Hods yielded exact error rates. Likewise, power was similar across methods. Resampling risk was L-660711 sodium salt side effects higher for methods (i), (iii) and (v). For comparable resampling risks, the method in which no permutations are done (iv) was the absolute fastest. All methods produced visually similar maps for the real data, with stronger effects being detected in the family-wise error rate corrected maps by (iii) and (v), and generally similar to the results seen in the reference set. Overall, for uncorrected p-values, method (iv) was found the best as long as symmetric errors can be assumed. In all other settings, including for familywise error corrected p-values, we recommend the tail approximation (iii). The methods considered are freely available in the tool PALM — Permutation Analysis of Linear Models. ?2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Article history: Received 12 March 2016 Revised 11 May 2016 Accepted 27 May 2016 Available online 7 June 2016 Keywords: Permutation tests Negative binomial distribution Tail approximation Gamma distribution Generalised Pareto distribution Low rank matrix completion Pearson type III distributionIntroduction Permutation tests allow exact control of error rates, with minimal assumptions. However, permutation tests are computationally intensive. For small, non-imaging datasets, recomputing a model thousands of times is seldom a problem, but for imaging applications, that involve testing at thousands of spatial points (voxels, vertices, faces, edges), large models that involve many subjects, LY2510924 cancer multiple measurements, pointwise (voxelwise) regressors, spatial statistics, as well as other sources of complexity, even with the availability of inexpensive computing power, the same procedure can be prohibitively slow. Strategies to accelerate the process include the use of efficient or optimised code, the use of parallel, multi-threaded, or distributed computing, and the use of graphics processing units (GPUs) (for example applications ofCorresponding author. E-mail address: [email protected] (A.M. Winkler). URL: http://www.fmrib.ox.ac.uk (A.M. Winkler).the latter, see Eklund et al., 2012, 2013; Hern dez et al., 2013). While these methods are attractive for increases in speed, none reduce the amount of tasks that effectively need to be executed, and the improvements in speed happen through more efficient use of resources available, or through the introduction of fpsyg.2017.00209 yet more resources. At a time in which Moore’s law (Moore, 1965) approaches physical limits (Waldrop, 2016), alternative methods to expedite computation are expected to gain prominence. Here we exploit properties of the statistics themselves and their distributions, which could be used to accelerate the evaluation of the test in order to accept or reject the null hypothesis in a fraction of the time that otherwise would be needed with a large number of permutations. The main tenet of these approaches is to obtain a reduction of the number of actual computations that need to be performed, such that acceleration can be obtained in addition to, or irrespective of, generic improvements of software or hardware. In particular, we discuss the following approaches: (i) performing a small number of shufflings (with no other change from the usual case of permutation tests); (ii) estimationhttp://dx.doi.org/10.1016/j.neuroimage.2016.05.068 1053-8119/?2016 The Authors. P.Hods yielded exact error rates. Likewise, power was similar across methods. Resampling risk was higher for methods (i), (iii) and (v). For comparable resampling risks, the method in which no permutations are done (iv) was the absolute fastest. All methods produced visually similar maps for the real data, with stronger effects being detected in the family-wise error rate corrected maps by (iii) and (v), and generally similar to the results seen in the reference set. Overall, for uncorrected p-values, method (iv) was found the best as long as symmetric errors can be assumed. In all other settings, including for familywise error corrected p-values, we recommend the tail approximation (iii). The methods considered are freely available in the tool PALM — Permutation Analysis of Linear Models. ?2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Article history: Received 12 March 2016 Revised 11 May 2016 Accepted 27 May 2016 Available online 7 June 2016 Keywords: Permutation tests Negative binomial distribution Tail approximation Gamma distribution Generalised Pareto distribution Low rank matrix completion Pearson type III distributionIntroduction Permutation tests allow exact control of error rates, with minimal assumptions. However, permutation tests are computationally intensive. For small, non-imaging datasets, recomputing a model thousands of times is seldom a problem, but for imaging applications, that involve testing at thousands of spatial points (voxels, vertices, faces, edges), large models that involve many subjects, multiple measurements, pointwise (voxelwise) regressors, spatial statistics, as well as other sources of complexity, even with the availability of inexpensive computing power, the same procedure can be prohibitively slow. Strategies to accelerate the process include the use of efficient or optimised code, the use of parallel, multi-threaded, or distributed computing, and the use of graphics processing units (GPUs) (for example applications ofCorresponding author. E-mail address: [email protected] (A.M. Winkler). URL: http://www.fmrib.ox.ac.uk (A.M. Winkler).the latter, see Eklund et al., 2012, 2013; Hern dez et al., 2013). While these methods are attractive for increases in speed, none reduce the amount of tasks that effectively need to be executed, and the improvements in speed happen through more efficient use of resources available, or through the introduction of fpsyg.2017.00209 yet more resources. At a time in which Moore’s law (Moore, 1965) approaches physical limits (Waldrop, 2016), alternative methods to expedite computation are expected to gain prominence. Here we exploit properties of the statistics themselves and their distributions, which could be used to accelerate the evaluation of the test in order to accept or reject the null hypothesis in a fraction of the time that otherwise would be needed with a large number of permutations. The main tenet of these approaches is to obtain a reduction of the number of actual computations that need to be performed, such that acceleration can be obtained in addition to, or irrespective of, generic improvements of software or hardware. In particular, we discuss the following approaches: (i) performing a small number of shufflings (with no other change from the usual case of permutation tests); (ii) estimationhttp://dx.doi.org/10.1016/j.neuroimage.2016.05.068 1053-8119/?2016 The Authors. P.