A Confidence Region Approach for Assessing Equivalence in Variability of Bioavailability

The problem for assessment of equivalence in variability of bioavailability between two drug products is considered. Similar to the method for assessing bioequivalence in average bioavailability proposed by Chow and Shao (1990), an exact confidence region approach is derived when the intersubject variance is known. When the intersubject variance is unknown, a large sample approximation is considered. The proposed method for assessing equivalence of variability of bioavailability appears to be asymptotically uncorrelated with that of Chow and Shao (1990) for average bioavailability. As a result, the proposed method in conjunction with the method proposed by Chow and Shao (1990) constitutes a confidence region approach for assessing population bioequivalence. An example concerning a bioequivalence trial with 24 healthy volunteers is provided to illustrate the proposed method.     

Interval Estimation for Ratios in Bioequivalence Trials

The problem for assessment of equivalence in variability of bioavailability between two drug products is considered. An exact confidence region for the ratio between intrasubject variabilities is derived when the intersubject variance is known. When the intersubject variance is unknown, a large sample approximation is considered. The proposed method for assessing equivalence in variability of bioavailability appears to be asymptotically uncorrelated with the sample mean ratio for average bioavailabilty. As a result, the proposed method in conjunction with the sample mean ratio method can be utilized for assessing population bioequivalence. An example concerning a bioequivalence trial with 24 healthy volunteers is presented to illustrate the proposed method.     

An Alternative Approach for the Assessment of Bioequivalence Between Two Formulations of a Drug

The problem of the assessment of bioequivalence between a test formulation (T) and a reference formulation (R) of a drug using a two-way crossover experiment is considered. To claim bioequivalence between two formulations, it is required by the United States Food and Drug Administration (FDA) to demonstrate that the true ratio of means μT/μR of pharmacokinetic parameters of concern falls within some reasonable limits (e.g., (80%, 120%)) with certain assurance. A commonly used approach is to construct an approximate 90% confidence interval for μT/μR and compare it with (80%, 120%). In this paper, an exact approach according to the FDA's criteria is proposed. The proposed procedure is derived by constructing an exact confidence region (an ellipse) for (μR, μT) and comparing it with the region bounded by μT = 0.8 μR and μT = 1.2 μR. Bioequivalence is concluded if the ellipse is within the critical region.     

On Assessment of Bioequivalence for Drugs with Negligible Plasma Levels

In bioavailability studies, bioequivalence between drug products is usually determined based on some pharmacokinetic responses such as area under the blood or plasma concentration-time curve and maximum concentration. For some drug products, however, we may have negligible plasma levels because their intended routes of administration. In this case, assessment of bioequivalence between drug products of this kind may be established using clinical endpoints such as therapeutic response and time to the onset of a therapeutic response. In this paper, we propose two procedures which modify the method of generalized estimating equations (Liang and Zeger, 1986) and the proportional hazard models for paired failure times to assess bioequivalence between two drug products under the structure of a standard two-sequence, two-period crossover design. An example concerning a bioequivalence trial for albuterol metered dose inhaler indicated for acute bronchospasm (Herson, 1991) is used to illustrate the proposed procedures.     

Testing the Equality of the Intra‐Subject Treatment Covariance Matrices in a Crossover Study

In a 2 × 2 crossover bioavailability study, the sets of estimates of the pharmacokinetic parameters quite often have a symmetric covariance structure between the two treatments. For testing the equality of the intra‐subject covariance matrices for the two treatments in such studies, we suggest in this paper some statistical tests. When the response vectors are bivariate, we propose an exact test. Since the statistical procedures depend on the assumption of a symmetric covariance structure between the two treatments, we put forth some statistical tests for this assumption. We then apply the discussed tests to real data from a crossover bioavailability trial.     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The calculation of probabilities in rejecting bioequivalence

Methods are given for deriving the lower and upper bounds of the probabilities for rejecting bioequivalence. Except when two formulations are nearly equal, the lower and upper bounds become very close and thus give an exact probability. The methods can be used in two- or higher-way crossover or parallel groups designs. One can also determine the sample size from the equations. The results are compared with the probabilities obtained by simulation.     

The empirical coverage of confidence intervals: Point estimates and confidence intervals for confidence levels

Many confidence intervals calculated in practice are potentially not exact, either because the requirements for the interval estimator to be exact are known to be violated, or because the (exact) distribution of the data is unknown. If a confidence interval is approximate, the crucial question is how well its true coverage probability approximates its intended coverage probability. In this paper we propose to use the bootstrap to calculate an empirical estimate for the (true) coverage probability of a confidence interval. In the first instance, the empirical coverage can be used to assess whether a given type of confidence interval is adequate for the data at hand. More generally, when planning the statistical analysis of future trials based on existing data pools, the empirical coverage can be used to study the coverage properties of confidence intervals as a function of type of data, sample size, and analysis scale, and thus inform the statistical analysis plan for the future trial. In this sense, the paper proposes an alternative to the problematic pretest of the data for normality, followed by selection of the analysis method based on the results of the pretest. We apply the methodology to a data pool of bioequivalence studies, and in the selection of covariance patterns for repeated measures data.     

A method for assessing the equivalence of repeated measures of muscle fatigue rates estimated from EMG power spectrum analysis

This report addresses some of the statistical problems that are encountered when the test/retest recording reliability of fatigue-related parameters of the EMG power spectrum is evaluated. It can be shown that some classical methods for reliability assessment such as correlational procedures are unsuitable for this purpose. Because the EMG power spectrum fatigue parameter depends on metabolic changes in muscle tissue, it is suggested that methods similar to those used in the evaluation of bioequivalence studies may be more appropriate in the assessment of such test/retest results.     

Impairment of proprioception after whiplash injury

Whiplash injury usually occurs in traffic accidents. Persons experienced this injury might have an impairment of proprioception clinically expressed as inability to determine the exact position of their heads. The aim of this study was to examine the loss of proprioception in people who had a whiplash injury. The study included 60 subjects with cervical spine injury, aged 20 to 50 years and 60 healthy volunteers matched by sex and age. The instrument used for cervical spine mobility assessment was the Cervical Measurement System (CMS), which determines the ability of subjects to return their head in the exact position as it was before they turned it 30 degrees left or right. Patients with cervical spine injury showed significant impairment of proprioception in comparison with healthy subjects (P < 0.001). The results support the hypothesis that subject with recent cervical spine injury have incorrect perception of their head position. Therefore, their rehabilitation should include the correction of proprioception and head coordination.     

Tests for establishing compatibility of an observed genotype distribution with Hardy-Weinberg equilibrium in the case of a biallelic locus

The classical chi(2)-procedure for the assessment of genetic equilibrium is tailored for establishing lack rather than goodness of fit of an observed genotype distribution to a model satisfying the Hardy-Weinberg law, and the same is true for the exact competitors to the large-sample procedure, which have been proposed in the biostatistical literature since the late 1930s. In this contribution, the methodology of statistical equivalence testing is adopted for the construction of tests for problems in which the assumption of approximate compatibility of the genotype distribution actually sampled with Hardy-Weinberg equilibrium (HWE) plays the role of the alternative hypothesis one aims to establish. The result of such a construction highly depends on the choice of a measure of distance to be used for defining an indifference zone containing those genotype distributions whose degree of disequilibrium shall be considered irrelevant. The first such measure proposed here is the Euclidean distance of the true parameter vector from that of a genotype distribution with identical allele frequencies being in strict HWE. The second measure is based on the (scalar) parameter of the distribution first introduced into the present context by Stevens (1938, Annals of Eugenics 8, 377-383). The first approach leads to a nonconditional test (which nevertheless can be carried out in a numerically exact way), the second to an exact conditional test shown to be uniformly most powerful unbiased (UMPU) for the associated pair of hypotheses. Both tests are compared in terms of the exact power attained against the class of those specific alternatives under which HWE is strictly satisfied.     

Effect of enantiomerism on the bioequivalence of a new ibuprofen 600-mg tablet formulation obtained by roller compaction

The bioequivalence of a new ibuprofen 600-mg film-coated tablet obtained by roller compaction was studied in a crossover study with 22 healthy volunteers. Bioequivalence was analyzed based on (a) the S-enantiomer, (b) the R-enantiomer, and (c) the sum of both enantiomers (representing the results of an achiral assay). The bioequivalence conclusion for ibuprofen products should be based not only on AUC and Cmax but also on tmax since tmax is related to the onset of action. However, it is not possible to ensure if bioequivalence has been demonstrated for tmax as regulators have not defined the acceptance range for the difference between medians of tmax in those cases, where tmax is clinically relevant. In this study, it was possible to conclude bioequivalence for tmax based on S-ibuprofen, though this conclusion might be questioned if the decision is based on R-ibuprofen or the achiral method.     

A two-stage procedure for bioequivalence studies

A two-stage experimental procedure for bioequivalence studies is proposed. The procedure is based on the idea that information from a first-stage experiment can be used to form a predictive distribution for the outcome of a second-stage experiment, thus permitting an assessment of the probability of a successful overall outcome. A systematic numerical study of a variety of possible strategies results in the identification of procedures that lead to substantial increases in efficiency compared with single-stage studies.     

Sensitivity analysis for m-estimates, tests, and confidence intervals in matched observational studies

Huber's m-estimates use an estimating equation in which observations are permitted a controlled level of influence. The family of m-estimates includes least squares and maximum likelihood, but typical applications give extreme observations limited weight. Maritz proposed methods of exact and approximate permutation inference for m-tests, confidence intervals, and estimators, which can be derived from random assignment of paired subjects to treatment or control. In contrast, in observational studies, where treatments are not randomly assigned, subjects matched for observed covariates may differ in terms of unobserved covariates, so differing outcomes may not be treatment effects. In observational studies, a method of sensitivity analysis is developed for m-tests, m-intervals, and m-estimates: it shows the extent to which inferences would be altered by biases of various magnitudes due to nonrandom treatment assignment. The method is developed for both matched pairs, with one treated subject matched to one control, and for matched sets, with one treated subject matched to one or more controls. The method is illustrated using two studies: (i) a paired study of damage to DNA from exposure to chromium and nickel and (ii) a study with one or two matched controls comparing side effects of two drug regimes to treat tuberculosis. The approach yields sensitivity analyses for: (i) m-tests with Huber's weight function and other robust weight functions, (ii) the permutational t-test which uses the observations directly, and (iii) various other procedures such as the sign test, Noether's test, and the permutation distribution of the efficient score test for a location family of distributions. Permutation inference with covariance adjustment is briefly discussed.     

Age-adjusted exact trend tests in the event of rare occurrences

Preclinical animal carcinogenicity studies are usually concerned with testing the statistical significance of a dose-response relationship. When the response consists of a rare event such as the development of a certain type of tumor, exact statistical methods are often employed. The exact randomization trend test based on the multivariate hypergeometric distribution is less powerful in the presence of treatment-related risks other than the specified response. Particularly, the loss of power becomes more pronounced when competing risks cause progressively higher mortality rates with increasing dose, which is usual in practice. An age-adjusted form of the randomization test is proposed to adjust for this effect. Permutational distribution for Peto's cause-of-death (COD) test is also explored and compared with its asymptotic counterpart by simulation. The use of COD information has been a controversial issue due to the subjectivity in the pathologists' determinations as well as for economic reasons. The proposed age-adjusted exact test does not require COD, and it is shown to compare favorably to the COD tests via an extensive Monte Carlo simulation. Applications of the methods to two real data sets are included.     

Establishing bioequivalence of racemic venlafaxine formulations using stereoselective assay method: Is it necessary?

A bioequivalence study for venlafaxine generic formulation was conducted as an open label, balanced, randomized, two‐way crossover, single‐dose study. In this study, a comparison of various pharmacokinetic parameters of venlafaxine hydrochloride 150 mg modified release capsules of Ranbaxy and EFEXOR®‐XR 150 mg capsules of Wyeth, in healthy, adult, male, human subjects under fasting condition was performed to conclude bioequivalence. Venlafaxine and its major active metabolite O‐desmethylvenlafaxine (ODV) are racemates. The “(S)‐(+)” and “(R)‐(−)” enantiomers of venlafaxine and ODV are established as being active. Hence, subject samples were analyzed using nonstereoselective and stereoselective assay methods. Both (S)‐(+) and (R)‐(−) enantiomers of venlafaxine and ODV showed similar absorption and disposition. The 90% confidence intervals for venlafaxine, (R)‐(−)‐venlafaxine as well as (S)‐(+)‐venlafaxine were within acceptance range concluding bioequivalence. The results obtained by stereoselective assay were comparable to the nonstereoselective analysis, as sum of concentrations of (S)‐(+)‐ and (R)‐(−)‐enantiomers of venlafaxine and ODV. The mean (S)‐(+)/(R)‐(−) ratios of the enantiomers of venlafaxine and ODV at various time points were consistent in the study subjects. Therefore, the estimation of venlafaxine and ODV using nonstereoselective assay method is effective in distinguishing formulation differences (if any) in bioequivalence studies in a cost‐effective manner. Chirality, 2011. © 2011 Wiley‐Liss, Inc.     

Exact Test of Uniform Association in a 2 × 3 Table

In a similar approach to Fisher's exact test of independence (null association) for a 2 × 2 table, this note gives an exact test of uniform association for a 2 × 3 table.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Exact Distribution of the Anderson-Kannemann Statistic

The Anderson-Kannemann test is a rank test for treatment effects in a randomized block design with K treatments and N blocks. In this paper, an algorithm for computing the exact distribution of the Anderson-Kannemann test statistic under the null hypothesis is deduced. Then, the exact distribution is compared with the asymptotic χ²-distribution, and it is shown that the exact distribution is approximated fairly well by the asymptotic distribution. Tables of the exact distribution are given for K = 3, N = 3(1)15; K = 4, N = 3(1)11; K = 5, N = 3(1)7; and K = 6, N = 3(1)5.