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The two one-sided tests (TOST) procedure is the common method for assessing bioequivalence between two different drugs or formulations of the same drug. Several solutions to Behrens-Fisher problem ...

We address sampling design of population pharmacokinetics (popPK) experiments in the context of two ongoing phase 2b efficacy trials that evaluate the efficacy of VRC01 (vs. placebo) in reducing the rate of HIV infection among 4625 participants. Blood samples are collected at up to 22 study visits from all participants for immediate HIV diagnosis as the primary trial outcome, and stored for future outcome-dependent marker measurements. A key secondary objective of the trials is to evaluate correlates of prevention efficacy among a sub-cohort of VRC01 recipients in terms of whether the current value of VRC01 serum concentration is associated with the instantaneous rate of HIV infection. To accomplish this, concentrations on a daily grid are estimated via non-linear mixed effects popPK modeling of observed 4-weekly concentrations. Given the impracticality of measuring concentrations in all stored blood samples, we devised a simulation-based sampling design framework to evaluate the impact of sub-cohort sample sizes (m) and sampling schemes of time-points on the accuracy and precision of the popPK model parameters. We accounted for specific study schedules and heterogeneity in participants’ characteristics and study adherence patterns. We found that with m = 120, reasonably unbiased and consistent estimates of most fixed and random effect terms could be obtained without complete sampling of all 22 time-points, even under low study adherence (about half of the 4-weekly visits missing per participant). The described simulation framework is not only novel in its application to popPK sampling design for studying correlates of prevention efficacy in a subcohort of the parent trial, but also flexible in accommodating real-life study setup options, and can be generalized to other single- or multiple-dose PK sampling design settings.

Researchers designing a clinical trial to demonstrate superiority or non-inferiority of a new treatment to an established control face an important choice: to conduct a randomized controlled trial ...

Biosimilar products are a relative newcomer in the pharmaceutical industry. It was only in 2006 that the first biosimilar product, Omnitrope[textregistered] (somatropin), a human growth hormone pro...

In randomized clinical trials to compare overdispersed count data between treatment groups, blinded sample size re-estimation (BSSR) is an effective approach to ensure power control even under poss...

A confirmatory randomized controlled trial can be severely underpowered if the variability and/or effect size estimated from a small-scale pilot study is not taken into account in the sample size c...

Clinical patient management is dynamic. It is not based on a single decision but on a sequence of decisions, with adjustments of therapy overtime, where adjustment are personalized to individual pa...

The two one-sided tests (TOST) procedure is widely applied for assessing bioequivalence between two different drugs in clinical studies. The same notion can be applied to construct TOST method for ...

An initial proposal was made to start 30 monkeys in the Run-In Period of a preclinical translational research study, to have 24 or more animals qualify for randomization in the subsequent Treatment...

In oncology trials, non-inferiority concepts (e.g., shortening duration of chemotherapy) have sometimes been evaluated without preliminary knowledge of an experimental treatment being non-inferior ...

The concepts and implementation of standardized mean differences and minimum effect tests have been emphasized in ANOVA. The corresponding processes and implications are, however, not yet well explicated in the context of ANCOVA. To enhance the usefulness of ANCOVA, this article describes the minimum effect tests of standardized contrast as a valuable alternative to the hypothesis testing of no effect difference and the conventional evaluation of unstandardized effects in ANCOVA. Power and sample size procedures are developed to accommodate covariate randomness and imbalance for randomized and nonrandomized designs. The data from a clinical study of comparing two treatments for gingivitis are used to illustrate the application of the suggested approaches. The emphases of numerical appraisal are on the merit of minimum effect detection in comparative analysis and the influence of covariate feature in power and sample size computation. The proposed power and sample size calculations improve upon approximate formulas by fully accounting for the stochastic property and intrinsic disparity of the covariate variables. Computer algorithms are available for calculating the p-value, power level, and sample size of one- and two-sided minimum effect tests.

For clinical trials in patients with asthma, chronic obstructive pulmonary disease, and relapsing-remitting multiple sclerosis, regulatory guidelines state that a noninferiority clinical trial is a...

Bioequivalence (BE) studies are most often conducted as crossover trials, and therefore establishing their required sample size necessitates specification of the within-person variance. Given that this specification is often difficult in practice, there has been great interest in recent years in the use of adaptive designs for BE trials. However, while numerous methods for this have now been presented, their focus has been solely on two-treatment BE studies. In some instances, it will be desired to incorporate more than a single test and reference formulation into a BE trial. It would therefore be useful to establish methodology for the design of adaptive multi-treatment BE trials, to acquire the benefits in the two-treatment setting in this more complex situation. Here, we achieve this for three-treatment studies by extending previously proposed designs for two-treatment trials. First, we discuss the additional design considerations that arise when multiple comparisons are made. Next, an extensive simulation study is employed to compare the performance of the proposed procedures. With this, we demonstrate that two-stage designs with desirable statistical operating characteristics can be readily identified for three-treatment BE trials. Supplementary materials for this article are available online.

Limited resources are a challenge when planning comparative effectiveness studies of multiple promising treatments, often prompting study planners to reduce the sample size to meet the financial constraints. The practical solution is often to increase the efficiency of this sample size by selecting a pair of treatments among the pool of promising treatments before the clinical trial begins. The problem with this approach is that the investigator may inadvertently leave out the most beneficial treatment. This paper demonstrates a possible solution to this problem by using Bayesian adaptive designs. We use a planned comparative effectiveness clinical trial of treatments for sialorrhea in amyotrophic lateral sclerosis as an example of the approach. Rather than having to guess at the two best treatments to compare based on limited data, we suggest putting more arms in the trial and letting response adaptive randomization (RAR) determine better arms. To ground this study relative to previous literature we first compare RAR, adaptive equal randomization (ER), arm(s) dropping, and a fixed design. Given the goals of this trial we demonstrate that we may avoid 'type III errors' - inadvertently leaving out the best treatment - with little loss in power compared to a two-arm design, even when choosing the correct two arms for the two-armed design. There are appreciable gains in power when the two arms are prescreened at random.

In clinical development of new drugs, investment increases over time—Phase III investment can be up to hundreds of millions of dollars. However, some drugs failed because the treatment effect in Ph...

Two-stage designs have been widely used in phase II clinical trials to evaluate whether a new treatment shows sufficient evidences of effectiveness to justify being tested in a phase III trial. The...

Over-dispersed count variables are frequently encountered in biomedical research. Despite extensive research in analytical methods, addressing over-dispersion in the design of clinical tria...

In this paper, we illustrate the method of designing a group-sequential randomized clinical trial based on the difference in restricted mean survival time (RMST). The procedure is based on theoretical formulations of Murray and Tsiatis (1999). We also present a numerical example in designing a cardiology surgical trial. Various practical considerations are discussed. R codes are provided in the Supplementary Materials. We conclude that the group-sequential design for RMST is a viable option in practice. A simulation study is performed to compare the proposed method to the Max-Combo and conventional log-rank tests. The simulation result shows that when there is a delayed treatment benefit and the proportional hazards assumption is untrue, the sequential design based on the RMST can be more efficient than that based on the log-rank test but less efficient than that based on the Max-Combo test. Compared with Max-Combo test, the RMST-based study design yield coherent estimand, statistical inference and result interpretation.

Analysis of covariance (ANCOVA) is commonly used in the analysis of randomized clinical trials to adjust for baseline covariates and improve the precision of the treatment effect estimate. We deriv...

While planning clinical trials, when simple formulas are unavailable to calculate sample size, statistical simulations are used instead. However, one has to spend much computation time obtaining adequately precise and accurate simulated sample size estimates, especially when there are many scenarios for the planning and/or the specified statistical method is complicated. In this article, we summarize the theoretical aspect of statistical simulation-based sample size calculation. Then, we propose a new simulation procedure for sample size calculation by fitting the probit model to simulation result data. From the theoretical and simulation-based evaluations, it is suggested that the proposed simulation procedure provide more efficient and accurate sample size estimates than ordinary algorithm-based simulation procedure especially when estimated sample sizes are moderate to large, therefore it would help to dramatically reduce the computational time required to conduct clinical trial simulations.

In the fixed sample size design, when the null hypothesis for the primary endpoint is rejected at trial completion, secondary endpoints sometimes are tested at the full alpha level in a prespecifie...

Futility analyses provide a mechanism to stop a trial early because of low likelihood to achieve its efficacy objective. They are usually motivated by ethical and economic purposes, so that stopping a trial with poor efficacy could save patients and resources for other promising trials. There are various methods to address futility analyses in the literature but most focus on equally spaced interim looks. We consider a constrained optimization framework where the timing and the futility boundary are decided jointly to balance the risks between stopping trials which should continue, and continuing trials which should stop. The average sample size is used as a key parameter, which is evaluated under different degrees of power loss. Alternative objective functions and constraints are compared to assess the operating characteristics of the optimal futility scheme. Numerical results for single and multiple futility looks are provided. Supplementary materials for this article are available online.

Demonstration of analytical similarity between a reference product and a biosimilar product is required as part of the biosimilarity approval process. A statistical two one-sided test (TOST) based on the difference of means proposed in the literature and recommended by the FDA for Tier 1 quality attributes is demonstrated to have a Type I error rate greater than the specified level, which cannot be corrected by increasing sample size. In this article, an alternative TOST based on the effect size is demonstrated to maintain the desired Type I error rate, and in some situations, provide greater power than the TOST based on the mean difference. Results are demonstrated both analytically and with computer simulations. An example with calculations is provided in the article. Supplementary materials for this article are avalilable online.

Univariate isotonic regression (IR) has been used for nonparametric estimation in dose–response and dose-finding studies. One undesirable property of IR is the prevalence of piecewise-constant stretches in its estimates, whereas the dose–response function is usually assumed to be strictly increasing. We propose a simple modification to IR, called centered isotonic regression (CIR). CIR's estimates are strictly increasing in the interior of the dose range. In the absence of monotonicity violations, CIR and IR both return the original observations. Numerical examination indicates that for sample sizes typical of dose–response studies and with realistic dose–response curves, CIR provides a substantial reduction in estimation error compared with IR when monotonicity violations occur. We also develop analytical interval estimates for IR and CIR, with good coverage behavior. An R package implements these point and interval estimates. Supplementary materials for this article are available online.

We introduce a “backward” Bayesian method to assist sponsors formulating early phase Go/No-Go criteria based on the ultimate efficacy or safety target, which is usually clearer for Phase 3. Derived from the definition of success for Phase 3, involving prior information and cost of later phases, this work presents the quantitative relationships among the following factors: previous and current study results, study designs (e.g., sample size, duration, or dose), true effect, target probability of success (PoS), expected financial loss, expected probability of terminating a potentially successful asset. An example is given to demonstrate how to accomplish these objectives for an exponential model describing the trajectory of weight loss. The expected loss and the probability of terminating a valuable compound are plotted against a range of criteria. The sponsors can then optimize the Go/No-Go criteria based on their tolerance for their objectives. This method can also be generalized to other nonlinea...

In many fields of research variables are often both continuous and restricted to be positive. We analyzed 70 endpoints that contained continuous and positive dependent variables from 6 clinical and 3 preclinical trials. The impact of data transformation and adjustment for baseline on the fit of the model was studied. On average, including baseline as a covariate decreases the necessary sample size to achieve a particular precision in the treatment effect estimate by about 70% as compared to a model that ignores baseline, or by 20% to 33% as compared to models that only adjust for baseline in the response. Additionally, log transformation of the endpoint (either the direct outcome or a baseline-adjusted outcome) appears to decrease the sample size needed on average by 20% to 35%. We draw three conclusions from this work. First, if a baseline is available, use of baseline as a covariate should always be undertaken. Second, although we recommend exploration of data from previous studies, percent chan...

In linear regression models, covariate-adjusted analysis is not expected to change the estimates of the treatment effect in the clinical trials with randomized treatment assignment but rather to increase the precision of the estimates. However, the covariate-adjusted treatment effect estimates are generally not equivalent to the unadjusted estimates in logistic regression analysis for binary clinical trial data. In this article, we report the results of a simulation study conducted to quantify the magnitude of difference between the estimands underlying the two estimators in logistic regression. The simulation results demonstrated that both unadjusted and adjusted analyses preserved Type I error at the nominal level. The covariate-adjusted analysis produced unbiased, larger treatment effect estimates, larger standard error, and increased power comparedwith the unadjusted analysiswhen the sample sizewas large. The unadjusted analysis resulted in biased estimates of treatment effect. Analysis results for five phase 3 diabetes trials of the same compound were consistent with the simulation findings. Therefore, covariate-adjusted analysis is recommended for evaluating binary outcomes in clinical data.

It is critical to use a precise estimate of treatment effect when drawing conclusions, evaluating benefit/risk, or designing a new study. Using data from all sources in an integrated data analysis/meta-analysis will help us move closer to meeting this need. Depending on the data sources and objectives, there are many approaches for integrated analyses. These include network meta-analysis, multivariate meta-analysis, model-based meta-analysis as well as methods of borrowing historical data. In this article, we discuss these methods with details for implementation and interpretation. In addition, we consider information adaptive repeated cumulative meta-analyses. We also discuss how to apply three approaches that take into account the variability of the overall treatment effect estimate obtained through integrated analysis to determine sample size for a new trial. Some computation and simulation results are provided.

In study designs for randomized clinical trials with a survival endpoint, the log-rank test is commonly used with the treatment effect under a proportional hazards assumption. Recently, treatment effects in cancer immunotherapy trials have exhibited a delayed effect pattern with late separation of survival curves, raising challenges to the use of conventional study design hypotheses and analysis. In particular, when a trial with interim analyses is designed using a group sequential method, the expected treatment effect from a log-rank test statistic varies across analysis times and differs from the parameter specified under the alternative hypothesis. In this article, we present statistical analytical work that formulates a design including interim analyses with a survival endpoint under a delayed treatment effect alternative. Closed-form solutions are provided for calculating power and sample size over varying study/follow-up times for the group sequential, delayed treatment effect design. The an...

ASTM Standard E2810 provides a methodology for establishing confidence in passing the USP Uniformity of Dosage Units (UDU) test, and provides acceptance limits for sample means and standard deviations that can be used as elements of lot release. These acceptance limits are quite conservative, however, due to both the nature and shape of an inverted triangular joint confidence region for the lot mean and standard deviation. We obtain improved (wider) acceptance limits by using a Bayesian approach that focuses on the posterior distribution of the probability of passing the USP UDU test. The Bayesian approach has good sampling properties, and the improvement in acceptance limits can be considerable. For example, for sample size of 10 units and sample means between 97 and 103, the Bayesian approach results in acceptance limits for sample standard deviations that are at least 25% greater than those in E2810. The impact of the improved acceptance limits is illustrated with operating characteristic c...

The underlying statistical exercise in some noninferiority studies is to test a composite null hypothesis of a nonzero difference in the means of two normally distributed populations. These noninferiority studies are often analyzed under equal variances assumption. Unequal variances frequently occur in practice. In this article, we propose a noncentral t-distribution-based generalized test for proving noninferiority in the presence of unequal variances. We provide a simple SAS IML code to calculate the new generalized p-value. We compare the new generalized p-value with that of the Student's central t-distribution-based Tsui–Weerahandi generalized test. We conclude that Tsui–Weerahandi generalized test is conservative.

The objective of pulse oximetry studies is to demonstrate that paired repeated measurements from a pulse oximeter (SpO2) and a co-oximeter (SaO2) agree sufficiently with respect to a prespecified agreement score (ARMS). The current methods for analyzing pulse oximetry method comparison studies are mostly descriptive. Sample size calculations in the context of a specific test of statistical hypothesis applied to the ARMS have been reported. The purpose of this article is to further study the distributional properties of the ARMS and recommend a test appropriate to this agreement score. Five tests are considered: Z-Wald test, Z-score test, chi-squared (score) test, chi-squared (Wald) test, and “Exact” test. We propose and evaluate via simulation studies a robust method for testing the hypothesis of the accuracy on a pulse oximeter. Overall, the Z-score test is superior to the other four tests, resulting in estimated α-levels that are closest to the nominal α-level. The chi-squared (Wald) test is liberal and...

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 In...

We discuss the decision-making frameworks for clinical trials with multiple co-primary endpoints in a group-sequential setting. The decision-making frameworks can account for flexibilities, such as a varying number of analyses, equally or unequally spaced increments of information, and fixed or adaptive Type I error allocation among endpoints. The frameworks can provide efficiency, that is, potentially fewer trial participants, than the fixed sample size designs. We investigate the operating characteristics of the decision-making frameworks and provide guidance on constructing efficient group-sequential strategies in clinical trials with multiple co-primary endpoints.

For designing single-arm phase II trials with time-to-event endpoints, a sample size formula is derived for the modified one-sample log-rank test under the proportional hazards model. The derived formula enables new methods for designing trials that allow a flexible choice of the underlying survival distribution. Simulation results showed that the proposed formula provides an accurate estimation of sample size. The sample size calculation has been implemented in an R function for the purpose of trial design. Supplementary materials for this article are available online.

In this article, we consider sample size calculations for combination drugs of two monotherapies that each has only one approved dose level. We modify the method of Laska and Meiner by employing the asymptotic joint distribution of test statistics to derive the power function and using unequal allocation to minimize the total sample sizes. Two cases are investigated. In the first case, each monotherapy has the same indication. A heuristic method, the method of Laska and Meiner, and the proposed method are compared in terms of the total sample sizes. We show that the proposed method produces the smallest total sample sizes. In the second case, each monotherapy has a different indication. While the method of Laska and Meiner cannot be applied in this case, we show that the proposed method can be employed and that it produces smaller total sample sizes than a heuristic method.

In this paper, three non-parametric test statistics are proposed to design single-arm phase II group sequential trials for monitoring survival probability. The small-sample properties of these test statistics are studied through simulations. Sample size formulas are derived for the fixed sample test. The Brownian motion property of the test statistics allowed us to develop a flexible group sequential design using a sequential conditional probability ratio test procedure (Xiong, 1995). An example is given to illustrate the trial design by using the proposed method.

This article develops methods of statistical monitoring of clinical trials with multiple co-primary endpoints, where success is defined as meeting both endpoints simultaneously. In practice, a group sequential design (GSD) method is used to stop trials early for promising efficacy, and conditional power (CP) is used for futility stopping rules. In this article, we show that stopping boundaries for the GSD with multiple co-primary endpoints should be the same as those for studies with single endpoints. Lan and Wittes proposed the B-value tool to calculate the CP of single endpoint trials and we extend this tool to calculate the CP for studies with multiple co-primary endpoints. We consider the cases of two-arm studies with co-primary normal and provide an example of implementation with simulated trial. A fixed-weight sample size reestimation approach based on CP is introduced.

With the advent of novel cancer therapies targeting molecular pathways or the immune system, time-to-event variables evaluated at longer times (e.g., 6 or 12 months) are increasingly used as the primary endpoint in randomized Phase II trials. Two-stage “Simon” designs have often been used in single-group phase II oncology trials to reduce the expected sample size under the null hypothesis. We develop optimal two-stage and three-stage randomized two-group designs with analogous optimality criteria. Accurate implementation of the Simon design can be difficult, and the difficulty increases with a longer-term endpoint, because it requires complete information from each accrued patient. Earlier work of Case and Morgan generalized the Simon design so it can be implemented without an accrual pause. We derive optimal designs that allow a brief preplanned pause (⩾ 0) in accrual before interim analyses to more efficiently use data from the accrued patients, and to allow focused data collection before the interim an...

We consider situations where a drug developer gets access to additional financial resources when a promising result has been observed in a pre-planned interim analysis during a clinical trial which should lead to the registration of the drug. First the option that the drug developer completely puts the additional resources into increasing the second stage sample size has been investigated. If investors invest the more the larger the observed interim effect, this may not be a reasonable strategy: Then additional sample sizes are applied when the conditional power is already very large and hardly any impact on the overall power can be expected. Nevertheless, further reducing the type II error rate in promising situations may be of interest for a drug developer. In a second step, sample size was based on a utility function including the reward of registration (which was allowed to depend on the observed effect size at the end of the trial) and sampling costs. Utility as a function of the sample size may have more than one local maximum, one of them at the lowest per group sample size. For small effects an optimal strategy could be to apply the smallest sample size accepted by regulators.

The most frequently seen adaptation of a clinical trial design in regulatory submission is adaptation of statistical information, for example, sample size or number of events. Such adaptation can be based solely on the clinical endpoint of interest or early biomarker. In this article, we articulate the technical merits and discuss challenges when statistical information based solely on the clinical endpoint is used as the design aspect for adaptation. We present the interplay between the weighted and unweighted adaptive Z-statistics with, versus without, additional criteria. We contrast Fisher's p-value product test and the modified version to the adaptive weighted Z-test to elucidate a way to minimize the potential heterogeneity of the observed treatment effects between stages in a two-stage adaptive design setting. Another framework pertains when one is using shorter-term biomarker data for adaptation of statistical information, where the final analysis is to test the null hypothesis of no treatment eff...

In long-term single cohort safety studies, the incremental incidence rate is a useful concept in evaluating risks related to adverse events of medical interest (EMI). We propose estimates for the incremental (say annual) incidence rates in the presence of independent random right censoring and describe their statistical properties. More importantly, we propose tests for the null hypothesis that the incremental incidence rates remain constant during the study against various alternative hypotheses, such as rates changing over the years or having a specific trend, say monotonic increasing or decreasing. In contrast with the existing methods that test constant hazard rates, the proposed method allows nonconstant hazard within each year; therefore, it is especially useful for EMIs that have a seasonal occurrence pattern. This article explores and characterizes properties of these tests under various alternatives via theory and simulations. A numerical example based on a real long-term single cohort safety stu...

Clinical trials on early stage Alzheimer’s disease (AD) are reaching a bottleneck because none of the current disease markers changes appreciably early in the disease process and therefore a huge sample is required to adequately power such trials. We propose a method to combine multiple markers so that the longitudinal rate of progression can be improved. The criterion is to maximize the probability that the combined marker will be decreased over time (assuming a negative mean slope for each marker). We propose estimates to the weights of markers in the optimum combination and a confidence interval estimate to the combined rate of progression through the maximum likelihood estimates and a bootstrap procedure. We conduct simulations to assess the performance of our estimates and compare our approach with the first principal component from a principal component analysis. The proposed method is applied to a real world sample of individuals with preclinical AD to combine measures from two cognitive domains. The combined cognitive marker is finally used to design future clinical trials on preclinical AD, demonstrating a significant improvement in reducing the sample sizes needed to power such trials when compared with individual markers alone.

In this article, we consider hypothesis testing and computationally feasible sample size determination for bivariate binary outcomes. The hypotheses are formulated as one-sided polygons, which allow flexible trade-offs between the two outcomes. Parameters are estimated by maximizing the likelihood. Hypothesis testing for each linear constraint is performed by the Wald, score, likelihood ratio, and exact tests. The overall hypothesis is then tested using either the union-intersection or intersection-union method. We propose methods to calculate both exact power functions and asymptotic power functions. Finite sample behaviors are evaluated by numerical examples. A data example is used for illustration.

In designing a clinical trial for comparing two or more treatments with respect to overall survival (OS), a proportional hazards assumption is commonly made. However, in many cancer clinical trials, patients pass through various disease states prior to death and because of this may receive treatments other than originally assigned. For example, patients may crossover from the control treatment to the experimental treatment at progression. Even without crossover, the survival pattern after progression may be very different than the pattern prior to progression. The proportional hazards assumption will not hold in these situations and the design power calculated on this assumption will not be correct. In this paper we describe a simple and intuitive multi-state model allowing for progression, death before progression, post-progression survival and crossover after progression and apply this model to the design of clinical trials for comparing the OS of two treatments. For given values of the parameters of the multi-state model, we simulate the required number of deaths to achieve a specified power and the distribution of time required to achieve the requisite number of deaths. The results may be quite different from those derived using the usual PH assumption.

Most phase II cancer trials evaluate total response (TR) rate using a single-arm open-label design. Simon's two-stage design minimizes either the expected or the total number of patients. In this article, we investigate the distribution of time to reaching trial conclusion (TRTC) of such two-stage trials, which considers the accrual rate, time to TR, futility, and efficacy decision rules. We develop a Monte Carlo algorithm for the distribution of TRTC for both one- and two-stage phase II designs. We further propose an optimal design that balances the needs of concluding the trial within the required time period and minimizing the expected sample size. Generalized recursive formulas are derived analytically in the Appendix, which are theoretically interesting but not recommended for practical use. In conclusion, a consideration of the TRTC can lead to an optimal design with a minimum increase in the expected number of patients but an increased confidence in reaching study conclusions within the required time.