statistics the minimum p -value, min$_{m}$$_{∈{}$$_{2}$$_{,...,m}$$_{max}$$_{}}$ p$_{m}$ , as well as the Fisher combined p -value, − ∑ m$_{max}$ m =2 log p$_{m}$ . These combined p -values are not p -values in themselves, but their null distribution can be easily obtained from the null distributions of the test statistics for fixed m s, as follows: (1) for each of B permutations, compute the test statistics for each m ∈ { 2 ,...,m$_{max}$ } ; (2) compute the p -value of each of the resulting statistics, so for each permutation, we have a set of p -values p$_{2}$,...,p$_{m}$$_{max}$ to combine; (3) combine the p -values for each of the B permutations. Choose B to be large enough for the desired accuracy of approximation of the quantiles of the null distribution of the combined p -values used for testing. Obviously, since the combined p -values are based on the ranks of the data, they are distribution-free. Since they do not require fixing m in advance, they are a practical alternative to the tests that require m as input. In order to examine how close this regularized score is to the optimal m (i.e., the m with highest power), we looked at the distribution of the m s with minimum p - values in 20000 data samples from the above-mentioned three simulation settings. For these settings, using the aggregation by maximization statistic, the median m of the minimal p -value was: 3 for the first setting, 9 for the second setting, and 33 for the third setting. Moreover, the first and third quartiles were 3 to 5 for the first setting, 7 to 14 for the second setting, and 19 to 60 for the third setting. We conclude that for these examples, the m that achieves the minimum p -values in most runs was remarkably close to the optimal m (which was 3, 10, and 34 for settings 1,2, and 3, respectively), suggesting that the power of the minimum p -value statistic is close to that of the statistic with optimal m . Indeed, the power of the minimum p -values in settings 1-3 using aggregation by maximum was 0.825, 0.799, and 0.785, whereas the power using the (unknown in practice) optimal m in settings 1-3 was 0.894, 0.86, and 0.88, respectively. Further empirical investigations detailed in Section 4 give additional support to this regularization method. 17