Sample size determination for proving equivalence based on the ratio of two means for normally distributed data.

Equivalence trials aim to demonstrate that two treatments do not differ by more than a prespecified clinically irrelevant amount. We consider the problem when equivalence is defined in terms of the ratio of population means and the original (untransformed) data are normally distributed. Application of the intersection-union principle to the test proposed by Sasabuchi results in a two one-sided tests procedure of size alpha. We give the associated 100 (1-2 alpha) per cent confidence interval and derive the exact methods for calculation of power and sample sizes for the parallel group design and the two-period cross-over. We present tables and figures of required sample sizes and achieved power.

Characteristics of transcranial Doppler signal enhancement using a phospholipid-containing echocontrast agent.

BACKGROUND AND PURPOSE: Ultrasound attenuation caused by the skull is a major limitation of transcranial Doppler. Echocontrast agents (EAs) may solve this problem. The aim of the present study was to investigate the characteristics of a new echocontrast agent (BY963) containing air bubbles stabilized by phospholipids. METHODS: Nine healthy volunteers received three different doses (2.5, 5.0, and 10 mL) of BY963 at an injection rate of 0.25 mL/s. The Doppler signal amplitude obtained from the middle cerebral artery was recorded with a 2-MHz pulsed-wave Doppler system. After complete decay of the signal enhancement, upward stroking of the veins of the upper arm was performed to evaluate the stability of the EA in the venous system. RESULTS: A dose-dependent increase of at least 30 dB in the Doppler signal amplitude lasted 19 to 47, 35 to 64, and 48 to 126 heart cycles (68% range) after 2.5, 5.0, and 10 mL EA, respectively. In 6 cases, there was a biphasic increase in EA enhancement. Upward stroking of the forearm, in general 12 to 18 minutes after administration, caused a Doppler signal enhancement of at least 30 dB in 6 cases. CONCLUSIONS: Each injection of BY963 caused a diagnostically relevant Doppler signal enhancement. A considerable amount of EA remained stable in the venous system for at least 12 minutes. The biphasic dose-response fits to models of dilution-indicator theory and indicates free recirculation, as well as a nonlinear washout curve.

Interobserver agreement: Cohen's kappa coefficient does not necessarily reflect the percentage of patients with congruent classifications.

A widely accepted approach to evaluate interrater reliability for categorical responses involves the rating of n subjects by at least 2 raters. Frequently, there are only 2 response categories, such as positive or negative diagnosis. The same approach is commonly used to assess the concordant classification by 2 diagnostic methods. Depending on whether one uses the percent agreement as such or corrected for that expected by chance, i.e. Cohen's kappa coefficient, one can get quite different values. This short communication demonstrates that Cohen's kappa coefficient of agreement between 2 raters or 2 diagnostic methods based on binary (yes/no) responses does not parallel the percentage of patients with congruent classifications. Therefore, it may be of limited value in the assessment of increases in the interrater reliability due to an improved diagnostic method. The percentage of patients with congruent classifications is of easier clinical interpretation, however, does not account for the percent of agreement expected by chance. We, therefore, recommend to present both, the percentage of patients with congruent classifications, and Cohen's kappa coefficient with 95% confidence limits.

Choice of student's t- or Wilcoxon-based confidence intervals for assessment of average bioequivalence.

An open question in the analysis of average bioequivalence is whether the nonparametric (Wilcoxon) or parametric (t) approaches to two one-sided tests is preferable. Previous work has made particular distributional assumptions as to the distribution of AUC and C(max). Instead, we simulate data according to a pharmacokinetic model for an immediate-release formulation. We find that both approaches have estimated level consistent with the nominal 5%. The only concern is a possible anticonservativeness of the parametric approach for C(max). Further, the nonparametric approach is consistently less powerful than the parametric for the cases studied.

Relative bioavailability of different valproic acid formulations.

Relative bioavailability of valproic acid after oral administration of 2 Convulsofin (test) tablets each containing 300 mg calcium valproate (263.4 mg valproic acid) was studied versus 2 references (2 dragees each of 300 mg calcium valproate, ref.A, 600 mg sodium valproate in liquid form (258.7 mg valproic acid), ref.B). The controlled, randomized, clinical trial was performed in 16 healthy volunteers (12 males, 4 females, body weight 58-100 kg, Broca index 0.85-1.15) according to a 3-period changeover design with 7 days wash-out between 2 periods. Valproic acid was measured in serum with a GC method. Pharmacokinetic evaluation was done by compartment free methods. Test was considered bioequivalent with ref.A or ref.B with reference to extent of absorption if the 90% confidence interval of their AUC ratio was within the range of 0.80-1.25, and with respect to rate of absorption if the 90% confidence intervals of Cmax/AUC ratios were within 0.70-1.43. The point estimators (90% confidence limits) of the AUC ratios of test/ref.A and test/ref.B were 0.952 (0.882-1.028) and 1.063 (0.989-1.141), respectively. The point estimators (90% confidence limits) of Cmax/AUC ratios were 1.005 (0.923-1.094, test/ref.A) and 0.915 (0.845-0.991, test/ref.B). The following Cmax ratios were calculated: 0.957 (0.866-1.057, test/ref.A) and 0.972 (0.886-1.067, test/ref.B). No serious and unexpected adverse events were observed during the clinical trial. Test was bioequivalent with the 2 reference formulations ref.A and ref.B with respect to extent and rate of absorption. However, according to the secondary criterion tmax test tablets were more rapidly bioavailable than ref.A dragees (tmax-difference: -2.6 (-4.8 to -0.3 h) but more slowly (tmax-difference:+0.8 (-1.3 to +2.9 h) than ref.B juice.

Reference tables for the intrasubject coefficient of variation in bioequivalence studies.

Bioequivalence studies are usually performed as crossover studies and, therefore, information on the intrasubject coefficient of variation is needed for sample size planning. However, this information is usually not accessible in publications on bioequivalence studies, and only the pooled inter- and intrasubject coefficient of variation for either test or reference formulation is reported. It is the purpose of the present communication to provide reference values of the intrasubject coefficient of variation for various previously investigated drugs. The presentation includes pertinent pharmacokinetic characteristics for immediate- and extended-release formulations in single- and multiple-dose crossover studies.

Presentation of the intrasubject coefficient of variation for sample size planning in bioequivalence studies.

Bioequivalence studies are generally performed as crossover studies and, therefore, information on the intrasubject coefficient of variation is needed for sample size planning. Unfortunately, this information is usually not presented in publications on bioequivalence studies, and only the pooled inter- and intrasubject coefficient of variation for either test or reference formulation is reported. Thus, the essential information for sample size planning of future studies is not made available to other researchers. In order to overcome such shortcomings, the presentation of results from bioequivalence studies should routinely include the intrasubject coefficient of variation. For the relevant coefficients of variation, theoretical background together with modes of calculation and presentation are given in this communication with particular emphasis on the multiplicative model.

Effect of changing the bioequivalence range from (0.80, 1.20) to (0.80, 1.25) on the power and sample size.

International harmonization of guidelines for bioequivalence assessment has led to a wide acceptance of the multiplicative model for the extent and rate characteristics AUC and Cmax and--in consistency with this--of the bioequivalence range (0.80, 1.25). The effect of this change from (0.80, 1.20) on the power of the two one-sided test procedure and the sample sizes based thereon is investigated as a function of the within-subject coefficient of variation (CV) and the ratio mu T/mu R of expected medians for test and reference. The relative reduction in sample size is practically zero for mu T/mu R < or = 0.9 and then gradually increases as mu T/mu R approaches 1.2. At mu T/mu R = 1, the reduction is up to 20%. For a fixed ratio mu T/mu R this reduction increases with the coefficient of variation, reaching a plateau at a CV of about 25%.

Sample size determination for bioequivalence assessment using a multiplicative model.

In bioequivalence studies Cmax and AUC serve as the primary pharmacokinetic characteristics of rate and extent of absorption. Based on pharmacokinetic relationships and on empirical evidence, the distribution of these characteristics corresponds to a multiplicative model, which implies a logarithmic normal distribution in the case of a parametric analysis. Hence, consideration is given to exact and approximate formulas of sample sizes in the case of a multiplicative model.

Sample size determination: extended tables for the multiplicative model and bioequivalence ranges of 0.9 to 1.11 and 0.7 to 1.43.

For the two-period crossover design and a multiplicative model (logarithmic normal distribution) the decision procedure of choice is based on the inclusion of the shortest 90%-confidence interval for the ratio of expected medians for test and reference in the equivalence range. This inclusion rule is equivalent to the two one-sided tests procedure. Sample sizes based on the power of the latter have been given by Diletti et al. [1991] for an equivalence range of 0.8 to 1.25. Corresponding tables for the tighter equivalence range of 0.9 to 1.11 as well as for the wider range of 0.7 to 1.43 are given in this amendment.

Presentation of results from bioequivalence studies.

Based on general guidelines and requirements for the design and analysis of bioequivalence studies, specific recommendations are made for the presentation of results, both in tabular and graphical form. This is done by means of two examples, one of a single-dose study and one of a multiple-dose study. The recommendations in this paper are twofold. Firstly, a complete and rather detailed presentation of results is given, which practically corresponds to the standard of research reports. Secondly, a subset of this is suggested for publication. It gives the essential results for bioequivalence assessment in a standardized form. From an editorial point of view, it would be highly appreciated if the papers submitted for publication were always accompanied by a complete presentation including the individual concentration/time data and the various steps of calculation. This would speed up peer review and ultimately improve and harmonize the standard of bioequivalence publications.

A distribution-free procedure for the statistical analysis of bioequivalence studies.

In bioequivalence assessment, the consumer risk of erroneously accepting bioequivalence is of primary concern. In order to control the consumer risk, the decision problem is formulated with bioinequivalence as hypothesis and bioequivalence as alternative. In the parametric approach, a split into two one-sided test problems and application of two-sample t-tests have been suggested. Rejection of both hypotheses at nominal alpha-level is equivalent to the inclusion of the classical (shortest) (1-2 alpha) 100%-confidence interval in the bioequivalence range. This paper demonstrates that the rejection of the two one-sided hypotheses at nominal alpha-level by means of nonparametric Mann-Whitney-Wilcoxon tests is equivalent to the inclusion of the corresponding distribution-free (1-2 alpha) 100%-confidence interval in the bioequivalence range. This distribution-free (nonparametric) approach needs weaker model assumptions and hence presents an alternative to the parametric approach.

Sample size determination for bioequivalence assessment by means of confidence intervals.

The statistical analysis of bioequivalence assessment has been consolidated in recent years through the work of Schuirmann [1987], Westlake [1988] and Hauschke et al. [1990], and this has been reflected in the CPMP Note for Guidance on Bioavailability and Bioequivalence and in the joint recommendations of the APV (International Association for Pharmaceutical Technology) and ZL (Central Laboratories of German Pharmacists) during a recent workshop in support of EC-Guidelines [Blume et al. 1990]. Since the decision procedure based on the inclusion of the shortest 90%-confidence interval in the bioequivalence range is the procedure of choice, and as this is equivalent to the two one-sided tests procedure, the sample size determination is based on the power of the latter. Following the approach of Phillips [1990] for the additive model, corresponding nomograms for the more relevant multiplicative model are given in this paper for various ratios of the expected means for test and reference and various coefficients of variation.

Sample size determination: extended tables for the multiplicative model and bioequivalence ranges of 0.9 to 1.11 and 0.7 to 1.43.

For the two-period crossover design and a multiplicative model (logarithmic normal distribution) the decision procedure of choice is based on the inclusion of the shortest 90%-confidence interval for the ratio of expected medians for test and reference in the equivalence range. This inclusion rule is equivalent to the two one-sided tests procedure. Sample sizes based on the power of the latter have been given by Diletti et al. [1991] for an equivalence range of 0.8 to 1.25. Corresponding tables for the tighter equivalence range of 0.9 to 1.11 as well as for the wider range of 0.7 to 1.43 are given in this amendment.

Presentation of results from bioequivalence studies.

Based on general guidelines and requirements for the design and analysis of bioequivalence studies, specific recommendations are made for the presentation of results, both in tabular and graphical form. This is done by means of two examples, one of a single-dose study and one of a multiple-dose study. The recommendations in this paper are twofold. Firstly, a complete and rather detailed presentation of results is given, which practically corresponds to the standard of research reports. Secondly, a subset of this is suggested for publication. It gives the essential results for bioequivalence assessment in a standardized form. From an editorial point of view, it would be highly appreciated if the papers submitted for publication were always accompanied by a complete presentation including the individual concentration/time data and the various steps of calculation. This would speed up peer review and ultimately improve and harmonize the standard of bioequivalence publications.

Sample size determination for bioequivalence assessment by means of confidence intervals.

The statistical analysis of bioequivalence assessment has been consolidated in recent years through the work of Schuirmann [1987], Westlake [1988] and Hauschke et al. [1990], and this has been reflected in the CPMP Note for Guidance on Bioavailability and Bioequivalence and in the joint recommendations of the APV (International Association for Pharmaceutical Technology) and ZL (Central Laboratories of German Pharmacists) during a recent workshop in support of EC-Guidelines [Blume et al. 1990]. Since the decision procedure based on the inclusion of the shortest 90%-confidence interval in the bioequivalence range is the procedure of choice, and as this is equivalent to the two one-sided tests procedure, the sample size determination is based on the power of the latter. Following the approach of Phillips [1990] for the additive model, corresponding nomograms for the more relevant multiplicative model are given in this paper for various ratios of the expected means for test and reference and various coefficients of variation.

A distribution-free procedure for the statistical analysis of bioequivalence studies.

In bioequivalence assessment, the consumer risk of erroneously accepting bioequivalence is of primary concern. In order to control the consumer risk, the decision problem is formulated with bioinequivalence as hypothesis and bioequivalence as alternative. In the parametric approach, a split into two one-sided test problems and application of two-sample t-tests have been suggested. Rejection of both hypotheses at nominal alpha-level is equivalent to the inclusion of the classical (shortest) (1-2 alpha) 100%-confidence interval in the bioequivalence range. This paper demonstrates that the rejection of the two one-sided hypotheses at nominal alpha-level by means of nonparametric Mann-Whitney-Wilcoxon tests is equivalent to the inclusion of the corresponding distribution-free (1-2 alpha) 100%-confidence interval in the bioequivalence range. This distribution-free (nonparametric) approach needs weaker model assumptions and hence presents an alternative to the parametric approach.

Validation of the in-vitro dissolution method used for a new sustained-release theophylline pellet formulation.

The in-vitro release method used within the scope of the galenical development of Euphylong pellets and subsequently employed on a routine basis is described. The suitability of the method is demonstrated by means of the validation procedure. The release system used is based on the standard USP XXI apparatus (paddle method) and consists of three essential components: a) dissolution apparatus (paddle model). b) measuring unit (spectrophotometer), c) data acquisition and analysis system. It is demonstrated that in-vitro release rates of theophylline from Euphylong pellets can be accurately investigated with other models as well, since the course of release is not affected by pH value, buffer capacity, surface tension, turbulence of the dissolution medium or agitation by the apparatus. The accuracy of the method as well as its precision and ruggedness are investigated and described. The influences arising from the withdrawal of specimens from the bulk product and of sampling from the release vessel are discussed. The accuracy of the analytical records generated by the computer system is shown and the ruggedness of the analytical program investigated under "worst case conditions". The documentation of the results obtained is described and examined with respect to reliability in the face of system and operating errors. All pertinent guidelines (GAP, FIP guidelines for dissolution testing, etc.) are taken into consideration and their relevance to the above investigation assessed.

Generalization of distribution--free confidence intervals for bioavailability ratios.

The confidence interval approach to bioavailability assessment depends first on selection of the confidence level, usually 95%, and then determination of the confidence limits for the expected bioavailability ratio AUC(Test)/AUC(Reference). In practice, however, it is sometimes of greater interest to know the probability that the expected bioavailability will fall below a critical value, for example 0.75, or within a clinically set bioequivalence range, for example 0.80 to 1.25. Up to now, posterior probability distributions have been suggested, based on classical analysis of variance (ANOVA) with its rather restrictive assumptions, including that of a (logarithmic) normal distribution. In this report, a distribution-free confidence interval based on the Wilcoxon signed-rank statistic has been generalized so that confidence probabilities can be obtained for any given confidence limits. In the case of unimodal and almost symmetrical sampling distributions, the results obtained are very similar to those of the ANOVA-based posterior probability distribution. However, skewed or multimodal sampling distributions are better reflected by the proposed distribution-free method, and more valid information is obtained in these cases, as demonstrated by examples.

Statistical analysis of bioavailability studies: parametric and nonparametric confidence intervals.

For a two-way cross-over design, which appears to be the most common experimental design in bioavailability studies, 95%-confidence limits for expected bioavailability can be obtained by classical analysis of variance (ANOVA). If symmetry of the confidence interval is desired about zero (differences) or unity (ratios) rather than about the corresponding point estimator, Westlake's modification can be used. Two nonparametric methods and their adaptations to bioavailability ratios are reviewed, one based on Wilcoxon's signed rank test (Tukey), and the other on Pitman's permutation test. The necessary assumptions and the merits of these procedures are discussed. The methods are illustrated by an example of a comparative bioavailability study. A FORTRAN program facilitating the procedures is available from the authors upon request.