Permutation tests for between-unit fixed effects in multivariate generalized linear mixed models
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A permutation testing approach in multivariate mixed models is presented. The solutions proposed allow for testing between-unit effect; they are exact under some assumptions, while approximated in the more general case. The classes of models comprised by this approach include generalized linear models, vector generalized additive models and other nonparametric models based on smoothing. Moreover it does not assume observations of different units to have the same distribution. The extensions to a multivariate framework are presented and discussed. The proposed multivariate tests exploit the dependence among variables, hence increasing the power with respect to other standard solutions (e.g. Bonferroni correction) which combine many univariate tests in an overall one. Examples are given of two applications to real data from psychological and ecological studies; a simulation study provides some insight into the unbiasedness of the tests and their power. The methods were implemented in the R package flip, freely available on CRAN.
KeywordsPermutation tests Mixed models Multivariate inference
The authors gratefully acknowledge Elisa Di Giorgio, Chiara Turati, Gianmarco Altoè and Francesca Simion for making their data available for the psychological example and Valerio Matozzo, Andrea Chinellato, Marco Munari, Monica Bressan and Maria Gabriella Marin for making their data available for the ecological example. The authors are also grateful to an associate editor and two referees for detailed comments that helped clarify the proofs and the presentation. LF was supported by grant from the University of Padua (Progetti di Ricerca di Ateneo 2011, project CPDA117517) and by the Cariparo Foundation Excellence grant 2011/2012.
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