Pilot–pivotal trials for average bioequivalence

Before carrying out a full scale bioequivalence trial, it is desirable to conduct a pilot trial to decide if a generic drug product shows promise of bioequivalence. The purpose of a pilot trial is to screen test formulations, and hence small sample sizes can be used. Based on the outcome of the pilot trial, one can decide whether or not a full scale pivotal trial should be carried out to assess bioequivalence. This article deals with the design of a pivotal trial, based on the evidence from the pilot trial. A two-stage adaptive procedure is developed in order to determine the sample size and the decision rule for the pivotal trial, for testing average bioequivalence using the two one-sided test (TOST). Numerical implementation of the procedure is discussed in detail, and the required tables are provided. Numerical results indicate that the required sample sizes could be smaller than that recommended by the FDA for a single trial, especially when the pilot study provides strong evidence in favor of bioequivalence.     

Likelihood approach for evaluating bioequivalence of highly variable drugs

Bioequivalence (BE) is required for approving a generic drug. The two one‐sided tests procedure (TOST, or the 90% confidence interval approach) has been used as the mainstream methodology to test average BE (ABE) on pharmacokinetic parameters such as the area under the blood concentration‐time curve and the peak concentration. However, for highly variable drugs (%CV > 30%), it is difficult to demonstrate ABE in a standard cross‐over study with the typical number of subjects using the TOST because of lack of power. Recently, the US Food and Drug Administration and the European Medicines Agency recommended similar but not identical reference‐scaled average BE (RSABE) approaches to address this issue. Although the power is improved, the new approaches may not guarantee a high level of confidence for the true difference between two drugs at the ABE boundaries. It is also difficult for these approaches to address the issues of population BE (PBE) and individual BE (IBE). We advocate the use of a likelihood approach for representing and interpreting BE data as evidence. Using example data from a full replicate 2 × 4 cross‐over study, we demonstrate how to present evidence using the profile likelihoods for the mean difference and standard deviation ratios of the two drugs for the pharmacokinetic parameters. With this approach, we present evidence for PBE and IBE as well as ABE within a unified framework. Our simulations show that the operating characteristics of the proposed likelihood approach are comparable with the RSABE approaches when the same criteria are applied. Copyright © 2014 John Wiley & Sons, Ltd.     

Bioequivalence and the pharmaceutical industry

Since the early 1990s, average bioequivalence (ABE) studies have served as the international regulatory standard for demonstrating that two formulations of drug product will provide the same therapeutic benefit and safety profile when used in the marketplace. Population (PBE) and individual (IBE) bioequivalence have been the subject of intense international debate since methods for their assessment were proposed in the late 1980s and since their use was proposed in United States Food and Drug Administration guidance in 1997. Guidance has since been proposed and finalized by the Food and Drug Administration for the implementation of such techniques in the pioneer and generic pharmaceutical industries. The current guidance calls for the use of replicate design and of cross‐over studies (cross‐overs with sequences TRTR, RTRT, where T is the test and R is the reference formulation) for selected drug products, and proposes restricted maximum likelihood and method‐of‐moments techniques for parameter estimation. In general, marketplace access will be granted if the products demonstrate ABE based on a restricted maximum likelihood model. Study sponsors have the option of using PBE or IBE if the use of these criteria can be justified to the regulatory authority. Novel and previously proposed SAS^®‐based approaches to the modelling of pharmacokinetic data from replicate design studies will be summarized. Restricted maximum likelihood and method‐of‐moments modelling results are compared and contrasted based on the analysis of data available from previously performed replicate design studies, and practical issues involved in the application of replicate designs to demonstrate ABE are characterized. It is concluded that replicate designs may be used effectively to demonstrate ABE for highly variable drug products. Statisticians should exercise caution in the choice of modelling procedure. Copyright © 2002 John Wiley & Sons, Ltd.     

Testing bioequivalence for multiple formulations with power and sample size calculations

Bioequivalence (BE) trials play an important role in drug development for demonstrating the BE between test and reference formulations. The key statistical analysis for BE trials is the use of two one‐sided tests (TOST), which is equivalent to showing that the 90% confidence interval of the relative bioavailability is within a given range. Power and sample size calculations for the comparison between one test formulation and the reference formulation has been intensively investigated, and tables and software are available for practical use. From a statistical and logistical perspective, it might be more efficient to test more than one formulation in a single trial. However, approaches for controlling the overall type I error may be required. We propose a method called multiplicity‐adjusted TOST (MATOST) combining multiple comparison adjustment approaches, such as Hochberg's or Dunnett's method, with TOST. Because power and sample size calculations become more complex and are difficult to solve analytically, efficient simulation‐based procedures for this purpose have been developed and implemented in an R package. Some numerical results for a range of scenarios are presented in the paper. We show that given the same overall type I error and power, a BE crossover trial designed to test multiple formulations simultaneously only requires a small increase in the total sample size compared with a simple 2 × 2 crossover design evaluating only one test formulation. Hence, we conclude that testing multiple formulations in a single study is generally an efficient approach. The R package MATOST is available at https://sites.google.com/site/matostbe/ . Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation assessments of statistical aspects of bioequivalence in the pharmaceutical industry

An Erratum has been published for this article in Pharmaceutical Statistics 2004; 3(3): 232 Since the early 1990s, average bioequivalence (ABE) has served as the international standard for demonstrating that two formulations of drug product will provide the same therapeutic benefit and safety profile. Population (PBE) and individual (IBE) bioequivalence have been the subject of intense international debate since methods for their assessment were proposed in the late 1980s. Guidance has been proposed by the Food and Drug Administration (FDA) for the implementation of these techniques in the pioneer and generic pharmaceutical industries. Hitherto no consensus among regulators, academia and industry has been established on the use of the IBE and PBE metrics. The need for more stringent bioequivalence criteria has not been demonstrated, and it is known that the PBE and IBE criteria proposed by the FDA are actually less stringent under certain conditions. The statistical properties of method of moments and restricted maximum likelihood modelling in replicate designs will be summarized, and the application of these techniques in the assessment of ABE, IBE and PBE will be considered based on a database of 51 replicate design studies and using simulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive designs for IVPT data with mixed scaled average bioequivalence

In vitro permeation tests (IVPT) offer accurate and cost-effective development pathways for locally acting drugs, such as topical dermatological products. For assessment of bioequivalence, the FDA draft guidance on generic acyclovir 5% cream introduces a new experimental design, namely the single-dose, multiple-replicate per treatment group design, as IVPT pivotal study design. We examine the statistical properties of its hypothesis testing method—namely the mixed scaled average bioequivalence (MSABE). Meanwhile, some adaptive design features in clinical trials can help researchers make a decision earlier with fewer subjects or boost power, saving resources, while controlling the impact on family-wise error rate. Therefore, we incorporate MSABE in an adaptive design combining the group sequential design and sample size re-estimation. Simulation studies are conducted to study the passing rates of the proposed methods—both within and outside the average bioequivalence limits. We further consider modifications to the adaptive designs applied for IVPT BE trials, such as Bonferroni's adjustment and conditional power function. Finally, a case study with real data demonstrates the advantages of such adaptive methods.     

Testing for bioequivalence of highly variable drugs from TR‐RT crossover designs with heterogeneous residual variances

Traditional bioavailability studies assess average bioequivalence (ABE) between the test (T) and reference (R) products under the crossover design with TR and RT sequences. With highly variable (HV) drugs whose intrasubject coefficient of variation in pharmacokinetic measures is 30% or greater, assertion of ABE becomes difficult due to the large sample sizes needed to achieve adequate power. In 2011, the FDA adopted a more relaxed, yet complex, ABE criterion and supplied a procedure to assess this criterion exclusively under TRR‐RTR‐RRT and TRTR‐RTRT designs. However, designs with more than 2 periods are not always feasible. This present work investigates how to evaluate HV drugs under TR‐RT designs. A mixed model with heterogeneous residual variances is used to fit data from TR‐RT designs. Under the assumption of zero subject‐by‐formulation interaction, this basic model is comparable to the FDA‐recommended model for TRR‐RTR‐RRT and TRTR‐RTRT designs, suggesting the conceptual plausibility of our approach. To overcome the distributional dependency among summary statistics of model parameters, we develop statistical tests via the generalized pivotal quantity (GPQ). A real‐world data example is given to illustrate the utility of the resulting procedures. Our simulation study identifies a GPQ‐based testing procedure that evaluates HV drugs under practical TR‐RT designs with desirable type I error rate and reasonable power. In comparison to the FDA's approach, this GPQ‐based procedure gives similar performance when the product's intersubject standard deviation is low (≤0.4) and is most useful when practical considerations restrict the crossover design to 2 periods.     

Controlling type I error in the reference‐scaled bioequivalence evaluation of highly variable drugs

Reference‐scaled average bioequivalence (RSABE) approaches for highly variable drugs are based on linearly scaling the bioequivalence limits according to the reference formulation within‐subject variability. RSABE methods have type I error control problems around the value where the limits change from constant to scaled. In all these methods, the probability of type I error has only one absolute maximum at this switching variability value. This allows adjusting the significance level to obtain statistically correct procedures (that is, those in which the probability of type I error remains below the nominal significance level), at the expense of some potential power loss. In this paper, we explore adjustments to the EMA and FDA regulatory RSABE approaches, and to a possible improvement of the original EMA method, designated as HoweEMA. The resulting adjusted methods are completely correct with respect to type I error probability. The power loss is generally small and tends to become irrelevant for moderately large (affordable in real studies) sample sizes.     

A Bayesian approach to assessing population bioequivalence in a 2 ×2 crossover design

A Bayesian testing procedure is proposed for assessment of the bioequivalence in both mean and variance, which ensures population bioequivalence under the normality assumption. We derive the joint posterior distribution of the means and variances in a standard 2 ×2 crossover experimental design and propose a Bayesian testing procedure for bioequivalence based on a Markov chain Monte Carlo method. The proposed method is applied to a real data set.     

A nearly unbiased test for individual bioequivalence problems using probability criteria

Statistical bioequivalence has recently attracted lots of attention. This is perhaps due to the importance of setting a reasonable criterion on the part of a regulatory agency such as the FDA in the US in regulating the manufacturing of drugs (especially generic drugs). Pharmaceutical companies are obviously interested in the criterion since a huge profit is involved. Various criteria and various types of bioequivalence have been proposed. At present, the FDA recommends testing for average bioequivalence. The FDA, however, is considering replacing average bioequivalence by individual bioequivalence. We focus on the criterion of individual bioequivalence proposed earlier by Anderson and Hauck (J. Pharmacokinetics and Biopharmaceutics 18 (1990) 259) and Wellek (Medizinische Informatik und Statistik, vol. 71, Springer, Berlin, 1989, pp. 95–99; Biometrical J. 35 (1993) 47). For their criterion, they proposed TIER (test of individual equivalence ratios). Other tests were also proposed by Phillips (J. Biopharmaceutical Statist. 3 (1993) 185), and Liu and Chow (J. Biopharmaceutical Statist. 7 (1997) 49). In this paper, we propose an alternative test, called nearly unbiased test, which is shown numerically to have power substantially larger than existing tests. We also show that our test works for various models including 2×3 and 2×4 crossover designs.     

Exact calculation of power and sample size in bioequivalence studies using two one‐sided tests

The number of subjects in a pharmacokinetic two‐period two‐treatment crossover bioequivalence study is typically small, most often less than 60. The most common approach to testing for bioequivalence is the two one‐sided tests procedure. No explicit mathematical formula for the power function in the context of the two one‐sided tests procedure exists in the statistical literature, although the exact power based on Owen's special case of bivariate noncentral t‐distribution has been tabulated and graphed. Several approximations have previously been published for the probability of rejection in the two one‐sided tests procedure for crossover bioequivalence studies. These approximations and associated sample size formulas are reviewed in this article and compared for various parameter combinations with exact power formulas derived here, which are computed analytically as univariate integrals and which have been validated by Monte Carlo simulations. The exact formulas for power and sample size are shown to improve markedly in realistic parameter settings over the previous approximations. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonparametric test for similarity of marginals—With applications to the assessment of population bioequivalence

In this paper we suggest a completely nonparametric test for the assessment of similar marginals of a multivariate distribution function. This test is based on the asymptotic normality of Mallows distance between marginals. It is also shown that the n out of n bootstrap is weakly consistent, thus providing a theoretical justification to the work in Czado, C. and Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280]. The test is extended to cross-over trials and is applied to the problem of population bioequivalence, where two formulations of a drug are shown to be similar up to a tolerable limit. This approach was investigated in small samples using bootstrap techniques in Czado, C., Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280], showing that the bias corrected and accelerated bootstrap yields a very accurate and powerful finite sample correction. A data example is discussed.     

On the assessment of variability in bioavailability/bioequivalence studies

A procedure is proposed for the assessment of bioequivalence of variabilities between two formulations in bioavailability/bioequivalence studies. This procedure is essentially a two one-sided Pitman-Morgan’s tests procedure which is based on the correlation between crossover differences and subject totals. The nonparametric version of the proposed test is also discussed. A dataset of AUC from a 2×2 crossover bioequivalence trial is presented to illustrate the proposed procedures.     

CONSTRUCTING MORE POWERFUL EXACT TESTS OF EQUIVALENCE FROM BINARY MATCHED PAIRS

In randomized clinical trials, it is often necessary to demonstrate that a new medical treatment does not substantially differ from a standard reference treatment. Formal testing of such ‘equivalence hypotheses’ is typically done by combining two one‐sided tests (TOST). A quite different strand of research has demonstrated that replacing nuisance parameters with a null estimate produces P‐values that are close to exact ( Lloyd 2008a ) and that maximizing over the residual dependence on the nuisance parameter produces P‐values that are exact and optimal within a class ( Röhmel & Mansmann 1999 ; Lloyd 2008a ). The three procedures – TOST, estimation and maximization of a nuisance parameter – can each be expressed as a transformation of an approximate P‐value. In this paper, we point out that TOST‐based P‐values will generally be conservative, even if based on exact and optimal one‐sided tests. This conservatism is avoided by applying the three transforms in a certain order – estimation followed by TOST followed by maximization. We compare this procedure with existing alternatives through a numerical study of binary matched pairs where the two treatments are compared by the difference of response rates. The resulting tests are uniformly more powerful than the considered competitors, although the difference in power can range from very small to moderate.     

Testing the similarity of two normal populations with application to the bioequivalence problem

The problem of testing the similarity of two normal populations is reconsidered, in this article, from a nonclassical point of view. We introduce a test statistic based on the maximum likelihood estimate of Weitzman's overlapping coefficient. Simulated critical points are provided for the proposed test for various sample sizes and significance levels. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing tests. Furthermore, Type-I error robustness of the proposed and the existing tests are studied via simulation studies when the underlying distributions are non-normal. Two data sets are analyzed for illustration purposes. Finally, the proposed test has been implemented to assess the bioequivalence of two drug formulations.     

Cross-over trials in drug development: theory and practice

It is maintained that much research into the design and analysis of cross-over trials has been of little practical relevance to drug development. The point is illustrated using three topics: the AB/BA design, bioequivalence and multi-period designs in two treatments. It is suggested that statisticians should pay more attention to the work of fellow scientists, in particular, in the field of pharmacokinetics, and also that the philosophical–inferential base employed in examining cross-over trials has often been too narrow.     

Comparison of models for average bioequivalence in replicated crossover designs

Average bioequivalence (ABE) has been the regulatory standard for bioequivalence (BE) since the 1990s. BE studies are commonly two-period crossovers, but may also use replicated designs. The replicated crossover will provide greater power for the ABE assessment. FDA has recommended that ABE analysis of replicated crossovers use a model which includes terms for separate within- and between-subject components for each formulation and which allows for a subject x formulation interaction component. Our simulation study compares the performance of four alternative mixed effects models: the FDA model, a three variance component model proposed by Ekbohm and Melander (EM), a random intercepts and slopes model (RIS) proposed by Patterson and Jones, and a simple model that contains only two variance components. The simple model fails (when not 'true') to provide adequate coverage and it accepts the hypothesis of equivalence too often. FDA and EM models are frequently indistinguishable and often provide the best performance with respect to coverage and probability of concluding BE. The RIS model concludes equivalence too often when both the within- and between-subject variance components differ between formulations. The FDA analysis model is recommended because it provides the most detail regarding components of variability and has a slight advantage over the EM model in confidence interval length.     

Nonparametric equivalence testing with respect to the median difference

The problem of comparing two independent groups of univariate data in the sense of testing for equivalence is considered for a fully nonparametric setting. The distribution of the data within each group may be a mixture of both a continuous and a discrete component, and no assumptions are made regarding the way in which the distributions of the two groups of data may differ from each other – in particular, the assumption of a shift model is avoided. The proposed equivalence testing procedure for this scenario refers to the median of the independent difference distribution, i.e. to the median of the differences between independent observations from the test group and the reference group, respectively. The procedure provides an asymptotic equivalence test, which is symmetric with respect to the roles of ‘test’ and ‘reference’. It can be described either as a two‐one‐sided‐tests (TOST) approach, or equivalently as a confidence interval inclusion rule. A one‐sided variant of the approach can be applied analogously to non‐inferiority testing problems. The procedure may be generalised to equivalence testing with respect to quantiles other than the median, and is closely related to tolerance interval type inference. Copyright © 2009 John Wiley & Sons, Ltd.     

Statistical evaluation of individual bioequivalence

Drug switchability requires the evidence of individual bioequivalence which -refers to the comparison of the closeness between the two distributions of the pharmacokinetic (PK) responses from the same subject obtained under the repeated administrations of the test and reference formulations. Advantages and drawbacks of the current statistical procedures for assessment of individual bioequivalence are discussed with emphasis on the aggregate-based criteria, An intersection-union test based on disaggregate criteria is proposed for the evaluation of individual bioequivalence. In addition, a modified aggregated criterion is suggested to overcome the drawbacks suffered by aggregate criteria. The relationships among different criteria are examined, and the performance of the procedures will be compared. A numerical example is given to illustrate the proposed procedures.     

Multiple Hypotheses Testing with Weights

In this paper we offer a multiplicity of approaches and procedures for multiple testing problems with weights. Some rationale for incorporating weights in multiple hypotheses testing are discussed. Various type-I error-rates and different possible formulations are considered, for both the intersection hypothesis testing and the multiple hypotheses testing problems. An optimal per family weighted error-rate controlling procedure a la Spjotvoll (1972) is obtained. This model serves as a vehicle for demonstrating the different implications of the approaches to weighting. Alternative approach es to that of Holm (1979) for family-wise error-rate control with weights are discussed, one involving an alternative procedure for family-wise error-rate control, and the other involving the control of a weighted family-wise error-rate. Extensions and modifications of the procedures based on Simes (1986) are given. These include a test of the overall intersec tion hypothesis with general weights, and weighted sequentially rejective procedures for testing the individual hypotheses. The false discovery rate controlling approach and procedure of Benjamini & Hochberg (1995) are extended to allow for different weights.