedure almost universally reduces the average bias in the point estimate, and consistently provides a confidence interval with coverage close to the nominal level. In contrast, the coverage of a naive 95% confidence interval is observed to range between 92 and 98%. Methodology is now readily available for the efficient design and analysis of group sequential SW-CRTs. In scenarios in which there are substantial ethical or financial reasons to terminate a SW-CRT as soon as possible, trialists should strongly consider a group sequential approach.

Primarily motivated by the drug development process, several publications have now presented methodology for the design of multi-arm multi-stage experiments with normally distributed outcome variables of known variance. Here, we extend these past considerations to allow the design of what we refer to as an abcd multi-arm multi-stage experiment. We provide a proof of how strong control of the a-generalised type-I familywise error-rate can be ensured. We then describe how to attain the power to reject at least b out of c false hypotheses, which is related to controlling the b-generalised type-II familywise error-rate. Following this, we detail how a design can be optimised for a scenario in which rejection of any d null hypotheses brings about termination of the experiment. We achieve this by proposing a highly computationally efficient approach for evaluating the performance of a candidate design. Finally, using a real clinical trial as a motivating example, we explore the effect of the design's control parameters on the statistical operating characteristics.

In small sample studies with binary outcome data, use of a normal approximation for hypothesis testing can lead to substantial inflation of the type-I error-rate. Consequently, exact statistical methods are necessitated, and accordingly, much research has been conducted to facilitate this. Recently, this has included methodology for the design of two-stage multi-arm studies utilising exact binomial tests. These designs were demonstrated to carry substantial efficiency advantages over a fixed sample design, but generally suffered from strong conservatism. An alternative classical means of small sample inference with dichotomous data is Fisher's exact test. However, this method is limited to single-stage designs when there are multiple arms. Therefore, here, we propose a two-stage version of Fisher's exact test, with the potential to stop early to accept or reject null hypotheses, which is applicable to multi-arm studies. In particular, we provide precise formulae describing the requirements for achieving weak or strong control of the familywise error-rate with this design. Following this, we describe how the design parameters may be optimised to confer desirable operating characteristics. For a motivating example based on a phase II clinical trial, we demonstrate that on average our approach is less conservative than corresponding optimal designs based on exact binomial tests.

This paper presents a set of Stata commands and Mata functions to evaluate different distributional quantities of the multivariate normal distribution, and a particular type of non-central multivariate t distribution. Specifically, their densities, distribution functions, equicoordinate quantiles, and pseudo-random vectors can be computed efficiently, either in the absence or presence of variable truncation.

Numerous publications have now addressed the principles of designing, analyzing, and reporting the results of, stepped-wedge cluster randomized trials. In contrast, there is little research available pertaining to the design and analysis of multi-arm stepped-wedge cluster randomized trials, utilized to evaluate the effectiveness of multiple experimental interventions. In this paper, we address this by explaining how the required sample size in these multi-arm trials can be ascertained when data are to be analyzed using a linear mixed model. We then go on to describe how the design of such trials can be optimized to balance between minimizing the cost of the trial, and minimizing some function of the covariance matrix of the treatment effect estimates. Using a recently commenced trial that will evaluate the effectiveness of sensor monitoring in an occupational therapy rehabilitation program for older persons after hip fracture as an example, we demonstrate that our designs could reduce the number of observations required for a fixed power level by up to 58%. Consequently, when logistical constraints permit the utilization of any one of a range of possible multi-arm stepped-wedge cluster randomized trial designs, researchers should consider employing our approach to optimize their trials efficiency.

Crossover designs are an extremely useful tool to investigators, whilst group sequential methods have proven highly proficient at improving the efficiency of parallel group trials. Yet, group sequential methods and crossover designs have rarely been paired together. One possible explanation for this could be the absence of a formal proof of how to strongly control the familywise error rate in the case when multiple comparisons will be made. Here, we provide this proof, valid for any number of initial experimental treatments and any number of stages, when results are analysed using a linear mixed model. We then establish formulae for the expected sample size and expected number of observations of such a trial, given any choice of stopping boundaries. Finally, utilising the four-treatment, four-period TOMADO trial as an example, we demonstrate group sequential methods in this setting could have reduced the trials expected number of observations under the global null hypothesis by over 33%.

Multi-arm multi-stage trial designs can bring notable gains in efficiency to the drug development process. However, for normally distributed endpoints, the determination of a design typically depends on the assumption that the patient variance in response is known. In practice, this will not usually be the case. To allow for unknown variance, previous research explored the performance of t-test statistics, coupled with a quantile substitution procedure for modifying the stopping boundaries, at controlling the familywise error-rate to the nominal level. Here, we discuss an alternative method based on Monte Carlo simulation that allows the group size and stopping boundaries of a multi-arm multi-stage t-test to be optimised according to some nominated optimality criteria. We consider several examples, provide R code for general implementation, and show that our designs confer a familywise error-rate and power close to the desired level. Consequently, this methodology will provide utility in future multi-arm multi-stage trials.

In a group sequential clinical trial, accumulated data are analysed at numerous time-points in order to allow early decisions about a hypothesis of interest. These designs have historically been recommended for their ethical, administrative and economic benefits. In this work, we discuss a collection of new Stata commands for computing the stopping boundaries and required group size of various classical group sequential designs, assuming a normally distributed outcome variable. Following this, we demonstrate how the performance of several designs can be compared graphically.

The ability to accurately estimate the sample size required by a stepped-wedge (SW) cluster randomized trial (CRT) routinely depends upon the specification of several nuisance parameters. If these parameters are mis-specified, the trial could be over-powered, leading to increased cost, or under-powered, enhancing the likelihood of a false negative. We address this issue here for cross-sectional SW-CRTs, analyzed with a particular linear mixed model, by proposing methods for blinded and unblinded sample size re-estimation (SSRE). Blinded estimators for the variance parameters of a SW-CRT analyzed using the Hussey and Hughes model are derived. Then, procedures for blinded and unblinded SSRE after any time period in a SW-CRT are detailed. The performance of these procedures is then examined and contrasted using two example trial design scenarios. We find that if the two key variance parameters were under-specified by 50%, the SSRE procedures were able to increase power over the conventional SW-CRT design by up to 29%, resulting in an empirical power above the desired level. Moreover, the performance of the re-estimation procedures was relatively insensitive to the timing of the interim assessment. Thus, the considered SSRE procedures can bring substantial gains in power when the underlying variance parameters are mis-specified. Though there are practical issues to consider, the procedure's performance means researchers should consider incorporating SSRE in to future SW-CRTs.