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Multiple Monte Carlo testing, with applications in spatial point processes

Abstract

The rank envelope test (Myllymäki et al. in J R Stat Soc B, doi:10.1111/rssb.12172, 2016) is proposed as a solution to the multiple testing problem for Monte Carlo tests. Three different situations are recognized: (1) a few univariate Monte Carlo tests, (2) a Monte Carlo test with a function as the test statistic, (3) several Monte Carlo tests with functions as test statistics. The rank test has correct (global) type I error in each case and it is accompanied with a p-value and with a graphical interpretation which determines subtests and distances of the used test function(s) which lead to the rejection at the prescribed significance level of the test. Examples of null hypotheses from point process and random set statistics are used to demonstrate the strength of the rank envelope test. The examples include goodness-of-fit test with several test functions, goodness-of-fit test for a group of point patterns, test of dependence of components in a multi-type point pattern, and test of the Boolean assumption for random closed sets. A power comparison to the classical multiple testing procedures is given.

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Acknowledgments

The authors would like to express great appreciation for the helpful comments they got from the anonymous reviewers. Mrkvička has been financially supported by the Grant Agency of Czech Republic (Project No. 16-03708S) and Mari Myllymäki by the Academy of Finland (Project Numbers 250860, 294162). Hahn’s research has been supported by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by the Villum Foundation. The authors thank William R. Kennedy, Gwen Wendelschafer-Crabb and Ioanna G. Panoutsopoulou for providing the ENF data and Torsten Mattfeldt for providing the intramembranous particle data. The rainforest data set origins from the Forest Dynamics Plot of Barro Colorado Island, which is made possible through the generous support of the U.S. National Science Foundation, the John D. and Catherine T. MacArthur Foundation, the Smithsonian Tropical Research Institute and through the hard work of over 100 people from 10 countries over the past two decades. The BCI Forest Dynamics Plot is part of the Center for Tropical Forest Science, a global network of large-scale demographic tree plots.

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Correspondence to Tomáš Mrkvička.

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Mrkvička, T., Myllymäki, M. & Hahn, U. Multiple Monte Carlo testing, with applications in spatial point processes. Stat Comput 27, 1239–1255 (2017). https://doi.org/10.1007/s11222-016-9683-9

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Keywords

  • Boolean model test
  • Envelope test
  • Extreme rank ordering
  • Goodness-of-fit test
  • Multi-type point process
  • Rank envelope test
  • Superposition hypothesis

Mathematics Subject Classification

  • 62G10
  • 62H15