ON STATISTICAL POWER FOR AVERAGE BIOEQUIVALENCE TESTING UNDER REPLICATED CROSSOVER DESIGNS

In its recent guidance on bioequivalence, the U.S. Food and Drug Administration (FDA) recommends a two-sequence, four-period (2×4)replicated crossover design be used for assessment of population and individual bioequivalence [FDA. Guidance for Industry on Statistical Approaches to Establishing Bioequivalence; Center for Drug Evaluation and Research, Food and Drug Administration: Rockville, MD, 2001]. The recommended replicated crossover design not only allows estimates of both the inter-subject and the intra-subject variabilities and the variability due to subject-by-formulation interaction, but also provides an assessment of average bioequivalence (ABE). In this article, power function for assessment of ABE under a general replicated crossover design (i.e., a 2×2mreplicated crossover design) based on the traditional analysis of variance model and the mixed effects model as suggested by the FDA are studied. It is found that the power of a 2×2mreplicated crossover design depends upon the variability due to subject-by-formulation interaction and the number of replicates. Based on the derived power function, formula for sample size calculation for assessment of ABE under a 2×2mreplicated crossover design is also provided.     

Statistical aspects of bioequivalence testing between two medicinal products.

A generic drug product (test product) is bioequivalent to an innovator product (reference product) when their bioavailabilities in the same molar dose are similar. Bioavailability is expressed by pharmacokinetic parameters such as the area under plasma concentration-time curve (AUC), the maximum plasma concentration (Cmax) and the time of maximum plasma concentration (tmax). The assessment of bioequivalence is carried out by in vivo bioequivalence studies. This paper examines and appraises design issues for performing a bioequivalence study: the use of crossover, parallel, replicated, and add-on designs; and the determination of sample size. In addition, it presents the valid statistical approaches for proving bioequivalence: average bioequivalence on transformed and untransformed data; parametric and non-parametric analyses; moment based individual bioequivalence; direct curve comparison metrics.     

On statistical power for average bioequivalence testing under replicated crossover designs

In its recent guidance on bioequivalence, the U.S. Food and Drug Administration (FDA) recommends a two-sequence, four-period (2 x 4) replicated crossover design be used for assessment of population and individual bioequivalence [FDA. Guidance for Industry on Statistical Approaches to Establishing Bioequivalence; Center for Drug Evaluation and Research, Food and Drug Administration: Rockville, MD, 2001]. The recommended replicated crossover design not only allows estimates of both the inter-subject and the intra-subject variabilities and the variability due to subject-by-formulation interaction, but also provides an assessment of average bioequivalence (ABE). In this article, power function for assessment of ABE under a general replicated crossover design (i.e., a 2 x 2m replicated crossover design) based on the traditional analysis of variance model and the mixed effects model as suggested by the FDA are studied. It is found that the power of a 2 x 2m replicated crossover design depends upon the variability due to subject-by-formulation interaction and the number of replicates. Based on the derived power function, formula for sample size calculation for assessment of ABE under a 2 x 2m replicated crossover design is also provided.     

ON SAMPLE SIZE CALCULATION BASED ON ODDS RATIO IN CLINICAL TRIALS

Sample size calculation formulas for testing equality, noninferiority, superiority, and equivalence based on odds ratio were derived under both parallel and one-arm crossover designs. An example concerning the study of odds ratio between a test compound (treatment) and a standard therapy (control) for prevention of relapse in subjects with schizophrenia and schizoaffective disorder is presented to illustrate the derived formulas for sample size calculation for various hypotheses under both a parallel design and a crossover design. Simulations were performed to assess the adequacy of the sample size calculation formulas. Simulation results were given at the end of the paper.     

On sample size calculation based on odds ratio in clinical trials

Sample size calculation formulas for testing equality, noninferiority, superiority, and equivalence based on odds ratio were derived under both parallel and one-arm crossover designs. An example concerning the study of odds ratio between a test compound (treatment) and a standard therapy (control) for prevention of relapse in subjects with schizophrenia and schizoaffective disorder is presented to illustrate the derived formulas for sample size calculation for various hypotheses under both a parallel design and a crossover design. Simulations were performed to assess the adequacy of the sample size calculation formulas. Simulation results were given at the end of the paper.     

Letter to the Editor: A Note on Sample Size Calculation in Bioequivalence Trials

Based on fundamental pharmacokinetic relationships, a multiplicative model is commonly used in bioequivalence trials. With regard to the parametric analysis, this implies the assumption of a lognormal distribution. Statistical methods for sample size calculation has been consolidated over the last years. Recently, methods for sample size calculation in the additive model, i.e., under normality assumption, were presented. Hence, these methods are reviewed from a statistical and regulatory point of view.     

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.     

Bioequivalence studies: biometrical concepts of alternative designs and pooled analysis.

A bioequivalence study compares the bioavailability between a test and a reference drug product in terms of the rate and extent of drug absorption. Area under the plasma concentration-time curve (AUC) and maximum plasma concentration (Cmax) are the pharmacokinetic parameters that serve as characteristics for the assessment of the extent and rate of absorption, respectively. The experimental design of a bioequivalence study is usually a crossover and rarely a parallel or a paired comparative. The statistical assessment of bioequivalence is based on the 90% confidence interval for the ratio of the test mean to the reference mean for AUC and Cmax The aims of this paper are to: (i) investigate alternative designs to a crossover design for conducting bioequivalence studies; (ii) propose the statistical analysis of different designs for bioequivalence studies on the same products; and (iii) discuss their usefulness for the approval of new generic drug products. For this purpose, three case studies are illustrated and analysed. The first case study concerns the investigation of the merits of a crossover design relative to a parallel group design for highly variable drugs using as an example a bioequivalence study of tamoxifen products. The second case study concerns the pooled statistical analysis of two bioequivalent studies of the same levodopa products. The analyses of the individual studies failed to meet the regulatory criteria for bioequivalence. The one study design was a paired comparative and the other one a crossover. Under some assumptions the crossover design may be considered as a paired comparative and the data from the two studies may be analysed together as a paired comparative design. The third case study concerns the statistical pooled analysis of two bioequivalent studies of the same clodronate products. The one study was a three-period crossover pilot study and it was used to identify the variability of the active substance. Then, this variability was used to determine the number of subjects for the main pivotal study which was a two-period crossover. The pilot study design was converted into a two-period crossover design and the data from the two studies were analysed together as a two-period crossover design. The original data of the studies were modified accordingly.     

临床试验Williams设计中样本含量的估算方法

在临床试验中,2组交叉设计应用已相当广泛,其样本含量估算方法也被研究者所熟悉。多组交叉试验由Williams首先提出,因此被称为Wil-liams设计。本文介绍基于Williams设计的样本含量估算方法,并提供示例分析,供研究者参考使用。     

Adaptive clinical endpoint bioequivalence studies with sample size re-estimation based on a nuisance parameter

ABSTRACT

A clinical endpoint bioequivalence (BE) study is often used to establish bioequivalence (BE) between a locally acting generic drug (T) and an innovator drug (R), which is a double-blind, randomized three-arm (T, R and placebo: P) parallel clinical trial. BE is established if two superiority tests (T vs. P, R vs. P) and one equivalence test (T vs. R) all pass. An accurate estimate of the nuisance parameter (e.g. variance) is vital in determining an accurate sample size to attain sufficient power. However, due to potential study design variations between NDA and Abbreviated NDA (ANDA) studies and high variability of clinical endpoints, variance may be over- or under-estimated, resulting in unnecessary extra costs or underpowered studies. Traditionally, clinical endpoint BE studies use a fixed study design. In this work, we propose four sample size re-estimation approaches based on a nuisance parameter and recommend one approach after comparing various operating characteristics by simulation.

The proposed adaptive design with sample size re-estimation provides a more accurate estimate of sample size without wasting resources or under-powering the study and controls the Type 1 error rate under a negligible level, both for the family-wise alpha and individual alpha for superiority and equivalence tests.     

A Note on Sample Size Determination for Bioequivalence Studies with Higher-Order Crossover Designs

Similar to Liu and Chow, approximate formulas for sample size determination are derived based on Schuirmann's two one-sided tests procedure for bioequiealence studies for the additive and the multiplicative models under various higher order crossover designs for comparing two formulations of a drug product. The higher order crossover designs under study include Balaam's design, the two-sequence dual design, and two four-period designs (with two and four sequences), which are commonly used for assessment of bioequivalence between formulations. The derived formulas are simple enough to be carried out with a pocket calculator. The number of subjects required for each of the four higher order designs are tabulated for selected powers and various parameter values.     

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.     

DROPOUTS IN LONGITUDINAL STUDIES: DEFINITIONS AND MODELS

In this paper, we consider statistical tests for inter-subject and total variabilities between treatments under crossover designs. Since estimators of variance components for inter-subject variability and total variability in crossover design are not independent, the usual F-test cannot be applied. Alternatively, we propose a test based on the concept of the extension of the modified large sample method to compare inter-subject variability and total variability between treatments under a 2×2mreplicated crossover design. An asymptotic power of the proposed test is derived. A sensitivity analysis is performed based on the asymptotic power to determine how the power changes with respect to various parameters such as inter-subject correlation and intra-class correlation. Also the two methods for sample size calculation for testing total variability under 2×4 crossover design are discussed. The method based on the Fisher–Cornish inversion shows better performance than the method based on the normal approximation. Several simulation studies were conducted to investigate the finite sample performance of the proposed test. Our simulation results show that the proposed test can control type I error satisfactorily.     

Extension to the use of tolerance intervals for the assessment of individual bioequivalence

For the determination of bioequivalence, researchers have recently shifted their emphasis from average bioequivalence alone to average and individual bioequivalence. Existing methods for assessing average bioequivalence were first developed for the standard 2 × 2 crossover design, but these methods are easily generalized to the two-treatment, ρ-period crossover designs (e.g., TRR, RTT, and TTRR, RRTT, TRRT, RTTR). With respect to individual bioequivalence, Westlake (1,2) implemented the use of parametric and distribution-free tolerance intervals for assessing individual bioequivalence. Anderson and Hauck (3) described what they call the test of individual equivalence ratios (TIER) for the same purpose. Note that these methods have been applied and/or developed only for the standard 2 × 2 crossover design. The present work extends the method of using parametric tolerance intervals for assessing individual bioequivalence.     

On assessment of bioequivalence under a higher-order crossover design

In bioavailability studies of two formulations of a drug, the standard two-sequence, two-period crossover design is usually considered to assess bioequivalence. The standard two-sequence, two-period crossover design, however, may not be useful when differential carryover effects are present. In addition, it does not provide independent estimates of intrasubject variabilities for the two formulations. To overcome these problems, alternatively, a higher-order crossover design may be considered. In this paper, we derive statistical methods based on Schuirmann's two one-sided tests procedure (1) for assessing bioequivalence for some commonly used higher-order crossover designs. Four designs, including Balaam's design, the two-sequence dual design, and two four-period designs (with two and four sequences), are considered. The relative merits of these designs as compared to the standard two-sequence, two-period design are discussed. Two examples concerning bioequivalence are used to illustrate the use of these methods.     

Crossover Designs With Two Binary Endpoints

ABSTRACT

We propose a crossover design for simultaneous significance testing of two binary endpoints, in which the AB/BA crossover design is carried out for each endpoint. An asymptotic α-level test is obtained by applying the intersection-union principle to the marginal Mainland–Gart tests. Power approximations and sample size calculation are derived and implemented in R programs. An adaptive design with sample size reestimation is also presented. We demonstrate the numerical accuracy of the proposed design through an extensive simulation study. Supplementary materials for this article are available online.     

Sample Size Calculations in Thorough QT Studies

An analysis of QTc data collected in four thorough QT studies conducted at Eli Lilly and Company was performed to estimate the variability of the QTc interval and to calculate the variance components related to time-to-time, day-to-day variability, etc. The results were used to develop a sample size calculation framework that enables clinical trial researchers to account for key features of their thorough QT studies, including study design (parallel and crossover designs), number of ECG replicates, number of post-baseline ECG recordings, and subject population (based on subject gender and age). The sample size calculation framework is illustrated using several popular study designs.     

Sample size calculations in thorough QT studies

An analysis of QTc data collected in four thorough QT studies conducted at Eli Lilly and Company was performed to estimate the variability of the QTc interval and to calculate the variance components related to time-to-time, day-to-day variability, etc. The results were used to develop a sample size calculation framework that enables clinical trial researchers to account for key features of their thorough QT studies, including study design (parallel and crossover designs), number of ECG replicates, number of post-baseline ECG recordings, and subject population (based on subject gender and age). The sample size calculation framework is illustrated using several popular study designs.     

Bioequivalence evaluation of two brands of amoxicillin/clavulanic acid 250/125 mg combination tablets in healthy human volunteers: use of replicate design approach

The purpose of this study was to apply a replicate design approach to a bioequivalence study of amoxicillin/clavulanic acid combination following a 250/125 mg oral dose to 23 subjects, and to compare the analysis of individual bioequivalence with average bioequivalence. This was conducted as a 2-treatment 2-sequence 4-period crossover study. Average bioequivalence was shown, while the results from the individual bioequivalence approach had no success in showing bioequivalence. In conclusion, the individual bioequivalence approach is a strong statistical tool to test for intra-subject variances and also subject-by-formulation interaction variance compared with the average bioequivalence approach.     

On power and sample size calculation for QT studies with recording replicates at given time point

The problem of the impact on power and sample size calculation for routine QT studies with ECG recording replicates under a parallel-group design and a crossover design is examined. Replicate ECGs are defined as single ECG recorded within several minutes of a nominal time (PhRMA, 2003). Formulas for sample size calculations with and without adjustment for covariates such as some pharmacokinetic responses (e.g., AUC or C(max)), which are known to be correlated to the QT intervals, were derived under both the parallel-group design and the crossover design. The results indicate that the approach of replicates may require a smaller sample size for achieving the same power when the correlation coefficient between the recording replicates (or repeated measures) is close to 0 (i.e., these replicate ECGs are almost independent). On the other hand, if the correlation coefficient is close to 1, then there is not much gain regardless of whether replicate ECGs are considered. In this paper, an approach to identifying optimal allocation between the number of subjects and the number of replicates per subject is proposed for achieving the maximum power under a fixed budget constraint. The proposed approach can also be applied to minimize the cost for a given power.