Abstract
We propose a procedure to test complicated ANOVA designs for functional data. The procedure is effective, flexible, easy to compute and does not require a heavy computational effort. It is based on the analysis of randomly chosen one-dimensional projections. The paper contains some theoretical results as well as some simulations and the analysis of some real data sets. Functional data include multidimensional data, so the paper contains a comparison between the proposed procedure and some usual MANOVA tests.
Similar content being viewed by others
References
Abramovich F, Angelini C (2005) Testing in mixed-effects FANOVA models. J Stat Plan Inference 136:4326–4348
Abramovich F, Antoniadis A, Sapatinas T, Vidakovic B (2004) Optimal testing in a fixed-effects functional analysis of variance model. Int J Wavelets Multiresolut Inf Process 2:323–349
Angelini C, Vidakovic B (2002) Some novel methods in wavelet data analysis: Wavelet ANOVA, F-test shrinkage and Γ-minimax wavelet shrinkage. Istituto per le Applicazioni del Calcolo “Mauro Picone.” Rapporti Tecnici RT 256/02
Antoniadis A, Sapatinas T (2007) Estimation and inference in functional mixed-effects models. Comput Stat Data Anal 51:4793–4813
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B 57:289–300
Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29:1165–1188
Brumback BA, Rice JA (1998) Smoothing spline models for the analysis of nested and crossed samples of curves. J Am Stat Assoc 93:961–994
Brunner E, Dette H, Munk A (1997) Box-type approximations in nonparametric factorial designs. J Am Stat Assoc 92:1494–1502
Cuesta-Albertos JA, Nieto-Reyes A (2007) An infinite dimensional depth: The random Tukey depth. Technical Report
Cuesta-Albertos JA, Nieto-Reyes A (2008) The random Tukey depth. Comput Stat Data Anal 52:4979–4988
Cuesta-Albertos JA, del Barrio T, Fraiman R, Matrán C (2007a) The random projection method in goodness of fit for functional data. Comput Stat Data Anal 51:4814–4831
Cuesta-Albertos JA, Fraiman R, Galves A, García J, Svarc M (2007b) Identifying rhythmic classes of languages using their sonority: a Kolmogorov–Smirnov approach. J Appl Stat 34:749–761
Cuesta-Albertos JA, Fraiman R, Ransford T (2007c) A sharp form of the Cramér–Wold theorem. J Theor Probab 20:201–209
Cuesta-Albertos JA, Cuevas A, Fraiman R (2008) On projection-based tests for spherical and compositional data. Stat Comput 19:367–380
Cuevas A, Fraiman R (2009) On depth measures and dual statistics. A methodology for dealing with general data. J Multivar Anal 100:753–766
Cuevas A, Febrero M, Fraiman R (2004) An ANOVA test for functional data. Comput Stat Data Anal 47:111–122
Fan J (1996) Tests of significance based on wavelet thresholding and Neyman’s truncation. J Am Stat Assoc 91:674–688
Fan J, Lin SK (1998) Test of significance when data are curves. J Am Stat Assoc 93:254–261
Faraway JJ (1997) Regression analysis for a functional response. J R Stat Soc B 64:887–898
Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer series in statistics. Springer, New York
Ferraty F, Vieu P, Viguier-Pla S (2007) Factor-based comparison of groups of curves. Comput Stat Data Anal 51:4903–4910
Fujikoshi Y, Himeno T, Wakaki H (2004) Asymptotic results of a high dimensional MANOVA and power comparisons when the dimension is large compared to the sample size. J Jpn Stat Soc 34:19–26
Gu C (2002) Smoothing spline ANOVA models. Springer, New York
Guo W (2002) Inference in smoothing spline analysis of variance. J R Stat Soc B 64:887–898
Krzanowski WJ (1988) Principles of multivariate analysis. A user’s perspective. Oxford University Press, Oxford
R Development Core Team (2007) R: a language and environment for statistical computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org
Ramsay JO, Silverman BW (2002) Applied functional data analysis. Methods and case studies. Springer, New York
Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New York
Schott JR (2007) Some high-dimensional tests for a one-way ANOVA. J Multivar Anal 98:1825–1839
Shohat JA, Tamarkin JD (1963) The problem of moments. American Mathematical Society, Providence
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Domingo Morales.
J.A. Cuesta-Albertos has been partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant MTM2008-0607-C02-02 and the Consejería de Educación y Cultura de la Junta de Castilla y León, grant PAPIJCL VA102/06.
M. Febrero-Bande has been partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant MTM2008-03010.
Rights and permissions
About this article
Cite this article
Cuesta-Albertos, J.A., Febrero-Bande, M. A simple multiway ANOVA for functional data. TEST 19, 537–557 (2010). https://doi.org/10.1007/s11749-010-0185-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-010-0185-3