Single-Arm Phase II Survival Trial Design Under the Proportional Hazards Model

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.     

Group Sequential Survival Trial Design and Monitoring Using the Log-Rank Test

For randomized group sequential survival trial designs with unbalanced treatment allocation, the widely used Schoenfeld formula is inaccurate, and the commonly used information time as the ratio of number of events at interim look to the number of events at the end of trial can be biased. In this article, a sample size formula for the two-sample log-rank test under the proportional hazards model is proposed that provides more accurate sample size calculation for unbalanced survival trial designs. Furthermore, a new information time is introduced for the sequential survival trials such that the new information time is more accurate than the traditional information time when the allocation of enrollments is unbalanced in groups. Finally, we demonstrate the monitoring process using the sequential conditional probability ratio test and compare it with two other well-known group sequential procedures. An example is given to illustrate unbalanced survival trial design using available software. Supplementary materials for this article are available online.     

Sample size computation for two-sample noninferiority log-rank test

When an experimental therapy is less extensive, less toxic, or less expensive than a standard therapy, we may want to prove that the former is not worse than the latter through a noninferiority trial. In this article, we discuss a modification of the log-rank test for noninferiority trials with survival endpoint and propose a sample size formula that can be used in designing such trials. Performance of our sample size formula is investigated through simulations. Our formula is applied to design a real clinical trial.     

Sample Size Computation for Two-Sample Noninferiority Log-Rank Test

When an experimental therapy is less extensive, less toxic, or less expensive than a standard therapy, we may want to prove that the former is not worse than the latter through a noninferiority trial. In this article, we discuss a modification of the log-rank test for noninferiority trials with survival endpoint and propose a sample size formula that can be used in designing such trials. Performance of our sample size formula is investigated through simulations. Our formula is applied to design a real clinical trial.     

Sample size calculation for testing differences between cure rates with the optimal log-rank test

In this article, sample size calculations are developed for use when the main interest is in the differences between the cure rates of two groups. Following the work of Ewell and Ibrahim, the asymptotic distribution of the weighted log-rank test is derived under the local alternative. The optimal log-rank test under the proportional distributions alternative is discussed, and sample size formulas for the optimal and standard log-rank tests are derived. Simulation results show that the proposed formulas provide adequate sample size estimation for trial designs and that the optimal log-rank test is more efficient than the standard log-rank test, particularly when both cure rates and percentages of censoring are small.     

Optimal two-stage log-rank test for randomized phase II clinical trials

Randomized controlled clinical trials are conducted to determine whether a new treatment is safe and efficacious compared to a standard therapy. We consider randomized clinical trials with right censored time to event endpoint, called survival time here. The two-sample log-rank test is popularly used to test if the experimental therapy has a longer survival distribution than the control therapy or not. We consider an early stopping for futility only or for both futility and efficacy. For planning such clinical trials, this article presents two-stage designs that are optimal in the sense that either the maximal sample size or the expected sample size when the experimental therapy is futile or superior is minimized under the given type I and II error rates. Optimal designs for a range of design parameters are tabulated and evaluated using simulations.     

临床随访研究中样本含量的估计

目的:介绍临床随访研究中生存分析资料的log-rank检验所需样本含量的估计法.方法:以离散性Markov链拟合生存过程,据此计算log-rank统计量的数学期望和方差,导出样本含量估计公式.结果:实例分析表明,该法能较好反映实际情况,应用灵活.结论:本法是一种有效、可行的样本含量估计法,值得推荐.     

Sample size considerations for historical control studies with survival outcomes

Historical control trials (HCTs) are frequently conducted to compare an experimental treatment with a control treatment from a previous study, when they are applicable and favored over a randomized clinical trial (RCT) due to feasibility, ethics and cost concerns. Makuch and Simon developed a sample size formula for historical control (HC) studies with binary outcomes, assuming that the observed response rate in the HC group is the true response rate. This method was extended by Dixon and Simon to specify sample size for HC studies comparing survival outcomes. For HC studies with binary and continuous outcomes, many researchers have shown that the popular Makuch and Simon method does not preserve the nominal power and type I error, and suggested alternative approaches. For HC studies with survival outcomes, we reveal through simulation that the conditional power and type I error over all the random realizations of the HC data have highly skewed distributions. Therefore, the sampling variability of the HC data needs to be appropriately accounted for in determining sample size. A flexible sample size formula that controls arbitrary percentiles, instead of means, of the conditional power and type I error, is derived. Although an explicit sample size formula with survival outcomes is not available, the computation is straightforward. Simulations demonstrate that the proposed method preserves the operational characteristics in a more realistic scenario where the true hazard rate of the HC group is unknown. A real data application of an advanced non-small cell lung cancer (NSCLC) clinical trial is presented to illustrate sample size considerations for HC studies in comparison of survival outcomes.     

Single-arm Phase II cancer survival trial designs

The current practice for designing single-arm Phase II trials with time-to-event endpoints is limited to using either a maximum likelihood estimate test under the exponential model or a naive approach based on dichotomizing the event time at a landmark time point. A trial designed under the exponential model may not be reliable, and the naive approach is inefficient. The modified one-sample log-rank test statistic proposed in this article fills the void. In general, the proposed test can be used to design single-arm Phase II survival trials under any parametric survival distribution. Simulation results showed that it preserves type I error well and provides adequate power for Phase II cancer trial designs with time-to-event endpoints.     

Sample size calculation for simulation-based multiple-testing procedures

In this article, we present a simple method to calculate sample size and power for a simulation-based multiple testing procedure which gives a sharper critical value than the standard Bonferroni method. The method is especially useful when several highly correlated test statistics are involved in a multiple-testing procedure. The formula for sample size calculation will be useful in designing clinical trials with multiple endpoints or correlated outcomes. We illustrate our method with a quality-of-life study for patients with early stage prostate cancer. Our method can also be used for comparing multiple independent groups.     

Sample Size Calculation for Simulation-Based Multiple-Testing Procedures

In this article, we present a simple method to calculate sample size and power for a simulation-based multiple testing procedure which gives a sharper critical value than the standard Bonferroni method. The method is especially useful when several highly correlated test statistics are involved in a multiple-testing procedure. The formula for sample size calculation will be useful in designing clinical trials with multiple endpoints or correlated outcomes. We illustrate our method with a quality-of-life study for patients with early stage prostate cancer. Our method can also be used for comparing multiple independent groups.     

A NONPARAMETRIC PROCEDURE OF THE SAMPLE SIZE DETERMINATION FOR SURVIVAL RATE TEST

Inthesurvivalstudiesofchronicdiseases,themostfrequentlyusedmethodsofdataanalysisarethenonparametricestimationsandcomparisonsofsurvivalratesatdifferenttimesbuttherehavebeennomatchedmethodsfordeterminingtherequiredsamplesizesofar.Insteadhavebeenoftenusedtheparametricmethodsbasedontheassumptionofexponentialdistributions[1～7]suchthattherelevantsamplesizesmaynotbesatisfiedwiththeprescribedpower.Thispaperreportsanonparametricprocedureofthesamplesizedeterminationforsurvivalratetest.1　SurvivalRateT…     

Notes on Testing Noninferiority in Ordinal Data Under the Parallel Groups Design

When testing the noninferiority of an experimental treatment to a standard (or control) treatment in a randomized clinical trial (RCT), we may come across the outcomes of patient response on an ordinal scale. We focus our discussion on testing noninferiority in ordinal data for an RCT under the parallel groups design. We develop simple test procedures based on the generalized odds ratio (GOR). We note that these test procedures not only can account for the information on the order of ordinal responses without assuming any specific parametric structural model, but also can be independent of any arbitrarily subjective scoring system. We further develop sample size determination based on the test procedure using the GOR. We apply Monte Carlo simulation to evaluate the performance of these test procedures and the accuracy of sample size calculation formula proposed here in a variety of situations. Finally, we employ the data taken from a trial comparing once-daily gatifloxican with three-times-daily co-amoxiclav in the treatment of community-acquired pneumonia to illustrate the use of these test procedures and sample size calculation formula.     

Nonparametric Sample Size Estimation for Sensitivity and Specificity with Multiple Observations per Subject

We propose a sample size calculation approach for the estimation of sensitivity and specificity of diagnostic tests with multiple observations per subjects. Many diagnostic tests such as diagnostic imaging or periodontal tests are characterized by the presence of multiple observations for each subject. The number of observations frequently varies among subjects in diagnostic imaging experiments or periodontal studies. Nonparametric statistical methods for the analysis of clustered binary data have been recently developed by various authors. In this paper, we derive a sample size formula for sensitivity and specificity of diagnostic tests using the sign test while accounting for multiple observations per subjects. Application of the sample size formula for the design of a diagnostic test is discussed. Since the sample size formula is based on large sample theory, simulation studies are conducted to evaluate the finite sample performance of the proposed method. We compare the performance of the proposed sample size formula with that of the parametric sample size formula that assigns equal weight to each observation. Simulation studies show that the proposed sample size formula generally yields empirical powers closer to the nominal level than the parametric method. Simulation studies also show that the number of subjects required increases as the variability in the number of observations per subject increases and the intracluster correlation increases.     

A simple formula for sample size calculation in equivalence studies

Bioequivalence and clinical equivalence can be claimed based on the two one-sided test approach or the confidence interval approach. Consequently the power function of the equivalence test can be derived from either noncentral t-distribution or central t-distribution. The sample size is then determined from the power function either by numerical method or closed formulas. In this paper, we propose a simple formula for sample size calculation based on central t-distribution. The proposed formula has better properties than those currently available and it can be easily applied in all equivalence studies.     

Many-to-One Comparison After Sample Size Reestimation for Trials with Multiple Treatment Arms and Treatment Selection

Sample size reestimation (SSRE) provides a useful tool to change the sample size when an interim look reveals that the original sample size is inadequate. To control the overall type I error, for testing one hypothesis, several approaches have been proposed to construct a statistic so that its distribution is independent to the SSRE under the null hypothesis. We considered a similar approach for comparisons between multiple treatment arms and placebo, allowing the change of sample sizes in all arms depending on interim information. A construction of statistics similar to that for a single hypothesis test is proposed. When the changes of sample sizes in different arms are proportional, we show that one-step and stepwise Dunnett tests can be used directly on statistics constructed in the proposed way. The approach can also be applied to clinical trials with SSRE and treatment selection at interim. The proposed approach is evaluated with simulations under different situations.     

Statistical Monitoring of Clinical Trials: Fundamentals for Investigators,L. A. Moyé

We consider collecting the measurements of a gold standard and two methods with error from each site of subjects. We propose an asymptotic test statistic to compare the concordance rates of two methods with the gold standard and a closed-form sample size formula. Through simulations, we show that the test statistic accurately controls the type I error in small sample sizes, and the sample size formula accurately maintains the power in various settings. The proposed test statistic and the sample size formula can be easily modified for other indices for agreement such as sensitivity and specificity. A real eye study is taken as an example.     

Sample size for comparing correlated concordance rates

We consider collecting the measurements of a gold standard and two methods with error from each site of subjects. We propose an asymptotic test statistic to compare the concordance rates of two methods with the gold standard and a closed-form sample size formula. Through simulations, we show that the test statistic accurately controls the type I error in small sample sizes, and the sample size formula accurately maintains the power in various settings. The proposed test statistic and the sample size formula can be easily modified for other indices for agreement such as sensitivity and specificity. A real eye study is taken as an example.     

Sample size determination for testing equality in a cluster randomized trial with noncompliance

For administrative convenience or cost efficiency, we may often employ a cluster randomized trial (CRT), in which randomized units are clusters of patients rather than individual patients. Furthermore, because of ethical reasons or patient's decision, it is not uncommon to encounter data in which there are patients not complying with their assigned treatments. Thus, the development of a sample size calculation procedure for a CRT with noncompliance is important and useful in practice. Under the exclusion restriction model, we have developed an asymptotic test procedure using a tanh(-1)(x) transformation for testing equality between two treatments among compliers for a CRT with noncompliance. We have further derived a sample size formula accounting for both noncompliance and the intraclass correlation for a desired power 1 - β at a nominal α level. We have employed Monte Carlo simulation to evaluate the finite-sample performance of the proposed test procedure with respect to type I error and the accuracy of the derived sample size calculation formula with respect to power in a variety of situations. Finally, we use the data taken from a CRT studying vitamin A supplementation to reduce mortality among preschool children to illustrate the use of sample size calculation proposed here.     

A SIMPLE WAY TO ESTIMATE THE MEDIAN TIME AND COMPARE SURVIVAL DISTRIBUTIONS IN ANALGESIC TRIALS UNDER INFORMATIVE CENSORING

We discuss the problem of estimating the median time and comparison of survival curves when data are nonrandomly censored in analgesic trials. In these trials patients experience post-surgical pain at the time of randomization. Time to onset of analgesia is measured by patient-administered stopwatches. An effective analgesic is one for which the median time to onset is “short.” The study design allows patients to remedicate if their pain persists, and this remedication prior to pain relief censors the time-to-onset measures. The time to onset for patients who remedicate is nonrandomly censored. Assuming noninformative censoring yields misleading results with the Kaplan-Meier method (for estimation of median time) and the log-rank test (for comparison of survival curves). This assumption can also obscure the superior effect of an effective analgesic over an ineffective one. We propose a simple and intuitive way to handle the nonrandomly censored data in analgesic trials in order to (a) estimate the median time to pain relief and (b) compare the survival distributions between treatments. The method proposed is applied to data collected from an acute pain clinical trial, and the results are discussed.