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.     

Increasing the sample size at interim for a two‐sample experiment without Type I error inflation

For the case of a one‐sample experiment with known variance σ²=1, it has been shown that at interim analysis the sample size (SS) may be increased by any arbitrary amount provided: (1) The conditional power (CP) at interim is ?50% and (2) there can be no decision to decrease the SS (stop the trial early). In this paper we verify this result for the case of a two‐sample experiment with proportional SS in the treatment groups and an arbitrary common variance. Numerous authors have presented the formula for the CP at interim for a two‐sample test with equal SS in the treatment groups and an arbitrary common variance, for both the one‐ and two‐sided hypothesis tests. In this paper we derive the corresponding formula for the case of unequal, but proportional SS in the treatment groups for both one‐sided superiority and two‐sided hypothesis tests. Finally, we present an SAS macro for doing this calculation and provide a worked out hypothetical example. In discussion we note that this type of trial design trades the ability to stop early (for lack of efficacy) for the elimination of the Type I error penalty. The loss of early stopping requires that such a design employs a data monitoring committee, blinding of the sponsor to the interim calculations, and pre‐planning of how much and under what conditions to increase the SS and that this all be formally written into an interim analysis plan before the start of the study. Copyright © 2009 John Wiley & Sons, Ltd.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Optimal sample size planning for the Wilcoxon–Mann–Whitney and van Elteren tests under cost constraints

Sampling cost is a crucial factor in sample size planning, particularly when the treatment group is more expensive than the control group. To either minimize the total cost or maximize the statistical power of the test, we used the distribution-free Wilcoxon–Mann–Whitney test for two independent samples and the van Elteren test for randomized block design, respectively. We then developed approximate sample size formulas when the distribution of data is abnormal and/or unknown. This study derived the optimal sample size allocation ratio for a given statistical power by considering the cost constraints, so that the resulting sample sizes could minimize either the total cost or the total sample size. Moreover, for a given total cost, the optimal sample size allocation is recommended to maximize the statistical power of the test. The proposed formula is not only innovative, but also quick and easy. We also applied real data from a clinical trial to illustrate how to choose the sample size for a randomized two-block design. For nonparametric methods, no existing commercial software for sample size planning has considered the cost factor, and therefore the proposed methods can provide important insights related to the impact of cost constraints.     

Sample‐size calculations for multi‐group comparison in population pharmacokinetic experiments

This paper describes an approach for calculating sample size for population pharmacokinetic experiments that involve hypothesis testing based on multi‐group comparison detecting the difference in parameters between groups under mixed‐effects modelling. This approach extends what has been described for generalized linear models and nonlinear population pharmacokinetic models that involve only binary covariates to more complex nonlinear population pharmacokinetic models. The structural nonlinear model is linearized around the random effects to obtain the marginal model and the hypothesis testing involving model parameters is based on Wald's test. This approach provides an efficient and fast method for calculating sample size for hypothesis testing in population pharmacokinetic models. The approach can also handle different design problems such as unequal allocation of subjects to groups and unbalanced sampling times between and within groups. The results obtained following application to a one compartment intravenous bolus dose model that involved three different hypotheses under different scenarios showed good agreement between the power obtained from NONMEM simulations and nominal power. Copyright © 2009 John Wiley & Sons, Ltd.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

Efficient sample size allocation with cost constraints for heterogeneous-variance group comparison

When conducting research with controlled experiments, sample size planning is one of the important decisions that researchers have to make. However, current methods do not adequately address this issue with regard to variance heterogeneity with some cost constraints for comparing several treatment means. This paper proposes a sample size allocation ratio in the fixed-effect heterogeneous analysis of variance when group variances are unequal and in cases where the sampling and/or variable cost has some constraints. The efficient sample size allocation is determined for the purpose of minimizing total cost with a designated power or maximizing the power with a given total cost. Finally, the proposed method is verified by using the index of relative efficiency and the corresponding total cost and the total sample size needed. We also apply our method in a pain management trial to decide an efficient sample size. Simulation studies also show that the proposed sample size formulas are efficient in terms of statistical power. SAS and R codes are provided in the appendix for easy application.     

Determination of hazard ratio for progression‐free survival considering the tumor assessment schedule in sample size calculation

Progression‐free survival is recognized as an important endpoint in oncology clinical trials. In clinical trials aimed at new drug development, the target population often comprises patients that are refractory to standard therapy with a tumor that shows rapid progression. This situation would increase the bias of the hazard ratio calculated for progression‐free survival, resulting in decreased power for such patients. Therefore, new measures are needed to prevent decreasing the power in advance when estimating the sample size. Here, I propose a novel calculation procedure to assume the hazard ratio for progression‐free survival using the Cox proportional hazards model, which can be applied in sample size calculation. The hazard ratios derived by the proposed procedure were almost identical to those obtained by simulation. The hazard ratio calculated by the proposed procedure is applicable to sample size calculation and coincides with the nominal power. Methods that compensate for the lack of power due to biases in the hazard ratio are also discussed from a practical point of view.     

Power and sample size calculation for paired right-censored data based on survival copula models

Sample size determination is essential during the planning phases of clinical trials. To calculate the required sample size for paired right-censored data, the structure of the within-paired correlations needs to be pre-specified. In this article, we consider using popular parametric copula models, including the Clayton, Gumbel, or Frank families, to model the distribution of joint survival times. Under each copula model, we derive a sample size formula based on the testing framework for rank-based tests and non-rank-based tests (i.e., logrank test and Kaplan–Meier statistic, respectively). We also investigate how the power or the sample size was affected by the choice of testing methods and copula model under different alternative hypotheses. In addition to this, we examine the impacts of paired-correlations, accrual times, follow-up times, and the loss to follow-up rates on sample size estimation. Finally, two real-world studies are used to illustrate our method and R code is available to the user.     

Shift in re‐randomization distribution with conditional randomization test

Proschan, Brittain, and Kammerman made a very interesting observation that for some examples of the unequal allocation minimization, the mean of the unconditional randomization distribution is shifted away from 0. Kuznetsova and Tymofyeyev linked this phenomenon to the variations in the allocation ratio from allocation to allocation in the examples considered in the paper by Proschan et al. and advocated the use of unequal allocation procedures that preserve the allocation ratio at every step. In this paper, we show that the shift phenomenon extends to very common settings: using conditional randomization test in a study with equal allocation. This phenomenon has the same cause: variations in the allocation ratio among the allocation sequences in the conditional reference set, not previously noted. We consider two kinds of conditional randomization tests. The first kind is the often used randomization test that conditions on the treatment group totals; we describe the variations in the conditional allocation ratio with this test on examples of permuted block randomization and biased coin randomization. The second kind is the randomization test proposed by Zheng and Zelen for a multicenter trial with permuted block central allocation that conditions on the within‐center treatment totals. On the basis of the sequence of conditional allocation ratios, we derive the value of the shift in the conditional randomization distribution for specific vector of responses and the expected value of the shift when responses are independent identically distributed random variables. We discuss the asymptotic behavior of the shift for the two types of tests. Copyright © 2013 John Wiley & Sons, Ltd.     

A convenient formula for sample size calculations in clinical trials with multiple co-primary continuous endpoints

The clinical efficacy of a new treatment may often be better evaluated by two or more co-primary endpoints. Recently, in pharmaceutical drug development, there has been increasing discussion regarding establishing statistically significant favorable results on more than one endpoint in comparisons between treatments, which is referred to as a problem of multiple co-primary endpoints. Several methods have been proposed for calculating the sample size required to design a trial with multiple co-primary correlated endpoints. However, because these methods require users to have considerable mathematical sophistication and knowledge of programming techniques, their application and spread may be restricted in practice. To improve the convenience of these methods, in this paper, we provide a useful formula with accompanying numerical tables for sample size calculations to design clinical trials with two treatments, where the efficacy of a new treatment is demonstrated on continuous co-primary endpoints. In addition, we provide some examples to illustrate the sample size calculations made using the formula. Using the formula and the tables, which can be read according to the patterns of correlations and effect size ratios expected in multiple co-primary endpoints, makes it convenient to evaluate the required sample size promptly.     

Optimal treatment allocation and study duration for trials with discrete-time survival endpoints

Studies on event occurrence may be conducted in experiments, where one or more treatment groups are compared to a control group. Most of the randomized trials are designed with equally sized groups, but this design is not always the best one. The statistical power of the study may be larger with unequal sample sizes, and researchers may want to place more participants in one group relative to the other due to resource constraints or costs. The optimal designs for discrete-time survival endpoints in trials with two groups, where different proportions of subjects in the experimental group are taken into account, can be studied using the generalized linear model. Applying a cost function, the optimal combination of the number of subjects and periods in the study and the optimal allocation ratio can be found. It is observed that the ratio of the recruitment costs in both groups, the ratio of the recruitment cost in the control group to the cost of obtaining a measurement, the size of the treatment effect, and the shape of the survival distribution have the greatest influence on the optimal design.     

Sample Size Calculation and Blinded Sample Size Recalculation in Clinical Trials Where the Treatment Effect is Measured by the Relative Risk

In clinical trials with binary endpoints, the required sample size does not depend only on the specified type I error rate, the desired power and the treatment effect but also on the overall event rate which, however, is usually uncertain. The internal pilot study design has been proposed to overcome this difficulty. Here, nuisance parameters required for sample size calculation are re-estimated during the ongoing trial and the sample size is recalculated accordingly. We performed extensive simulation studies to investigate the characteristics of the internal pilot study design for two-group superiority trials where the treatment effect is captured by the relative risk. As the performance of the sample size recalculation procedure crucially depends on the accuracy of the applied sample size formula, we firstly explored the precision of three approximate sample size formulae proposed in the literature for this situation. It turned out that the unequal variance asymptotic normal formula outperforms the other two, especially in case of unbalanced sample size allocation. Using this formula for sample size recalculation in the internal pilot study design assures that the desired power is achieved even if the overall rate is mis-specified in the planning phase. The maximum inflation of the type I error rate observed for the internal pilot study design is small and lies below the maximum excess that occurred for the fixed sample size design.     

Sample size calculation for the one‐sample log‐rank test

In this paper, an exact variance of the one‐sample log‐rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population. Copyright © 2014 John Wiley & Sons, Ltd.     

Distribution of the two‐sample t‐test statistic following blinded sample size re‐estimation

We consider the blinded sample size re‐estimation based on the simple one‐sample variance estimator at an interim analysis. We characterize the exact distribution of the standard two‐sample t‐test statistic at the final analysis. We describe a simulation algorithm for the evaluation of the probability of rejecting the null hypothesis at given treatment effect. We compare the blinded sample size re‐estimation method with two unblinded methods with respect to the empirical type I error, the empirical power, and the empirical distribution of the standard deviation estimator and final sample size. We characterize the type I error inflation across the range of standardized non‐inferiority margin for non‐inferiority trials, and derive the adjusted significance level to ensure type I error control for given sample size of the internal pilot study. We show that the adjusted significance level increases as the sample size of the internal pilot study increases. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of sample size formulae for 2 × 2 cross‐over designs applied to bioequivalence studies

We consider the comparison of two formulations in terms of average bioequivalence using the 2 × 2 cross‐over design. In a bioequivalence study, the primary outcome is a pharmacokinetic measure, such as the area under the plasma concentration by time curve, which is usually assumed to have a lognormal distribution. The criterion typically used for claiming bioequivalence is that the 90% confidence interval for the ratio of the means should lie within the interval (0.80, 1.25), or equivalently the 90% confidence interval for the differences in the means on the natural log scale should be within the interval (?0.2231, 0.2231). We compare the gold standard method for calculation of the sample size based on the non‐central t distribution with those based on the central t and normal distributions. In practice, the differences between the various approaches are likely to be small. Further approximations to the power function are sometimes used to simplify the calculations. These approximations should be used with caution, because the sample size required for a desirable level of power might be under‐ or overestimated compared to the gold standard method. However, in some situations the approximate methods produce very similar sample sizes to the gold standard method. Copyright © 2005 John Wiley & Sons, Ltd.     

Study design of single‐arm phase II immunotherapy trials with long‐term survivors and random delayed treatment effect

In the traditional study design of a single‐arm phase II cancer clinical trial, the one‐sample log‐rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long‐term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single‐arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one‐sample log‐rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log‐rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.     

Design and analysis of a 3‐arm noninferiority trial with a prespecified margin for the hazard ratio

A 3‐arm trial design that includes an experimental treatment, an active reference treatment, and a placebo is useful for assessing the noninferiority of an experimental treatment. The inclusion of a placebo arm enables the assessment of assay sensitivity and internal validation, in addition to the testing of the noninferiority of the experimental treatment compared with the reference treatment. In 3‐arm noninferiority trials, various statistical test procedures have been considered to evaluate the following 3 hypotheses: (i) superiority of the experimental treatment over the placebo, (ii) superiority of the reference treatment over the placebo, and (iii) noninferiority of the experimental treatment compared with the reference treatment. However, hypothesis (ii) can be insufficient and may not accurately assess the assay sensitivity for the noninferiority of the experimental treatment compared with the reference treatment. Thus, demonstrating that the superiority of the reference treatment over the placebo is greater than the noninferiority margin (the nonsuperiority of the reference treatment compared with the placebo) can be necessary. Here, we propose log‐rank statistical procedures for evaluating data obtained from 3‐arm noninferiority trials to assess assay sensitivity with a prespecified margin Δ. In addition, we derive the approximate sample size and optimal allocation required to minimize the total sample size and that of the placebo treatment sample size, hierarchically.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

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.     

Testing multiple primary endpoints in clinical trials with sample size adaptation

In this paper, we propose a design that uses a short‐term endpoint for accelerated approval at interim analysis and a long‐term endpoint for full approval at final analysis with sample size adaptation based on the long‐term endpoint. Two sample size adaptation rules are compared: an adaptation rule to maintain the conditional power at a prespecified level and a step function type adaptation rule to better address the bias issue. Three testing procedures are proposed: alpha splitting between the two endpoints; alpha exhaustive between the endpoints; and alpha exhaustive with improved critical value based on correlation. Family‐wise error rate is proved to be strongly controlled for the two endpoints, sample size adaptation, and two analysis time points with the proposed designs. We show that using alpha exhaustive designs greatly improve the power when both endpoints are effective, and the power difference between the two adaptation rules is minimal. The proposed design can be extended to more general settings. Copyright © 2015 John Wiley & Sons, Ltd.