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.     

On equivalence of two variances of a bivariate normal vector

A test for assessing the equivalence of two variances of a bivariate normal vector is constructed. It is uniformly more powerful than the two one-sided tests procedure and the power improvement is substantial. Numerical studies show that it has a type I error close to the test level at most boundary points of the null hypothesis space. One can apply this test to paired difference experiments or 2×2 crossover designs to compare the variances of two populations with two correlated samples. The application of this test on bioequivalence in variability is presented. We point out that bioequivalence in intra-variability implies bioequivalence in variability, however, the latter has a larger power.     

On the test of liu and chow's procedure for assessing equivalence in variability of bioavailability

Similar to Schuirmann's two one-sided tests procedure for assessment of bioequivalence in average bioavailability (Schuirmann,), Liu and Chow proposed a two one-sided tests procedure for assessment of equivalence of variability of bioavailability. Their procedure is derived based on the correlation between crossover differences and subject totals. In this paper, we examined the performance of their test procedure in terms of its test size and power for various situations where the intersubject variability and the intrasubject variability of the test drug product are relatively larger, similar, and smaller than that of the intrasubject variability of the reference drug product.     

A More Powerful Average Bioequivalence Analysis for the 2 × 2 Crossover

The 2 × 2 crossover is commonly used to establish average bioequivalence of two treatments. In practice, the sample size for this design is often calculated under a supposition that the true average bioavailabilities of the two treatments are almost identical. However, the average bioequivalence analysis that is subsequently carried out does not reflect this prior belief and this leads to a loss in efficiency. We propose an alternate average bioequivalence analysis that avoids this inefficiency. The validity and substantial power advantages of our proposed method are illustrated by simulations, and two numerical examples with real data are provided.     

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.     

Influence analysis on crossover design experiment in bioequivalence studies

Crossover designs are commonly used in bioequivalence studies. However, the results can be affected by some outlying observations, which may lead to the wrong decision on bioequivalence. Therefore, it is essential to investigate the influence of aberrant observations. Chow and Tse in 1990 discussed this issue by considering the methods based on the likelihood distance and estimates distance. Perturbation theory provides a useful tool for the sensitivity analysis on statistical models. Hence, in this paper, we develop the influence functions via the perturbation scheme proposed by Hampel as an alternative approach on the influence analysis for a crossover design experiment. Moreover, the comparisons between the proposed approach and the method proposed by Chow and Tse are investigated. Two real data examples are provided to illustrate the results of these approaches. Our proposed influence functions show excellent performance on the identification of outlier/influential observations and are suitable for use with small sample size crossover designs commonly used in bioequivalence studies. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Controlling type 1 error rate for sequential,bioequivalence studies with crossover designs

Sample size reestimation in a crossover, bioequivalence study can be a useful adaptive design tool, particularly when the intrasubject variability of the drug formulation under investigation is not well understood. When sample size reestimation is done based on an interim estimate of the intrasubject variability and bioequivalence is tested using the pooled estimate of intrasubject variability, type 1 error inflation will occur. Type 1 error inflation is caused by the pooled estimate being a biased estimator of the intrasubject variability. The type 1 error inflation and bias of the pooled estimator of variability are well characterized in the setting of a two‐arm, parallel study. The purpose of this work is to extend this characterization to the setting of a crossover, bioequivalence study with sample size reestimation and to propose an estimator of the intrasubject variability that will prevent type 1 error inflation.     

Assessment of individual bioequivalence using sufficient bootstrap procedure

This paper proposes a sufficient bootstrap method, which uses only the unique observations in the resamples, to assess the individual bioequivalence under 2 × 4 randomized crossover design. The finite sample performance of the proposed method is illustrated by extensive Monte Carlo simulations as well as a real‐experimental data set, and the results are compared with those obtained by the traditional bootstrap technique. Our records reveal that the proposed method is a good competitor or even better than the classical percentile bootstrap confidence limits.     

Viewpoint: observations on scaled average bioequivalence

The two one-sided test procedure (TOST) has been used for average bioequivalence testing since 1992 and is required when marketing new formulations of an approved drug. TOST is known to require comparatively large numbers of subjects to demonstrate bioequivalence for highly variable drugs, defined as those drugs having intra-subject coefficients of variation greater than 30%. However, TOST has been shown to protect public health when multiple generic formulations enter the marketplace following patent expiration. Recently, scaled average bioequivalence (SABE) has been proposed as an alternative statistical analysis procedure for such products by multiple regulatory agencies. SABE testing requires that a three-period partial replicate cross-over or full replicate cross-over design be used. Following a brief summary of SABE analysis methods applied to existing data, we will consider three statistical ramifications of the proposed additional decision rules and the potential impact of implementation of scaled average bioequivalence in the marketplace using simulation. It is found that a constraint being applied is biased, that bias may also result from the common problem of missing data and that the SABE methods allow for much greater changes in exposure when generic-generic switching occurs in the marketplace.     

A joint model for incomplete data in crossover trials

Crossover designs are popular in early phases of clinical trials and in bioavailability and bioequivalence studies. Assessment of carryover effects, in addition to the treatment effects, is a critical issue in crossover trails. The observed data from a crossover trial can be incomplete because of potential dropouts. A joint model for analyzing incomplete data from crossover trials is proposed in this article; the model includes a measurement model and an outcome dependent informative model for the dropout process. The informative-dropout model is compared with the ignorable-dropout model as specific cases of the latter are nested subcases of the proposed joint model. Markov chain sampling methods are used for Bayesian analysis of this model. The joint model is used to analyze depression score data from a clinical trial in women with late luteal phase dysphoric disorder. Interestingly, carryover effect is found to have a strong effect in the informative dropout model, but it is less significant when dropout is considered ignorable.     

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.     

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.     

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.     

The evaluation of bioequivalence using nonparametric procedures

Applications of nonparametric methods to the evaluation of bioequiv-alence for two treatments are presented for independent samples and for a crossover design. Included are procedures for testing for equivalence in location, in dispersion, and in general. Also presented are procedures for the calculation of confidence limits. A general strategy for the evaluation of bioequivalence is developed which involves both hypothesis testing and the calculation of confidencelimits for parameters which characterize departures from equivalene.     

On testing the bioequivalence of several treatments using the measure of distance

A studentized range test is proposed to test the hypothesis of bioequivalence of normal means in terms of a standardized distance among means. A least favourable configuration (LFC) of means to guarantee the maximum level at a null hypothesis and an LFC of means to guarantee the minimum power at an alternative hypothesis are obtained. This level and power of the test are fully independent of the unknown means and variances. For a given level, the critical value of the test under a null hypothesis can be determined. Furthermore, if the power under an alternative is also required at a given level, then both the critical value and the required sample size for an experiment can be simultaneously determined. In situations where the common population variance is unknown and the bioequivalence is the actual distance between means without standardization, a two-stage sampling procedure can be employed to find these solutions.     

Sample size determination for testing equality in frequency data under an incomplete block crossover design

When there are more than two treatments under comparison, we may consider the use of the incomplete block crossover design (IBCD) to save the number of patients needed for a parallel groups design and reduce the duration of a crossover trial. We develop an asymptotic procedure for simultaneously testing equality of two treatments versus a control treatment (or placebo) in frequency data under the IBCD with two periods. We derive a sample size calculation procedure for the desired power of detecting the given treatment effects at a nominal-level and suggest a simple ad hoc adjustment procedure to improve the accuracy of the sample size determination when the resulting minimum required number of patients is not large. We employ Monte Carlo simulation to evaluate the finite-sample performance of the proposed test, the accuracy of the sample size calculation procedure, and that with the simple ad hoc adjustment suggested here. We use the data taken as a part of a crossover trial comparing the number of exacerbations between using salbutamol or salmeterol and a placebo in asthma patients to illustrate the sample size calculation procedure.     

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.     

The impact of missing values in the concentration–time curve on the assessment of bioequivalence

A complication that may arise in some bioequivalence studies is that of ‘incomplete subject profiles’, caused by missing values that occur at one or more sampling points in the concentration–time curve for some study subjects. We assess the impact of incomplete subject profiles on the assessment of bioequivalence in a standard two‐period crossover design. The specific aim of the investigation is to assess the impact of four different patterns of missing concentration values on the coverage level of a 90% nominal two‐sided confidence interval for the ratio of geometric means and then to consider the impact on the probability of concluding bioequivalence. An overall conclusion from the results is that random missingness – that is, missingness for reasons unrelated to the bioavailability of the formulation involved or, more generally, to any aspect of the study design and conduct – has a damaging effect on the study conclusions only when the number of missing values is fairly large. On the other hand, a missingness pattern that potentially has a very damaging effect on the study conclusions is that which arises when values are missing ‘late’ in the concentration–time curve. Copyright © 2005 John Wiley & Sons, Ltd     

Implementation of an adaptive group sequential design in a bioequivalence study

The study design was a multi-center, multiple-dose, randomized, open-label, 2 x 2 crossover study in patients with advanced solid tumors. Each patient was randomized to receive the test formulation or the reference formulation of the drug. The primary objective of the study was to demonstrate the bioequivalence of the test formulation T relative to the reference formulation R. The primary pharmacokinetic endpoints were AUC and Cmax. Since there were different bioequivalence criteria, different endpoints, with different and highly variable coefficients of variation, an adaptive design with a stopping rule for early establishing the bioequivalence as well as early stopping for futility with a flexible information-based monitoring based on error spending approach was implemented to manage uncertainty in assumptions of variability and expected slow enrollment rates.