Monte Carlo (MC) permutation test is considered the gold standard for statistical hypothesis testing, especially when standard parametric assumptions are not clear or likely to fail. However, in modern data science settings where a large number of hypothesis tests need to be performed simultaneously, it is rarely used due to its prohibitive computational cost. In genome-wide association studies, for example, the number of hypothesis tests @math is around @math while the number of MC samples @math for each test could be greater than @math , totaling more than @math = @math samples. In this paper, we propose Adaptive MC multiple Testing (AMT) to estimate MC p-values and control false discovery rate in multiple testing. The algorithm outputs the same result as the standard full MC approach with high probability while requiring only @math samples. This sample complexity is shown to be optimal. On a Parkinson GWAS dataset, the algorithm reduces the running time from 2 months for full MC to an hour. The AMT algorithm is derived based on the theory of multi-armed bandits.
It is a common practice to use resampling methods such as the bootstrap for calculating the p-value for each test when performing large scale multiple testing. The precision of the bootstrap p-values and that of the false discovery rate (FDR) relies on the number of bootstraps used for testing each hypothesis. Clearly, the larger the number of bootstraps the better the precision. However, the required number of bootstraps can be computationally burdensome, and it multiplies the number of tests to be performed. Further adding to the computational challenge is that in some applications the calculation of the test statistic itself may require considerable computation time. As technology improves one can expect the dimension of the problem to increase as well. For instance, during the early days of microarray technology, the number of probes on a cDNA chip was less than 10,000. Now the Affymetrix chips come with over 50,000 probes per chip. Motivated by this important need, we developed a simple adaptive bootstrap methodology for large scale multiple testing, which reduces the total number of bootstrap calculations while ensuring the control of the FDR. The proposed algorithm results in a substantial reduction in the number of bootstrap samples. Based on a simulation study we found that, relative to the number of bootstraps required for the Benjamini-Hochberg (BH) procedure, the standard FDR methodology which was the proposed methodology achieved a very substantial reduction in the number of bootstraps. In some cases the new algorithm required as little as 1 6th the number of bootstraps as the conventional BH procedure. Thus, if the conventional BH procedure used 1,000 bootstraps, then the proposed method required only 160 bootstraps. This methodology has been implemented for time-course dose-response data in our software, ORIOGEN, which is available from the authors upon request. We are concerned with a situation in which we would like to test multiple hypotheses with tests whose p-values cannot be computed explicitly but can be approximated using Monte Carlo simulation. This scenario occurs widely in practice. We are interested in obtaining the same rejections and non-rejections as the ones obtained if the p-values for all hypotheses had been available. The present article introduces a framework for this scenario by providing a generic algorithm for a general multiple testing procedure. We establish conditions that guarantee that the rejections and non-rejections obtained through Monte Carlo simulations are identical to the ones obtained with the p-values. Our framework is applicable to a general class of step-up and step-down procedures, which includes many established multiple testing corrections such as the ones of Bonferroni, Holm, Sidak, Hochberg or Benjamini–Hochberg. Moreover, we show how to use our framework to improve algorithms available in the literature in such a way as to yield theoretical guarantees on their results. These modifications can easily be implemented in practice and lead to a particular way of reporting multiple testing results as three sets together with an error bound on their correctness, demonstrated exemplarily using a real biological dataset. Multiple hypothesis testing is widely used to evaluate scientific studies involving statistical tests. However, for many of these tests, p values are not available and are thus often approximated using Monte Carlo tests such as permutation tests or bootstrap tests. This article presents a simple algorithm based on Thompson Sampling to test multiple hypotheses. It works with arbitrary multiple testing procedures, in particular with step-up and step-down procedures. Its main feature is to sequentially allocate Monte Carlo effort, generating more Monte Carlo samples for tests whose decisions are so far less certain. A simulation study demonstrates that for a low computational effort, the new approach yields a higher power and a higher degree of reproducibility of its results than previously suggested methods. Motivation: In molecular biology, as in many other scientific fields, the scale of analyses is ever increasing. Often, complex Monte Carlo simulation is required, sometimes within a large-scale multiple testing setting. The resulting computational costs may be prohibitively high. Results: We here present MCFDR, a simple, novel algorithm for false discovery rate (FDR) modulated sequential Monte Carlo (MC) multiple hypothesis testing. The algorithm iterates between adding MC samples across tests and calculating intermediate FDR values for the collection of tests. MC sampling is stopped either by sequential MC or based on a threshold on FDR. An essential property of the algorithm is that it limits the total number of MC samples whatever the number of true null hypotheses. We show on both real and simulated data that the proposed algorithm provides large gains in computational efficiency. Availability: MCFDR is implemented in the Genomic HyperBrowser ( http: hyperbrowser.uio.no mcfdr), a web-based system for genome analysis. All input data and results are available and can be reproduced through a Galaxy Pages document at: http: hyperbrowser.uio.no mcfdr u sandve p mcfdr. Contact: geirksa@ifi.uio.no Multiple testing is often carried out in practice using approximated p-values obtained, for instance, via bootstrap or permutation tests. We are interested in allocating a pre-specified total number of samples (that is draws from a bootstrap distribution or permutations) to all hypotheses in order to approximate their p-values in an optimal way, in the sense that the allocation minimizes the total expected number of misclassified hypotheses. By a misclassified hypothesis we refer to a decision on single hypotheses which differs from the one obtained if all p-values were known analytically. Neither using a constant number of samples per p-value estimate nor more sophisticated approaches available in the literature guarantee the computation of an optimal allocation in the above sense. This article derives the optimal allocation of a finite total number of samples to a finite number of hypotheses tested using the Bonferroni correction. Simulation studies show that a simple sampling algorithm based on Thompson Sampling asympotically mimics this optimal allocation. Software packages usually report the results of statistical tests using p-values. Users often interpret these by comparing them to standard thresholds, e.g. 0.1 , 1 and 5 , which is sometimes reinforced by a star rating (***, **, *). In this article, we consider an arbitrary statistical test whose p-value p is not available explicitly, but can be approximated by Monte Carlo samples, e.g. by bootstrap or permutation tests. The standard implementation of such tests usually draws a fixed number of samples to approximate p. However, the probability that the exact and the approximated p-value lie on different sides of a threshold (the resampling risk) can be high, particularly for p-values close to a threshold. We present a method to overcome this. We consider a finite set of user-specified intervals which cover [0,1] and which can be overlapping. We call these p-value buckets. We present algorithms that, with arbitrarily high probability, return a p-value bucket containing p. We prove that for both a bounded resampling risk and a finite runtime, overlapping buckets need to be employed, and that our methods both bound the resampling risk and guarantee a finite runtime for such overlapping buckets. To interpret decisions with overlapping buckets, we propose an extension of the star rating system. We demonstrate that our methods are suitable for use in standard software, including for low p-values occurring in multiple testing settings, and that they can be computationally more efficient than standard implementations.