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.     

Carryover negligibility and relevance in bioequivalence studies

The carryover effect is a recurring issue in the pharmaceutical field. It may strongly influence the final outcome of an average bioequivalence study. Testing a null hypothesis of zero carryover is useless: not rejecting it does not guarantee the non‐existence of carryover, and rejecting it is not informative of the true degree of carryover and its influence on the validity of the final outcome of the bioequivalence study. We propose a more consistent approach: even if some carryover is present, is it enough to seriously distort the study conclusions or is it negligible? This is the central aim of this paper, which focuses on average bioequivalence studies based on 2 × 2 crossover designs and on the main problem associated with carryover: type I error inflation. We propose an equivalence testing approach to these questions and suggest reasonable negligibility or relevance limits for carryover. Finally, we illustrate this approach on some real datasets. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

A robust variable screening method for high-dimensional data

In practice, the presence of influential observations may lead to misleading results in variable screening problems. We, therefore, propose a robust variable screening procedure for high-dimensional data analysis in this paper. Our method consists of two steps. The first step is to define a new high-dimensional influence measure and propose a novel influence diagnostic procedure to remove those unusual observations. The second step is to utilize the sure independence screening procedure based on distance correlation to select important variables in high-dimensional regression analysis. The new influence measure and diagnostic procedure that we developed are model free. To confirm the effectiveness of the proposed method, we conduct simulation studies and a real-life data analysis to illustrate the merits of the proposed approach over some competing methods. Both the simulation results and the real-life data analysis demonstrate that the proposed method can greatly control the adverse effect after detecting and removing those unusual observations, and performs better than the competing methods.     

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.     

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.     

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.     

CONSTRUCTION OF CROSSOVER DESIGNS WITH CORRELATED ERRORS

Crossover experiments are widely used, particularly where a sequence of treatments is given to subjects. Correlations between observations on the same subject are therefore likely and should be considered in both the design and analysis of crossover experiments. This paper presents an algorithm for the generation of efficient crossover designs with autoregressive and linear variance structures. The algorithm has been implemented as a module in the experimental design generation package CycDesigN (Release 3.0; CycSoftware, Hamilton, New Zealand). Output from the algorithm is compared with earlier work. Some results are given from the analysis of a crossover experiment assuming correlated errors.     

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.     

Bayesian estimation and influence diagnostics of generalized partially linear mixed-effects models for longitudinal data

This paper develops a Bayesian approach to obtain the joint estimates of unknown parameters, nonparametric functions and random effects in generalized partially linear mixed models (GPLMMs), and presents three case deletion influence measures to identify influential observations based on the φ-divergence, Cook's posterior mean distance and Cook's posterior mode distance of parameters. Fisher's iterative scoring algorithm is developed to evaluate the posterior modes of parameters in GPLMMs. The first-order approximation to Cook's posterior mode distance is presented. The computationally feasible formulae for the φ-divergence diagnostic and Cook's posterior mean distance are given. Several simulation studies and an example are presented to illustrate our proposed methodologies.     

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.     

Practical approaches for design and analysis of clinical trials of infertility treatments: crossover designs and the Mantel–Haenszel method are recommended

Crossover designs have some advantages over standard clinical trial designs and they are often used in trials evaluating the efficacy of treatments for infertility. However, clinical trials of infertility treatments violate a fundamental condition of crossover designs, because women who become pregnant in the first treatment period are not treated in the second period. In previous research, to deal with this problem, some new designs, such as re‐randomization designs, and analysis methods including the logistic mixture model and the beta‐binomial mixture model were proposed. Although the performance of these designs and methods has previously been evaluated in large‐scale clinical trials with sample sizes of more than 1000 per group, the actual sample sizes of infertility treatment trials are usually around 100 per group. The most appropriate design and analysis for these moderate‐scale clinical trials are currently unclear. In this study, we conducted simulation studies to determine the appropriate design and analysis method of moderate‐scale clinical trials for irreversible endpoints by evaluating the statistical power and bias in the treatment effect estimates. The Mantel–Haenszel method had similar power and bias to the logistic mixture model. The crossover designs had the highest power and the smallest bias. We recommend using a combination of the crossover design and the Mantel–Haenszel method for two‐period, two‐treatment clinical trials with irreversible endpoints. Copyright © 2015 John Wiley & Sons, Ltd.     

Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions

A completely nonparametric approach to population bioequivalence in crossover trials has been suggested by Munk and Czado (1999). It is based on the Mallows (1972) metric as a nonparametric distance measure which allows the comparison between the entire distribution functions of test and reference formulations. It was shown that a separation between carry-over and period effects is not possible in the nonparametric setting. However when carry-over effects can be excluded, treatment effects can be assessed when period effects are or not. Munk and Czado (1999) proved bootstrap limit laws of the corresponding test statistics because estimation of the limiting variance of the test statistic is very cumbersome. The purpose of this paper is to investigate the small sample behavior of various bootstrap methods and to compare it with the asymptotic test obtained by estimation of the limiting variance. The percentile (PC) and bias correct- ed and accelerated (BCA) bootstrap were compared for multivariate normal and nonnormal populations. From the simulation results presented, the BCA bootstrap is found to be less conservative and provides higher power compared to the PC bootstrap, especially when skewed multivariate populations are present.     

Blinded sample size re‐estimation in crossover bioequivalence trials

In drug development, bioequivalence studies are used to indirectly demonstrate clinical equivalence of a test formulation and a reference formulation of a specific drug by establishing their equivalence in bioavailability. These studies are typically run as crossover studies. In the planning phase of such trials, investigators and sponsors are often faced with a high variability in the coefficients of variation of the typical pharmacokinetic endpoints such as the area under the concentration curve or the maximum plasma concentration. Adaptive designs have recently been considered to deal with this uncertainty by adjusting the sample size based on the accumulating data. Because regulators generally favor sample size re‐estimation procedures that maintain the blinding of the treatment allocations throughout the trial, we propose in this paper a blinded sample size re‐estimation strategy and investigate its error rates. We show that the procedure, although blinded, can lead to some inflation of the type I error rate. In the context of an example, we demonstrate how this inflation of the significance level can be adjusted for to achieve control of the type I error rate at a pre‐specified level. Furthermore, some refinements of the re‐estimation procedure are proposed to improve the power properties, in particular in scenarios with small sample sizes. Copyright © 2014 John Wiley & Sons, Ltd.     

Confidence intervals for ratios of AUCs in the case of serial sampling: a comparison of seven methods

Pharmacokinetic studies are commonly performed using the two-stage approach. The first stage involves estimation of pharmacokinetic parameters such as the area under the concentration versus time curve (AUC) for each analysis subject separately, and the second stage uses the individual parameter estimates for statistical inference. This two-stage approach is not applicable in sparse sampling situations where only one sample is available per analysis subject similar to that in non-clinical in vivo studies. In a serial sampling design, only one sample is taken from each analysis subject. A simulation study was carried out to assess coverage, power and type I error of seven methods to construct two-sided 90% confidence intervals for ratios of two AUCs assessed in a serial sampling design, which can be used to assess bioequivalence in this parameter.