Sample Size Calculation for Comparing Two Poisson or Negative Binomial Rates in Noninferiority or Equivalence Trials

Poisson and negative binomial models are frequently used to analyze count data in clinical trials. While several sample size calculation methods have recently been developed for superiority tests for these two models, similar methods for noninferiority and equivalence tests are not available. When a noninferiority or equivalence trial is designed to compare Poisson or negative binomial rates, an appropriate method is needed to estimate the sample size to ensure the trial is properly powered. In this article, several sample size calculation methods for noninferiority and equivalence tests have been derived based on Poisson and negative binomial models. All of these methods accounted for potential over-dispersion that commonly exists in count data obtained from clinical trials. The precision of these methods was evaluated using simulations. Supplementary materials for this article are available online.     

配对二项数据等效性／非劣效性评价的样本含量估计和假设检验

新药及医疗器械临床试验中，有时会涉及到两比较组采用配对设计获得的二项反应数据（配对二项数据）的等效性／非劣效性问题。两独立组率之间等效性／非劣效试验的样本含量估计及假设检验方法已较为成熟，但对于配对二项数据两组率之间的等效性／非劣效性试验的样本含量估计及假设检验方法还应用不多。本文介绍了一种渐进的基于约束极大似然估计的方法用于配对二项数据两组率之间的等效性／非劣效性试验的样本含量估计和假设检验，借助一个超声诊断仪临床试验的例子阐明了本方法的应用，还就有关实际问题进行了讨论。     

Clinical trials for authorized biosimilars in the European Union: a systematic review

AimIn 2006, Omnitrope (by Sandoz) was the first approved biosimilar in Europe. To date, 21 biosimilars for seven different biologics are on the market. The present study compared the clinical trials undertaken to obtain market authorization.MethodsWe summarized the findings of a comprehensive review of all clinical trials up to market authorization of approved biosimilars, using the European public assessment reports (EPARs) published by the European Medicines Agency (EMA). The features compared were, among others, the number of patients enrolled, the number of trials, the types of trial design, choice of endpoints and equivalence margins for pharmacokinetic (PK)/pharmacodynamic (PD) and phase III trials.ResultsThe variability between the clinical development strategies is high. Some differences are explainable by the characteristics of the product; if, for example, the PD marker can be assumed to predict the clinical outcome, no efficacy trials might be necessary. However, even for products with the same reference product, the sample size, endpoints and statistical models are not always the same.ConclusionsThere seems to be flexibility for sponsors regarding the decision as to how best to prove biosimilarity.     

Statistical Considerations for NonInferiority/Equivalence Trials in Vaccine Development

Noninferiority/equivalence designs are often used in vaccine clinical trials. The goal of these designs is to demonstrate that a new vaccine, or new formulation or regimen of an existing vaccine, is similar in terms of effectiveness to the existing vaccine, while offering such advantages as easier manufacturing, easier administration, lower cost, or improved safety profile. These noninferiority/equivalence designs are particularly useful in four common types of immunogenicity trials: vaccine bridging trials, combination vaccine trials, vaccine concomitant use trials, and vaccine consistency lot trials. In this paper, we give an overview of the key statistical issues and recent developments for noninferiority/equivalence vaccine trials. Specifically, we cover the following topics: (i) selection of study endpoints; (ii) formulation of the null and alternative hypotheses; (iii) determination of the noninferiority/equivalence margin; (iv) selection of efficient statistical methods for the statistical analysis of noninferiority/equivalence vaccine trials, with particular emphasis on adjustment for stratification factors and missing pre- or post-vaccination data; and (v) the calculation of sample size and power.     

Adjustment for unbalanced sample size for analytical biosimilar equivalence assessment

ABSTRACT

Large sample size imbalance is not uncommon in the biosimilar development. At the beginning of a product development, sample sizes of a biosimilar and a reference product may be limited. Thus, a sample size calculation may not be feasible. During the development stage, more batches of reference products may be added at a later stage to have a more reliable estimate of the reference variability. On the other hand, we also need a sufficient number of biosimilar batches in order to have a better understanding of the product. Those challenges lead to a potential sample size imbalance. In this paper, we show that large sample size imbalance may increase the power of the equivalence test in an unfavorable way, giving higher power for less similar products when the sample size of biosimilar is much smaller than that of the reference product. Thus, it is necessary to make some sample size imbalance adjustments to motivate sufficient sample size for biosimilar as well. This paper discusses two adjustment methods for the equivalence test in analytical biosimilarity studies. Please keep in mind that sufficient sample sizes for both biosimilar and reference products (if feasible) are desired during the planning stage.     

On sample size calculation for comparing survival curves under general hypothesis testing

The log-rank test is commonly used to test the equivalence of two survival distributions under right censoring. Jung et al. (2005) proposed a modified log-rank test for noninferiority trials and its corresponding sample size calculation. In this article, we extend the use of the modified log-rank test for clinical trials with various types of nonconventional study objectives and propose its sample size calculation under general null and alternative hypotheses. The proposed formula is so flexible that we can specify any survival distributions and accrual pattern. The proposed methods are illustrated with designing real clinical trials. Through simulations, the modified log-rank test and the derived formula for sample size calculation are shown to have satisfactory small sample performance.     

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.     

Choice of δ Noninferiority Margin and Dependency of the Noninferiority Trials∗

For a two-arm active control clinical trial designed to test for noninferiority of the test treatment to the active control standard treatment, data of historical studies were often used. For example, with a cross-trial comparison approach (also called synthetic approach or λ-margin approach), the trial is conducted to test the hypothesis that the mean difference or the ratio between the current test product and the active control is no larger than a certain portion of the mean difference or no smaller that a certain portion of the ratio of the active control and placebo obtained in the historical data when the positive response indicates treatment effective. For a generalized historical control approach (also known as confidence interval approach or δ -margin approach), the historical data is often used to determine a fixed value noninferiority margin δ for all trials involving the active control treatment. The regulatory agency usually requires that the clinical trials of two different test treatments need to be independent and in most regular cases, it also requires to have two independent positive trials of the same test treatment in order to provide confirmatory evidence of the efficacy of the test product. Because of the nature of information (historical data) shared in active-controlled trials, the independency assumption of the trials is not satisfied in general. The correlation between two noninferiority tests has been examined which showed that it is an increasing function of (1 ? λ ) when the response variable is normally distributed. In this article, we examine the relationship between the correlation of the two test statistics and the choice of the noninferiority margin, δ as well as the sample sizes and variances under the normality assumption. We showed that when δ is determined by the lower limit of the confidence interval of the adjusted effect size of the active control treatment (μ_C ? μ_P) using data from historical studies, dependency of the two noninferiority tests can be very high. In order to control the correlation under 15%, the overall sample size of the historical studies needs to be at least five times of the current active control trial.     

Choice of delta noninferiority margin and dependency of the noninferiority trials

For a two-arm active control clinical trial designed to test for noninferiority of the test treatment to the active control standard treatment, data of historical studies were often used. For example, with a cross-trial comparison approach (also called synthetic approach or lambda-margin approach), the trial is conducted to test the hypothesis that the mean difference or the ratio between the current test product and the active control is no larger than a certain portion of the mean difference or no smaller that a certain portion of the ratio of the active control and placebo obtained in the historical data when the positive response indicates treatment effective. For a generalized historical control approach (also known as confidence interval approach or delta -margin approach), the historical data is often used to determine a fixed value noninferiority margin delta for all trials involving the active control treatment. The regulatory agency usually requires that the clinical trials of two different test treatments need to be independent and in most regular cases, it also requires to have two independent positive trials of the same test treatment in order to provide confirmatory evidence of the efficacy of the test product. Because of the nature of information (historical data) shared in active-controlled trials, the independency assumption of the trials is not satisfied in general. The correlation between two noninferiority tests has been examined which showed that it is an increasing function of (1 - lambda ) when the response variable is normally distributed. In this article, we examine the relationship between the correlation of the two test statistics and the choice of the noninferiority margin, delta as well as the sample sizes and variances under the normality assumption. We showed that when delta is determined by the lower limit of the confidence interval of the adjusted effect size of the active control treatment (muC - muP) using data from historical studies, dependency of the two noninferiority tests can be very high. In order to control the correlation under 15%, the overall sample size of the historical studies needs to be at least five times of the current active control trial.     

Power and sample size determination for noninferiority trials using an exact method

Noninferiority studies are frequently conducted to justify the development of new drugs and vaccines that have been shown to offer better safety profiles, easier administration, or lower cost while maintaining similar efficacy as compared to the standard treatment. Recently, exact methods have been developed to address the concern that existing asymptotic methods for analyzing and planning noninferiority may fail because of small sample size or because of skewed or sparse data structure. In this paper, we explore the use of exact methods in determining sample size and power for noninferiority studies that focus on the difference of two proportions. The methodology for sample size and power calculations is developed based on an exact unconditional test of noninferiority. We illustrate this exact method using a clinical trial example in childhood nephroblastoma and briefly discuss the optimal sample-size allocation strategy. This exact unconditional method performs very well in various scenarios and compares favorably to its asymptotic counterpart in terms of sensitivity. Therefore, it is a very desirable tool for planning noninferiority trials, especially in situations where asymptotic methods are likely to fail.     

非劣性/等效性试验的样本含量估计及统计推断

就近年应用逐渐增多的非劣性/等效性试验中涉及的一些关键统计学问题进行详细介绍，其中包括设计过程中的非劣性/等效性界值的确定、样本含量的估计方法和统计推断过程中的检验假设建立、检验统计量计算以及可信区间计算方法。结合7个有针对性的应用实例有助于对相关事项的理解和在非劣性/等效性试验时进行参照。     

SAMPLE SIZE AND OPTIMAL DESIGNS IN STRATIFIED COMPARATIVE TRIALS TO ESTABLISH THE EQUIVALENCE OF TREATMENT EFFECTS AMONG TWO ETHNIC GROUPS

When a new investigational medicine is intended to be applied to populations with different ethnic backgrounds, a stratified comparative phase III trial using ethnic groups as strata may be conducted to assess the influence of ethnic factors on clinical outcomes of this new medicine. In this paper, based on a binomial model with odds ratio as the measure of the treatment effect, we derive the score test and the associated sample size formula for establishing the equivalence/noninferiority of the treatment effects of a medicine among two ethnic groups. A simplified test together with its sample size formula are also given. Taking into account the sample size, cost, and power of testing, respectively, we derive the optimal design parameters, i.e., the allocation among treatment groups and ethnic groups, based on the simplified test.     

Using the Instrumental Variables Estimator to Analyze Noninferiority Trials with Noncompliance

The standard intent-to-treat (ITT) approach can lead to erroneous conclusions about treatment efficacy in noninferiority trials with noncompliance. Per-protocol and as-treated analyses are also known to result in biased comparisons of treatment effects. Alternative statistical methods are therefore needed to better address the effects of noncompliance in noninferiority trials. In this paper, we consider the use of the instrumental variables (IV) estimator in a noninferiority trial with a binary outcome and evaluate the performance of this approach in comparison to other conventional analytic methods. Unlike the ITT, per-protocol, and as-treated approaches, the IV method provides an unbiased estimate of the average causal effect of treatment among the subgroup of compliers and maintains the nominal type I error rate, but does increase the sample size requirements of the study as the expected proportion of noncompliers increases. Further development of the IV estimator for more general patterns of noncompliance would be useful and would encourage broader application of this method in noninferiority trials.     

Group sequential design and analysis of clinical equivalence assessment for generic nonsystematic drug products

Clinical trials with therapeutical endpoints are designed with three arms to demonstrate both the efficacy and the equivalence of the test generic treatment and the reference treatment. A generic drug product is determined to be equivalent to the reference drug product if the ratio or difference between the mean responses is bounded within the pre-specified equivalence limits. Often the trials are oversized for the placebo arm. For improvement, we propose a group sequential design with hierarchical testing for the purpose of terminating the placebo arm before testing equivalence between the test and the reference treatments. The hierarchical feature of the proposal will reduce the sample size of the placebo arm and provide treatments to patients in a more efficient manner in a clinical trial setting. After dropping the placebo arm, the option of allocating the planned but unused sample size from the placebo group to the test and reference groups will increase the sample size and power of the equivalence test without inflating the type I error rate by delaying spending it.     

Sample size for a noninferiority clinical trial with time-to-event data in the presence of competing risks

The analysis and planning methods for competing risks model have been described in the literature in recent decades, and noninferiority clinical trials are helpful in current pharmaceutical practice. Analytical methods for noninferiority clinical trials in the presence of competing risks (NiCTCR) were investigated by Parpia et al., who indicated that the proportional sub-distribution hazard (SDH) model is appropriate in the context of biological studies. However, the analytical methods of the competing risks model differ from those appropriate for analyzing noninferiority clinical trials with a single outcome; thus, a corresponding method for planning such trials is necessary. A sample size formula for NiCTCR based on the proportional SDH model is presented in this paper. The primary endpoint relies on the SDH ratio. A total of 120 simulations and an example based on a randomized controlled trial verified the empirical performance of the presented formula. The results demonstrate that the empirical power of sample size formulas based on the Weibull distribution for noninferiority clinical trials with competing risks can reach the targeted power.     

非虚假设多中心配对临床试验

目的:将配对设计加以扩展,以适应非虚假设和分层。方法:将多中心临床试验的每个中心看作1层,取层样本分数为权计算平均治疗-对照差。将其期望与最小可识别差量比较建立非虚假设,按其方差构造分层配对设计基本关系式,进而推导出样本量公式,检验统计量,和功效函数。将这些用于临床试验的设计、执行和分析。以Monte Carlo方法展示观测功效。结果:这些在最小可识别差量取零时还原为对应经典统计学方法,其观测功效和期望功效吻合,所需样本量小于对应的两组设计。结论:这种临床试验直观高效,适于建立试药对于对照药的临床优效性、非劣效性或等效性,并附有实例描述用法。     

Three-Arm Noninferiority Trials with a Prespecified Margin for Inference of the Difference in the Proportions of Binary Endpoints

The design of a three-arm trial including the experimental treatment, an active reference treatment, and a placebo is recommended as a useful approach to the assessment of noninferiority of the experimental treatment. The inclusion of the placebo arm enables the assessment of assay sensitivity and internal validation, in addition to testing the noninferiority of the experimental treatment to the reference. Generally, the acceptable noninferiority margin Δ has been defined as the maximum clinically irrelevant difference between treatments in many two-arm noninferiority trials. However, many articles have considered a design in which the noninferiority margin Δ is relatively defined as a prespecified fraction f of the unknown effect size of the reference treatment. Therefore, these methods cannot be applied to cases where the margin is defined as a prespecified difference between treatments. In this article, we propose score-based statistical procedures for a three-arm noninferiority trial with a prespecified margin Δ for inference of the difference in the proportions of binary endpoints. In addition, we derive the approximate sample size and optimal allocation to minimize the total sample size and that of the placebo arm. A randomized controlled trial on major depressive disorder based on the difference in the proportions of remission is used to demonstrate our proposed method.     

Group-sequential three-arm noninferiority clinical trial designs

We discuss group-sequential three-arm noninferiority clinical trial designs that include active and placebo controls for evaluating both assay sensitivity and noninferiority. We extend two existing approaches, the fixed margin and fraction approaches, into a group-sequential setting with two decision-making frameworks. We investigate the operating characteristics including power, Type I error rate, maximum, and expected sample sizes, as design factors vary. In addition, we discuss sample size recalculation and its impact on the power and Type I error rate via a simulation study.     

Exact and Approximate Power and Sample Size Calculations for Analysis of Covariance in Randomized Clinical Trials With or Without Stratification

ABSTRACT

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 derive the exact power formulas for testing a homogeneous treatment effect in superiority, noninferiority, and equivalence trials under both unstratified and stratified randomizations, and for testing the overall treatment effect and treatment × stratum interaction in the presence of heterogeneous treatment effects when the covariates excluding the intercept, treatment, and prestratification factors are normally distributed. These formulas also work very well for nonnormal covariates. The sample size methods based on the normal approximation or the asymptotic variance generally underestimate the required size. We adapt the recently developed noniterative and two-step sample size procedures to the above tests. Both methods take into account the nonnormality of the t statistic, and the lower order variance term commonly ignored in the sample size estimation. Numerical examples demonstrate the excellent performance of the proposed methods particularly in small samples. We revisit the topic on the prestratification versus post-stratification by comparing their relative efficiency and power. Supplementary materials for this article are available online.     

A note on sample size calculation for mean comparisons based on noncentral t-statistics

One-sample and two-sample t-tests are commonly used in analyzing data from clinical trials in comparing mean responses from two drug products. During the planning stage of a clinical study, a crucial step is the sample size calculation, i.e., the determination of the number of subjects (patients) needed to achieve a desired power (e.g., 80%) for detecting a clinically meaningful difference in the mean drug responses. Based on noncentral t-distributions, we derive some sample size calculation formulas for testing equality, testing therapeutic noninferiority/superiority, and testing therapeutic equivalence, under the popular one-sample design, two-sample parallel design, and two-sample crossover design. Useful tables are constructed and some examples are given for illustration.