E fnhum.2014.00074 number of permutations, and gamma slightly slower than few permutations. This considers a voxelwise fit, for uncorrected p-values; if only corrected p-values are needed, the time needed for the single fit of the GPD or gamma for the distribution of extremum statistic is negligible. The negative binomial and, specially, low rank matrix completion were the Tyrphostin AG 490 price slowest. Low rank, however, fpsyg.2017.00209 is expected to perform better in settings where there are more tests to be performed (more voxels) than those used in the simulations and real data, and with a relatively smaller sample size (Table 4). Noise distribution and shuffling strategy The performance of the various methods was similar in terms of error rates, power, resampling risk, and speed, regardless of the errors being Gaussian or Weibull (skewed). However, as expected given its assumptions, the method in which no permutations are used did not produce correct results that could be compared with the purchase Quizartinib reference set if the reference set used sign-flippings (for either error distribution), or if the errors were skewed (regardless of the shuffling strategy, i.e., permutations, signflippings, or permutations with sign-flippings). Spatial statistics The behaviour for spatial statistics followed the same trends as for the voxelwise, non-spatial statistics, in terms of error rates, power, agreement with the reference set, and resampling risk. Multivariate statistics and non-parametric combination Likewise, the results for CMV and for NPC followed similar trends as above, with error rates controlled exactly, and yielding similar power as the reference set, as evidenced by the results of Phase I shown in the Supplementary Material. Real data All methods yielded visually similar maps for the real data, with smaller p-values observable with more permutations for the methods that use permutations, or more exceedances for the negative binomial. In the TFCE, FWER-corrected maps, stronger effects of interest could be revealed by tail and gamma methods for equivalent J of few permutations. These results are remarkably similar to the results seen in the reference set, even using about a hundred times fewer permutations, with proportional increases in speed, as summarised in Figs. 3 and 4, and shown in greater detail in the Supplementary Material. The timings refer to the implementation available in PALM, as described at the end of the paper. The acceleration methods worked similarly, and yielded similar increases in speed, for the two levels of smoothing considered. Discussion Assumptions All six methods presented are non-parametric in the sense that they do not depend on the distribution of the test statistic. Some of the methods can still be said to be parametric in that certain parameters need to be estimated, such as for the gamma or for the generalised Pareto distribution, although they remain non-parametric in that theA.M. Winkler et al. / NeuroImage 141 (2016) 502?Fig. 2. Balance between resampling risk when compared to a reference set of J=50000 permutations and the respective running time, with the data simulated for Phase II (hence, 100 repetitions, Gaussian noise). Some methods have parameters that could be varied: few permutations, tail approximation and gamma approximation use a certain number of permutations that varied in the simulations as J=40,60,100,200,300,500,1000,2000,5000. The negative binomial distribution uses a fixed upper limit on the number of permutations (set as J=50000) and a numbe.E fnhum.2014.00074 number of permutations, and gamma slightly slower than few permutations. This considers a voxelwise fit, for uncorrected p-values; if only corrected p-values are needed, the time needed for the single fit of the GPD or gamma for the distribution of extremum statistic is negligible. The negative binomial and, specially, low rank matrix completion were the slowest. Low rank, however, fpsyg.2017.00209 is expected to perform better in settings where there are more tests to be performed (more voxels) than those used in the simulations and real data, and with a relatively smaller sample size (Table 4). Noise distribution and shuffling strategy The performance of the various methods was similar in terms of error rates, power, resampling risk, and speed, regardless of the errors being Gaussian or Weibull (skewed). However, as expected given its assumptions, the method in which no permutations are used did not produce correct results that could be compared with the reference set if the reference set used sign-flippings (for either error distribution), or if the errors were skewed (regardless of the shuffling strategy, i.e., permutations, signflippings, or permutations with sign-flippings). Spatial statistics The behaviour for spatial statistics followed the same trends as for the voxelwise, non-spatial statistics, in terms of error rates, power, agreement with the reference set, and resampling risk. Multivariate statistics and non-parametric combination Likewise, the results for CMV and for NPC followed similar trends as above, with error rates controlled exactly, and yielding similar power as the reference set, as evidenced by the results of Phase I shown in the Supplementary Material. Real data All methods yielded visually similar maps for the real data, with smaller p-values observable with more permutations for the methods that use permutations, or more exceedances for the negative binomial. In the TFCE, FWER-corrected maps, stronger effects of interest could be revealed by tail and gamma methods for equivalent J of few permutations. These results are remarkably similar to the results seen in the reference set, even using about a hundred times fewer permutations, with proportional increases in speed, as summarised in Figs. 3 and 4, and shown in greater detail in the Supplementary Material. The timings refer to the implementation available in PALM, as described at the end of the paper. The acceleration methods worked similarly, and yielded similar increases in speed, for the two levels of smoothing considered. Discussion Assumptions All six methods presented are non-parametric in the sense that they do not depend on the distribution of the test statistic. Some of the methods can still be said to be parametric in that certain parameters need to be estimated, such as for the gamma or for the generalised Pareto distribution, although they remain non-parametric in that theA.M. Winkler et al. / NeuroImage 141 (2016) 502?Fig. 2. Balance between resampling risk when compared to a reference set of J=50000 permutations and the respective running time, with the data simulated for Phase II (hence, 100 repetitions, Gaussian noise). Some methods have parameters that could be varied: few permutations, tail approximation and gamma approximation use a certain number of permutations that varied in the simulations as J=40,60,100,200,300,500,1000,2000,5000. The negative binomial distribution uses a fixed upper limit on the number of permutations (set as J=50000) and a numbe.

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