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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramovich F, Angelini C (2006) Testing in mixed-effects fanova models. J Stat Plan Infer 136(12):4326–4348
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
Anderson MJ, Ter Braak CJ (2003) Permutation tests for multi-factorial analysis of variance. J Stat Comput Simul 73(2):85–113
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
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/
Ferraty F, Vieu P, Viguier-Pla S (2007) . Factor-based comparison of groups of curves 51:4903–4910
Freedman D, Lane D (1983) . A nonstochastic interpretation of reported significance levels 1:292–98
Hahn U (2012) A studentized permutation test for the comparison of spatial point patterns. Am Stat Assoc J 107(498):754–764
Legendre P, Anderson MJ (1999) Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecol Monogr 69(1):1–24
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
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
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
Ramsay J, Silverman B (2006) Functional data analysis, 2nd edn. Springer Series in Statistics, Springer
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
Acknowledgements
The project has been financially supported by the Grant Agency of Czech Republic (Project No. 19-04412S).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
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.
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:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-019-09756-y