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.     

Use of the Fieller-Hinkley distribution of the ratio of random variables in testing for noninferiority

We address the noninferiority assessment problem defined in terms of the ratio of population means in a parallel group design analysis of variance setting. The sample ratio as a point estimate of the corresponding population ratio has been considered. It has been shown that the Fieller-Hinkley distribution of the ratio of two correlated normally distributed random variables readily provide a technique for constructing confidence intervals comparable to the bootstrap percentile and Fieller's confidence intervals. A finite parameter space based level alpha test of an inferiority hypothesis formulated in terms of a fixed margin has been derived. We illustrate our approach using the forced vital capacity (FVC) data. We claim that it is easy to construct and straight forward to interpret our bootstrap equivalent confidence intervals that are used to assess noninferiority. We discuss appropriate methods for calculation of sample sizes.     

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。     

Confidence-Interval Construction for Rate Ratio in Matched-Pair Studies with Incomplete Data

Matched-pair design is often used in clinical trials to increase the efficiency of establishing equivalence between two treatments with binary outcomes. In this article, we consider such a design based on rate ratio in the presence of incomplete data. The rate ratio is one of the most frequently used indices in comparing efficiency of two treatments in clinical trials. In this article, we propose 10 confidence-interval estimators for the rate ratio in incomplete matched-pair designs. A hybrid method that recovers variance estimates required for the rate ratio from the confidence limits for single proportions is proposed. It is noteworthy that confidence intervals based on this hybrid method have closed-form solution. The performance of the proposed confidence intervals is evaluated with respect to their exact coverage probability, expected confidence interval width, and distal and mesial noncoverage probability. The results show that the hybrid Agresti–Coull confidence interval based on Fieller’s theorem performs satisfactorily for small to moderate sample sizes. Two real examples from clinical trials are used to illustrate the proposed confidence intervals.     

Use of the Fieller–Hinkley Distribution of the Ratio of Random Variables in Testing for Noninferiority

We address the noninferiority assessment problem defined in terms of the ratio of population means in a parallel group design analysis of variance setting. The sample ratio as a point estimate of the corresponding population ratio has been considered. It has been shown that the Fieller–Hinkley distribution of the ratio of two correlated normally distributed random variables readily provide a technique for constructing confidence intervals comparable to the bootstrap percentile and Fieller's confidence intervals. A finite parameter space based level α test of an inferiority hypothesis formulated in terms of a fixed margin has been derived. We illustrate our approach using the forced vital capacity (FVC) data. We claim that it is easy to construct and straight forward to interpret our bootstrap equivalent confidence intervals that are used to assess noninferiority. We discuss appropriate methods for calculation of sample sizes.     

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.     

Combining Risk Difference and Risk Ratio in Noninferiority Trials of Safety

Prior to marketing, the long-term safety profile of a new therapy is often uncertain. One recommendation for premarket safety studies is to compare the new therapy to an appropriate control to determine whether the 95% confidence interval of the risk ratio is entirely less than a prespecified threshold (e.g., 1.8). The restriction to the risk ratio, however, has consequences that may not be intended. Risk difference may be a more appropriate measure of risk in this setting when event rates are very low. We propose using a suitable combination of risk ratio and risk difference in demonstrating noninferiority.     

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.     

A Bootstrap-Based Test for Establishing Noninferiority in Clinical Trials

A randomized, active-control clinical trial setting with the objective of testing noninferiority for a continuous response variable is considered. Noninferiority margin is based on the concept of preserving a certain fraction of the active control effect. Noninferiority is established if the ratio of the lower (upper) limit of the two-sided 95% confidence interval for the treatment difference to the estimated mean of the active control is greater (less) than a certain fraction. The nominal significance level is not maintained by the above confidence interval-based noninferiority test. We use bootstrapping to derive an accurate lower (upper) limit of the same confidence interval, which approximates the nominal significance level better and improves the power.     

A bootstrap-based test for establishing noninferiority in clinical trials

A randomized, active-control clinical trial setting with the objective of testing noninferiority for a continuous response variable is considered. Noninferiority margin is based on the concept of preserving a certain fraction of the active control effect. Noninferiority is established if the ratio of the lower (upper) limit of the two-sided 95% confidence interval for the treatment difference to the estimated mean of the active control is greater (less) than a certain fraction. The nominal significance level is not maintained by the above confidence interval-based noninferiority test. We use bootstrapping to derive an accurate lower (upper) limit of the same confidence interval, which approximates the nominal significance level better and improves the power.     

Adjusting for Unknown Bias in Noninferiority Clinical Trials

Evaluation of noninferiority is based on ruling out a threshold for what would constitute unacceptable loss of efficacy of an experimental treatment relative to an active comparator “Standard.” This threshold, the “noninferiority margin,” is often based on preservation of a percentage of Standard's effect. To obtain an estimate of this effect to be used in the development of the “noninferiority margin,” data are needed from earlier trials comparing Standard to Placebo if the noninferiority trial does not have a Placebo arm. This approach often provides a biased overestimate of Standard's true effect in the setting of the current noninferiority study. We describe two commonly used noninferiority margin methods that adjust for this bias, the two-confidence interval (95-95), and the Synthesis margins. However, the added “variance inflation” adjustment made by 95-95 margin diminishes with increasing information from historical trial(s), and the Synthesis margin is based on a strong assumption that the relative bias is known. We introduce an alternative “Bias-adjusted” margin addressing vulnerabilities of each by attenuating the estimate and by accounting for uncertainty in the true level of bias. Examples and asymptotic estimates of noninferiority hypothesis rejection rates in the proportional hazards setting are used to compare methods.     

Sample size for comparing negative binomial rates in noninferiority and equivalence trials with unequal follow-up times

We derive the sample size formulae for comparing two negative binomial rates based on both the relative and absolute rate difference metrics in noninferiority and equivalence trials with unequal follow-up times, and establish an approximate relationship between the sample sizes required for the treatment comparison based on the two treatment effect metrics. The proposed method allows the dispersion parameter to vary by treatment groups. The accuracy of these methods is assessed by simulations. It is demonstrated that ignoring the between-subject variation in the follow-up time by setting the follow-up time for all individuals to be the mean follow-up time may greatly underestimate the required size, resulting in underpowered studies. Methods are provided for back-calculating the dispersion parameter based on the published summary results.     

Test Statistics and Confidence Intervals to Establish Noninferiority between Treatments with Ordinal Categorical Data

The problem for establishing noninferiority is discussed between a new treatment and a standard (control) treatment with ordinal categorical data. A measure of treatment effect is used and a method of specifying noninferiority margin for the measure is provided. Two Z-type test statistics are proposed where the estimation of variance is constructed under the shifted null hypothesis using U-statistics. Furthermore, the confidence interval and the sample size formula are given based on the proposed test statistics. The proposed procedure is applied to a dataset from a clinical trial. A simulation study is conducted to compare the performance of the proposed test statistics with that of the existing ones, and the results show that the proposed test statistics are better in terms of the deviation from nominal level and the power.     

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.     

Bayesian Sample Size Estimates for One Sample Test in Clinical Trials With Dichotomous and Countable Outcomes

This article presents a Bayesian approach to sample size determination in binomial and Poisson clinical trials. It uses exact methods and Bayesian methodology. Our sample size estimations are based on power calculations under the one-sided alternative hypothesis that a new treatment is better than a control by a clinically important margin. The method resembles a standard frequentist problem formulation and, in the case of conjugate prior distributions with integer parameters, is similar to the frequentist approach. We evaluate Type I and II errors through the use of credible limits in Bayesian models and through the use of confidence limits in frequentist models. Particularly, for conjugate priors with integer parameters, credible limits are identical to frequentist confidence limits with adjusted numbers of events and sample sizes. We consider conditions under which the minimal Bayesian sample size is less than the frequentist one and vice versa.     

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。     

Test for the consistency of noninferiority from multiple clinical trials

A testing procedure is proposed to assess the consistency of noninferiority from a collection of trials based on simultaneous t lower confidence bounds or Scheffé's lower confidence bounds. Methods for simultaneous inferences on pairwise or many-to-one comparisons among multiple noninferiority trials are also discussed. To avoid bias due to subjective trial exclusion a tuning parameter k is embedded into the testing procedure to provide flexibility to quantify the "consistency of noninferiority" when the total number of trials is large. The size and power of the proposed test are discussed. The method is illustrated using simulations and real data analysis.     

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.