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1.  FastANOVA: an Efficient Algorithm for Genome-Wide Association Study 
Studying the association between quantitative phenotype (such as height or weight) and single nucleotide polymorphisms (SNPs) is an important problem in biology. To understand underlying mechanisms of complex phenotypes, it is often necessary to consider joint genetic effects across multiple SNPs. ANOVA (analysis of variance) test is routinely used in association study. Important findings from studying gene-gene (SNP-pair) interactions are appearing in the literature. However, the number of SNPs can be up to millions. Evaluating joint effects of SNPs is a challenging task even for SNP-pairs. Moreover, with large number of SNPs correlated, permutation procedure is preferred over simple Bonferroni correction for properly controlling family-wise error rate and retaining mapping power, which dramatically increases the computational cost of association study.
In this paper, we study the problem of finding SNP-pairs that have significant associations with a given quantitative phenotype. We propose an efficient algorithm, FastANOVA, for performing ANOVA tests on SNP-pairs in a batch mode, which also supports large permutation test. We derive an upper bound of SNP-pair ANOVA test, which can be expressed as the sum of two terms. The first term is based on single-SNP ANOVA test. The second term is based on the SNPs and independent of any phenotype permutation. Furthermore, SNP-pairs can be organized into groups, each of which shares a common upper bound. This allows for maximum reuse of intermediate computation, efficient upper bound estimation, and effective SNP-pair pruning. Consequently, FastANOVA only needs to perform the ANOVA test on a small number of candidate SNP-pairs without the risk of missing any significant ones. Extensive experiments demonstrate that FastANOVA is orders of magnitude faster than the brute-force implementation of ANOVA tests on all SNP pairs.
PMCID: PMC2951741  PMID: 20945829
Association study; ANOVA test
2.  Screening and Replication using the Same Data Set: Testing Strategies for Family-Based Studies in which All Probands Are Affected 
PLoS Genetics  2008;4(9):e1000197.
For genome-wide association studies in family-based designs, we propose a powerful two-stage testing strategy that can be applied in situations in which parent-offspring trio data are available and all offspring are affected with the trait or disease under study. In the first step of the testing strategy, we construct estimators of genetic effect size in the completely ascertained sample of affected offspring and their parents that are statistically independent of the family-based association/transmission disequilibrium tests (FBATs/TDTs) that are calculated in the second step of the testing strategy. For each marker, the genetic effect is estimated (without requiring an estimate of the SNP allele frequency) and the conditional power of the corresponding FBAT/TDT is computed. Based on the power estimates, a weighted Bonferroni procedure assigns an individually adjusted significance level to each SNP. In the second stage, the SNPs are tested with the FBAT/TDT statistic at the individually adjusted significance levels. Using simulation studies for scenarios with up to 1,000,000 SNPs, varying allele frequencies and genetic effect sizes, the power of the strategy is compared with standard methodology (e.g., FBATs/TDTs with Bonferroni correction). In all considered situations, the proposed testing strategy demonstrates substantial power increases over the standard approach, even when the true genetic model is unknown and must be selected based on the conditional power estimates. The practical relevance of our methodology is illustrated by an application to a genome-wide association study for childhood asthma, in which we detect two markers meeting genome-wide significance that would not have been detected using standard methodology.
Author Summary
The current state of genotyping technology has enabled researchers to conduct genome-wide association studies of up to 1,000,000 SNPs, allowing for systematic scanning of the genome for variants that might influence the development and progression of complex diseases. One of the largest obstacles to the successful detection of such variants is the multiple comparisons/testing problem in the genetic association analysis. For family-based designs in which all offspring are affected with the disease/trait under study, we developed a methodology that addresses this problem by partitioning the family-based data into two statistically independent components. The first component is used to screen the data and determine the most promising SNPs. The second component is used to test the SNPs for association, where information from the screening is used to weight the SNPs during testing. This methodology is more powerful than standard procedures for multiple comparisons adjustment (i.e., Bonferroni correction). Additionally, as only one data set is required for screening and testing, our testing strategy is less susceptible to study heterogeneity. Finally, as many family-based studies collect data only from affected offspring, this method addresses a major limitation of previous methodologies for multiple comparisons in family-based designs, which require variation in the disease/trait among offspring.
PMCID: PMC2529406  PMID: 18802462
3.  Testing Haplotype-Environment Interactions Using Case-Parent Triads 
Human Heredity  2010;70(1):23-33.
Joint analysis of multiple SNP markers can be informative, but studying joint effects of haplotypes and environmental exposures is challenging. Population structure can involve both genes and exposures and a case-control study is susceptible to bias from either source of stratification. We propose a procedure that uses case-parent triad data and, though not fully robust, resists bias from population structure.
Our procedure assumes that haplotypes under study have no influence on propensity to exposure. Then, under a no-interaction null hypothesis (multiplicative scale), transmission of a causative haplotype from parents to affected offspring might show distortion from Mendelian proportions but should be independent of exposure. We used this insight to develop a permutation test of no haplotype-by-exposure interaction.
Simulations showed that our proposed test respects the nominal Type I error rate and provides good power under a variety of scenarios. We illustrate by examining whether SNP variants in GSTP1 modify the association between maternal smoking and oral clefting.
Our procedure offers desirable features: no need for haplotype estimation, validity under unspecified genetic main effects, tolerance to Hardy-Weinberg disequilibrium, ability to handle missing genotypes and a relatively large number of SNPs. Simulations suggest resistance to bias due to exposure-related population stratification.
PMCID: PMC2912643  PMID: 20413979
Haplotype-environment interaction; Gene-environment interaction; Case-parent triad; Permutation test; Non-parametreic test; Population stratification
4.  Accounting for multiple comparisons in a genome-wide association study (GWAS) 
BMC Genomics  2010;11:724.
As we enter an era when testing millions of SNPs in a single gene association study will become the standard, consideration of multiple comparisons is an essential part of determining statistical significance. Bonferroni adjustments can be made but are conservative due to the preponderance of linkage disequilibrium (LD) between genetic markers, and permutation testing is not always a viable option. Three major classes of corrections have been proposed to correct the dependent nature of genetic data in Bonferroni adjustments: permutation testing and related alternatives, principal components analysis (PCA), and analysis of blocks of LD across the genome. We consider seven implementations of these commonly used methods using data from 1514 European American participants genotyped for 700,078 SNPs in a GWAS for AIDS.
A Bonferroni correction using the number of LD blocks found by the three algorithms implemented by Haploview resulted in an insufficiently conservative threshold, corresponding to a genome-wide significance level of α = 0.15 - 0.20. We observed a moderate increase in power when using PRESTO, SLIDE, and simpleℳ when compared with traditional Bonferroni methods for population data genotyped on the Affymetrix 6.0 platform in European Americans (α = 0.05 thresholds between 1 × 10-7 and 7 × 10-8).
Correcting for the number of LD blocks resulted in an anti-conservative Bonferroni adjustment. SLIDE and simpleℳ are particularly useful when using a statistical test not handled in optimized permutation testing packages, and genome-wide corrected p-values using SLIDE, are much easier to interpret for consumers of GWAS studies.
PMCID: PMC3023815  PMID: 21176216
5.  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
6.  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
7.  Holm multiple correction for large-scale gene-shape association mapping 
BMC Genetics  2014;15(Suppl 1):S5.
Linkage Disequilibrium (LD) is a powerful approach for the identification and characterization of morphological shape, which usually involves multiple genetic markers. However, multiple testing corrections substantially reduce the power of the associated tests. In addition, the principle component analysis (PCA), used to quantify the shape variations into several principal phenotypes, further increases the number of tests. As a result, a powerful multiple testing correction for simultaneous large-scale gene-shape association tests is an essential part of determining statistical significance. Bonferroni adjustments and permutation tests are the most popular approaches to correcting for multiple tests within LD based Quantitative Trait Loci (QTL) models. However, permutations are extremely computationally expensive and may mislead in the presence of family structure. The Bonferroni correction, though simple and fast, is conservative and has low power for large-scale testing.
We propose a new multiple testing approach, constructed by combining an Intersection Union Test (IUT) with the Holm correction, which strongly controls the family-wise error rate (FWER) without any additional assumptions on the joint distribution of the test statistics or dependence structure of the markers. The power improvement for the Holm correction, as compared to the standard Bonferroni correction, is examined through a simulation study. A consistent and moderate increase in power is found under the majority of simulated circumstances, including various sample sizes, Heritabilities, and numbers of markers. The power gains are further demonstrated on real leaf shape data from a natural population of poplar, Populus szechuanica var tietica, where more significant QTL associated with morphological shape are detected than under the previously applied Bonferroni adjustment.
The Holm correction is a valid and powerful method for assessing gene-shape association involving multiple markers, which not only controls the FWER in the strong sense but also improves statistical power.
PMCID: PMC4118635  PMID: 25079623
Bonferroni; Holm; QTL mapping; LD; Multiple correction
8.  Suggestive evidence for association of the circadian genes PERIOD3 and ARNTL with bipolar disorder 
Bipolar affective disorder (BPAD) is suspected to arise in part from malfunctions of the circadian system, a system which enables adaptation to a daily and seasonally cycling environment. Genetic variations altering functions of genes involved with the input to the circadian clock, in the molecular feedback loops constituting the circadian oscillatory mechanism itself, or in the regulatory output systems could influence BPAD as a result. Several human circadian system genes have been identified and localized recently, and a comparison with linkage hotspots for BPAD has revealed some correspondences.
We have assessed evidence for linkage and association involving polymorphisms in ten circadian clock genes (ARNTL, CLOCK, CRY2, CSNK1ε, DBP, GSK3β, NPAS2, PER1, PER2, and PER3) to BPAD. Linkage analysis in 52 affected families showed suggestive evidence for linkage to CSNK1ε. This finding was not substantiated in the association study. 52 SNPs in ten clock genes were genotyped in 185 parent proband triads. Single SNP TDT analyses showed no evidence for association to BPAD. However, more powerful haplotype analyses suggest two candidates deserving further studies. Haplotypes in ARNTL and PER3 were found to be significantly associated with BPAD via single-gene permutation tests (PG=0.025 and 0.008, respectively). The most suggestive haplotypes in PER3 showed a Bonferroni-corrected p-value of PGC=0.07. These two genes have previously been implicated in circadian rhythm sleep disorders and affective disorders.
With correction for the number of genes considered and tests conducted, these data do not provide statistically significant evidence for association. However, the trends for ARNTL and PER3 are suggestive of their involvement in bipolar disorder and warrant further study in a larger sample.
PMCID: PMC2651679  PMID: 16528748
manic-depressive illness; genetic linkage; genetic association; PER3; BMAL1
9.  Detecting purely epistatic multi-locus interactions by an omnibus permutation test on ensembles of two-locus analyses 
BMC Bioinformatics  2009;10:294.
Purely epistatic multi-locus interactions cannot generally be detected via single-locus analysis in case-control studies of complex diseases. Recently, many two-locus and multi-locus analysis techniques have been shown to be promising for the epistasis detection. However, exhaustive multi-locus analysis requires prohibitively large computational efforts when problems involve large-scale or genome-wide data. Furthermore, there is no explicit proof that a combination of multiple two-locus analyses can lead to the correct identification of multi-locus interactions.
The proposed 2LOmb algorithm performs an omnibus permutation test on ensembles of two-locus analyses. The algorithm consists of four main steps: two-locus analysis, a permutation test, global p-value determination and a progressive search for the best ensemble. 2LOmb is benchmarked against an exhaustive two-locus analysis technique, a set association approach, a correlation-based feature selection (CFS) technique and a tuned ReliefF (TuRF) technique. The simulation results indicate that 2LOmb produces a low false-positive error. Moreover, 2LOmb has the best performance in terms of an ability to identify all causative single nucleotide polymorphisms (SNPs) and a low number of output SNPs in purely epistatic two-, three- and four-locus interaction problems. The interaction models constructed from the 2LOmb outputs via a multifactor dimensionality reduction (MDR) method are also included for the confirmation of epistasis detection. 2LOmb is subsequently applied to a type 2 diabetes mellitus (T2D) data set, which is obtained as a part of the UK genome-wide genetic epidemiology study by the Wellcome Trust Case Control Consortium (WTCCC). After primarily screening for SNPs that locate within or near 372 candidate genes and exhibit no marginal single-locus effects, the T2D data set is reduced to 7,065 SNPs from 370 genes. The 2LOmb search in the reduced T2D data reveals that four intronic SNPs in PGM1 (phosphoglucomutase 1), two intronic SNPs in LMX1A (LIM homeobox transcription factor 1, alpha), two intronic SNPs in PARK2 (Parkinson disease (autosomal recessive, juvenile) 2, parkin) and three intronic SNPs in GYS2 (glycogen synthase 2 (liver)) are associated with the disease. The 2LOmb result suggests that there is no interaction between each pair of the identified genes that can be described by purely epistatic two-locus interaction models. Moreover, there are no interactions between these four genes that can be described by purely epistatic multi-locus interaction models with marginal two-locus effects. The findings provide an alternative explanation for the aetiology of T2D in a UK population.
An omnibus permutation test on ensembles of two-locus analyses can detect purely epistatic multi-locus interactions with marginal two-locus effects. The study also reveals that SNPs from large-scale or genome-wide case-control data which are discarded after single-locus analysis detects no association can still be useful for genetic epidemiology studies.
PMCID: PMC2759961  PMID: 19761607
10.  Rapid Testing of SNPs and Gene–Environment Interactions in Case–Parent Trio Data Based on Exact Analytic Parameter Estimation 
Biometrics  2011;68(3):766-773.
Case–parent trio studies concerned with children affected by a disease and their parents aim to detect single nucleotide polymorphisms (SNPs) showing a preferential transmission of alleles from the parents to their affected offspring. A popular statistical test for detecting such SNPs associated with disease in this study design is the genotypic transmission/disequilibrium test (gTDT) based on a conditional logistic regression model, which usually needs to be fitted by an iterative procedure. In this article, we derive exact closed-form solutions for the parameter estimates of the conditional logistic regression models when testing for an additive, a dominant, or a recessive effect of a SNP, and show that such analytic parameter estimates also exist when considering gene–environment interactions with binary environmental variables. Because the genetic model underlying the association between a SNP and a disease is typically unknown, it might further be beneficial to use the maximum over the gTDT statistics for the possible effects of a SNP as test statistic. We therefore propose a procedure enabling a fast computation of the test statistic and the permutation-based p-value of this MAX gTDT. All these methods are applied to whole-genome scans of the case–parent trios from the International Cleft Consortium. These applications show our procedures dramatically reduce the required computing time compared to the conventional iterative methods allowing, for example, the analysis of hundreds of thousands of SNPs in a few minutes instead of several hours.
PMCID: PMC3387527  PMID: 22150644
Conditional logistic regression; Family-based design; Genome-wide association studies; Genotypic transmission/disequilibrium test; International Cleft Consortium; MAX test
11.  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
12.  ParaHaplo: A program package for haplotype-based whole-genome association study using parallel computing 
Since more than a million single-nucleotide polymorphisms (SNPs) are analyzed in any given genome-wide association study (GWAS), performing multiple comparisons can be problematic. To cope with multiple-comparison problems in GWAS, haplotype-based algorithms were developed to correct for multiple comparisons at multiple SNP loci in linkage disequilibrium. A permutation test can also control problems inherent in multiple testing; however, both the calculation of exact probability and the execution of permutation tests are time-consuming. Faster methods for calculating exact probabilities and executing permutation tests are required.
We developed a set of computer programs for the parallel computation of accurate P-values in haplotype-based GWAS. Our program, ParaHaplo, is intended for workstation clusters using the Intel Message Passing Interface (MPI). We compared the performance of our algorithm to that of the regular permutation test on JPT and CHB of HapMap.
ParaHaplo can detect smaller differences between 2 populations than SNP-based GWAS. We also found that parallel-computing techniques made ParaHaplo 100-fold faster than a non-parallel version of the program.
ParaHaplo is a useful tool in conducting haplotype-based GWAS. Since the data sizes of such projects continue to increase, the use of fast computations with parallel computing--such as that used in ParaHaplo--will become increasingly important. The executable binaries and program sources of ParaHaplo are available at the following address:
PMCID: PMC2774321  PMID: 19845960
13.  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
14.  A hidden Markov random field model for genome-wide association studies 
Biostatistics (Oxford, England)  2009;11(1):139-150.
Genome-wide association studies (GWAS) are increasingly utilized for identifying novel susceptible genetic variants for complex traits, but there is little consensus on analysis methods for such data. Most commonly used methods include single single nucleotide polymorphism (SNP) analysis or haplotype analysis with Bonferroni correction for multiple comparisons. Since the SNPs in typical GWAS are often in linkage disequilibrium (LD), at least locally, Bonferroni correction of multiple comparisons often leads to conservative error control and therefore lower statistical power. In this paper, we propose a hidden Markov random field model (HMRF) for GWAS analysis based on a weighted LD graph built from the prior LD information among the SNPs and an efficient iterative conditional mode algorithm for estimating the model parameters. This model effectively utilizes the LD information in calculating the posterior probability that an SNP is associated with the disease. These posterior probabilities can then be used to define a false discovery controlling procedure in order to select the disease-associated SNPs. Simulation studies demonstrated the potential gain in power over single SNP analysis. The proposed method is especially effective in identifying SNPs with borderline significance at the single-marker level that nonetheless are in high LD with significant SNPs. In addition, by simultaneously considering the SNPs in LD, the proposed method can also help to reduce the number of false identifications of disease-associated SNPs. We demonstrate the application of the proposed HMRF model using data from a case–control GWAS of neuroblastoma and identify 1 new SNP that is potentially associated with neuroblastoma.
PMCID: PMC2800164  PMID: 19822692
Empirical Bayes; False discovery; Iterative conditional model; Linkage disequilibrium
15.  The null distributions of test statistics in genomewide association studies 
Statistics in biosciences  2009;1(2):214-227.
Genomewide association (GWA) studies assay hundreds of thousands of single nucleotide polymorphisms (SNPs) simultaneously across the entire genome and associate them with diseases, other biological or clinical traits. The association analysis usually tests each SNP as an independent entity and ignores the biological information such as linkage disequilibrium. Although the Bonferroni correction and other approaches have been proposed to address the issue of multiple comparisons as a result of testing many SNPs, there is a lack of understanding of the distribution of an association test statistic when an entire genome is considered together. In other words, there are extensive efforts in hypothesis testing, and almost no attempt in estimating the density under the null hypothesis. By estimating the true null distribution, we can apply the result directly to hypothesis testing; better assess the existing approaches of multiple comparisons; and evaluate the impact of linkage disequilibrium on the GWA studies. To this end, we estimate the empirical null distribution of an association test statistic in GWA studies using simulated population data. We further propose a convenient and accurate method based on adaptive spline to estimate the empirical value in GWA studies and validate our findings using a real data set. Our method enables us to fully characterize the null distribution of an association test that not only can be used to test the null hypothesis of no association, but also provide important information about the impact of density of the genetic markers on the significance of the tests. Our method does not require users to perform computationally intensive permutations, and hence provides a timely solution to an important and difficult problem in GWA studies.
PMCID: PMC2826849  PMID: 20190869
critical value; generalized extreme-value distribution; genomewide association
16.  A Simple Scalable Association Hypothesis Test Combining Gene-wide Evidence From Multiple Polymorphisms 
In single-nucleotide polymorphism (SNP) scans, SNP-phenotype association hypotheses are tested, however there is biological interpretation only for genes that span multiple SNPs. We demonstrate and validate a method of combining gene-wide evidence using data for high-density lipoprotein cholesterol (HDLC).
In a family based study (N=1782 from 482 families), we used 1000 phenotype-permuted datasets to determine the correlation of z-test statistics for 592 SNP-HDLC association tests comprising 14 genes previously reported to be associated with HDLC. We generated gene-wide p-values using the distribution of the sum of correlated z-statistics.
Of the 14 genes, CETP was significant (p=4.0×10−5 <0.05/14), while PLTP was significant at the borderline (p=6.7×10−3 <0.1/14). These p-values were confirmed using empirical distributions of the sum of χ2 association statistics as a gold standard (2.9×10−6 and 1.8×10−3, respectively). Genewide p-values were more significant than Bonferroni-corrected p-value for the most significant SNP in 11 of 14 genes (p=0.023). Genewide p-values calculated from SNP correlations derived for 20 simulated normally distributed phenotypes reproduced those derived from the 1000 phenotype-permuted datasets were correlated with the empirical distributions (Spearman correlation = 0.92 for both).
We have validated a simple scalable method to combine polymorphism-level evidence into gene-wide statistical evidence. High-throughput gene-wide hypothesis tests may be used in biologically interpretable genomewide association scans. Genewide association tests may be used to meaningfully replicate findings in populations with different linkage disequilibrium structure, when SNP-level replication is not expected.
PMCID: PMC3969754  PMID: 24688980
Bonferroni; hypothesis tests; combining evidence
17.  Prioritize and Select SNPs for Association Studies with Multi-Stage Designs 
Journal of Computational Biology  2008;15(3):241-257.
Large-scale whole genome association studies are increasingly common, due in large part to recent advances in genotyping technology. With this change in paradigm for genetic studies of complex diseases, it is vital to develop valid, powerful, and efficient statistical tools and approaches to evaluate such data. Despite a dramatic drop in genotyping costs, it is still expensive to genotype thousands of individuals for hundreds of thousands single nucleotide polymorphisms (SNPs) for large-scale whole genome association studies. A multi-stage (or two-stage) design has been a promising alternative: in the first stage, only a fraction of samples are genotyped and tested using a dense set of SNPs, and only a small subset of markers that show moderate associations with the disease will be genotyped in later stages. Multi-stage designs have also been used in candidate gene association studies, usually in regions that have shown strong signals by linkage studies. To decide which set of SNPs to be genotyped in the next stage, a common practice is to utilize a simple test (such as a χ2 test for case-control data) and a liberal significance level without corrections for multiple testing, to ensure that no true signals will be filtered out. In this paper, I have developed a novel SNP selection procedure within the framework of multi-stage designs. Based on data from stage 1, the method explicitly explores correlations (linkage disequilibrium) among SNPs and their possible interactions in determining the disease phenotype. Comparing with a regular multi-stage design, the approach can select a much reduced set of SNPs with high discriminative power for later stages. Therefore, not only does it reduce the genotyping cost in later stages, it also increases the statistical power by reducing the number of tests. Combined analysis is proposed to further improve power, and the theoretical significance level of the combined statistic is derived. Extensive simulations have been performed, and results have shown that the procedure can reduce the number of SNPs required in later stages, with improved power to detect associations. The procedure has also been applied to a real data set from a genome-wide association study of the sporadic amyotrophic lateral sclerosis (ALS) disease, and an interesting set of candidate SNPs has been identified.
PMCID: PMC3326652  PMID: 18352819
genome wide association studies; SNP selection; two-stage design
18.  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
19.  Genome-wide association studies of rheumatoid arthritis data via multiple hypothesis testing methods for correlated tests 
BMC Proceedings  2009;3(Suppl 7):S38.
Genome-wide association studies often involve testing hundreds of thousands of single-nucleotide polymorphisms (SNPs). These tests may be highly correlated because of linkage disequilibrium among SNPs. Multiple testing correction ignoring the correlation among markers, as is done in the Bonferroni procedure, can cause loss of power. Several multiple testing adjustment methods accounting for correlations among tests have been developed and have shown improved power compared to the Bonferroni procedure. These methods include a Monte Carlo (MC) method and a method of computing p-values adjusted for correlated tests. The objective of this study is to apply these two multiple testing methods to genome-wide association study of the Genetic Analysis Workshop 16 rheumatoid arthritis data from the North American Rheumatoid Arthritis Consortium, to compare the performance of these two methods to the Bonferroni procedure in identifying susceptibility loci underlying rheumatoid arthritis, and to discuss the strengths and weaknesses of these methods. The results show that both the MC method and p-values adjusted for correlated tests method identified more significant SNPs, thus potentially have higher power than the corresponding Bonferroni methods using the same test statistics as in the MC method and p-values adjusted for correlated tests, respectively. Simulation studies demonstrate that the MC method may have slightly higher power than the p-values adjusted for correlated tests method.
PMCID: PMC2795936  PMID: 20018029
20.  Mass Univariate Analysis of Event-Related Brain Potentials/Fields II: Simulation Studies 
Psychophysiology  2011;48(12):1726-1737.
Mass univariate analysis is a relatively new approach for the study of ERPs/ERFs. It consists of many statistical tests and one of several powerful corrections for multiple comparisons. Multiple comparison corrections differ in their power and permissiveness. Moreover, some methods are not guaranteed to work or may be overly sensitive to uninteresting deviations from the null hypothesis. Here we report the results of simulations assessing the accuracy, permissiveness, and power of six popular multiple comparison corrections (permutation-based control of the family-wise error rate: FWER, weak control of FWER via cluster-based permutation tests, permutation based control of the generalized FWER, and three false discovery rate control procedures) using realistic ERP data. In addition, we look at the sensitivity of permutation tests to differences in population variance. These results will help researchers apply and interpret these procedures.
PMCID: PMC4059014  PMID: 21895684
21.  Joint Testing of Genotypic and Gene-Environment Interaction Identified Novel Association for BMP4 with Non-Syndromic CL/P in an Asian Population Using Data from an International Cleft Consortium 
PLoS ONE  2014;9(10):e109038.
Non-syndromic cleft lip with or without cleft palate (NSCL/P) is a common disorder with complex etiology. The Bone Morphogenetic Protein 4 gene (BMP4) has been considered a prime candidate gene with evidence accumulated from animal experimental studies, human linkage studies, as well as candidate gene association studies. The aim of the current study is to test for linkage and association between BMP4 and NSCL/P that could be missed in genome-wide association studies (GWAS) when genotypic (G) main effects alone were considered.
Methodology/Principal Findings
We performed the analysis considering G and interactions with multiple maternal environmental exposures using additive conditional logistic regression models in 895 Asian and 681 European complete NSCL/P trios. Single nucleotide polymorphisms (SNPs) that passed the quality control criteria among 122 genotyped and 25 imputed single nucleotide variants in and around the gene were used in analysis. Selected maternal environmental exposures during 3 months prior to and through the first trimester of pregnancy included any personal tobacco smoking, any environmental tobacco smoke in home, work place or any nearby places, any alcohol consumption and any use of multivitamin supplements. A novel significant association held for rs7156227 among Asian NSCL/P and non-syndromic cleft lip and palate (NSCLP) trios after Bonferroni correction which was not seen when G main effects alone were considered in either allelic or genotypic transmission disequilibrium tests. Odds ratios for carrying one copy of the minor allele without maternal exposure to any of the four environmental exposures were 0.58 (95%CI = 0.44, 0.75) and 0.54 (95%CI = 0.40, 0.73) for Asian NSCL/P and NSCLP trios, respectively. The Bonferroni P values corrected for the total number of 117 tested SNPs were 0.0051 (asymptotic P = 4.39*10−5) and 0.0065 (asymptotic P = 5.54*10−5), accordingly. In European trios, no significant association was seen for any SNPs after Bonferroni corrections for the total number of 120 tested SNPs.
Our findings add evidence from GWAS to support the role of BMP4 in susceptibility to NSCL/P originally identified in linkage and candidate gene association studies.
PMCID: PMC4193821  PMID: 25303326
22.  Association analyses of the MAS-QTL data set using grammar, principal components and Bayesian network methodologies 
BMC Proceedings  2011;5(Suppl 3):S8.
It has been shown that if genetic relationships among individuals are not taken into account for genome wide association studies, this may lead to false positives. To address this problem, we used Genome-wide Rapid Association using Mixed Model and Regression and principal component stratification analyses. To account for linkage disequilibrium among the significant markers, principal components loadings obtained from top markers can be included as covariates. Estimation of Bayesian networks may also be useful to investigate linkage disequilibrium among SNPs and their relation with environmental variables.
For the quantitative trait we first estimated residuals while taking polygenic effects into account. We then used a single SNP approach to detect the most significant SNPs based on the residuals and applied principal component regression to take linkage disequilibrium among these SNPs into account. For the categorical trait we used principal component stratification methodology to account for background effects. For correction of linkage disequilibrium we used principal component logit regression. Bayesian networks were estimated to investigate relationship among SNPs.
Using the Genome-wide Rapid Association using Mixed Model and Regression and principal component stratification approach we detected around 100 significant SNPs for the quantitative trait (p<0.05 with 1000 permutations) and 109 significant (p<0.0006 with local FDR correction) SNPs for the categorical trait. With additional principal component regression we reduced the list to 16 and 50 SNPs for the quantitative and categorical trait, respectively.
GRAMMAR could efficiently incorporate the information regarding random genetic effects. Principal component stratification should be cautiously used with stringent multiple hypothesis testing correction to correct for ancestral stratification and association analyses for binary traits when there are systematic genetic effects such as half sib family structures. Bayesian networks are useful to investigate relationships among SNPs and environmental variables.
PMCID: PMC3103207  PMID: 21624178
Genetic epidemiology  2010;34(7):739-746.
Meta-analysis has become a key component of well-designed genetic association studies due to the boost in statistical power achieved by combining results across multiple samples of individuals and the need to validate observed associations in independent studies. Meta-analyses of genetic association studies based on multiple SNPs and traits are subject to the same multiple testing issues as single-sample studies, but it is often difficult to adjust accurately for the multiple tests. Procedures such as Bonferroni may control the type I error rate but will generally provide an overly harsh correction if SNPs or traits are correlated. Depending on study design, availability of individual-level data, and computational requirements, permutation testing may not be feasible in a meta-analysis framework. In this paper we present methods for adjusting for multiple correlated tests under several study designs commonly employed in meta-analyses of genetic association tests. Our methods are applicable to both prospective meta-analyses in which several samples of individuals are analyzed with the intent to combine results, and retrospective meta-analyses, in which results from published studies are combined, including situations in which 1) individual-level data are unavailable, and 2) different sets of SNPs are genotyped in different studies due to random missingness or two-stage design. We show through simulation that our methods accurately control the rate of type I error and achieve improved power over multiple testing adjustments that do not account for correlation between SNPs or traits.
PMCID: PMC3070606  PMID: 20878715
meta-analysis; association study; multiple testing; SNPs
24.  Simultaneous Analysis of All SNPs in Genome-Wide and Re-Sequencing Association Studies 
PLoS Genetics  2008;4(7):e1000130.
Testing one SNP at a time does not fully realise the potential of genome-wide association studies to identify multiple causal variants, which is a plausible scenario for many complex diseases. We show that simultaneous analysis of the entire set of SNPs from a genome-wide study to identify the subset that best predicts disease outcome is now feasible, thanks to developments in stochastic search methods. We used a Bayesian-inspired penalised maximum likelihood approach in which every SNP can be considered for additive, dominant, and recessive contributions to disease risk. Posterior mode estimates were obtained for regression coefficients that were each assigned a prior with a sharp mode at zero. A non-zero coefficient estimate was interpreted as corresponding to a significant SNP. We investigated two prior distributions and show that the normal-exponential-gamma prior leads to improved SNP selection in comparison with single-SNP tests. We also derived an explicit approximation for type-I error that avoids the need to use permutation procedures. As well as genome-wide analyses, our method is well-suited to fine mapping with very dense SNP sets obtained from re-sequencing and/or imputation. It can accommodate quantitative as well as case-control phenotypes, covariate adjustment, and can be extended to search for interactions. Here, we demonstrate the power and empirical type-I error of our approach using simulated case-control data sets of up to 500 K SNPs, a real genome-wide data set of 300 K SNPs, and a sequence-based dataset, each of which can be analysed in a few hours on a desktop workstation.
Author Summary
Tests of association with disease status are normally conducted one SNP at a time, ignoring the effects of all other genotyped SNPs. We developed a computationally efficient method to simultaneously analyse all SNPs, either in a genome-wide association (GWA) study, or a fine-mapping study based on re-sequencing and/or imputation. The method selects a subset of SNPs that best predicts disease status, while controlling the type-I error of the selected SNPs. This brings many advantages over standard single-SNP approaches, because the signal from a particular SNP can be more clearly assessed when other SNPs associated with disease status are already included in the model. Thus, in comparison with single-SNP analyses, power is increased and the false positive rate is reduced because of reduced residual variation. Localisation is also greatly improved. We demonstrate these advantages over the widely used single-SNP Armitage Trend Test using GWA simulation studies, a real GWA dataset, and a sequence-based fine-mapping simulation study.
PMCID: PMC2464715  PMID: 18654633
25.  Asymptotic Tests of Association with Multiple SNPs in Linkage Disequilibrium 
Genetic epidemiology  2009;33(6):497-507.
We consider detecting associations between a trait and multiple SNPs in linkage disequilibrium (LD). To maximize the use of information contained in multiple SNPs while minimizing the cost of large degrees of freedom (DF) in testing multiple parameters, we first theoretically explore the sum test derived under a working assumption of a common association strength between the trait and each SNP, testing on the corresponding parameter with only one DF. Under the scenarios that the association strengths between the trait and the SNPs are close to each other (and in the same direction), as considered by Wang and Elston (2007), we show with simulated data that the sum test was powerful as compared to several existing tests; otherwise, the sum test might have much reduced power. To overcome the limitation of the sum test, based on our theoretical analysis of the sum test, we propose five new tests that are closely related to each other and are shown to consistently perform similarly well across a wide range of scenarios. We point out the close connection of the proposed tests to the Goeman test. Furthermore, we derive the asymptotic distributions of the proposed tests so that p-values can be easily calculated, in contrast to the use of computationally demanding permutations or simulations for the Goeman test. A distinguishing feature of the five new tests is their use of a diagonal working covariance matrix, rather than a full covariance matrix as used in the usual Wald or score test. We recommend the routine use of two of the new tests, along with several other tests, to detect disease associations with multiple linked SNPs.
PMCID: PMC2732754  PMID: 19170135
genome-wide association study; logistic regression; multilocus analysis; permutation; single-locus analysis; SNP

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