Ublished by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).A.M. Winkler et al. / NeuroImage 141 (2016) 502?of the p-value as jasp.12117 a parameter of a negative binomial distribution; (iii) fitting of a generalised Pareto distribution to the tail of the empirical permutation distribution; (iv) computing the p-values based on the expected moments of the empirical distribution, approximated from a gamma distribution; (v) direct fitting of a gamma distribution to the empirical distribution; and (vi) shuffling of a reduced number of points (e.g., voxels), with completion of the remainder using low rank matrix theory. Details of each are provided in the Theory section. Very few of such acceleration strategies have been investigated or used in brain imaging. The tail approximation was considered by Ge et al. (2012) for an imaging genetics application in which, due to the sheer volume of data, conventional permutation tests were not considered feasible. A variant of many possible algorithms for low rank matrix completion was proposed by Hinrichs et al. (2013). The fitting of a gamma distribution without the need for permutations was proposed recently for a range of statistics by Minas and Montana (2014). For inference for support-vector machine problems, Gaonkar and Davatzikos (2012) suggested an analytical approximation to the permutation distribution of each component of the hyperplane that separates classes. Here we aim to study, evaluate, and in some cases propose, solutions that can accelerate permutation tests for the general linear model (GLM), considering aspects that are specially scan/nsw074 relevant to imaging, such as the multiplicity of tests and the use of spatial statistics. In particular, we make the following main contributions: (I) show how a connection between Pillai’s trace and the popular univariate t statistic allows the direct computation of the p-values from the permutation distribution, even without performing actual permutations, (II) use the moments of the empirical permutation distribution for the fit of a gamma distribution, and (III) propose a novel low rank matrix completion algorithm, writing the test statistic as the product of two matrices that can be sampled sparsely, and allowing exact recovery of what otherwise would be an approximation. Overview of the paper In the Theory section we begin by briefly reviewing the uni- and multivariate GLM, their assessment using permutation tests, andintroduce the Ensartinib msds Notation used throughout the paper. The six different acceleration strategies are then presented in sequence, followed by certain aspects related to spatial statistics and multiple testing correction in the context of these methods. In the Evaluation and Results sections we assess the performance of these different methods on both synthetic and real data. In the Discussion we provide recommendations for general circumstances. A summary of the acceleration strategies is provided in Table 1. Fig. 1 illustrates four of them. Theory Notation and general aspects At each spatial point of an image representation of the brain, consider a general linear model (GLM) (Scheff? 1959) expressed as: Y ?M ? ??where Y is the N ?K matrix of observed data, with N observations of K distinct (possibly non-independent) variables, M is the full-rank N ?R design matrix of explanatory BLU-554 web variables (i.e., effects of interest and possibly nuisance effects), is the R?K matrix of regression coeffic.Ublished by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).A.M. Winkler et al. / NeuroImage 141 (2016) 502?of the p-value as jasp.12117 a parameter of a negative binomial distribution; (iii) fitting of a generalised Pareto distribution to the tail of the empirical permutation distribution; (iv) computing the p-values based on the expected moments of the empirical distribution, approximated from a gamma distribution; (v) direct fitting of a gamma distribution to the empirical distribution; and (vi) shuffling of a reduced number of points (e.g., voxels), with completion of the remainder using low rank matrix theory. Details of each are provided in the Theory section. Very few of such acceleration strategies have been investigated or used in brain imaging. The tail approximation was considered by Ge et al. (2012) for an imaging genetics application in which, due to the sheer volume of data, conventional permutation tests were not considered feasible. A variant of many possible algorithms for low rank matrix completion was proposed by Hinrichs et al. (2013). The fitting of a gamma distribution without the need for permutations was proposed recently for a range of statistics by Minas and Montana (2014). For inference for support-vector machine problems, Gaonkar and Davatzikos (2012) suggested an analytical approximation to the permutation distribution of each component of the hyperplane that separates classes. Here we aim to study, evaluate, and in some cases propose, solutions that can accelerate permutation tests for the general linear model (GLM), considering aspects that are specially scan/nsw074 relevant to imaging, such as the multiplicity of tests and the use of spatial statistics. In particular, we make the following main contributions: (I) show how a connection between Pillai’s trace and the popular univariate t statistic allows the direct computation of the p-values from the permutation distribution, even without performing actual permutations, (II) use the moments of the empirical permutation distribution for the fit of a gamma distribution, and (III) propose a novel low rank matrix completion algorithm, writing the test statistic as the product of two matrices that can be sampled sparsely, and allowing exact recovery of what otherwise would be an approximation. Overview of the paper In the Theory section we begin by briefly reviewing the uni- and multivariate GLM, their assessment using permutation tests, andintroduce the notation used throughout the paper. The six different acceleration strategies are then presented in sequence, followed by certain aspects related to spatial statistics and multiple testing correction in the context of these methods. In the Evaluation and Results sections we assess the performance of these different methods on both synthetic and real data. In the Discussion we provide recommendations for general circumstances. A summary of the acceleration strategies is provided in Table 1. Fig. 1 illustrates four of them. Theory Notation and general aspects At each spatial point of an image representation of the brain, consider a general linear model (GLM) (Scheff? 1959) expressed as: Y ?M ? ??where Y is the N ?K matrix of observed data, with N observations of K distinct (possibly non-independent) variables, M is the full-rank N ?R design matrix of explanatory variables (i.e., effects of interest and possibly nuisance effects), is the R?K matrix of regression coeffic.