A Nonparametric Graphical Tests of Significance in Functional GLM

Methodology and Computing in Applied Probability (2019)Cite this article

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.

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References

  1. Abramovich F, Angelini C (2006) Testing in mixed-effects fanova models. J Stat Plan Infer 136(12):4326–4348

    MathSciNet  Article  Google Scholar 

  2. 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

    MathSciNet  Article  MATH  Google Scholar 

  3. Anderson MJ, Ter Braak CJ (2003) Permutation tests for multi-factorial analysis of variance. J Stat Comput Simul 73(2):85–113

    MathSciNet  Article  Google Scholar 

  4. 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

    MathSciNet  Article  MATH  Google Scholar 

  5. 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 

  6. Ferraty F, Vieu P, Viguier-Pla S (2007) . Factor-based comparison of groups of curves 51:4903–4910

    Google Scholar 

  7. Freedman D, Lane D (1983) . A nonstochastic interpretation of reported significance levels 1:292–98

    Google Scholar 

  8. Hahn U (2012) A studentized permutation test for the comparison of spatial point patterns. Am Stat Assoc J 107(498):754–764

    MathSciNet  Article  Google Scholar 

  9. 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 

  10. 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

    MathSciNet  Article  MATH  Google Scholar 

  11. 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]

  12. 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

    MathSciNet  Article  MATH  Google Scholar 

  13. Nichols TE, Holmes AP (2001) Nonparametric permutation tests for functional neuroimaging: a primer with examples. Human brain mapping

  14. 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 

  15. Ramsay J, Silverman B (2006) Functional data analysis, 2nd edn. Springer Series in Statistics, Springer

    Google Scholar 

  16. 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 

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Acknowledgements

The project has been financially supported by the Grant Agency of Czech Republic (Project No. 19-04412S).

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  1. 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

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  1. Tomáš Mrkvička
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  2. Tomáš Roskovec
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  3. Michael Rost
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Correspondence to Tomáš Mrkvička.

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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
Full size table
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
Full size table

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
Full size table

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Mrkvička, T., Roskovec, T. & Rost, M. A Nonparametric Graphical Tests of Significance in Functional GLM. Methodol Comput Appl Probab (2019). https://doi.org/10.1007/s11009-019-09756-y

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Keywords

  • Functional ANCOVA
  • Freedman-Lane algorithm
  • Global envelope test
  • Groups comparison
  • Permutation test

Mathematics Subject Classification (2010)

  • 62H15
  • 62G10