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1.  EPEPT: A web service for enhanced P-value estimation in permutation tests 
BMC Bioinformatics  2011;12:411.
In computational biology, permutation tests have become a widely used tool to assess the statistical significance of an event under investigation. However, the common way of computing the P-value, which expresses the statistical significance, requires a very large number of permutations when small (and thus interesting) P-values are to be accurately estimated. This is computationally expensive and often infeasible. Recently, we proposed an alternative estimator, which requires far fewer permutations compared to the standard empirical approach while still reliably estimating small P-values [1].
The proposed P-value estimator has been enriched with additional functionalities and is made available to the general community through a public website and web service, called EPEPT. This means that the EPEPT routines can be accessed not only via a website, but also programmatically using any programming language that can interact with the web. Examples of web service clients in multiple programming languages can be downloaded. Additionally, EPEPT accepts data of various common experiment types used in computational biology. For these experiment types EPEPT first computes the permutation values and then performs the P-value estimation. Finally, the source code of EPEPT can be downloaded.
Different types of users, such as biologists, bioinformaticians and software engineers, can use the method in an appropriate and simple way.
PMCID: PMC3277916  PMID: 22024252
2.  Moment based gene set tests 
BMC Bioinformatics  2015;16(1):132.
Permutation-based gene set tests are standard approaches for testing relationships between collections of related genes and an outcome of interest in high throughput expression analyses. Using M random permutations, one can attain p-values as small as 1/(M+1). When many gene sets are tested, we need smaller p-values, hence larger M, to achieve significance while accounting for the number of simultaneous tests being made. As a result, the number of permutations to be done rises along with the cost per permutation. To reduce this cost, we seek parametric approximations to the permutation distributions for gene set tests.
We study two gene set methods based on sums and sums of squared correlations. The statistics we study are among the best performers in the extensive simulation of 261 gene set methods by Ackermann and Strimmer in 2009. Our approach calculates exact relevant moments of these statistics and uses them to fit parametric distributions. The computational cost of our algorithm for the linear case is on the order of doing |G| permutations, where |G| is the number of genes in set G. For the quadratic statistics, the cost is on the order of |G|2 permutations which can still be orders of magnitude faster than plain permutation sampling. We applied the permutation approximation method to three public Parkinson’s Disease expression datasets and discovered enriched gene sets not previously discussed. We found that the moment-based gene set enrichment p-values closely approximate the permutation method p-values at a tiny fraction of their cost. They also gave nearly identical rankings to the gene sets being compared.
We have developed a moment based approximation to linear and quadratic gene set test statistics’ permutation distribution. This allows approximate testing to be done orders of magnitude faster than one could do by sampling permutations.
We have implemented our method as a publicly available Bioconductor package, npGSEA (
Electronic supplementary material
The online version of this article (doi:10.1186/s12859-015-0571-7) contains supplementary material, which is available to authorized users.
PMCID: PMC4419444  PMID: 25928861
GSEA; Expression analysis; Permutation tests; ROAST
3.  Rapid and Accurate Multiple Testing Correction and Power Estimation for Millions of Correlated Markers 
PLoS Genetics  2009;5(4):e1000456.
With the development of high-throughput sequencing and genotyping technologies, the number of markers collected in genetic association studies is growing rapidly, increasing the importance of methods for correcting for multiple hypothesis testing. The permutation test is widely considered the gold standard for accurate multiple testing correction, but it is often computationally impractical for these large datasets. Recently, several studies proposed efficient alternative approaches to the permutation test based on the multivariate normal distribution (MVN). However, they cannot accurately correct for multiple testing in genome-wide association studies for two reasons. First, these methods require partitioning of the genome into many disjoint blocks and ignore all correlations between markers from different blocks. Second, the true null distribution of the test statistic often fails to follow the asymptotic distribution at the tails of the distribution. We propose an accurate and efficient method for multiple testing correction in genome-wide association studies—SLIDE. Our method accounts for all correlation within a sliding window and corrects for the departure of the true null distribution of the statistic from the asymptotic distribution. In simulations using the Wellcome Trust Case Control Consortium data, the error rate of SLIDE's corrected p-values is more than 20 times smaller than the error rate of the previous MVN-based methods' corrected p-values, while SLIDE is orders of magnitude faster than the permutation test and other competing methods. We also extend the MVN framework to the problem of estimating the statistical power of an association study with correlated markers and propose an efficient and accurate power estimation method SLIP. SLIP and SLIDE are available at
Author Summary
In genome-wide association studies, it is important to account for the fact that a large number of genetic variants are tested in order to adequately control for false positives. The simplest way to correct for multiple hypothesis testing is the Bonferroni correction, which multiplies the p-values by the number of markers assuming the markers are independent. Since the markers are correlated due to linkage disequilibrium, this approach leads to a conservative estimate of false positives, thus adversely affecting statistical power. The permutation test is considered the gold standard for accurate multiple testing correction, but is often computationally impractical for large association studies. We propose a method that efficiently and accurately corrects for multiple hypotheses in genome-wide association studies by fully accounting for the local correlation structure between markers. Our method also corrects for the departure of the true distribution of test statistics from the asymptotic distribution, which dramatically improves the accuracy, particularly when many rare variants are included in the tests. Our method shows a near identical accuracy to permutation and shows greater computational efficiency than previously suggested methods. We also provide a method to accurately and efficiently estimate the statistical power of genome-wide association studies.
PMCID: PMC2663787  PMID: 19381255
4.  Efficient p-value estimation in massively parallel testing problems 
Biostatistics (Oxford, England)  2008;9(4):601-612.
We present a new method to efficiently estimate very large numbers of p-values using empirically constructed null distributions of a test statistic. The need to evaluate a very large number of p-values is increasingly common with modern genomic data, and when interaction effects are of interest, the number of tests can easily run into billions. When the asymptotic distribution is not easily available, permutations are typically used to obtain p-values but these can be computationally infeasible in large problems. Our method constructs a prediction model to obtain a first approximation to the p-values and uses Bayesian methods to choose a fraction of these to be refined by permutations. We apply and evaluate our method on the study of association between 2-way interactions of genetic markers and colorectal cancer using the data from the first phase of a large, genome-wide case–control study. The results show enormous computational savings as compared to evaluating a full set of permutations, with little decrease in accuracy.
PMCID: PMC2536722  PMID: 18304995
Bayesian testing; Genome-wide association studies; Interaction effects; Permutation distribution; p-value distribution; Random Forest
5.  Efficient Calculation of Empirical P-values for Genome-Wide Linkage Analysis Through Weighted Permutation 
Behavior genetics  2008;39(1):91-100.
Linkage analysis in multivariate or longitudinal context presents both statistical and computational challenges. The permutation test can be used to avoid some of the statistical challenges, but it substantially adds to the computational burden. Utilizing the distributional dependencies between π̂ (defined as the proportion of alleles at a locus that are identical by descent (IBD) for a pairs of relatives, at a given locus) and the permutation test we report a new method of efficient permutation. In summary, the distribution of π̂ for a sample of relatives at locus x is estimated as a weighted mixture of π̂ drawn from a pool of ‘representative’ π̂ distributions observed at other loci. This weighting scheme is then used to sample from the distribution of the permutation tests at the representative loci to obtain an empirical P-value at locus x (which is asymptotically distributed as the permutation test at loci x). This weighted mixture approach greatly reduces the number of permutation tests required for genome-wide scanning, making it suitable for use in multivariate and other computationally intensive linkage analyses. In addition, because the distribution of π̂ is a property of the genotypic data for a given sample and is independent of the phenotypic data, the weighting scheme can be applied to any phenotype (or combination of phenotypes) collected from that sample. We demonstrate the validity of this approach through simulation.
PMCID: PMC4010137  PMID: 18810631
Empirical significance level; Mixture distribution; Linkage analysis
6.  PRESTO: Rapid calculation of order statistic distributions and multiple-testing adjusted P-values via permutation for one and two-stage genetic association studies 
BMC Bioinformatics  2008;9:309.
Large-scale genetic association studies can test hundreds of thousands of genetic markers for association with a trait. Since the genetic markers may be correlated, a Bonferroni correction is typically too stringent a correction for multiple testing. Permutation testing is a standard statistical technique for determining statistical significance when performing multiple correlated tests for genetic association. However, permutation testing for large-scale genetic association studies is computationally demanding and calls for optimized algorithms and software. PRESTO is a new software package for genetic association studies that performs fast computation of multiple-testing adjusted P-values via permutation of the trait.
PRESTO is an order of magnitude faster than other existing permutation testing software, and can analyze a large genome-wide association study (500 K markers, 5 K individuals, 1 K permutations) in approximately one hour of computing time. PRESTO has several unique features that are useful in a wide range of studies: it reports empirical null distributions for the top-ranked statistics (i.e. order statistics), it performs user-specified combinations of allelic and genotypic tests, it performs stratified analysis when sampled individuals are from multiple populations and each individual's population of origin is specified, and it determines significance levels for one and two-stage genotyping designs. PRESTO is designed for case-control studies, but can also be applied to trio data (parents and affected offspring) if transmitted parental alleles are coded as case alleles and untransmitted parental alleles are coded as control alleles.
PRESTO is a platform-independent software package that performs fast and flexible permutation testing for genetic association studies. The PRESTO executable file, Java source code, example data, and documentation are freely available at .
PMCID: PMC2483288  PMID: 18620604
7.  An adaptive permutation approach for genome-wide association study: evaluation and recommendations for use 
BioData Mining  2014;7:9.
Permutation testing is a robust and popular approach for significance testing in genomic research, which has the broad advantage of estimating significance non-parametrically, thereby safe guarding against inflated type I error rates. However, the computational efficiency remains a challenging issue that limits its wide application, particularly in genome-wide association studies (GWAS). Because of this, adaptive permutation strategies can be employed to make permutation approaches feasible. While these approaches have been used in practice, there is little research into the statistical properties of these approaches, and little guidance into the proper application of such a strategy for accurate p-value estimation at the GWAS level.
In this work, we advocate an adaptive permutation procedure that is statistically valid as well as computationally feasible in GWAS. We perform extensive simulation experiments to evaluate the robustness of the approach to violations of modeling assumptions and compare the power of the adaptive approach versus standard approaches. We also evaluate the parameter choices in implementing the adaptive permutation approach to provide guidance on proper implementation in real studies. Additionally, we provide an example of the application of adaptive permutation testing on real data.
The results provide sufficient evidence that the adaptive test is robust to violations of modeling assumptions. In addition, even when modeling assumptions are correct, the power achieved by adaptive permutation is identical to the parametric approach over a range of significance thresholds and effect sizes under the alternative. A framework for proper implementation of the adaptive procedure is also generated.
While the adaptive permutation approach presented here is not novel, the current study provides evidence of the validity of the approach, and importantly provides guidance on the proper implementation of such a strategy. Additionally, tools are made available to aid investigators in implementing these approaches.
PMCID: PMC4070098  PMID: 24976866
8.  Systematic permutation testing in GWAS pathway analyses: identification of genetic networks in dilated cardiomyopathy and ulcerative colitis 
BMC Genomics  2014;15:622.
Genome wide association studies (GWAS) are applied to identify genetic loci, which are associated with complex traits and human diseases. Analogous to the evolution of gene expression analyses, pathway analyses have emerged as important tools to uncover functional networks of genome-wide association data. Usually, pathway analyses combine statistical methods with a priori available biological knowledge. To determine significance thresholds for associated pathways, correction for multiple testing and over-representation permutation testing is applied.
We systematically investigated the impact of three different permutation test approaches for over-representation analysis to detect false positive pathway candidates and evaluate them on genome-wide association data of Dilated Cardiomyopathy (DCM) and Ulcerative Colitis (UC). Our results provide evidence that the gold standard - permuting the case–control status – effectively improves specificity of GWAS pathway analysis. Although permutation of SNPs does not maintain linkage disequilibrium (LD), these permutations represent an alternative for GWAS data when case–control permutations are not possible. Gene permutations, however, did not add significantly to the specificity. Finally, we provide estimates on the required number of permutations for the investigated approaches.
To discover potential false positive functional pathway candidates and to support the results from standard statistical tests such as the Hypergeometric test, permutation tests of case control data should be carried out. The most reasonable alternative was case–control permutation, if this is not possible, SNP permutations may be carried out. Our study also demonstrates that significance values converge rapidly with an increasing number of permutations. By applying the described statistical framework we were able to discover axon guidance, focal adhesion and calcium signaling as important DCM-related pathways and Intestinal immune network for IgA production as most significant UC pathway.
PMCID: PMC4223581  PMID: 25052024
DCM; UC; GWAS; Permutation tests; Pathway analysis
9.  Accurate Assignment of Significance to Neuropeptide Identifications Using Monte Carlo K-Permuted Decoy Databases 
PLoS ONE  2014;9(10):e111112.
In support of accurate neuropeptide identification in mass spectrometry experiments, novel Monte Carlo permutation testing was used to compute significance values. Testing was based on k-permuted decoy databases, where k denotes the number of permutations. These databases were integrated with a range of peptide identification indicators from three popular open-source database search software (OMSSA, Crux, and X! Tandem) to assess the statistical significance of neuropeptide spectra matches. Significance p-values were computed as the fraction of the sequences in the database with match indicator value better than or equal to the true target spectra. When applied to a test-bed of all known manually annotated mouse neuropeptides, permutation tests with k-permuted decoy databases identified up to 100% of the neuropeptides at p-value < 10−5. The permutation test p-values using hyperscore (X! Tandem), E-value (OMSSA) and Sp score (Crux) match indicators outperformed all other match indicators. The robust performance to detect peptides of the intuitive indicator “number of matched ions between the experimental and theoretical spectra” highlights the importance of considering this indicator when the p-value was borderline significant. Our findings suggest permutation decoy databases of size 1×105 are adequate to accurately detect neuropeptides and this can be exploited to increase the speed of the search. The straightforward Monte Carlo permutation testing (comparable to a zero order Markov model) can be easily combined with existing peptide identification software to enable accurate and effective neuropeptide detection. The source code is available at
PMCID: PMC4201571  PMID: 25329667
10.  A fast algorithm for determining bounds and accurate approximate p-values of the rank product statistic for replicate experiments 
BMC Bioinformatics  2014;15(1):367.
The rank product method is a powerful statistical technique for identifying differentially expressed molecules in replicated experiments. A critical issue in molecule selection is accurate calculation of the p-value of the rank product statistic to adequately address multiple testing. Both exact calculation and permutation and gamma approximations have been proposed to determine molecule-level significance. These current approaches have serious drawbacks as they are either computationally burdensome or provide inaccurate estimates in the tail of the p-value distribution.
We derive strict lower and upper bounds to the exact p-value along with an accurate approximation that can be used to assess the significance of the rank product statistic in a computationally fast manner. The bounds and the proposed approximation are shown to provide far better accuracy over existing approximate methods in determining tail probabilities, with the slightly conservative upper bound protecting against false positives. We illustrate the proposed method in the context of a recently published analysis on transcriptomic profiling performed in blood.
We provide a method to determine upper bounds and accurate approximate p-values of the rank product statistic. The proposed algorithm provides an order of magnitude increase in throughput as compared with current approaches and offers the opportunity to explore new application domains with even larger multiple testing issue. The R code is published in one of the Additional files and is available at
Electronic supplementary material
The online version of this article (doi:10.1186/s12859-014-0367-1) contains supplementary material, which is available to authorized users.
PMCID: PMC4245829  PMID: 25413493
Rank product statistic; p-value distribution; Transcriptomics; Proteomics; Metabolomics
11.  Double-check: validation of diagnostic statistics for PLS-DA models in metabolomics studies 
Metabolomics  2011;8(Suppl 1):3-16.
Partial Least Squares-Discriminant Analysis (PLS-DA) is a PLS regression method with a special binary ‘dummy’ y-variable and it is commonly used for classification purposes and biomarker selection in metabolomics studies. Several statistical approaches are currently in use to validate outcomes of PLS-DA analyses e.g. double cross validation procedures or permutation testing. However, there is a great inconsistency in the optimization and the assessment of performance of PLS-DA models due to many different diagnostic statistics currently employed in metabolomics data analyses. In this paper, properties of four diagnostic statistics of PLS-DA, namely the number of misclassifications (NMC), the Area Under the Receiver Operating Characteristic (AUROC), Q2 and Discriminant Q2 (DQ2) are discussed. All four diagnostic statistics are used in the optimization and the performance assessment of PLS-DA models of three different-size metabolomics data sets obtained with two different types of analytical platforms and with different levels of known differences between two groups: control and case groups. Statistical significance of obtained PLS-DA models was evaluated with permutation testing. PLS-DA models obtained with NMC and AUROC are more powerful in detecting very small differences between groups than models obtained with Q2 and Discriminant Q2 (DQ2). Reproducibility of obtained PLS-DA models outcomes, models complexity and permutation test distributions are also investigated to explain this phenomenon. DQ2 and Q2 (in contrary to NMC and AUROC) prefer PLS-DA models with lower complexity and require higher number of permutation tests and submodels to accurately estimate statistical significance of the model performance. NMC and AUROC seem more efficient and more reliable diagnostic statistics and should be recommended in two group discrimination metabolomic studies.
Electronic supplementary material
The online version of this article (doi:10.1007/s11306-011-0330-3) contains supplementary material, which is available to authorized users.
PMCID: PMC3337399  PMID: 22593721
PLS-DA; AUROC; DQ2; Q2; Misclassifications; Diagnostic statistics; Metabolomics; Life Sciences; Developmental Biology; Cell Biology; Biochemistry, general; Biomedicine general; Molecular Medicine
12.  MCPerm: A Monte Carlo Permutation Method for Accurately Correcting the Multiple Testing in a Meta-Analysis of Genetic Association Studies 
PLoS ONE  2014;9(2):e89212.
Traditional permutation (TradPerm) tests are usually considered the gold standard for multiple testing corrections. However, they can be difficult to complete for the meta-analyses of genetic association studies based on multiple single nucleotide polymorphism loci as they depend on individual-level genotype and phenotype data to perform random shuffles, which are not easy to obtain. Most meta-analyses have therefore been performed using summary statistics from previously published studies. To carry out a permutation using only genotype counts without changing the size of the TradPerm P-value, we developed a Monte Carlo permutation (MCPerm) method. First, for each study included in the meta-analysis, we used a two-step hypergeometric distribution to generate a random number of genotypes in cases and controls. We then carried out a meta-analysis using these random genotype data. Finally, we obtained the corrected permutation P-value of the meta-analysis by repeating the entire process N times. We used five real datasets and five simulation datasets to evaluate the MCPerm method and our results showed the following: (1) MCPerm requires only the summary statistics of the genotype, without the need for individual-level data; (2) Genotype counts generated by our two-step hypergeometric distributions had the same distributions as genotype counts generated by shuffling; (3) MCPerm had almost exactly the same permutation P-values as TradPerm (r = 0.999; P<2.2e-16); (4) The calculation speed of MCPerm is much faster than that of TradPerm. In summary, MCPerm appears to be a viable alternative to TradPerm, and we have developed it as a freely available R package at CRAN:
PMCID: PMC3931718  PMID: 24586601
13.  A Computationally Efficient Hypothesis Testing Method for Epistasis Analysis using Multifactor Dimensionality Reduction 
Genetic epidemiology  2009;33(1):87-94.
Multifactor dimensionality reduction (MDR) was developed as a nonparametric and model-free data mining method for detecting, characterizing, and interpreting epistasis in the absence of significant main effects in genetic and epidemiologic studies of complex traits such as disease susceptibility. The goal of MDR is to change the representation of the data using a constructive induction algorithm to make nonadditive interactions easier to detect using any classification method such as naïve Bayes or logistic regression. Traditionally, MDR constructed variables have been evaluated with a naïve Bayes classifier that is combined with 10-fold cross validation to obtain an estimate of predictive accuracy or generalizability of epistasis models. Traditionally, we have used permutation testing to statistically evaluate the significance of models obtained through MDR. The advantage of permutation testing is that it controls for false-positives due to multiple testing. The disadvantage is that permutation testing is computationally expensive. This is in an important issue that arises in the context of detecting epistasis on a genome-wide scale. The goal of the present study was to develop and evaluate several alternatives to large-scale permutation testing for assessing the statistical significance of MDR models. Using data simulated from 70 different epistasis models, we compared the power and type I error rate of MDR using a 1000-fold permutation test with hypothesis testing using an extreme value distribution (EVD). We find that this new hypothesis testing method provides a reasonable alternative to the computationally expensive 1000-fold permutation test and is 50 times faster. We then demonstrate this new method by applying it to a genetic epidemiology study of bladder cancer susceptibility that was previously analyzed using MDR and assessed using a 1000-fold permutation test.
PMCID: PMC2700860  PMID: 18671250
Extreme Value Distribution; Permutation Testing; Power; Type I Error; Bladder Cancer; Data Mining
14.  FastPval: a fast and memory efficient program to calculate very low P-values from empirical distribution 
Bioinformatics  2010;26(22):2897-2899.
Motivation: Resampling methods, such as permutation and bootstrap, have been widely used to generate an empirical distribution for assessing the statistical significance of a measurement. However, to obtain a very low P-value, a large size of resampling is required, where computing speed, memory and storage consumption become bottlenecks, and sometimes become impossible, even on a computer cluster.
Results: We have developed a multiple stage P-value calculating program called FastPval that can efficiently calculate very low (up to 10−9) P-values from a large number of resampled measurements. With only two input files and a few parameter settings from the users, the program can compute P-values from empirical distribution very efficiently, even on a personal computer. When tested on the order of 109 resampled data, our method only uses 52.94% the time used by the conventional method, implemented by standard quicksort and binary search algorithms, and consumes only 0.11% of the memory and storage. Furthermore, our method can be applied to extra large datasets that the conventional method fails to calculate. The accuracy of the method was tested on data generated from Normal, Poison and Gumbel distributions and was found to be no different from the exact ranking approach.
Availability: The FastPval executable file, the java GUI and source code, and the java web start server with example data and introduction, are available at
Supplementary information: Supplementary data are available at Bioinformatics online and
PMCID: PMC2971576  PMID: 20861029
15.  Increasing power for voxel-wise genome-wide association studies: The random field theory, least square kernel machines and fast permutation procedures 
NeuroImage  2012;63(2):858-873.
Imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains. Although imaging genetics has attracted growing attention and interest, most brain-wide genome-wide association studies focus on voxel-wise single-locus approaches, without taking advantage of the spatial information in images or combining the effect of multiple genetic variants. In this paper we present a fast implementation of voxel- and cluster-wise inferences based on the random field theory to fully use the spatial information in images. The approach is combined with a multi-locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms (SNP) with imaging traits. A fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p-value estimates based on parametric tail approximation. We explored the relation between 448,294 single nucleotide polymorphisms and 18,043 genes in 31,662 voxels of the entire brain across 740 elderly subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Structural MRI scans were analyzed using tensor-based morphometry (TBM) to compute 3D maps of regional brain volume differences compared to an average template image based on healthy elderly subjects. We find method to be more sensitive compared with voxel-wise single-locus approaches. A number of genes were identified as having significant associations with volumetric changes. The most associated gene was GRIN2B, which encodes the N-methyl-d-aspartate (NMDA) glutamate receptor NR2B subunit and affects both the parietal and temporal lobes in human brains. Its role in Alzheimer's disease has been widely acknowledged and studied, suggesting the validity of the approach. The various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains.
PMCID: PMC3635688  PMID: 22800732
Imaging genetics; Association study; Random field theory; Voxel-wise inference; Cluster size inference; Nonstationarity; Least square kernel machines; Permutation; Pareto distribution; Parametric tail approximation; ADNI; MRI; Tensor-based morphometry; Alzheimer's disease; GRIN2B
16.  Permutation inference for the general linear model 
Neuroimage  2014;92(100):381-397.
Permutation methods can provide exact control of false positives and allow the use of non-standard statistics, making only weak assumptions about the data. With the availability of fast and inexpensive computing, their main limitation would be some lack of flexibility to work with arbitrary experimental designs. In this paper we report on results on approximate permutation methods that are more flexible with respect to the experimental design and nuisance variables, and conduct detailed simulations to identify the best method for settings that are typical for imaging research scenarios. We present a generic framework for permutation inference for complex general linear models (glms) when the errors are exchangeable and/or have a symmetric distribution, and show that, even in the presence of nuisance effects, these permutation inferences are powerful while providing excellent control of false positives in a wide range of common and relevant imaging research scenarios. We also demonstrate how the inference on glm parameters, originally intended for independent data, can be used in certain special but useful cases in which independence is violated. Detailed examples of common neuroimaging applications are provided, as well as a complete algorithm – the “randomise” algorithm – for permutation inference with the glm.
•Permutation for the GLM in the presence of nuisance or non-independence.•A generalised statistic that performs well even under heteroscedasticity.•Permutation and/or sign-flipping, exchangeability blocks and variance groups.•The “randomise” algorithm, as well as various practical examples.
PMCID: PMC4010955  PMID: 24530839
Permutation inference; Multiple regression; General linear model; Randomise
17.  A permutation-based multiple testing method for time-course microarray experiments 
BMC Bioinformatics  2009;10:336.
Time-course microarray experiments are widely used to study the temporal profiles of gene expression. Storey et al. (2005) developed a method for analyzing time-course microarray studies that can be applied to discovering genes whose expression trajectories change over time within a single biological group, or those that follow different time trajectories among multiple groups. They estimated the expression trajectories of each gene using natural cubic splines under the null (no time-course) and alternative (time-course) hypotheses, and used a goodness of fit test statistic to quantify the discrepancy. The null distribution of the statistic was approximated through a bootstrap method. Gene expression levels in microarray data are often complicatedly correlated. An accurate type I error control adjusting for multiple testing requires the joint null distribution of test statistics for a large number of genes. For this purpose, permutation methods have been widely used because of computational ease and their intuitive interpretation.
In this paper, we propose a permutation-based multiple testing procedure based on the test statistic used by Storey et al. (2005). We also propose an efficient computation algorithm. Extensive simulations are conducted to investigate the performance of the permutation-based multiple testing procedure. The application of the proposed method is illustrated using the Caenorhabditis elegans dauer developmental data.
Our method is computationally efficient and applicable for identifying genes whose expression levels are time-dependent in a single biological group and for identifying the genes for which the time-profile depends on the group in a multi-group setting.
PMCID: PMC2772858  PMID: 19832992
18.  Estimation of Significance Thresholds for Genomewide Association Scans 
Genetic epidemiology  2008;32(3):227-234.
The question of what significance threshold is appropriate for genomewide association studies is somewhat unresolved. Previous theoretical suggestions have yet to be validated in practice, whereas permutation testing does not resolve a discrepancy between the genomewide multiplicity of the experiment and the subset of markers actually tested. We used genotypes from the Wellcome Trust Case-Control Consortium to estimate a genomewide significance threshold for the UK Caucasian population. We subsampled the genotypes at increasing densities, using permutation to estimate the nominal P-value for 5% family-wise error. By extrapolating to infinite density, we estimated the genomewide significance threshold to be about 7.2×10−8. To reduce the computation time, we considered Patterson’s eigenvalue estimator of the effective number of tests, but found it to be an order of magnitude too low for multiplicity correction. However, by fitting a Beta distribution to the minimum P-value from permutation replicates, we showed that the effective number is a useful heuristic and suggest that its estimation in this context is an open problem. We conclude that permutation is still needed to obtain genomewide significance thresholds, but with subsampling, extrapolation and estimation of an effective number of tests, the threshold can be standardized for all studies of the same population.
PMCID: PMC2573032  PMID: 18300295
multiple testing; permutation test; Bonferroni; eigenvalue
19.  Estimation of significance thresholds for genomewide association scans 
Genetic Epidemiology  2008;32(3):227-234.
The question of what significance threshold is appropriate for genomewide association studies is somewhat unresolved. Previous theoretical suggestions have yet to be validated in practice, whereas permutation testing does not resolve a discrepancy between the genomewide multiplicity of the experiment and the subset of markers actually tested. We used genotypes from the Wellcome Trust Case-Control Consortium to estimate a genomewide significance threshold for the UK Caucasian population. We subsampled the genotypes at increasing densities, using permutation to estimate the nominal P-value for 5% family-wise error. By extrapolating to infinite density, we estimated the genomewide significance threshold to be about 7.2 × 10−8. To reduce the computation time, we considered Patterson's eigenvalue estimator of the effective number of tests, but found it to be an order of magnitude too low for multiplicity correction. However, by fitting a Beta distribution to the minimum P-value from permutation replicates, we showed that the effective number is a useful heuristic and suggest that its estimation in this context is an open problem. We conclude that permutation is still needed to obtain genomewide significance thresholds, but with subsampling, extrapolation and estimation of an effective number of tests, the threshold can be standardized for all studies of the same population.
PMCID: PMC2573032  PMID: 18300295
multiple testing; permutation test; Bonferroni; eigenvalue
20.  Fast and Accurate Approximation to Significance Tests in Genome-Wide Association Studies 
Genome-wide association studies commonly involve simultaneous tests of millions of single nucleotide polymorphisms (SNP) for disease association. The SNPs in nearby genomic regions, however, are often highly correlated due to linkage disequilibrium (LD, a genetic term for correlation). Simple Bonferonni correction for multiple comparisons is therefore too conservative. Permutation tests, which are often employed in practice, are both computationally expensive for genome-wide studies and limited in their scopes. We present an accurate and computationally efficient method, based on Poisson de-clumping heuristics, for approximating genome-wide significance of SNP associations. Compared with permutation tests and other multiple comparison adjustment approaches, our method computes the most accurate and robust p-value adjustments for millions of correlated comparisons within seconds. We demonstrate analytically that the accuracy and the efficiency of our method are nearly independent of the sample size, the number of SNPs, and the scale of p-values to be adjusted. In addition, our method can be easily adopted to estimate false discovery rate. When applied to genome-wide SNP datasets, we observed highly variable p-value adjustment results evaluated from different genomic regions. The variation in adjustments along the genome, however, are well conserved between the European and the African populations. The p-value adjustments are significantly correlated with LD among SNPs, recombination rates, and SNP densities. Given the large variability of sequence features in the genome, we further discuss a novel approach of using SNP-specific (local) thresholds to detect genome-wide significant associations. This article has supplementary material online.
PMCID: PMC3226809  PMID: 22140288
Genome-wide association study; Multiple comparison; Poisson approximation
21.  Comparison of artificial neural network analysis with other multimarker methods for detecting genetic association 
BMC Genetics  2007;8:49.
Debate remains as to the optimal method for utilising genotype data obtained from multiple markers in case-control association studies. I and colleagues have previously described a method of association analysis using artificial neural networks (ANNs), whose performance compared favourably to single-marker methods. Here, the perfomance of ANN analysis is compared with other multi-marker methods, comprising different haplotype-based analyses and locus-based analyses.
Of several methods studied and applied to simulated SNP datasets, heterogeneity testing of estimated haplotype frequencies using asymptotic p values rather than permutation testing had the lowest power of the methods studied and ANN analysis had the highest power. The difference in power to detect association between these two methods was statistically significant (p = 0.001) but other comparisons between methods were not significant. The raw t statistic obtained from ANN analysis correlated highly with the empirical statistical significance obtained from permutation testing of the ANN results and with the p value obtained from the heterogeneity test.
Although ANN analysis was more powerful than the standard haplotype-based test it is unlikely to be taken up widely. The permutation testing necessary to obtain a valid p value makes it slow to perform and it is not underpinned by a theoretical model relating marker genotypes to disease phenotype. Nevertheless, the superior performance of this method does imply that the widely-used haplotype-based methods for detecting association with multiple markers are not optimal and efforts could be made to improve upon them. The fact that the t statistic obtained from ANN analysis is highly correlated with the statistical significance does suggest a possibility to use ANN analysis in situations where large numbers of markers have been genotyped, since the t value could be used as a proxy for the p value in preliminary analyses.
PMCID: PMC1940019  PMID: 17640352
22.  Multiple testing corrections for imputed SNPs 
Genetic epidemiology  2011;35(3):154-158.
Multiple testing corrections are an active research topic in genetic association studies, especially for genome-wide association studies (GWAS), where tests of association with traits are conducted at millions of imputed SNPs with estimated allelic dosages now. Failure to address multiple comparisons appropriately can introduce excess false positive results and make subsequent studies following up those results inefficient. Permutation tests are considered the gold standard in multiple testing adjustment; however, this procedure is computationally demanding, especially for GWAS. Notably, the permutation thresholds for the huge number of estimated allelic dosages in real data sets have not been reported. Although many researchers have recently developed algorithms to rapidly approximate the permutation thresholds with accuracy similar to the permutation test, these methods have not been verified with estimated allelic dosages. In this study, we compare recently published multiple testing correction methods using 2.5M estimated allelic dosages. We also derive permutation significance levels based on 10,000 GWAS results under the null hypothesis of no association. Our results show that the simpleM method works well with estimated allelic dosages and gives the closest approximation to the permutation threshold while requiring the least computation time.
PMCID: PMC3055936  PMID: 21254223
multiple testing; genome-wide association studies; imputed SNPs; allelic dosages
23.  Comments on the analysis of unbalanced microarray data 
Bioinformatics  2009;25(16):2035-2041.
Motivation: Permutation testing is very popular for analyzing microarray data to identify differentially expressed (DE) genes; estimating false discovery rates (FDRs) is a very popular way to address the inherent multiple testing problem. However, combining these approaches may be problematic when sample sizes are unequal.
Results: With unbalanced data, permutation tests may not be suitable because they do not test the hypothesis of interest. In addition, permutation tests can be biased. Using biased P-values to estimate the FDR can produce unacceptable bias in those estimates. Results also show that the approach of pooling permutation null distributions across genes can produce invalid P-values, since even non-DE genes can have different permutation null distributions. We encourage researchers to use statistics that have been shown to reliably discriminate DE genes, but caution that associated P-values may be either invalid, or a less-effective metric for discriminating DE genes.
Supplementary information: Supplementary data are available at Bioinformatics online.
PMCID: PMC2732368  PMID: 19528084
24.  An empirical study using permutation-based resampling in meta-regression 
Systematic Reviews  2012;1:18.
In meta-regression, as the number of trials in the analyses decreases, the risk of false positives or false negatives increases. This is partly due to the assumption of normality that may not hold in small samples. Creation of a distribution from the observed trials using permutation methods to calculate P values may allow for less spurious findings. Permutation has not been empirically tested in meta-regression. The objective of this study was to perform an empirical investigation to explore the differences in results for meta-analyses on a small number of trials using standard large sample approaches verses permutation-based methods for meta-regression.
We isolated a sample of randomized controlled clinical trials (RCTs) for interventions that have a small number of trials (herbal medicine trials). Trials were then grouped by herbal species and condition and assessed for methodological quality using the Jadad scale, and data were extracted for each outcome. Finally, we performed meta-analyses on the primary outcome of each group of trials and meta-regression for methodological quality subgroups within each meta-analysis. We used large sample methods and permutation methods in our meta-regression modeling. We then compared final models and final P values between methods.
We collected 110 trials across 5 intervention/outcome pairings and 5 to 10 trials per covariate. When applying large sample methods and permutation-based methods in our backwards stepwise regression the covariates in the final models were identical in all cases. The P values for the covariates in the final model were larger in 78% (7/9) of the cases for permutation and identical for 22% (2/9) of the cases.
We present empirical evidence that permutation-based resampling may not change final models when using backwards stepwise regression, but may increase P values in meta-regression of multiple covariates for relatively small amount of trials.
PMCID: PMC3351721  PMID: 22587815
25.  A Bayesian approach to efficient differential allocation for resampling-based significance testing 
BMC Bioinformatics  2009;10:198.
Large-scale statistical analyses have become hallmarks of post-genomic era biological research due to advances in high-throughput assays and the integration of large biological databases. One accompanying issue is the simultaneous estimation of p-values for a large number of hypothesis tests. In many applications, a parametric assumption in the null distribution such as normality may be unreasonable, and resampling-based p-values are the preferred procedure for establishing statistical significance. Using resampling-based procedures for multiple testing is computationally intensive and typically requires large numbers of resamples.
We present a new approach to more efficiently assign resamples (such as bootstrap samples or permutations) within a nonparametric multiple testing framework. We formulated a Bayesian-inspired approach to this problem, and devised an algorithm that adapts the assignment of resamples iteratively with negligible space and running time overhead. In two experimental studies, a breast cancer microarray dataset and a genome wide association study dataset for Parkinson's disease, we demonstrated that our differential allocation procedure is substantially more accurate compared to the traditional uniform resample allocation.
Our experiments demonstrate that using a more sophisticated allocation strategy can improve our inference for hypothesis testing without a drastic increase in the amount of computation on randomized data. Moreover, we gain more improvement in efficiency when the number of tests is large. R code for our algorithm and the shortcut method are available at .
PMCID: PMC2718927  PMID: 19558706

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