Skip to main content
SpringerLink
Log in

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

  • Research Paper
  • Published:
Pharmaceutical Research Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Abbreviations

θ :

GMR of bioavailability characteristic

ABE:

Average bioequivalence

ANOVA:

Analysis of variance

BayesN :

Non-robust Bayesian model

BayesT :

Robust Bayesian model

CI:

Confidence interval

CL:

Confidence limit

eCDF:

Empirical cumulative distribution function

GMR:

Geometric means ratio

HPD:

Highest posterior density

LML:

Log-marginal likelihood

PK:

Pharmacokinetic

REML:

Restricted maximum likelihood

RMSE:

Root mean square error

TOST:

Two one-sided test

References

  1. FDA. Statistical approaches to establishing bioequivalence; 2001. Available at: https://www.fda.gov/ [accessed on 8 March 2020].

  2. CHMP. Guideline on the investigation of bioequivalence; 2010. Available at: https://www.ema.europa.eu/en/ [accessed on 8 March 2020].

  3. Meyners M. Equivalence tests–a review. Food Qual Prefer. 2012;26(2):231–45.

    Article  Google Scholar 

  4. Schall R, Endrenyi L, Ring A. Residuals and outliers in replicate design crossover studies. J Biopharm Stat. 2010;20(4):835–49.

    Article  Google Scholar 

  5. Schall R. The empirical coverage of confidence intervals: point estimates and confidence intervals for confidence levels. Biom J. 2012;54(4):537–51.

    Article  Google Scholar 

  6. Chen ML, Blume H, Beuerle G, Mehta M, Potthast H, Brandt A, et al. Summary report of second EU-FEPS/AAPS conference on global harmonization in bioequivalence. Eur J Pharm Sci. 2019;127:24–8.

    Article  CAS  Google Scholar 

  7. Chow SC, Tse SK. Outlier detection in bioavailability/bioequivalence studies. Stat Med. 1990;9(5):549–58.

    Article  CAS  Google Scholar 

  8. Liu JP, Weng CS. Detection of outlying data in bioavailability/bioequivalence studies. Stat Med. 1991;10(9):1375–89.

    Article  CAS  Google Scholar 

  9. Wang W, Chow SC. Examining outlying subjects and outlying records in bioequivalence trials. J Biopharm Stat. 2003;13(1):43–56.

    Article  Google Scholar 

  10. Ramsay T, Elkum N. A comparison of four different methods for outlier detection in bioequivalence studies. J Biopharm Stat. 2004;15(1):43–52.

    Article  Google Scholar 

  11. Zellner A. Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. J Am Stat Assoc. 1976;71(354):400–5.

    Google Scholar 

  12. Lange KL, Little RJA, Taylor JMG. Robust statistical modeling using the t distribution. J Am Stat Assoc. 1989;84(408):881–96.

    Google Scholar 

  13. Pinheiro JC, Liu C, Wu YN. Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution. J Comput Graph Stat. 2001;10(2):249–76.

    Article  Google Scholar 

  14. Wang WL, Fan TH. Estimation in multivariate t linear mixed models for multiple longitudinal data. Stat Sin. 2011;21(4):1857–80.

    Article  Google Scholar 

  15. Matos LA, Prates MO, Chen MH, Lachos VH. Likelihood-based inference for mixed-effects models with censored response using the multivariate-t distribution. Stat Sin. 2013;23(3):1323–45.

    Google Scholar 

  16. Azzalini A, Capitanio A. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J Roy Stat Soc B. 2003;65(2):367–89.

    Article  Google Scholar 

  17. Azzalini A, Genton MG. Robust likelihood methods based on the skew-t and related distributions. Int Stat Rev. 2008;76(1):106–29.

    Article  Google Scholar 

  18. Fernández C, Steel MFJ. On Bayesian modeling of fat tails and skewness. J Am Stat Assoc. 1998;93(441):359–71.

    Google Scholar 

  19. Yellowlees A, Bursa F, Fleetwood KJ, Charlton S, Hirst KJ, Sun R, et al. The appropriateness of robust regression in addressing outliers in an anthrax vaccine potency test. Bioscience. 2016;66(1):63–72.

    Article  Google Scholar 

  20. Castro LM, Wang WL, Lachos VH, Inácio de Carvalho V, Bayes CL. Bayesian semiparametric modeling for HIV longitudinal data with censoring and skewness. Stat Methods Med Res. 2019;28(5):1457–76.

    Article  Google Scholar 

  21. De Souza RM, Achcar JA, Martinez EZ, Mazucheli J. The use of asymmetric distributions in average bioequivalence. Stat Med. 2016;35(15):2525–42.

    Article  Google Scholar 

  22. Ghosh P, Ntzoufras I. Testing population and individual bioequivalence: a hierarchical Bayesian approach. Department of Mathematics and Statistics, Georgia State University, Atlanta, GA & Department of Statistics, Athens University of Economics and Business, Greece; 2005.

  23. Chow SC. Bioavailability and bioequivalence in drug development. Wiley Interdiscip Rev Comput Stat. 2014;6(4):304–12.

    Article  Google Scholar 

  24. Arnold BC, Groeneveld RA. Measuring skewness with respect to the mode. Am Stat. 1995;49(1):34–8.

    Google Scholar 

  25. SAS Institute. SAS/IML user’s guide, Version 9.4. SAS Institute Cary, NC; 2013.

  26. Robert CP. The Bayesian choice. 2nd ed. New York: Springer-Verlag; 2007.

    Google Scholar 

  27. Gelman A. Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 2006;1(3):515–33. https://doi.org/10.1214/06-BA117A.

    Article  Google Scholar 

  28. Lunn D, Jackson C, Best N, Thomas A, Spiegelhalter D. The BUGS Book: A Practical Introduction to Bayesian Analysis. CRC Press; 2012.

  29. Juárez MA, Steel MFJ. Model-based clustering of non-Gaussian panel data based on skew-t distributions. J Bus Econ Stat. 2010;28(1):52–66.

    Article  Google Scholar 

  30. Carpenter B, Gelman A, Hoffman MD, Lee D, Goodrich B, Betancourt M, et al. Stan: A probabilistic programming language. J Stat Softw 2017;76(1).

  31. Stan Development Team. rstan: R interface to Stan; 2020. R package Version 2.19.3. Available from: http://CRAN.R-project.org/package=rstan

  32. R Core Team. R: A language and environment for statistical computing. Vienna, Austria; 2018. Available from: https://www.R-project.org/

  33. Ntzoufras I. Bayesian modeling using WinBUGS. Hoboken: John Wiley & Sons, Inc.; 2009.

    Book  Google Scholar 

  34. Gronau QF, Singmann H, Wagenmakers EJ. bridgesampling: an R package for estimating normalizing constants. J Stat Softw. 2020;92(10):1–29.

    Article  Google Scholar 

  35. Labes D, Schütz H. Inflation of type I error in the evaluation of scaled average bioequivalence, and a method for its control. Pharm Res. 2016;33(11):2805–14.

    Article  CAS  Google Scholar 

  36. Tothfalusi L, Endrenyi L, Arieta AG. Evaluation of bioequivalence for highly variable drugs with scaled average bioequivalence. Clin Pharmacokinet. 2009;48(11):725–43.

    Article  CAS  Google Scholar 

Download references

ACKNOWLEDGMENTS AND DISCLOSURES

The datasets supporting this study’s findings are available on reasonable request from the corresponding author. The data are not publicly available due to privacy and ethics restrictions. The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. Department of Statistics, University of Pretoria, Pretoria, South Africa

    Divan Aristo Burger

  2. Department of Mathematical Statistics and Actuarial Science, University of the Free State, Bloemfontein, South Africa

    Robert Schall & Sean van der Merwe

  3. IQVIA, Biostatistics, Bloemfontein, South Africa

    Robert Schall

Authors
  1. Divan Aristo Burger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Schall
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sean van der Merwe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Divan Aristo Burger.

Additional information

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is based upon research supported by the South Africa National Research Foundation and South Africa Medical Research Council (South Africa DST-NRF-SAMRC SARChI Research Chair in Biostatistics, grant number 114613); and the Research Development Programme 219/2018 at the University of Pretoria, South Africa.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (ZIP 1.06 MB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burger, D.A., Schall, R. & van der Merwe, S. A robust method for the assessment of average bioequivalence in the presence of outliers and skewness. Pharm Res 38, 1697–1709 (2021). https://doi.org/10.1007/s11095-021-03110-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11095-021-03110-z

KEY WORDS

Search

Navigation