A Nonparametric Graphical Tests of Significance in Functional GLM (original) (raw)
Abstract
A new nonparametric graphical test of significance of a covariate in functional GLM is proposed. Our approach is especially interesting due to its functional graphical interpretation of the results. As such, it is able to find not only if the factor of interest is significant but also which functional domain is responsible for the potential rejection. In the case of functional multi-way main effect ANOVA or functional main effect ANCOVA models it is able to find which groups differ (and where they differ), in the case of functional factorial ANOVA or functional factorial ANCOVA models it is able to find which combination of levels (which interactions) differ (and where they differ). The described tests are extensions of global envelope tests in the GLM models. It applies Freedman-Lane algorithm for the permutation of functions, and as such, it approximately achieves the desired significance level.
Access this article
Subscribe and save
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime Subscribe now
Buy Now
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Instant access to the full article PDF.
Similar content being viewed by others
References
- Abramovich F, Angelini C (2006) Testing in mixed-effects fanova models. J Stat Plan Infer 136(12):4326–4348
Article MathSciNet Google Scholar - Anderson MJ, Robinson J (2001) Permutation tests for linear models. Austr New Zealand J Stat 43(1):75–88. https://doi.org/10.1111/1467-842X.00156
Article MathSciNet MATH Google Scholar - Anderson MJ, Ter Braak CJ (2003) Permutation tests for multi-factorial analysis of variance. J Stat Comput Simul 73(2):85–113
Article MathSciNet Google Scholar - Cuesta-Albertos JA, Febrero-Bande M (2010) A simple multiway anova for functional data. TEST 19(3):537–557. https://doi.org/10.1007/s11749-010-0185-3
Article MathSciNet MATH Google Scholar - Febrero-Bande M, Oviedo de la Fuente M (2012) Statistical computing in functional data analysis: the R package fda.usc. J Stat Softw 51(4):1–28. http://www.jstatsoft.org/v51/i04/
Article Google Scholar - Ferraty F, Vieu P, Viguier-Pla S (2007) . Factor-based comparison of groups of curves 51:4903–4910
Google Scholar - Freedman D, Lane D (1983) . A nonstochastic interpretation of reported significance levels 1:292–98
Google Scholar - Hahn U (2012) A studentized permutation test for the comparison of spatial point patterns. Am Stat Assoc J 107(498):754–764
Article MathSciNet Google Scholar - Legendre P, Anderson MJ (1999) Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecol Monogr 69(1):1–24
Article Google Scholar - Mrkvička T, Myllymäki M, Hahn U (2017) Multiple monte carlo testing, with applications in spatial point processes. Stat Comput 27(5):1239–1255. https://doi.org/10.1007/s11222-016-9683-9
Article MathSciNet MATH Google Scholar - Mrkvička T, Myllymäki M, Jílek M, Hahn U (2018) A one-way anova test for functional data with graphical interpretation, arXiv:https://arxiv.org/abs/1612.03608 [stat.ME]
- Myllymäki M, Mrkvička T, Grabarnik P, Seijo H, Hahn U (2017) Global envelope tests for spatial processes. J R Stat Soc: Series B (Stat Methodol) 79 (2):381–404. https://doi.org/10.1111/rssb.12172
Article MathSciNet MATH Google Scholar - Nichols TE, Holmes AP (2001) Nonparametric permutation tests for functional neuroimaging: a primer with examples. Human brain mapping
- Pantazis D, Nichols TE, Baillet S, Leahy RM (2005) A comparison of random field theory and permutation methods for the statistical analysis of meg data. NeuroImage 25(2):383–394. http://www.sciencedirect.com/science/article/pii/S1053811904005671
Article Google Scholar - Ramsay J, Silverman B (2006) Functional data analysis, 2nd edn. Springer Series in Statistics, Springer
MATH Google Scholar - Winkler AM, Ridgway GR, Webster MA, Smith SM, Nichols TE (2014) Permutation inference for the general linear model. NeuroImage 92:381–397. http://www.sciencedirect.com/science/article/pii/S1053811914000913
Article Google Scholar
Acknowledgements
The project has been financially supported by the Grant Agency of Czech Republic (Project No. 19-04412S).
Author information
Authors and Affiliations
- Department of Applied Mathematics and Informatics, Faculty of Economics, University of South Bohemia, Studentská 13, 37005, Ceské Budejovice, Czech Republic
Tomáš Mrkvička, Tomáš Roskovec & Michael Rost
Authors
- Tomáš Mrkvička
You can also search for this author inPubMed Google Scholar - Tomáš Roskovec
You can also search for this author inPubMed Google Scholar - Michael Rost
You can also search for this author inPubMed Google Scholar
Corresponding author
Correspondence toTomáš Mrkvička.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Table 4 The estimated probabilities of rejecting of factor of interest in main effect FGLM with two categorical factors and Brownian motion error
Table 5 The estimated probabilities of rejecting of factor of interest in main effect two factor FGLM with continuous factor of interest and Brownian motion error
Let us consider the previous simulation design, where the i.i.d. error term e(t) would be replaced by the Brownian motion. The difference is that with i.i.d. error used in previous sections the variance is constant, but with the Brownian motion, it is increasing in dependence on t. This may cause some trouble, since the bigger variance for bigger t means different sensitivity for effects influencing values close to t = 0, such as parameter i and effects that influence the values close to t = 1 such as parameter j, see Fig. ??.
The standard deviation of the Brownian motion e(1) was kept ten times bigger than the standard deviation of the i.i.d. error, since then the increments in our discrete Brownian motion has the same standard deviation as the i.i.d. error and we get comparable results.
We present three tables in the same spirit as in the main text. The results here are calculated from 100 simulations only since we did not have enough time to finish the whole study. The full study will appear in the final version.
The estimated levels of significance are slightly liberal for the procedures using the Freedman-Lane algorithm. The powers of our tests are again much bigger than the powers of the other two tests. Even more, in some cases, the difference between these tests is more significant than for the i.i.d. error rate and in other cases, the difference between these tests is similar as for the i.i.d. error rate.
Table 6 The estimated probabilities of rejecting of effect of interactions in factorial two factor FGLM with Brownian motion error
Rights and permissions
About this article
Cite this article
Mrkvička, T., Roskovec, T. & Rost, M. A Nonparametric Graphical Tests of Significance in Functional GLM.Methodol Comput Appl Probab 23, 593–612 (2021). https://doi.org/10.1007/s11009-019-09756-y
- Received: 01 February 2019
- Revised: 15 October 2019
- Accepted: 20 November 2019
- Published: 16 December 2019
- Issue Date: June 2021
- DOI: https://doi.org/10.1007/s11009-019-09756-y