GeneBased Tests of Association
Genomewide association studies (GWAS) are now used routinely to identify SNPs associated with complex human phenotypes. In several cases, multiple variants within a gene contribute independently to disease risk. Here we introduce a novel GeneWide Significance (GWiS) test that uses greedy Bayesian model selection to identify the independent effects within a gene, which are combined to generate a stronger statistical signal. Permutation tests provide pvalues that correct for the number of independent tests genomewide and within each genetic locus. When applied to a dataset comprising 2.5 million SNPs in up to 8,000 individuals measured for various electrocardiography (ECG) parameters, this method identifies more validated associations than conventional GWAS approaches. The method also provides, for the first time, systematic assessments of the number of independent effects within a gene and the fraction of diseaseassociated genes housing multiple independent effects, observed at 35%–50% of loci in our study. This method can be generalized to other study designs, retains power for lowfrequency alleles, and provides genebased pvalues that are directly compatible for pathwaybased metaanalysis.
Published in the journal:
. PLoS Genet 7(7): e32767. doi:10.1371/journal.pgen.1002177
Category:
Research Article
doi: 10.1371/journal.pgen.1002177
Summary
Genomewide association studies (GWAS) are now used routinely to identify SNPs associated with complex human phenotypes. In several cases, multiple variants within a gene contribute independently to disease risk. Here we introduce a novel GeneWide Significance (GWiS) test that uses greedy Bayesian model selection to identify the independent effects within a gene, which are combined to generate a stronger statistical signal. Permutation tests provide pvalues that correct for the number of independent tests genomewide and within each genetic locus. When applied to a dataset comprising 2.5 million SNPs in up to 8,000 individuals measured for various electrocardiography (ECG) parameters, this method identifies more validated associations than conventional GWAS approaches. The method also provides, for the first time, systematic assessments of the number of independent effects within a gene and the fraction of diseaseassociated genes housing multiple independent effects, observed at 35%–50% of loci in our study. This method can be generalized to other study designs, retains power for lowfrequency alleles, and provides genebased pvalues that are directly compatible for pathwaybased metaanalysis.
Introduction
Traditional singleSNP GWAS methods have been remarkably successful in identifying genetic associations, including those for various ECG parameters in recent studies of PR interval (the beginning of the P wave to the beginning of the QRS interval) [1], QRS interval (depolarization of both ventricles) [2] and QT interval (the start of the Q wave to the end of the T wave) [3]–[5]. Much of this success has relied upon increasing sample size through metaanalyses across multiple cohorts, rather than through the use of novel analytical methods to increase power.
One analytical approach, genebased tests proposed during the initial development of GWAS [6], has natural appeal. First, variations in proteincoding and adjacent regulatory regions are more likely to have functional relevance. Second, genebased tests allow for direct comparison between different populations, despite the potential for different linkage disequilibrium (LD) patterns and/or functional alleles. Third, these analyses can account for multiple independent functional variants within a gene, with the potential to greatly increase the power to identify disease/traitassociated genes.
Despite these appealing properties, genebased and related multimarker association tests have generally underperformed singlelocus tests when assessed with real data [7], [8]. A general drawback of methods that attempt to exploit the structure of LD to reduce the number of tests, for example through principal component analysis, is the loss of power to detect lowfrequency alleles. Methods that consider multiple independent effects often require that the number of effects be prespecified [9], which loses power when the tested and true model are different.
Multilocus tests often have the additional practical drawback of being highly CPU and memory intensive. Several methods use Bayesian statistics to drive a bruteforce sum or Monte Carlo sample over models [10], [11], but again often restrict the search to one or twomarker associations. In general, the computational costs have made these approaches infeasible for genomewide applications.
The GeneWide Significance (GWiS) test addresses these problems by performing model selection simultaneously with parameter estimation and significance testing in a computational framework that is feasible for genomewide SNP data (see Methods). Model selection, defined as identifying the best tagging SNP for each independent effect within a gene, uses the Bayesian model likelihood as the test statistic [12]–[14]. Our innovation is to use gene regions to impose a structured search through locally optimal models, which is computationally efficient and matches the biological intuition that the presence of one causal variant within a gene increases the likelihood of additional causal effects. Models are penalized based on the effective number of independent SNPs within a gene and the number of SNPs in the model, akin to a multipletesting correction. The Schwarzian Bayesian Information Criterion corrects for the difference between the full model likelihood and the easily computed maximum likelihood estimate [15]. This method has greater power than current methods for genomewide association studies and provides a principled alternative to ad hoc followup analyses to identify additional independent association signals in loci with genomewide significant primary associations.
Results
Reference genotype and phenotype data
The ECG parameters PR interval, QRS interval and QT interval are ideal test cases because recent largescale GWAS studies have established known positive associations. These traits are all clinically relevant, with increased PR interval associated with increased risk of atrial fibrillation and stroke [16], and both increased QRS and QT intervals associated with mortality and sudden cardiac death [17]–[20]. We assessed the ability of standard methods and GWiS to rediscover these known positives using data from only the Atherosclerosis Risk in Communities (ARIC) cohort, which contributes 15% of the total sample size for QRS, 25% for PR, and 50% for QT (Table 1).
The SNPs were assigned to genes based on the NCBI Homo sapiens genome build 35.1 reference assembly [21]. Gene boundaries were defined by the most transcriptional start site and transcriptional end position for any transcript annotated to a gene, yielding 25,251 nonredundant transcribed gene regions. Incorporating additional flanking sequence increases coverage of more distant regulatory elements, which increases power, but also increases the number of SNPs tested, which decreases power. Expression quantitative trait loci (eQTL) mapping in humans has shown that most cisregulatory SNPs are within 100 kb of the transcribed region [22], [23], with quantitative estimates that of large effect eQTNs (functional nucleotides that create eQTLs) are within 20 kb of the transcribed region [24]. We report results for 20 kb flanking regions; the performance ranking is robust to flanking by up to 100 kb (Table S1). SNPs within these regions are then assigned to one or more genes. Of the approximately 2.5 million genotyped and imputed SNPs, about 1.4 million are assigned to at least one gene. The median number of SNPs per gene is 43 and the mean is 72 (Table 1), reflecting a skewed distribution with many small genes having few SNPs.
The “gold standard” known positives rely on previously published metaanalyses of PR interval [1], QRS interval [2] and QT interval [4], [5]. We first identify goldstandard SNPs having . Any gene within 200 kb of a goldstandard SNP is classified as a known positive, and known positives within a 200 kb window are merged into a single locus, yielding 38 known positive genebased loci. This procedure was followed to ensure that each association signal results in a single locus as opposed to being split between adjacent loci, which could result in overcounting.
Other methods
The minSNP test uses the pvalue for the best single SNP within a gene. The minSNPP test converts this SNPbased pvalue to a genebased pvalue by performing permutation tests within each gene. BIMBAM averages the Bayes Factors for subsets of SNPs within a gene, with restriction to singleSNP models recommended for genomewide applications [10]. Because the Bayes Factor sum is dominated by the single best term, results for BIMBAM are very similar to minSNPP. The Versatile GeneBased Test for Genomewide Association (VEGAS) [25] is a recent multivariate method that sums the association signal from all the SNPs within a gene and corrects the sum for LD to generate a test statistic. The terms summed by VEGAS are asymptotically equivalent to the negative logarithms of the Bayes Factors summed by BIMBAM. LASSO regression, or L1 regularized regression, is a multivariate method that combines sparse model selection and parameter optimzation [26]–[28], with promising recent applications to GWAS [29]. See Methods for more details.
Simulated data and power
Power calculations used genotypes from the ARIC population to ensure realistic LD. Phenotypes were then simulated for genetic models with one or more causal variants within a gene. GWiS was the bestperforming method, with an advantage growing as more independent effects are present (Figure 1a). Theoretically, GWiS should have lower power than singleSNP tests when the true model is a single effect; according to the “no free lunch theorem”, this loss of power cannot be avoided [30]. The performance of GWiS therefore depends on the genetic architecture of a disease or trait: higher power if genes house multiple independent causal variants, and lower power if each gene has only a single causal variant. In practice, the loss of power was so slight as to be virtually undetectable.
Of the other methods, minSNPP and BIMBAM had similar performance that degraded as the true model included more SNPs. The VEGAS test did not perform well, presumably because the sum over all SNPs creates a bias to find causal variants in LD blocks represented by many SNPs and to miss variants in LD blocks with few SNPs. In the absence of LD, with genotypes and phenotype simulated using PLINK [31], VEGAS performs better (Figure S1). The LASSO method performed worst.
The advantage of GWiS arises in part from better power to detect associations with lowfrequency alleles (Figure 1b). GWiS, minSNPP, and BIMBAM have roughly constant power for a given variance explained, regardless of minor allele frequency. In contrast, both VEGAS and LASSO suffer from a twofold loss of power when minor allele frequencies drop from 50% to 5%. VEGAS may lose power because these lowfrequency SNPs lack correlation with other SNPs, reducing the contribution to the VEGAS sum statistic. The LASSO penalty shrinks the regression coefficient, which may adversely affect SNPs with large regression coefficients that balance low minor allele frequencies.
Simulated data and model size
The model size selected by GWiS and LASSO was evaluated by simulation (Figure 2). These simulations also used the ARIC population to supply realistic LD, with genes selected at random with replacement from chromosome 1. In chromosome 1, the number of SNPs in a gene ranges from 1 to over 1000, and the number of independent effects ranges from 1 to over 100, similar to the distributions in the genome as a whole (Figure S2). A subset of SNPs within a gene had causal effects assigned (“True ”), phenotypes were simulated to mimic weak and strong genebased signals, and then models were selected by GWiS and LASSO. Model selection to retain a subset of SNPs (“Estimated ”) was performed both for the full genotype data and for the genotype data with the causal SNPs all removed.
GWiS provides a better estimate of the true model size than LASSO, assessed from the of estimated versus true . With causal SNPs kept, for GWiS is substantially higher, 0.65 versus 0.47 at low power (Figure 2a, 2c) and 0.81 versus 0.60 at high power (Figure 2b, 2d). GWiS also performs better when causal SNPs are removed, 0.55 versus 0.33 at low power and 0.60 versus 0.39 at high power. GWiS also provides a conservative estimate of the model size, with the ratio of estimated to true size ranging from a worstcase of 44% (low power, causal SNPs removed) to a bestcase of 81% (high power, causal SNPs kept) over the four scenarios examined. In contrast, LASSO is prone to overpredict the size of the model, with a worstcase of models that are on average 33% too large (high power, causal SNPs kept, Figure 2d).
Removing a causal SNP results in GWiS predicting a smaller model, with the ratio of estimated to true dropping from 0.55 to 0.44 for low power and from 0.85 to 0.81 for high power. These reductions in model size are highly significant (p for both, Wilcoxon pair test) and counter a concern that the absence of a causal variant from a marker set will inflate the model size by introducing multiple markers that are partially correlated with the untyped causal variant.
These results demonstrate that the model size returned by GWiS is conservative for causal variants with small effects, and approaches the true model size for causal variants with large effects.
Application to ECG data
We then obtained pvalues from GWiS, minSNP, minSNPP, BIMBAM, VEGAS, and LASSO for the ARIC data. Permutations of phenotype data holding genotypes fixed [32] provided thresholds for genomewide significance for each method (Table S2). Due to LD across genes, a strong signal in one gene can lead to a neighboring gene reaching genomewide significance. This effect is well known, and scoring these as false positives would unduly penalize traditional univariate tests. Instead, neighboring genes reaching genomewide significance were merged, and overlap (even partial) with a known positive was scored as a true positive.
GWiS outperformed all other methods in the comparison (Figure 3 and Table 2). GWiS identifies 6 of 38 known genes or loci as genomewide significant. In contrast, BIMBAM identifies 5 known positives; minSNP, minSNPP and VEGAS identify 4; and LASSO identifies 2. Loci identified by the other methods are all subsets of the 6 found by GWiS. None of the methods produced any false positives at genomewide significance.
Due to the limited size of the ARIC cohort relative to the studies that generated the known positives, no method was expected to find all 38 known loci to be genomewide significant. Nevertheless, known positives should still rank high among the top predictions of each method, assessed by the ranks of the known positives at 40% recall (Figure S3). We found that GWiS, minSNP, minSNPP, BIMBAM, and VEGAS were equally effective in ranking known positives (MannWhitney rank sum pvalues for any pairwise comparison). LASSO performed below the other methods (pvalue for a pairwise comparison of LASSO to any other method). Top associations (up to 100 false positives) from each method are provided for PR interval, QRS interval, and QT interval (Tables S3, S4, S5).
While our conclusions are based on cardiovascular phenotypes, the results suggest that GWiS will have an advantage when causal genes have multiple effects. When an association is sufficiently strong to be found by a univariate test, GWiS is generally able to identify it. Beyond these association, GWiS is also able to detect genes that are genomewide significant, but where no single effect is large enough to be significant by univariate tests. The association of QRS interval with SCN5ASCN10A is a striking example: 4 independent effects are found by GWiS (pvalue = ) but the association is not genomewide significant by univariate methods (pvalue = for minSNPP) (Figure 4). A common followup strategy for singleSNP methods is to search for secondary associations in the same locus as a strong primary association. These results for ARIC together with results above for simulated data (Figure 2) demonstrate that GWiS performs this task well. While this feature is present in previous followup methods for candidate loci [11], [33], [34], it is absent from methods generally used for primary analysis of GWAS data.
Of the 38 known positives, 20 have GWiS models with at least one SNP (regardless of genomewide significance), and 7 of these are predicted to have multiple independent effects (Figure 5). These results suggest that the genetic architecture of ECG traits supports the hypothesis underlying GWiS. Moreover, for QT interval where the power is greatest to identify known positives (the ARIC sample size is 50% of the GWAS discovery cohorts), 5 of the 10 loci identified by GWiS are predicted to have multiple independent effects.
Discussion
In summary, we describe a new method for genebased tests of association. By gathering multiple independent effects into a single test, GWiS has greater power than conventional tests to identify genes with multiple causal variants. GWiS also retains power for lowfrequency minor alleles that are increasingly important for personal genetics, a feature not shared by other multiSNP tests.
Furthermore, GWiS provides an accurate, conservative estimate for the number of independent effects within a gene or region. Currently there are no standard criteria for establishing the genomewide significance of a weak second association in a gene whose strongest effect is genomewide significant. While the number of effects can be provided by existing Bayesian methods [34], their computational expense has limited their applicability to candidate regions, and they are not routinely used. By providing a computationally efficient alternative to existing methods, GWiS provides a new capability to estimate the number of effects as part of primary GWAS data analysis. Demonstrated effectiveness on real data may lead to more widespread use of this type of analysis. Applied to cardiovascular phenotypes relevant to sudden cardiac death and atrial fibrillation, GWiS indicates that 35 to 50% of all known loci contain multiple independent genetic effects.
The test we describe includes a prior on models designed to be unaffected by SNP density, in particular by the number of SNPs that are wellcorrelated with a causal variant. The priors on regression parameters are essentially uniform, with the benefit of eliminating any useradjustable parameters. A theoretical drawback is that the priors are improper [35], [36]. Theoretical concerns are mitigated, however, because improper priors pose no challenge for model selection, and our permutation procedure ensures uniform pvalues under the null.
Bayesian methods can be computationally expensive. GWiS minimizes computation by evaluating only the locally optimal models of increasing size in a greedy forward search. This appears to be an approximation compared to previous Bayesian methods that sum over all models. Previous Bayesian methods entail their own approximations, however, because the search space must either be truncated at 1 or 2 SNPs, heavily pruned, or lightly sampled using Monte Carlo. Our results demonstrate that the approximations used by GWiS provide greater computational efficiency than approximations used in previous Bayesian frameworks, with no loss of statistical power. GWiS currently calculates pvalues, rather than Bayesian evidence provided by other Bayesian methods. If Bayesian evidence is desired, an intriguing alternative to Bayesian postprocessing of candidate loci might be to use the Bayes Factor from the most likely alternative model identified by GWiS as a proxy for the sum over all alternatives to the null model. This may be an accurate approximation because, in practice, the Bayes Factor for the most likely model from GWiS dominates all other Bayes Factors in the sum.
The GWiS framework, using gene annotations to structure Bayesian model selection, may be applied to casecontrol data by encoding phenotypes as 1 (case) versus 0 (control), a reasonable approach when effects are small. More fundamental extensions to logistic regression, Transmission Disequilibrium Tests (TDTs), and other tests and designs should be possible and may yield further improvements. Moreover, similar genebased structured searches can be applied to genetic models to include explicit interaction terms [14]. The Bayesian format also permits incorporation of prior information about the possible functional effects of SNPs [37], [38], and disease linkage [39], [40]. Finally, the genebased pvalues provide a natural entry to gene annotations and pathwaybased gene set enrichment analysis [41]–[43].
Materials and Methods
Ethics statement
This research involves only the study of existing data with information recorded in such a manner that the subjects cannot be identified directly or through identifiers linked to the subjects.
Known positives
Known positive associations are taken from published genomewide significant SNP associations (pvalue ) [1], [2], [4], [5]. Genes within 200 kb of any genomewide significant SNP are scored as known positives. Finally, genes within 200 kb that are both positive are merged into a single known positive locus to avoid overcounting.
Study cohort
The ARIC study includes 15,792 men and women from four communities in the US (Jackson, Mississippi; Forsyth County, North Carolina; Washington County, Maryland; suburbs of Minneapolis, Minnesota) enrolled in 198789 and prospectively followed [44]. ECGs were recorded using MAC PC ECG machines (Marquette Electronics, Milwaukee, Wisconsin) and initially processed by the Dalhousie ECG program in a central laboratory at the EPICORE Center (University of Alberta, Edmonton, Alberta, Canada) but during later phases of the study using the GE Marquette 12SL program (2001 version) (GE Marquette, Milwaukee, Wisconsin) at the EPICARE Center (Wake Forest University, WinstonSalem, North Carolina). All ECGs were visually inspected for technical errors and inadequate quality. Genotype data sets were cleaned initially by discarding SNPs with HardyWeinberg equilibrium violations at p , minor allele frequencies , or call rates . Imputation with HapMap CEU reference panel version 22 was then performed, and all imputed SNPs were retained for analysis, included imputed SNPs with minorallele frequencies as low as 0.001. These cleaned data sets contributed to the metaanalysis to yield the known positives, and full descriptions of phenotype and sample data cleaning are available elsewhere [1], [2], [4]. Regional association plots were generated using a modified version of “make.fancy.locus.plot” [45].
Conventional multiple regression
The phenotype vector Y for N individuals is an vector of trait values. The genotype matrix X has N rows and P columns, one for each of P genotyped markers assumed to be biallelic SNPs. For simplicity, the vector Y and each column of X are standardized to have zero mean. A standard regression model estimates the phenotype vector as , where b is a vector of regression coefficients and e is a vector of residuals assumed to be independent and normally distributed with mean 0 and variance . The log probability of the phenotypes given these parameters is(1)
The maximum likelihood estimators (MLEs) are and , where denotes the transpose of . The total sumofsquares (SST) is , and the sumofsquares of the model (SSM) is . The sumofsquares of the errors or residuals (SSE) is(2)
A conventional multiple regression approach uses the Fstatistic to decide whether adding a new SNP improves the model significantly,(3)for a model with K SNPs, distributed as under the null. This approach fails, however, when the best SNPs are selected from the much larger number of M total SNPs, because the statistic does not account for the selection process.
Bayesian model selection
A model M is defined as the subset of SNPs in a gene with P total SNPs that are permitted to have nonzero regression coefficients. For each gene, GWiS attempts to find the subset that maximizes the model probability , where each of the P columns of X corresponds to a SNP assigned to the gene. In the absence of association, the null model with = 0 usually maximizes the probability, indicating no association. When a model with maximizes the probability, an association is possible, and permutation tests provide a pvalue. According to Bayes rule,(4)
The factor is modelindependent and can be ignored.
The prior probability of the model, , assumes that each of the P SNPs within the gene has an identical probability of being associated with the trait. This probability, denoted f, is unknown, and is integrated out with a uniform prior. The prior is also designed to make the model probability insensitive to SNP density: it should be unaffected if an existing SNP is replicated to create a new SNP marker with identical genotypes. We do this by replacing the number of SNPs within a gene with an effective number of tests, , calculated from the local LD within a gene. Correlations between SNPs make the effective number of tests smaller than the number of SNPs. The model prior based on the effective number of tests is(5)or for integer values. As the effective number of tests, , whose calculation is described below, is generally noninteger, we use the standard Beta function rather than factorials.
The remaining factor in Eq. 4 is(6)
The integration limits and prefactor ensure normalization. We assume that these limits are sufficiently large to permit a steepest descents approximation as in Schwarzian BIC model selection [15]. First, assuming that the genotypes are centered, the genotype covariance matrix is , where indicates matrix transpose as before, and diagonal elements for SNP with allele frequency . Provided that is much greater than each component of , the integral over is approximately(7)where the sumsquarederror SSE is . Provided that the limit is much greater than the maximum likelihood value , the integral over can be approximated as(8)where is the standard Gamma function. To avoid the cost of Gamma function evaluations, we instead use the steepest descents approximation,(9)
The loglikelihood is then(10)
As in the BIC approximation, we retain only terms that depend on the model and are of order or greater. Thus we replace by , and . For historical reasons, we also included a factor of in the prior for model size, to yield the asymptotic approximation(11)
The strategy of GWiS is therefore to find the model that maximizes the objective function(12)
The terms involving provide a Bayesian penalty for model performance, but also make this an NPhard optimization problem. We have adopted two efficient deterministic heuristics for approximate optimization. First is a greedy forward search, essentially Bayesian regularized forward regression, in which the SNP giving the maximal increase to the model likelihood is added to the model sequentially until all remaining SNPs decrease the likelihood. The second is a similar heuristic, except that the initial model searches through all subsets of 2 SNPs or 3 SNPs. We adopted this subset search to permit the possibility that all = 1 models are worse than the = 0 null, whereas a more complex model with or 3 has higher score. In practice, all associations identified by subset selection were also identified by greedy forward search. We therefore used the greedy forward search for computational efficiency.
GWiS is designed to select a single model for each gene. An alternative related approach would be to test for the posterior probability of the null model, , against all other models, + + + , using our model selection procedure either to choose the locally best model of each size or to include multiple models (which could suffer from a systematic bias favoring SNPs in large LD blocks). This is in fact the strategy of BIMBAM, which attempts to systematically evaluate all terms up to a given model size. Unfortunately, the number of terms increases exponentially fast with model size, and the bruteforce approach does not scale to genomewide applications. Monte Carlo searches over models have also been difficult to apply genomewide. Our work suggests that approximations that limit the search for fixed model size can be accurate, and further that the probabilities of models that are too large are expected to decrease exponentially fast, permitting the sum to be pruned and truncated. We have observed in practice that the model with the most likely value of dominates the sum, and similarly for BIMBAM that the single SNP with the best Bayes Factor dominates the sumofBayesFactors test statistic. These results suggest that the results of a more computationally expensive sum over all models would be largely consistent with the results of GWiS method. Furthermore, the Bayes Factor for the most likely model could provide a proxy for the Bayesian evidence.
Effective number of tests
The effective number of tests is an established concept in GWAS to provide a multipletesting correction for correlated markers. While the exact correction can be established by permutation tests, faster approximate methods can perform well [46]–[49]. While we use a fast procedure, a final permutation test ensures that pvalues are uniform under the null.
The method we adopt is based on multiple linear regression of SNPs on SNPs. The genotype vector for each SNP i is standardized to have zero mean. Correlations between all pairs of SNPs i and j are initialized as . Each SNPs weight is initialized to 1, and the number of effective tests T is initialized to 0. The SNP i with maximum weight is identified, and the following updates are executed:(13)
This process continues until all weights are equal to zero. When SNPs with maximum weight are tied (as occurs for the first SNP processed), the SNP with lowest genomic coordinate is selected to ensure reproducibility; we have ensured that this method is robust to other methods for breaking ties, including random selection. For simplicity, the correlations are not updated (the update rule would be ), which may lead to an overestimate for T. Model selection may therefore have a conservative bias. The pvalues are not affected, however, because they are calculated by permutation tests as described below.
The effective number of tests implies a trivial renormalization of the model prior, (Eq. 5), that does not affect the test statistic. Letting be the total number of markers, be the effective number tests, and be the size of the model, our prior gives each model of size the weight . If and are identical, there are models of this size, and the total weight of all models of size is . Since can range from 0 to , the sum is normalized. But when is larger than , the sum of all models of size is , which is . The sum from to is therefore . A normalization of 1 can be recovered by including an overall normalization factor, . The explicit prior for models of size is , which is normalized to 1. Since is modelindependent, it does not contribute to the test statistic.
Pvalues and genomewide significance
We use two stages of permutation tests: the first stage converts the GWiS test statistic into a pvalue that is uniform under the null; the second stage establishes the pvalue threshold for genomewide significance.
The first stage is conducted genebygene. We permute the trait array using the FisherYates shuffle algorithm [50], [51] and use the permuted trait to calculate the test statistics using the same procedure as for the original trait. Specifically, the model size is optimized independently for each permutation, with most permutations correctly choosing = 0. For S successes (loglikelihoods greater than or equal to the unpermuted phenotype data) out of Q permutations, the empirical pvalue is S/Q. To save computation, permutations are ended when . Furthermore, once a finding is genomewide significant, there is no practical need for additional permutations. For genebased tests (GWiS, minSNPP, BIMBAM, and VEGAS), the pvalue for genomewide significance depends on the number of genes tested (rather than the number of SNPs), for humans. We therefore also terminate permutations after Q = 1 million trials, regardless of S. In these cases, for purposes of ranking, a parametric pvalue is estimated for GWiS as(14)
The first factor is the parametric pvalue for the F statistic from the MLE fit, and the second term is the combinatorial factor for the number of possible models of the same size.
While these pvalues are uniform under the null, the threshold for genomewide significance requires a second set of permutations. To establish genomewide significance thresholds, in the second stage we permuted the ARIC phenotype for each trait 100 times, ran GWiS for the permuted phenotypes on the entire genome, and recorded the best genomewide pvalue from each of the 100 permutations. We then combined the results from each trait to obtain an empirical distribution of the best genomewide pvalue under the null. We then estimated the p = 0.05 genomewide significance threshold as the 15th best pvalue of the 300. This procedure was performed for GWiS, minSNP, minSNPP, LASSO, and VEGAS to obtain genomewide significance thresholds for each. Since minSNPP and BIMBAM are both uniform under the null, we used the genomewide significance threshold calculated for minSNPP, , for BIMBAM to avoid additional computional cost (Table S2). The threshold for GWiS is more stringent, , presumably because of the locus merging procedure described below. Changes in the genomewide significance thresholds of up to 50% would not affect any of the reported results.
Hierarchical analysis of genetic loci
In a region with a strong association and LD, GWiS can generate significant pvalues for multiple genes in a region. A hierarchical version of GWiS is used to distinguish between two possibilities. First, through LD, a strong association in one gene may cause a weaker association signal in a second gene. In this case, only the strong association should be reported. Second, the causal variant may not be localized in a single gene; for example, the best SNP tags are assigned to multiple genes. In this case, the individual genes should be merged into a single associated locus. The hierarchical procedure is as follows.

Identify all genes with GWiS , and use transitive clustering to merge into a locus all genes whose transcript boundaries are within 200 kb.

Run GWiS on the merged locus (including a recalculation of the number of effective tests within the locus) and identify the SNPs selected by the GWiS model. If genes at either end of the locus have no GWiS SNPs, trim these genes from the locus. Repeat this step until no more trimming is possible. If only a single gene remains, accept it with its original pvalue as the only association in the region. Otherwise, proceed to step 3.

Use a permutation test to calculate the pvalue for the merged locus from step 2. Assign it a pvalue equal to the minimum of the pvalues from the individual genes, and the pvalue from its own permutation. Regardless of the pvalue used, retain the entire trimmed region as an associated locus.
The trimming in step 2 handles the first possibility, a strong association in one gene that causes a weaker association in a neighbor. The rationale for accepting the smallest pvalue in step 3 is the case of a single SNP assigned to multiple genes. The merged region will have a less significant pvalue than any single gene, and it does not seem reasonable to incur such a drastic penalty for gene overlap.
Univariate tests: minSNP and minSNPP
For these tests, SNPs are assigned to gene regions as before. The pvalue for each SNP is then calculated using the Fstatistic as the test statistic, with empirical pvalues from permutation to ensure correct pvalues for SNPs with low minor allele frequencies. The minSNP method assigns a gene the pvalue of its best SNP. Selection of the best pvalue out of many leads to nonuniform pvalues under the null. It is standard to reduce this bias by scaling pvalues by a Bonferroni correction based on the number of SNPs or number of estimated tests. Instead, we perform genebygene permutation tests using the best F statistic for SNPs within the gene as the test statistic. As with GWiS, if 1 million permutations do not lead to one success, the association is clearly genomewide significant and we use the Bonferronicorrected pvalue for ranking purposes.
BIMBAM
The Bayesian Imputationbased Association Mapping (BIMBAM) is a Bayesian genebased method [10]. BIMBAM calculates the Bayes Factor for a model and then averages the Bayes Factors for all models within a gene to obtain a test statistic. Because 1SNP models were found to have as much power as 2SNP models, and because 2SNP models are not computationally feasible for genomewide analysis, BIMBAM by default restricts its sum to all 1SNP models within a gene [10]. The Bayes Factor for a single SNP is(15)
The design matrix has first column s and second column equal to the dosages of SNP in the individuals; is the phenotypic mean; ; the matrix is diagonal with diagonal terms ; and contains the regression coefficients . We used the recommended value relative to the phenotypic standard deviation. The test statistic for a gene with SNPs is . As with other methods, we used genebygene permutations to convert this statistic into a pvalue that is uniform under the null. Up to 1 million permutations were used, stopping after 10 successes.
The sufficient statistics used by BIMBAM are identical to minSNP and minSNPP, yet we found that the runtime of the public implementation was much slower, taking 270 sec for 1000 permutations of a gene with 135 SNPs across 8000 individuals. By improving memory management and optimizing computations, we improved the timing to 14 sec per 1000 permutations, a 19fold speedup. This implementation is included in our Supplementary Materials.
VEGAS
The Versatile GeneBased Test for Genomewide Association (VEGAS) [25] is a recently proposed method that considers the SNPs within a gene as candidates for association study. VEGAS assigns SNPs to each of the autosomal genes using the UCSC genome browser hg18 assembly. The gene boundaries are defined as of the and UTRs. Single SNP pvalues are used to compute a genebased test statistic for each gene and significance of each gene is evaluated using simulations from a multivariate normal distribution with mean 0 and covariance matrix being the pairwise LD values between the SNPs from HapMap Phase 2. As a result the method avoids permutations in calculating per gene pvalues, although permutations are required to obtain the genomewide significance threshold.
LASSO regression
LASSO regression is a recent method for combined model selection and parameter estimation that maps L1 regularized regression onto a computationally tractable quadratic optimization problem [26]–[28]. Applications to GWAS are attractive because it is possible to perform model selection on an entire chromosome. We therefore implemented a recent LASSO procedure developed specifically for GWAS [29].
To reduce computational cost, univariate pvalues are estimated from parametric tests, and genebased SNPs with are retained (we have confirmed that this computational constraint does not lose any known positive associations). Incremental model selection was performed by Least Angle Regression [27] using the R lars package [52]. The LASSO parameter was determined using 5fold cross validation. All genes with at least one SNP selected were identified, and selected genes overlapping other selected genes (including flanking regions) were merged into single loci.
As suggested previously, we used the Selection Index to rank genes and as the test statistic for a permutation pvalue [29]. To obtain the Selection Index, the MLE loglikelihood is calculated for the full model and for a reduced model with a subset of SNPs removed. Twice the loglikelihood difference is interpreted as a statistic, and the Selection Index is defined as the corresponding pvalue for a distribution with the number of removed SNPs as the degrees of freedom. Due to the LASSO model selection procedure, the Selection Index is not distributed as a under the null, and permutation tests are used to establish genomewide significance levels.
Simulations: power
For each true model size of to 8, we performed a series of simulations by picking 1000 genes from chromosome 1 randomly with replacement, using genotype data from the ARIC population of approximately 8000 individuals. For each gene, we selected “causal” SNPs that have from regression with other “causal” SNPs within the gene. A gene had to have at least SNPs to be picked for models of size to ensure enough remaining SNPs after the removal of the causal SNPs to permit a model of the true size.
We attempted to distribute the total population variance explained, , equally across the SNPs. The covariance matrix for the SNPs calculated from the population is denoted , with understood to be . The coefficient for SNP in the model was set to(16)which ensures that . The phenotype for an individual with genotype rowvector was then calculated as , with again the population average of and drawn from a standard normal distribution.
The power was calculated as (number of genes that are genomewide significant)/1000, and the error of the estimate was calculated using 95% exact binomial confidence intervals. The pvalue thresholds were taken directly from genomewide permutations (Table 2).
Simulations: model size
Phenotypes that were used to estimate the model size were generated by assigning each “causal” SNP the same power of 0.1 and 0.8. The population variance explained for each SNP was calculated as , in which is the quantile of the standard normal for uppertail cumulative probability of , and is the quantile for lowertail probability power. We chose to be , the commonly used genomewide significance threshold for univariate tests. The effect of SNP is then , in which is the genotype covariance matrix. The simulated phenotypes are then , with drawn from a standard normal distribution. In this test we control for the variance explained by the SNP, not by the gene, and therefore do not rescale the regression coefficients to account for LD. For each ranging from 0 to 10, we repeated these steps using ARIC genotype data for 100 genes chosen at random from chromosome 1.
Only GWiS and LASSO give model size estimates. GWiS directly reports the model size as the number of independent effects within a gene and LASSO reports the model size as the number of selected SNPs within a gene. We ran both methods using the simulated data with LD. We also tested both scenarios when the causal SNPs were kept or removed from gene.
Performance evaluation
Gene associations were scored as true positives if the gene (or merged locus) overlapped with a known association, and as false positives if no overlap exists. Only the first hit to a known association spanning several genes was counted.
The primary evaluation criterion is the ability to identify known positive associations at genomewide significance. The genomewide significance threshold was determined separately for each method (see above), and no method gave any false positives at its appropriate threshold.
A secondary criterion was the ability to enrich highly ranked loci for known associations, regardless of genomewide significance. This criterion was assessed through precisionrecall curves, with precision = TP/(TP+FP), recall = TP/(TP+FN), and true positives (TP), false positives (FP), and false negatives (FN) defined as a function of the number of predictions considered.
Small differences in precision and recall may not be statistically significant. To estimate statistical significance, we performed a MannWhitney rank sum test for the ranks of the known associations at 40% recall for GWiS, minSNP, minSNPP, and LASSO.
Implementation
GWiS runs efficiently in memory and CPU time, roughly equivalent to other genomewide tests that require permutations (Table 3). Computational times are greater for real data because real associations with small pvalues require more permutations. LASSO required far less computational resources, but also prefiltered the SNPs and had the worst performance. Genomewide studies can be finished within around 100 hours. Low memory requirements allow GWiS to run in parallel on multiple CPUs. The GWiS source code implementing GWiS, minSNP, minSNPP, and BIMBAM is available under an open source GNU General Public License as Supplementary Material, also from the authors' website (www.baderzone.org), and is being incorporated into PLINK [31].
Supporting Information
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Štítky
Genetika Reprodukční medicínaČlánek vyšel v časopise
PLOS Genetics
2011 Číslo 7
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