Kybernetika 56 no. 3, 432-458, 2020

A one-way ANOVA test for functional data with graphical interpretation

Tomáš Mrkvička, Mari Myllymäki, Milan Jílek and Ute HahnDOI: 10.14736/kyb-2020-3-0432


A new functional ANOVA test, with a graphical interpretation of the result, is presented. The test is an extension of the global envelope test introduced by Myllymäki et al. (2017, Global envelope tests for spatial processes, J. R. Statist. Soc. B 79, 381-404, doi: 10.1111/rssb.12172). The graphical interpretation is realized by a global envelope which is drawn jointly for all samples of functions. If a mean function computed from the empirical data is out of the given envelope, the null hypothesis is rejected with the predetermined significance level $\alpha$. The advantages of the proposed one-way functional ANOVA are that it identifies the domains of the functions which are responsible for the potential rejection. We introduce two versions of this test: the first gives a graphical interpretation of the test results in the original space of the functions and the second immediately offers a post-hoc test by identifying the significant pair-wise differences between groups. The proposed tests rely on discretization of the functions, therefore the tests are also applicable in the multidimensional ANOVA problem. In the empirical part of the article, we demonstrate the use of the method by analyzing fiscal decentralization in European countries.


global envelope test, groups comparison, permutation test, Europe, fiscal decentralization, nonparametrical methods


62H15, 62G10


  1. F. Abramovich and C. Angelini: Testing in mixed-effects FANOVA models. J. Statist. Planning Inference 136 (2006), 4326-4348.   DOI:10.1016/j.jspi.2005.06.002
  2. B. Ackerman: The rise of world constitutionalism. Virginia Law Rev. 83 (1997), 771-797.   DOI:10.2307/1073748
  3. A. Alesina and E. Spolaore: On the number and size of nations. Quarterly J. Econom. 112 (1997), 1027-1056.   DOI:10.1162/003355300555411
  4. M. Arzaghi and J. V. Henderson: Why countries are fiscally decentralizing. J. Public Econom. 89 (2005), 1157-1189.   DOI:10.1016/j.jpubeco.2003.10.009
  5. P. Bolton and G. Roland: The breakup of nations: A political economy analysis. Quarterly J. Econom. 112 (1997), 1057-1090.   DOI:10.1162/003355300555420
  6. F. Cerniglia: Decentralization in the public sector: quantitative aspects in federal and unitary countries. J. Pol. Model. 25 (2003), 749-776.   DOI:10.1016/s0161-8938(03)00069-3
  7. H. Choi and M. Reimherr: A geometric approach to confidence regions and bands for functional parameters. J. Royal Statist. Soc.: Series B (Statist. Methodology) 80 (2018), 239-260.   DOI:10.1111/rssb.12239
  8. D. D. Cox and J. S. Lee: Pointwise testing with functional data using the Westfall-Young randomization method. Biometrika 95 (2008), 621-634.   DOI:10.1093/biomet/asn021
  9. J. Cuesta-Albertos and M. Febrero-Bande: A simple multiway ANOVA for functional data. Test 19 (2010), 537-557.   DOI:10.1007/s11749-010-0185-3
  10. A. Cuevas, M. Febrero and R. Fraiman: An anova test for functional data. Computat. Statist. Data Analysis 47 (2004), 111-122.   DOI:10.1016/j.csda.2003.10.021
  11. B. Ermini and R. santolini: does globalization matter on fiscal decentralization? New evidence from the OECD. Global Econom. Rev. 43 (2014), 153-183.   DOI:10.1080/1226508x.2014.920240
  12. Eurostat 2018: Government revenue, Expenditure and Main Aggregates (gov_10a_main).    CrossRef
  13. M. Febrero-Bande and M. {Oviedo de la Fuente}: Statistical computing in functional data analysis: The {R} package {fda.usc}. J. Statist. Software 51 (2012), 1-28.   DOI:10.18637/jss.v051.i04
  14. F. Ferraty, P. Vieu and S. Viguier-Pla: Factor-based comparison of groups of curves. Computat. Statist. Data Analysis 51 (2007), 4903-4910.   DOI:10.1016/j.csda.2006.10.001
  15. T. Górecki and L. Smaga: A comparison of tests for the one-way ANOVA problem for functional data. Comput. Statist. 30 (2015), 987-1010.   DOI:10.1007/s00180-015-0555-0
  16. T. Górecki and L. Smaga: fdANOVA: Analysis of Variance for Univariate and Multivariate Functional Data, R package version 0.1.0, 2017.    CrossRef
  17. U. Hahn: A studentized permutation test for the comparison of spatial point patterns. J. Amer. Statist. Assoc. 107 (2012), 754-764.   DOI:10.1080/01621459.2012.688463
  18. N. B. Loosmore and E. D. Ford: Statistical inference using the {G} or {K} point pattern spatial statistics. Ecology 87 (2006), 1925-1931.   DOI:10.1890/0012-9658(2006)87[1925:siutgo];2
  19. T. Mrkvička, M. Myllymäki and U. Hahn: Multiple Monte Carlo testing, with applications in spatial point processes. Statist. Comput. 27 (2017), 1239-1255.   DOI:10.1007/s11222-016-9683-9
  20. M. Myllymäki and T. Mrkvička: GET: Global envelopes in {R}. arXiv:1911.06583 [stat.ME], 2019.   CrossRef
  21. M. Myllymäki, T. Mrkvička, P. Grabarnik, H. Seijo and U. Hahn: Global envelope tests for spatial processes. J. Royal Statist. Soc. B 79 (2017), 381-404.   DOI:10.1111/rssb.12172
  22. N.-N. Narisetty V. J. and Nair: Extremal depth for functional data and applications. J. Amer. Statist. Assoc. 111 (2016), 1705-1714.   DOI:10.1080/01621459.2015.1110033
  23. T. E.. Nichols and E. Holmes: Nonparametric permutation tests for functional neuroimaging: A primer with examples. Human Brain M 15 (2001), 1-25.   DOI:10.1002/hbm.1058
  24. Oates, E. Wallace and E: Toward A second-generation theory of fiscal federalism. Int. Tax Public Finance 12 (2005), 349-373.   DOI:10.1007/s10797-005-1619-9
  25. D. Pantazis, T. E. Nichols, S. Baillet and R. M. Leahya: A comparison of random field theory and permutation methods for the statistical analysis of MEG data. Neuroi 25 (2005), 383-394.   DOI:10.1016/j.neuroimage.2004.09.040
  26. A. Pini, S. Vantini, B. M. Colosimo and M. Grasso: Domain-selective functional analysis of variance for supervised statistical profile monitoring of signal data. J. Royal Statist. Soc.: Series C (Appl. Statist.) 67 (2001), 55-81.   DOI:10.1111/rssc.12218
  27. R Core Team: R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Vienna 2019.   CrossRef
  28. J. Ramsay and B. Silverman: Functional Data Analysis. Second edition. Springer Series in Statistics, Springer 2006.   DOI:10.1007/978-1-4757-7107-7
  29. D. Rodrik: Why do more open economies have bigger governments? J. Polit. Economy 106 (1998), 997-1032.   DOI:10.1086/250038
  30. J. Sedova, H. Lipovska and Fischer: Fiscal autonomy in the secessionist regions. In: Current Trends in Public Sector Research, Masaryk University, Brno 2017.   CrossRef
  31. P. B. Spahn: Contract Federalims. Edward Elgar Publishing Limited, Book Section 7, Cheltenham 2015, pp. 144-160.   CrossRef
  32. D. Stegarescu: Public sector decentralisation: Measurement concepts and recent international trends. Fiscal Stud. 26 (2005), 301-333.   DOI:10.1111/j.1475-5890.2005.00014.x
  33. D. Stegarescu: The effects of economic and political integration on fiscal decentralization: Evidence from OECD countries. Canadian J. Econom. / Revue Canadienne d'Economique 42 (2009), 694-718.   DOI:10.1111/j.1540-5982.2009.01524.x
  34. D. H. Vo: New Fiscal Decentralization Indices. The University of Western Australia Discussion Paper 08.14, 93, 2008.   CrossRef
  35. D. H. Vo: The economics of fiscal decentralization. J. Econom. Surveys 24 (2010), 657-679.   DOI:10.1111/j.1467-6419.2009.00600.x
  36. O. Vsevolozhskaya, M. Greenwood and D. Holodov: Pairwise comparison of treatment levels in functional analysis of variance with application to erythrocyte hemolysis. Ann. Appl. Statist. 8 (2014), 905-925.   DOI:10.1214/14-aoas723
  37. J.-T. Zhang: Analysis of Variance for the functional data. Chapman and Hall, 2014.   DOI:10.1201/b15005