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Parametric and nonparametric bootstrap methods for general MANOVA

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  • Konietschke, Frank
  • Bathke, Arne C.
  • Harrar, Solomon W.
  • Pauly, Markus

Abstract

We develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens–Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.

Suggested Citation

  • Konietschke, Frank & Bathke, Arne C. & Harrar, Solomon W. & Pauly, Markus, 2015. "Parametric and nonparametric bootstrap methods for general MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 291-301.
    • Handle: RePEc:eee:jmvana:v:140:y:2015:i:c:p:291-301
    DOI: 10.1016/j.jmva.2015.05.001
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    References listed on IDEAS

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    1. Bathke, Arne C. & Harrar, Solomon W. & Madden, Laurence V., 2008. "How to compare small multivariate samples using nonparametric tests," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4951-4965, July.
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    6. Xu, Li-Wen & Yang, Fang-Qin & Abula, Aji’erguli & Qin, Shuang, 2013. "A parametric bootstrap approach for two-way ANOVA in presence of possible interactions with unequal variances," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 172-180.
    7. Holger Dette & Efstathios Paparoditis, 2009. "Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 831-857, September.
    8. Van Aelst, Stefan & Willems, Gert, 2013. "Fast and Robust Bootstrap for Multivariate Inference: The R Package FRB," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 53(i03).
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    10. Solomon Harrar & Arne Bathke, 2012. "A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(1), pages 135-165, February.
    11. Liu, Chunxu & Bathke, Arne C. & Harrar, Solomon W., 2011. "A nonparametric version of Wilks' lambda--Asymptotic results and small sample approximations," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1502-1506, October.
    12. Krishnamoorthy, K. & Yu, Jianqi, 2004. "Modified Nel and Van der Merwe test for the multivariate Behrens-Fisher problem," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 161-169, January.
    13. Chung, EunYi & Romano, Joseph P., 2016. "Multivariate and multiple permutation tests," Journal of Econometrics, Elsevier, vol. 193(1), pages 76-91.
    14. Harrar, Solomon W. & Bathke, Arne C., 2008. "Nonparametric methods for unbalanced multivariate data and many factor levels," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1635-1664, September.
    15. Girón, Fco. Javier & del Castillo, Carmen, 2010. "The multivariate Behrens-Fisher distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2091-2102, October.
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    Cited by:

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    7. Zimmermann, Georg & Pauly, Markus & Bathke, Arne C., 2020. "Multivariate analysis of covariance with potentially singular covariance matrices and non-normal responses," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    8. Friedrich, Sarah & Konietschke, Frank & Pauly, Markus, 2017. "A wild bootstrap approach for nonparametric repeated measurements," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 38-52.
    9. H. V. Kulkarni & S. M. Patil, 2021. "Uniformly implementable small sample integrated likelihood ratio test for one-way and two-way ANOVA under heteroscedasticity and normality," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(2), pages 273-305, June.
    10. Friedrich, Sarah & Brunner, Edgar & Pauly, Markus, 2017. "Permuting longitudinal data in spite of the dependencies," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 255-265.
    11. Roth, Lucas & Lowitzsch, Jens & Yildiz, Özgür, 2023. "Which (co-)ownership types in renewables are associated with the willingness to adopt energy-efficient technologies and energy-conscious behaviour? Data from German households," Energy Policy, Elsevier, vol. 180(C).
    12. Smaga, Łukasz, 2015. "Wald-type statistics using {2}-inverses for hypothesis testing in general factorial designs," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 215-220.
    13. Stefano Bonnini & Getnet Melak Assegie & Kamila Trzcinska, 2024. "Review about the Permutation Approach in Hypothesis Testing," Mathematics, MDPI, vol. 12(17), pages 1-29, August.
    14. Markus Pauly & Maria Umlauft & Ali Ünlü, 2018. "Resampling-Based Inference Methods for Comparing Two Coefficients Alpha," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 203-222, March.
    15. Dickhaus, Thorsten & Sirotko-Sibirskaya, Natalia, 2019. "Simultaneous statistical inference in dynamic factor models: Chi-square approximation and model-based bootstrap," Computational Statistics & Data Analysis, Elsevier, vol. 129(C), pages 30-46.
    16. Baumeister, Marléne & Ditzhaus, Marc & Pauly, Markus, 2024. "Quantile-based MANOVA: A new tool for inferring multivariate data in factorial designs," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    17. Watanabe, Hiroki & Hyodo, Masashi & Nakagawa, Shigekazu, 2020. "Two-way MANOVA with unequal cell sizes and unequal cell covariance matrices in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    18. Marc Ditzhaus & Roland Fried & Markus Pauly, 2021. "QANOVA: quantile-based permutation methods for general factorial designs," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(4), pages 960-979, December.
    19. Dennis Dobler & Sarah Friedrich & Markus Pauly, 2020. "Nonparametric MANOVA in meaningful effects," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 997-1022, August.

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