Sample size determination for a three-arm equivalence trial of Poisson and negative binomial responses

ABSTRACT

Assessing equivalence or similarity has drawn much attention recently as many drug products have lost or will lose their patents in the next few years, especially certain best-selling biologics. To claim equivalence between the test treatment and the reference treatment when assay sensitivity is well established from historical data, one has to demonstrate both superiority of the test treatment over placebo and equivalence between the test treatment and the reference treatment. Thus, there is urgency for practitioners to derive a practical way to calculate sample size for a three-arm equivalence trial. The primary endpoints of a clinical trial may not always be continuous, but may be discrete. In this paper, the authors derive power function and discuss sample size requirement for a three-arm equivalence trial with Poisson and negative binomial clinical endpoints. In addition, the authors examine the effect of the dispersion parameter on the power and the sample size by varying its coefficient from small to large. In extensive numerical studies, the authors demonstrate that required sample size heavily depends on the dispersion parameter. Therefore, misusing a Poisson model for negative binomial data may easily lose power up to 20%, depending on the value of the dispersion parameter.     

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。     

Review and evaluation of methods for computing confidence intervals for the ratio of two proportions and considerations for non-inferiority clinical trials

This article reviews several methods for forming confidence intervals for a risk ratio of two independent binomial proportions (which are both less than 0.50) and evaluates their statistical performance. These methods include use of a Taylor Series expansion to estimate variance, solutions to a quadratic equation, and maximum likelihood methods. In addition, for improvement of the properties of the methods based on large sample approximations, situations where either binomial count was less than or equal to 3 were managed conservatively by having an exact confidence interval for the odds ratio become the confidence interval for its risk ratio counterpart. Methods were initially evaluated by computing confidence limits for certain cases. Second, simulations were used to identify the better methods for controlling the Type I error rate while maintaining power. Last, relationships between methods were evaluated by calculating the percent of disagreement in the decision made regarding non inferiority. Methods in the group using a Taylor Series expansion in variance estimation perform similarly to the Pearson method preferred in the literature. In addition, the group of methods using a Taylor Series expansion are most easily computed. Applications of these findings are discussed for ratios that arise in randomized clinical trials that are conducted to show noninferiority of a new medical product to a reference control. Consideration is given as well to sample size calculations for noninferiority clinical trials.     

A Bayesian equivalency test for two independent binomial proportions

In clinical trials, it is often necessary to perform an equivalence study. The equivalence study requires actively denoting equivalence between two different drugs or treatments. Since it is not possible to assert equivalence that is not rejected by a superiority test, statistical methods known as equivalency tests have been suggested. These methods for equivalency tests are based on the frequency framework; however, there are few such methods in the Bayesian framework. Hence, this article proposes a new index that suggests the equivalency of binomial proportions, which is constructed based on the Bayesian framework. In this study, we provide two methods for calculating the index and compare the probabilities that have been calculated by these two calculation methods. Moreover, we apply this index to the results of actual clinical trials to demonstrate the utility of the index.     

Demonstrating Biosimilarity via Equivalence in Clinical Trials

The opportunities for biosimilar medicines have stimulated much legal and regulatory debates and actions around the world, most notably the passing of the Biologics Price Competition and Innovation Act in the United States in 2009. A key difference between the development of a biosimilar product versus a generic chemical entity is the requirement for well-controlled clinical studies to demonstrate similarity in efficacy and safety. The main objective of this article is to extend the clinical study design methods commonly used in noninferiority trials to equivalence trials within the context of biosimilar product development. We extend the synthesis method to the equivalence setting and provide sample size considerations. We show that while an equivalence trial in general requires a larger sample size than a noninferiority trial, the difference may not be substantial depending on the significance level required for the equivalence trial.     

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.     

Weighted Rank Tests for Noninferiority Hypotheses Based on Paired Survival Times

Motivated by the design and analysis of a specific type of Phase II cancer clinical trials, we derive in this article a class of weighted rank tests for the testing of noninferiority hypotheses with a prespecified margin based on paired survival times. Monte Carlo simulations are performed to evaluate the performance of the proposed tests. Sample size calculation when these tests are used for the analysis is also discussed. Procedures developed are also applied to a Phase II clinical trial of metastatic breast cancer.     

ANALYZING COUNTS,DURATIONS, AND RECURRENCES IN CLINICAL TRIALS

In 1985 the (Byar and Blackard, Urology, Vol. X, 556–561, 1978) data set on bladder cancer became available to researchers. Since then, a number of studies have made use of it. However, none of these has fully utilized all of the data nor have they developed a methodology in which it is possible to estimate models of the number of recurrences and durations that are consistent with each other.

The purpose of this research is to determine which, if any, of the two drugs used in the trial, pyridoxine and theotepa, were effective and, by example, illustrate procedures that could be useful in the analysis of other clinical trial data sets.

First, the number of recurrences is modeled as a count using the Poisson and negative binomial distributions with covariates. Then Poisson models are tested on the durations. Finally, durations and tumor counts per recurrence are fitted to more general autoregressive Wiebull and negative binomial distributions, respectively.

Poisson models for durations are rejected in favor of the more general autoregressive models. The data on durations and tumor counts are shown to be more reliable from an inference point of view than the data on the number of recurrences.

The data on durations and tumor counts show quite conclusively that both drugs are effective in treating bladder cancer, a result that differs from what others have found.     

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.     

Analysis of Zero-Inflated Count Data From Clinical Trials With Potential Dropouts

Counts of prespecified events are important endpoints for many safety and efficacy clinical trials. The conventional Poisson model might not be ideal due to three potential issues: (1) overdispersion arising from intra-subject correlation, (2) zero inflation when the prespecified event is rare, and (3) missing observations due to early dropouts. Negative binomial (NB), Poisson hurdle (PH), and negative binomial hurdle (NBH) models are more appropriate for overdispersed and/or zero-inflated count data. An offset can be included in these models to adjust for differential exposure duration due to early dropouts. In this article, we propose new link functions for the hurdle part of a PH/NBH model to facilitate testing for zero-inflation and model selection. The proposed link function particularly improves the model fit of a NBH model when an offset is included to adjust for differential exposure. A simulation study is conducted to compare the existing and proposed models, which are then applied to data from two clinical trials to demonstrate application and interpretation of these methods.     

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.     

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.     

Power Analyses for Negative Binomial Models with Application to Multiple Sclerosis Clinical Trials

We use negative binomial (NB) models for the magnetic resonance imaging (MRI)-based brain lesion count data from parallel group (PG) and baseline versus treatment (BVT) trials for relapsing remitting multiple sclerosis (RRMS) patients, and describe the associated likelihood ratio (LR), score, and Wald tests. We perform power analyses and sample size estimation using the simulated percentiles of the exact distribution of the test statistics for the PG and BVT trials. When compared to the corresponding nonparametric test, the LR test results in 30–45% reduction in sample sizes for the PG trials and 25–60% reduction for the BVT trials.     

Analyzing counts, durations, and recurrences in clinical trials.

In 1985 the (Byar and Blackard, Urology, Vol. X, 556-561, 1978) data set on bladder cancer became available to researchers. Since then, a number of studies have made use of it. However, none of these has fully utilized all of the data nor have they developed a methodology in which it is possible to estimate models of the number of recurrences and durations that are consistent with each other. The purpose of this research is to determine which, if any, of the two drugs used in the trial, pyridoxine and theotepa, were effective and, by example, illustrate procedures that could be useful in the analysis of other clinical trial data sets. First, the number of recurrences is modeled as a count using the Poisson and negative binomial distributions with covariates. Then Poisson models are tested on the durations. Finally, durations and tumor counts per recurrence are fitted to more general autoregressive Wiebull and negative binomial distributions, respectively. Poisson models for durations are rejected in favor of the more general autoregressive models. The data on durations and tumor counts are shown to be more reliable from an inference point of view than the data on the number of recurrences. The data on durations and tumor counts show quite conclusively that both drugs are effective in treating bladder cancer, a result that differs from what others have found.     

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.     

Bayes Models of Clinical Trials with Dichotomous Outcomes and Sample Size Determination

Bayesian methods are used in the clinical trial environment to reduce sample sizes and/or increase power. The Beta distribution is a natural prior for binomial models. Under the empirical Bayes approach, the parameters of this distribution are the maximum likelihood estimator of the marginal beta-binomial distribution. This distribution is defined by a combination of gamma functions. Because of factorial growth of these functions, the straightforward numerical search for the maximum likelihood solution is frequently impractical given available software. In this article, we consider some simplifications to the marginal beta-binomial distribution, which are easily computationally tractable and very precise. Using empirical Bayes priors is restricted to the case of complete exchangeability of historical trials as opposed to the current trial. In order to reflect some difference between the historical studies and the current studies, we introduce an adjustment to the maximum likelihood estimate. The exchangeability is measured by the confidence interval for the historical rate of events. With this prior, the formula for the sample size calculation is completely defined.     

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.     

Inferiority Index and Margin in Noninferiority Trials

The specification of a margin in a noninferiority trial is often subject to challenge and disagreement. One reason for the contentious discussion is the concern about the validity of the constancy and assay sensitivity assumptions that accompany all such trials. But the main reason for disagreement is the subjectivity associated with the different current procedures that are used to determine the noninferiority margin. This article introduces the concept of an inferiority index between two distributions and establishes its link to an effect measure. Through this relationship, the inferiority index can be used as a standard measure to assess the degree of tightness of any given noninferiority margin and, in conjunction with clinical knowledge and available historical data, it can also guide the selection of a margin. Specifically, this relationship is investigated under survival, normal, and binomial distributions. A general theorem is given that establishes the asymptotic normality of a test statistic for the noninferiority hypothesis defined by the noninferiority margin so derived. Some examples from noninferiority trials are used to illustrate the usefulness of the inferiority index.     

Considering Over-Dispersion in the Sample Size Calculation for Clinical Trials With Repeated Count Measurements

Abstract

Over-dispersed count variables are frequently encountered in biomedical research. Despite extensive research in analytical methods, addressing over-dispersion in the design of clinical trials has received much less attention. In this study, we propose to directly incorporate over-dispersion into sample size calculation for clinical trials where a count outcome is repeatedly measured on each subject. The proposed method is applicable to the comparison of slopes as well as time-averaged responses. It is easy to compute and flexible enough to account for unbalanced randomization, arbitrary missing patterns, and different correlation structures. We show that sample size requirement is proportional to over-dispersion, which highlights the danger of ignoring over-dispersion in experimental design. Simulation results demonstrate that the proposed sample size calculation methods maintain the nominal levels of power and Type I error over a wide range of scenarios. Application example to an epileptic trial is presented. 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.