Detection of outlying data in bioavailability/bioequivalence studies

A robust method for the assessment of average bioequivalence in the presence of outliers and skewness

PURPOSE

In this paper, we propose a robust Bayesian method for the assessment of average bioequivalence based on data from conventional crossover studies. We evaluate and motivate empirically the need for robust methods in bioequivalence studies by comparing the results of robust and conventional statistical methods in a large data pool of bioequivalence studies.

METHODS

Robustness of the statistical methodology is achieved by replacing the normal distributions for residuals in the linear mixed model with skew-t distributions. In this way, the statistical model can accommodate skew and heavy-tailed data, particularly outliers, yielding robust statistical inference without the need for excluding outliers from the analysis. We performed a simulation study to investigate and compare the performance of the robust and conventional models.

RESULTS

Our study shows that in some trials, the distribution of residuals is skew and heavy-tailed. In the presence of outliers, the 90% confidence intervals for the ratio of geometric means tend to be narrower for the robust methods than for the conventional method. Our simulation study shows that the robust method has suitable frequentist properties and yields more precise confidence intervals and higher statistical power than the conventional maximum likelihood method when outliers are present in the data.

CONCLUSIONS

As a sensitivity analysis, we recommend the fit of robust models for handling outliers that are occasionally encountered in crossover design bioequivalence data.

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.

Residuals and outliers in replicate design crossover studies

Outliers in bioequivalence trials may arise through various mechanisms, requiring different interpretation and handling of such data points. For example, regulatory authorities might permit exclusion from analysis of outliers caused by product or process failure, while exclusion of outliers caused by subject-by-treatment interaction generally is not acceptable. In standard 2 x 2 crossover studies it is not possible to distinguish between relevant types of outliers based on statistical criteria alone. However, in replicate design (2-treatment, 4-period) crossover studies three types of outliers can be distinguished: (i) Subject outliers are usually unproblematic, at least regarding the analysis of bioequivalence, and may require no further action; (ii) Subject-by-formulation outliers may affect the outcome of the bioequivalence test but generally cannot simply be removed from analysis; and (iii) Removal of single-data-point outliers from analysis may be justified in certain cases. As a very simple but effective diagnostic tool for the identification and classification of outliers in replicate design crossover studies we propose to calculate and plot three types of residual corresponding to the three different types of outliers that can be distinguished. The residuals are obtained from four mutually orthogonal linear contrasts of the four data points associated with each subject. If preferred, outlier tests can be applied to the resulting sets of residuals after suitable standardization.

Bioequivalence testing by statistical shape analysis

Bioequivalence testing has been traditionally centered in summary variables such as AUC, C (max) and t (max) which filter out the intrinsic information conveyed by discrete sequential concentration-time observations. Comparing entire concentration-time profiles between test and reference formulations for bioequivalence purposes provides stronger evidence about either their similarity or their discrepancy. The Kullback-Leibler information criterion (KLIC) may be computed for each concentration-time across all subjects between formulations of the same drug, with a standard crossover study design. It has been shown that if properly scaled it follow a chi-squared distribution and dependent p-values may be computed in order to construct a bioequivalence criterion. Extensive simulations and real data were used to compare it with the current standard procedures. This statistical shape analysis method may provide important clinical and regulatory advantages.

A COMPARISON OF FOUR DIFFERENT METHODS FOR OUTLIER DETECTION IN BIOEQUIVALENCE STUDIES

Bioequivalence studies, required by law whenever a new formulation of an existing drug product is introduced to the market, are designed to test whether the bioavailability, defined as the rate and extent to which a substance reaches systemic circulation, is equivalent for each of two or more formulations. Detection and treatment of outlying data in bioequivalence studies are practically important, because inclusion or deletion of potential outlying data may lead to a different conclusion concerning bioequivalence. A review of the literature reveals that four different methods have been proposed for detecting outliers in bioavailability/bioequivalence studies. We present the results of an extensive computer simulation testing the small sample performance of these four testing methods, the results of which indicate that one of these, the estimates distance test, is substantially more powerful than the alternatives.

Average bioequivalence evaluation: general methods for pilot trials

In clinical development of a bioequivalent (BE) drug product, a two-step strategy is commonly adopted. In the first step, a pilot BE trial is conducted to evaluate the acceptability of the test drug product as a candidate for further evaluation in a subsequent pivotal BE trial. In the second step, a full-scale pivotal BE trial is conducted to formally establish bioequivalence. The objective and criterion of a pilot BE trial are different from those of a pivotal BE trial. In practice, however, a pilot BE trial is often inappropriately designed and analyzed based on the criterion for a pivotal BE trial. One main reason is the lack of well-established design and analysis methods for a pilot BE trial. To close this gap in practice, this study proposes a Pilot Acceptance Range method specfically constructed for analyzing a pilot BE trial within the framework of a two-step strategy. For designing a crossover pilot BE trial, this paper derives the power function and provides an easy-to-use method for determining the sample size.

Sample size calculation for simulation-based multiple-testing procedures

In this article, we present a simple method to calculate sample size and power for a simulation-based multiple testing procedure which gives a sharper critical value than the standard Bonferroni method. The method is especially useful when several highly correlated test statistics are involved in a multiple-testing procedure. The formula for sample size calculation will be useful in designing clinical trials with multiple endpoints or correlated outcomes. We illustrate our method with a quality-of-life study for patients with early stage prostate cancer. Our method can also be used for comparing multiple independent groups.

Examining outlying subjects and outlying records in bioequivalence trials

The problem of detecting outliers in bioequivalence trials is considered. We formulate the problem as a hypothesis-testing problem under a mean-shift model and propose a test procedure based on the likelihood function. The test statistic has two components: one is to detect whether a specific pharmacokinetic measurement of a subject for certain formulation/drug product is an outlying value; the other is to test whether a subject as a whole is an outlying subject (with unusual high or low bioavailability for all formulations/drug products). Under normality assumption, the proposed procedure is most powerful. The small sample distribution of the proposed test statistic is derived. A numerical example illustrates the use of the procedure. The proposed test is then compared in a simulation study against the Hotelling T2 test, recommended by Liu and Weng (1991) for the use of outlier detection in bioequivalence studies. The results from the simulation study show that the proposed test is more powerful than the Hotelling T2 test.

Sensitivity of parametric link functions in generalized linear models

A common method of choosing the link function in generalized linear models is to specify a parametric link family indexed by unknown parameters. The maximum likelihood estimates of such link parameters, however, may often depend on one or several extreme observations. Diagnostics are derived to assess the sensitivity of the parametric link analysis. Two examples demonstrate that the proposed diagnostics can identify jointly influential observations on the link even when masking is present.

Outlier detection in psychiatric epidemiology

Outliers may be of interest in their own right or they may merely be distractions from the point in question, and a hindrance to generalisation. In the medical field, including psychiatry, concise summary statistics and parsimonious models have tended to be the main aim of many studies. However, partly motivated by the requirements of medical audit and other political and financial considerations, the detection of outliers as an end in itself is becoming a subject of interest in the public domain, and this may extend from medicine in general to psychiatry. The recognition of uncertainty in ranks using statistical methods can place the labels "best" and "worst", when applied to hospitals and consultants, into perspective, and here new developments in Bayesian methods will be important. Other areas of current development include computer intensive methods for multiple, multivariate outliers and for outlier detection tailored to specific situations such as correlational models in factor analysis and reliability studies, and in meta-analysis. These areas are likely to be of particular interest to psychiatric epidemiologists because of the complex nature of their data.

On likelihood distance for outliers detection

The likelihood distance has been widely used to detect outlying observations in data analysis. Cook and Weisberg (5) suggested that the likelihood distance may be compared to a chi 2 distribution for large samples. In this paper, we show that use of the chi 2 distribution is inappropriate. The results indicate that the likelihood distance does not follow an asymptotically chi 2 distribution. Instead, it converges to 0 in probability as the sample size increases. We show that for a nondegenerate limiting distribution, a multiplication factor related to the sample size n is needed. In general, the limiting distribution of this modified statistic is model-dependent.

The impact of outlying subjects on decision of bioequivalence

As a consequence of a hearing on bioequivalence conducted by the Food and Drug Administration in 1986, the identification and the treatment of a potential outlier in bioequivalence trials has become an important issue in the assessment of bioequivalence because the exclusion of a statistically identified outlier may lead to a totally different conclusion on bioequivalence. In this paper, we examine the impact of a statistically identified outlying subject on the decision of bioequivalence through a simulation study under the structure of a standard two-way crossover design based on interval hypotheses for bioequivalence. The Hotelling T2 test suggested by Liu and Weng (1) is used for detection of an outlying subject.

Robust and bootstrap testing procedures for bioequivalence

A common problem encountered in bioequivalence studies is the presence of outliers. In this situation, the two one-sided t-tests proposed by Schuirmann fail to provide reasonable power for concluding bioequivalence. In contrast, our proposed 2 beta trimmed-t procedure has the following advantages: (1) it has higher efficiency for nonnormal symmetric distributions, (2) it is resistant to outliers, and (3) it is relatively easy to compute. Two bootstrap procedures introduced here provide further justification for the proposed trimmed t-test procedure. Results from Monte Carlo studies illustrate the power of the proposed procedures under various distributional assumptions for a 2 x 2 crossover trial.