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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Genet Epidemiol. Author manuscript; available in PMC 2013 January 1.
Published in final edited form as:
PMCID: PMC3310255

Using the Gene Ontology to Scan Multi-Level Gene Sets for Associations in Genome Wide Association Studies


Gene-set analyses have been widely used in gene expression studies, and some of the developed methods have been extended to genome wide association studies (GWAS). Yet, complications due to linkage disequilibrium (LD) among single nucleotide polymorphisms (SNPs), and variable numbers of SNPs per gene and genes per gene-set, have plagued current approaches, often leading to ad hoc “fixes”. To overcome some of the current limitations, we developed a general approach to scan GWAS SNP data for both gene-level and gene-set analyses, building on score statistics for generalized linear models, and taking advantage of the directed acyclic graph structure of the gene ontology when creating gene-sets. However, other types of gene-set structures can be used, such as the popular Kyoto Encyclopedia of Genes and Genomes (KEGG). Our approach combines SNPs into genes, and genes into gene-sets, but assures that positive and negative effects of genes on a trait do not cancel. To control for multiple testing of many gene-sets, we use an efficient computational strategy that accounts for LD and provides accurate step-down adjusted p-values for each gene-set. Application of our methods to two different GWAS provide guidance on the potential strengths and weaknesses of our proposed gene-set analyses.

Keywords: gene-sets, genome wide association, pathways, score statistics


Genome wide association studies (GWAS) that survey ½ to 1 million single nucleotide polymorphisms (SNPs) for their associations with traits have had an enormous impact on the understanding of some complex traits. More critical, perhaps, is that they provide a glimpse at the large number of genes likely involved in the genetic basis of many common traits. As of July, 2011, there have been 957 published GWAS with individual single-nucleotide polymorphisms (SNPs) associated with over 165 traits, with single-SNP p-values < 5E-8 1. Yet, many of the associated SNPs explain only a small fraction of the expected heritability 2; 3. Several recent studies suggest that a very large number of SNPs, most with very small effects on traits, would be required to explain a large fraction of the heritability of some traits 4; 5. In this case, extremely large sample sizes would be needed to reliably detect the individual SNPs.

In contrast to thinking of effects of individual SNPs, there is growing experimental evidence that sets of genes, possibly in overlapping pathways, tend to function as sub-networks having a larger effect on a complex trait than individual genes. A good example was shown for tissue-specific gene expression data combined with genotypes measured on a segregating mouse population 6. The different sub-networks were enriched for a number of biological processes and were interpreted to represent key functional units that underlie processes specific to the different cell types. This suggests that common diseases might result from many genes that are highly interconnected in networks. Following these ideas, and stimulated by gene expression studies, it is anticipated that grouping SNPs into genes, and then genes into sets of similar function, might facilitate biological interpretations of GWAS, and perhaps account for genetic heterogeneity, as well as improve statistical power to detect relevant genes 7. The benefit of grouping genes into sets would be greatest if many genes in a given set are associated with a trait — by averaging over the scores for genes in a set, the signal for association with a trait can be larger for the average than that for individual gene-scores, because of the reduced variability of the average.

A variety of gene-set methods have been widely used for gene expression studies 8; 9. But, we take a different strategy for GWAS. To understand our strategy, and appreciate why it differs from many methods used for gene expression studies, it is helpful to briefly review gene-set analyses developed for gene expression studies. A typical gene expression study compares the expression of tens of thousands of genes between two groups, searching for differential expression, resulting in a list of p-values for the different genes. A common approach to analyze a set of genes is to choose a cut-off of statistical significance, and then cross-classify the discrete indicator of significance with whether the genes are in a particular set or not, creating a 2×2 contingency table. The fraction of significant genes can then be compared between those in the set versus the remainder not in the set, using Fisher's exact test. An alterative approach that does not require a cut-off is to order the p-values from smallest to largest (or perhaps some other measure of differential expression), and then test whether the genes in a set tend to clump toward smaller p-values (Gene Set Enrichment Analysis – GSEA). Many other approaches have been developed for lists of p-values, such as adaptively truncating the p-values in a set in order to emphasize the smaller p-values 10, or ad hoc methods that attempt to account for the number of SNPs per gene and the correlation structure among SNPs by use of arbitrary correlation thresholds and heuristic adjustments for both linkage disequilibrium and the number of SNPs 11.

An advantage of many of the proposed methods is that they require only lists of p-values, making them appealing for pooling across multiple studies in meta analyses. Because of their simplicity, they have been adapted and used for GWAS 12-19. However, there are important limitations, well discussed elsewhere 7-9; 20. Some of the major issues relevant for GWAS follow. First, by comparing genes in the set versus those not in the set, for their associations with a trait, the null hypothesis focuses on “competition” between the set and its complement. Lack of enrichment does not mean that the genes in the set are not associated with the trait, but rather that they are associated, on average, as much as those not in the set. Second, when genes are correlated, the Type-I error rate can be inflated (related to whether the sample or the genes are the random units under the null hypothesis). Third, ”size-biased” sampling haunts analyses in two ways; at the gene-level and at the set-level. At the gene-level, most statistical tests require a single p-value per gene, so when there are multiple p-values for a gene, a common strategy is to choose the smallest p-value to represent the gene 13. This, however, results in a bias towards detecting associations with genes having more p-values (e.g., larger genes having more SNPs). The magnitude of this bias depends not only on the number of SNPs in a gene, but also on the correlations among the SNPs — weaker correlations allow more extreme statistics. This extreme p-value approach might also have weak power when a gene contains multiple SNPs that are jointly associated with a trait, yet individually have small effects. Size-bias at the set-level occurs when the number of genes varies over sets: sets that have a large number of genes can be biased if only the most extreme p-value among all genes in a set is used to represent the set. For further discussion about these issues, and references, see 7. Fourth, most gene-set methods focus on one set at a time (comparing a set to its complement), and so the resulting p-values across multiple sets are correlated, making it difficult to accurately evaluate the Type-I error rates across all gene-set tests. Bonferroni correction based on the number of sets tested would be overly conservative.

Because of the above concerns, our strategy takes a “direct” approach to test the association of each gene-set with a trait; the null hypothesis is that the gene-set is not associated with a trait, not the competitive hypothesis discussed above. This seems more relevant for GWAS that are expected to have a much smaller number of statistical associations than gene expression studies that typically have many differentially expressed genes. Our approach allows SNPs to be mapped to multiple genes, and directly accounts for correlations among SNPs, both within and between genes. By doing so we are able to normalize genes, and gene-sets, to avoid size-biased sampling. This also avoids using arbitrary cut-offs to choose SNPs with low correlations, or to assign SNPs to haplotype blocks 21. Furthermore, we consider all possible gene-sets, and provide a simulation-based method for the entire genome that is computationally efficient and accounts for the testing of many gene-sets. Our methods also allow for overlapping gene-sets, which can occur when genes map to multiple gene-sets, or when sets can be nested or overlap when grouping genes according to their function. These new methods resolve some of the unsolved analytic issues for GWAS gene-set analyses that have been highlighted elsewhere22.

A variety of public resources are widely used to create gene-sets, such as Kyoto Encyclopedia of Genes and Genomes (KEGG 23), BioCarta, The Reactome Project, NetPath, Pathway Interaction Databases, and the Gene Ontology (GO 24). A complete list of over 300 resources for biological pathways can be found at the Pathguide website ( Because the GO has the largest amount of information and is well structured, we primarily use it to create gene-sets. However, our approach is sufficiently general that it could easily be adapted to gene-sets created from other sources of information, as we illustrate with KEGG, or even gene-sets created from multiple sources (e.g., GO + KEGG + others).

The GO provides structured information on the properties of the products of genes, using three general domains: cellular components (the parts of a cell or its extracellular environment), molecular function (the activities of gene products at the molecular level), and biological process (sets of molecular events with a defined beginning and end). For each of these domains, there are well-defined terms that describe gene product properties. For example, a single gene product might be described by a molecular function term that describes its enzymatic activity, a biological process term that describes a sequence of molecular activities, and a cellular component term that describes where the activities occur, such as a cell membrane. Furthermore, terms can be related to each other. The more specific terms are called child terms, and these are linked to more general terms, called parent terms. If every child term had only one parent term, the terms would form a hierarchy. But, because a child term can have more than one parent, the terms are related in a directed acyclic graph (DAG). Consistent with terminology for graphs, we refer to the GO terms in the DAG as vertices, and the connections between them as edges. The majority of edges are “is_a” relationships, meaning that a child is a subtype of its parents. Other terms are “part_of” (meaning a child is necessarily part of its parent, so the parent must exist when the child exists), “has_part” (meaning a parent necessarily has its child as a part, so the child must exist when the parent exists), and terms related to how a gene product regulates another gene product. All directions go from child to parent, except for “has_part”, which is directed from parent to child.

A number of methods have been developed that capitalize on the GO data. Holmans, like others, used a list of p-values from a GWAS 25. By applying an arbitrary threshold to determine statistical significance, a gene was categorized as significant if it had at least one significant SNP: then, a count of the number of significant genes was used in gene-set analyses. The gene-sets were created by mapping genes to the GO vertices, so that each vertex is a gene-set. Additional sets were created by recursively building larger sets by uniting child vertices with parent vertices. The null distribution of the statistics was simulated by randomly sampling the SNP p-values from the list of all p-values. This process, however, assumes that the level of linkage disequilibrium among SNPs is approximately equal among all genes and among all GO categories, which seems unlikely. Analyses based on p-values are limited by the inability to appropriately account for the underlying correlation structure among SNPs, making it questionable whether Holman's approach adequately corrects for LD among SNPs, variable gene sizes, and overlapping genes.

Others have also capitalized on the GO DAG structure, but mainly to control the family-wise error rate (FWER) when testing gene-sets using a list of p-values as input. One approach is “bottom-up”, which starts with the most specific terms, the “child-vertices”, and applies a FWER control for these vertices. The procedure moves up to parent vertices only if the null hypothesis for the child vertex can be rejected. Another approach is “top-down”, which starts with the root vertices and moves down to child-vertices only if the root can be rejected. Meinhausen developed this procedure to control the FWER when hypotheses can be structured in a hierarchy, by starting at the highest level of a hierarchy, and continuing down to the next level of the hierarchy until no tests are significant 26. To apply this to GO, the DAG would have to be transformed to a hierarchical tree structure (requiring duplication of terms so each child has only one parent). However, this approach would not be useful for GWAS, because the procedure would likely stop at the highest GO vertex that contains all genes. Another alternative is to start testing at a specific “focus level” of the GO graph, which determines the level of specificity of the GO terms used in the analyses 27. This approach controls the FWER by a sequentially rejective multiple testing procedure that exploits the GO DAG structure. Starting at a specified level, the procedure sequentially moves away (up and down the DAG) until no gene-sets are statistically significant. However, this approach does not account for the correlations among SNPs. Furthermore, for the typical agnostic GWAS, it is difficult to specify a specific focus level. Another approach developed for gene-expression studies is to model the joint distribution of p-values using a hidden Markov model after transforming the DAG into a hierarchical tree 28. A limitation of these methods for GWAS data is that they are based on a list of p-values, and so they cannot account for the correlation structure for SNPs.

As we emphasize throughout, accounting for SNP correlations is impossible when only a list of p-values is available, and at best a crude approximation when external reference correlation structures are used to infer correlations in the GWAS data that generated the p-values. Because of this, we use the subject-level genotypes to directly account for SNP correlations. This allows us to avoid unrealistic assumptions or arbitrary thresholds, as well as develop more powerful methods for gene-level and gene-set analyses. While developing our methods, we recognize the balance between computer-intensive methods required to fit sophisticated statistical models and the need to be able to handle large amounts of data that can have complex data structures, as well as the need to compute simulation-based p-values. To achieve this balance, we focus on score statistics, which are rapid to compute and tend to be optimal for alternative hypotheses that are “local” to the null hypothesis, which in our context means small gene effects, which is expected for many GWAS.

The paper begins with a detailed description of our methods to score SNPs, genes, and gene-sets, along with how gene-sets are created from the GO structure. We then apply these new method to several GWAS, contrasting results from gene-set analyses with traditional single-SNP analyses. Simulations are used to explore and highlight some of the statistical properties of our proposed methods, although a thorough simulation that evaluates the power of our new methods across a broad spectrum of possible genetic architectures is beyond the scope of this initial presentation. We then discuss strengths and weaknesses of gene-set analyses, and areas for potential research directions.


Mapping SNPs, Genes, and GO Terms

To map SNPs to genes, and genes to the GO structures, we used publically available physical maps for both HapMap SNPs and known genes, as well as the GO data files. These files and the steps we used to process them are described in Appendix A. A few points are worth emphasizing. For a SNP to be included in the gene-set analyses, SNP genotypes for cases and controls must be available, the SNP must map to within a user-specified distance to known genes, and the gene(s) it maps to must map to the GO structure. To create gene-sets, we use a recursive method so that genes are included in a gene-set if the genes map to a specific GO term or any of its descendant GO terms. At the highest level, such as a single DAG root, the gene-set would be all genes that map to the entire GO structure. At the lowest level, such as a leaf, the gene-set would be only those genes that map to the leaf. See Appendix A for more details.

Scoring SNPs

We first describe how we score individual SNPs for their association with a trait, and then describe how we combine these scores into scores for genes, and then scores for gene-sets. To score the effects of SNPs on a trait, we use score statistics from regression models. To simplify our exposition, we focus on case-control data using logistic regression, but then discuss simple ways to use our methods for traits that can be analyzed with generalized linear models, such as quantitative traits using linear regression. Let yi have values of 1 for cases and 0 for controls, and let xij represent the score for the jth SNP of the ith subject (i = 1,. . .n, ;j1=, . . . ,m). We let xij count the number of minor alleles for a SNP (xij = 0,1,2 ), although other genotype scoring could be used. With this set up, the log-likelihood for a logistic regression model with only the jth SNP is

equation M1


equation M2

The score statistic for βj can be found by taking the first partial derivative of ln like with respect to βj, and then setting βj = 0. The contribution of the ith subject to this score is Uif = (yipoi)xif, where poi is the null probability P(yi = 1|α,βj = 0). As a side note, it is important to recognize that adjusting covariates could be included in the analyses by including them in the logistic regression model and using them, along with the maximum likelihood estimates of their regression coefficients, to compute poi. Then, the efficient score statistic to test whether βj = 0 can be computed as

equation M3

Note that the denominator of zj is asymptotically equivalent to the square-root of the observed information for βj, adjusted for estimating the intercept.

To apply our methods to other types of traits, such as quantitative, one simply needs to scale the Uij scores by the appropriate dispersion parameter. To see this, note that the log-likelihood for a subject's trait yi based on a generalized linear model (GLM) for exponential family data can be expressed as ln L = [b(η)]/a(ϕ) + c(y, ϕ), where a, b, and c are known functions, ϕ is a dispersion parameter, and η = Z′β; Z denotes a vector of covariates to be adjusted out, and β the vector of corresponding regression coefficients. The expected value of the trait is = f–1(Z′β), where f is the link function. Then, for a GLM, the corresponding score is Uif = [(yiỹi)/a(ϕ)]xif. For binomial and Poisson data, a(ϕ) = 1, giving the results we present above for logistic regression. For quantitative traits that have a normal distribution, equation M4, the mean-squared error for the residuals, (yii), after regressing out the covariates. This works because under the null hypothesis the expected value of equation M5 is equal to Fisher's information, the variance of equation M6.

Scoring Genes

We now need to consider how to combine the zj scores for all SNPs in a single gene to create a gene-level score. Some guidance can be gleaned from the literature. Critically, Newton et al. 29 compared two different ways to score sets of genes for gene expression studies. One way was to consider the fraction of genes in a set that had p-values below a threshold. The other way was to average scores across genes in a set. Their simulations showed that power was greater for the averaging method when a set contains a large fraction of associated genes, each with small effects. In contrast, thresholding was more powerful when there was a small fraction of genes with large effects. For GWAS, we expect many genes with small effects, so averaging scores seems appropriate. Furthermore, for gene-set analysis of expression data, others have found that squaring the statistics to allow for both positive and negative directions of the genetic effects and then averaging these squared terms gave greater power than other popular methods, including the enrichment scoring method of the GSEA 8. They also found that squared statistics gave similar results as using absolute values of the statistics. Because of this, we too use averages of squared statistics to score genes. So, to create gene-level scores, we use the average of equation M7 across all SNPs in a gene,

equation M8

Here, there are m SNPs in gene g, and the vector Zg contains the zj scores for all SNPs in gene g. By using vector notation, we emphasize that the gene score is a quadratic form. It should be noted that this quadratic form is equivalent to a logistic kernel-machine test when using a linear kernel 30. This opens opportunities to consider alternative ways to score genes, as discussed in Appendix B. Under the null hypothesis, it is well known that Zg has an asymptotic multivariate normal distribution with mean vector 0 and correlation (and covariance) matrix R. This correlation matrix is based on the observed information matrix, equation M9, where the first element of Xi is the intercept and the remaining elements are the doses of the minor alleles for the m SNP genotypes. After accounting for estimating α in the score statistic and assuming that there are no adjusting covariates (poi is the same for all subjects), it can be shown that R is equal to the correlation matrix for the xij's.

Now, because ZgMVN(0,R), the moments of the quadratic form are known to be equation M10 and equation M11, where tr(A) means trace of matrix A31. This implies that the expected value of a gene score is E[gene] = 1, and its variance is Var(gene)=2tr(RR)/m2. It is important to recognize that this allows us to compute the variance of the gene-scores using all the SNPs within a gene while accounting for the correlations among them. To illustrate, when there is no LD among SNPs within a gene, Var(gene)=2/m. At the other extreme, if all SNPs within a gene were in complete LD, so that equation M12, then Var(gene)=2. As an alternative to averaging equation M13, we considered averaging the absolute value, |zj|, but avoided this because the resulting truncated multivariate normal distribution would require numerical evaluation of multivariate integrals 32.

Scoring Gene-Sets

To score a gene-set, we use the weighted average of the gene-scores within a set,

equation M14

where wj = 1/Var(genej). Under the null, E[genej] = 1, so E[set]=1. The variance of the set-score is

equation M15

Like the methods used to derive Var(gene), the covariance between two gene scores can be shown to be

equation M16

This covariance depends on the matrix of cross-correlations (Rif) of SNPs for genes i and j, and the number of SNPs for gene i (mj). and gene j(mj). These derivations allow us to measure the correlation structure among genes in a set so we can normalize the scores for gene-sets by subtracting the null expectation and dividing by the standard error of the set-score,

equation M17

This approach directly accounts for the correlation among SNPs, eliminating the need to specify arbitrary correlation thresholds for including or excluding SNPs, as well accounts for the number of SNPs per gene and the number of genes per set. This approach also accounts for situations when a SNP maps to multiple genes within the same set, a likely occurrence when a gene has alternative splicing.

After applying our methods to real data and evaluating its properties by simulations, we found a few transformations that aided the distributional properties of zset. First, the scores from logistic regression, Uif = (yipoi)xif, are based on binomial yi and trinomial xij, and so asymptotic normality of the SNP scores might not hold if n is not very large. Centering the xij's about their sample means helped to prevent the mean of the set scores to depend on the set size. Second, because we are using quadratic statistics (i.e.., equation M18) to score genes, sets with few genes tended to have positively skewed values of zset. To reduce this skewness, we used the square-root transformation for the gene scores, equation M19. Using a 2nd order Taylor-series approximation, the null expected value of this square-root transformation is E[gene*]≈1 – Var(gene)/8 and its variance is Var(gene*)≈Var(gene)/4. Likewise, the covariance between two transformed gene-scores is equation M20. Hence, the final set-scores are based on the weighted average of the transformed gene* scores, along with E[gene*] and equation M21 to create zset based on the square-root transformation.

When applying our methods, a critical issue is to decide on the maximum number of genes in a set. Without any restrictions, the largest set will contain all genes which might not be sensible. One approach would be to limit the depth of the GO DAG structure. However, this could still result in sets containing many genes, potentially washing out the association signal by many random non-associated genes. For this reason, we allow the user to specify the maximum number of genes in a set so that the scan for gene-set associations would only cover sets that meet this constraint. For the applications we illustrate, we set the maximum set-size to be 30. Avoiding large sets not only avoids diluting out signals of association from many non-associated genes, but it also speeds the computations by reducing the number of covariance terms to compute for genes in the same set.

Controlling for Population Structure or Other Covariates

As many have emphasized, it is important to correct for systematic genomic inflation of test statistics 22, such as by using the eigenvectors of the independent SNPs from a GWA study panel as adjusting covariates in logistic regression models 33; 34. A way to perform this adjustment for our gene-set scanning approach is to use adjusted Uij scores

equation M22

where equation M23 is the logistic regression predicted status based on the eigenvectors for the ith subject and the maximum likelihood estimates, equation M24, of the eigenvector regression coefficients, and rij is the residual from linear regression of the jth SNP on the eigenvectors. As discussed elsewhere 35, this method works because the residuals are uncorrelated with the eigenvectors and testing the residuals is equivalent to testing the allele dosage score adjusted for the eigenvectors.

Missing Data

To account for missing genotypes, it is most informative to impute them using powerful methods and then use the allele dosage scores for the missing genotypes 36. Absent this imputation, one could set the Uij score to 0 for missing genotypes, which is equivalent to replacing missing xij values with their sample means.

Computing p-values

To determine the null distribution of the set-score statistics, zset, we could permute the trait over all subjects, preserving the correlation structure among the SNPs, and then recompute all statistics. This, however, is computationally intensive, particularly due to the need to compute Cov(genei,genej). Lin37 proposed a more efficient method, capitalizing on the multivariate normal distribution of the score-statistics from logistic regression. Suppose that we place the Uij scores for all SNPs into a vector for the ith subject, Ui. If n were very large, equation M25. We could approximate the null distribution by simply multiplying the observed vector Ui by a standard normal random variable (scalar), si ~ N(0,1), to create equation M26. Conditional on the observed data, this new vector also leads to a multivariate normal statistic, equation M27, with mean 0 and the same covariance matrix as that for the observed statistic, equation M28. Because we standardized U scores by dividing by the square root of their variances, we work with the correlation matrix, R, for the x allele doses. This means that we only need to compute the R matrix for the observed data once. Furthermore, the entire R matrix is not used — only sub-matrices that correspond to genes in the same set. Hence, to further speed calculations, we only compute Cov(genei,genej) at the first time it is needed, and then store it for future use. This allows us to avoid computing Cov(genei,genej) for genes that are never in the same set, dramatically reducing the computational burden. For the simulations, we only need to create equation M29 for all n subjects in each simulation loop, and then use this equation M30 for all subsequent calculations to create scores for genes, and scores for gene-sets, but use the once-calculated terms for Cov(genei,genej).

Another benefit of Lin's approach is a simulation-based step-down procedure to compute adjusted p-values that control the FWER. This step-down procedure first orders the observed statistics (e.g., gene-sets) from largest to smallest, denoted z(1),z(2),...,z(m). The corresponding hypotheses are labeled H(1), H(2),...,H(m). Starting with H(1), it is rejected if equation M31, where sk is the simulated statistic and α is the Type-I error rate. If H(1) rejects, H(2) is tested, and it is rejected if equation M32, and so forth. This means that H(j) is tested only if all prior hypotheses H(1), H(2),...,H(j–1) are rejected. By noting that the p-value is the α - level test at which the null hypothesis would be rejected, we can use this step-down approach so the p-values for the ordered statistics, z(1),z(2),...,z(m), are monotonically increasing, and the FWER is controlled. Two important points are: 1) the p-value for z(1) is simply how frequent the maximum simulated statistic is at least as large at the maximum observed statistic, as one would desire for the most extreme statistic, and 2) this approach accounts for the correlations among the statistics, avoiding the conservative approximation when using the Bonferroni correction.

Application to GWAS Data

1) GWAS for Musculoskeletal Adverse Events from Breast Cancer Treatment

Aromatase inhibitors are well established adjuvant therapies for postmenopausal women with early-stage breast cancer, yet about half of the patients receiving these types of treatments have joint-related musculoskeletal complaints, which likely contributes to decreased compliance. A GWAS was performed to identify SNPs associated with musculoskeletal adverse events (MSAEs) in women treated with aromatase inhibitors for early breast cancer 38. A nested case-control design was used to select patients enrolled onto the MA.27 phase III trial that compared the aromatase inhibitors anastrozole and exemestane. Cases (n=293) were defined as patients with MSAEs. Controls (n=585) did not experience any of the MSAEs, were followed for at least two years, and had at least six months longer follow-up than their matched case. Genotyping was performed with the Illumina Human610- Quad BeadChip. After quality control checks, a total of 551,358 SNPs were analyzed. The smallest P-values (< 1E-4) occurred for 14 SNPs that were on chromosomes 1-4, 8, 9, 14, 15, 22, and X. The smallest P-values were for 4 SNPs on chromosome 14 (P-values 2.23E-06 to 6.67E-07). The gene closest to these 4 SNPs is the T-cell leukemia 1A (TCL1A), 926-7000 bp distance from the SNPs. Further details of the study design and main results can be found in the primary publication 38. The analyses presented here include only the SNPs measured on the Illumina platform. Imputed SNPs were not included.

2) GWAS for Breast Cancer Risk among High-Risk Women in Breast Cancer Prevention Trials

The selective estrogen receptor modulators (SERM) tamoxifen and raloxifene have been evaluated for their ability to prevent breast cancer in high-risk women in large randomized clinical trials 39; 40. To further reduce the risk of breast cancer, indentifying SNPs and genes associated with breast cancer occurrence in these women would provide important leads to better understand why some women develop breast cancer despite SERM therapy, and might even lead to better selection of patients for preventive therapy. To achieve this aim, cases (n=594) and controls (n=1,172) were selected from the tamoxifen arm in the NSABP prevention trial P-1 and from the tamoxifen and raloxifene arms in the NSABP prevention trial P-2. Cases were women who experienced an invasive breast cancer or ductal carcinoma in-situ (DCIS); controls had neither. Genotyping was performed with the Illumina Human610- Quad BeadChip. After quality control checks, a total of 547,356 SNPs were analyzed. The smallest P-values (< 1E-4) occurred for 11 SNPs that were on chromosomes 3, 4, 8, 9, 13, and 16. The smallest P-value was for a SNP on chromosome 16 (P-value 2.12E-06), near the ZNF423 zinc fingerprint gene. Further details of the study design and main results can be found in the primary publication 41.


Results from MA.27 Analyses

Results from gene-set analyses for the MA.27 study are presented in Table 1. Although no gene-sets achieved statistical significance at the global level, it is intriguing that the most extreme gene-set statistic occurred for the GO cellular component term relating to actinomysin, the actin part of any complex of actin, myosin, and accessory proteins. The single gene that maps to this GO term is ACTC1 actin, a highly conserved protein found in muscle tissues, making it a suspect for musculoskeletal adverse events in the MA.27 study. The second most extreme gene-set for cellular component relates to the acrosome, an organelle in sperm cells. Although this is non-sensible for musculoskeletal adverse events in women, one of the three genes in this gene-set codes the L protein of the mitochondrial glycine cleavage system. Other gene-sets related to mitochondrial cellular components and biological processes had extreme gene-set statistics, suggesting that mitochondria could be related to musculoskeletal adverse events.

Table 1
MA27 GO gene-set scan results.

Gene-level analyses for MA.27 are presented in Table 2. The most extreme z-score was for a microRNA, which are short non-coding RNAs involved in post-translational regulation of gene expression. The second most extreme statistic was for a hypothetical protein (predicted gene with unknown function). It is important to see that the TCL1B gene identified by single-SNP analyses had the 3rd largest gene-level statistic. This gene is near the TCL1A gene, which was near the most significant single-SNP test. In fact, the gene-level statistics for these two gene were quite similar (z-scores of 4.313 for TCL1B and 3.897 for TCL1A), and each had approximately 40 SNPs mapping to the genes. But, based on p-values, TCL1A ranked 25 from the top. This is partly because of the 40 SNPs mapping to TCL1A, some have weak associations and others very strong, resulting in a reduced average gene-level score. It is possible that the SNPs in this region, found by the strongest single-SNP tests, have functional roles for both genes, or are in LD with functional SNPs.

Table 2
MA27 gene-level scan results.

To contrast the single-SNP analyses with the gene-set analyses, we examined how the most striking single SNP result contributed to the gene-set results. The SNP rs7158782 near the TCL1A gene had a p-value of 6.67E-07. This gene maps to the GO term “stem cell maintenance” (ID 0019827), within the biological process domain. There were 23 other genes in the gene-set for this GO term, with a gene-set z-statistic = -0.3, suggesting no evidence of the genes in this set to be associated with MSAE's. This illustrates a potential limitation of gene-set analyses when only a single gene in a large set is associated with a trait.

We also considered why the actin gene was not revealed in the single-SNP analyses. There were 43 SNPs in the actin gene region with single-SNP p-values ranging 0.00034 to 0.9728. Only 7 SNPs had p-values < 0.02; their distance spanned 24 kb, with 4 other intervening SNPs having p-values ranging 0.26 to 0.86. Based on single-SNP analyses, these SNPs individually were not near the top of the list of interesting SNPs. This illustrates that single-SNP analyses might miss important statistical associations when their individual signals are small, and by combining over a gene, or a gene-set, there is potential to strengthen the statistical association signal.

Results from NSABP Analyses

Results from gene-set analyses for the NSABP study are presented in Table 3. None of the gene-set analyses achieved global statistical significance, and the most extreme gene-set z-score was for biological process related to activation of Janus kinase activity, a process which activates the JAK protein by introducing a phosphate group to a tyrosine residue of a JAK protein. The two genes in this gene-set are IL6R interleukin 6 receptor, a cytokine that regulates cell growth and differentiation and plays an important role in immune response, and GH1 growth hormone, which plays an important role in growth control. The second most extreme z-statistic for biological process was for the GO term virus-host interaction, which included 21 genes. Many of these genes are involved in immune response (e.g. THO complex) or regulation of gene expression or growth hormones. This set also includes SMAD3, a signal transducer and transcriptional modulator that mediates signaling pathways. In particular, this protein is a transcriptional modulator activated by transforming growth factor-beta and is thought to play a role in the regulation of carcinogenesis. This suggests that immune-related genes and those related to growth factor regulation would be worth while to pursue as candidates for associations with breast cancer occurrence among women treated with tamoxifen or raloxifene.

Table 3
NSABP GO gene-set scan results.

Gene-level analyses for NSABP are presented in Table 4. The gene with the most extreme statistic relates to a mitochondrial ribosomal protein, but currently interpreted as a pseudo-gene. The single-SNP analyses led to the ZNF423 gene, which has the second largest z-score for the gene-level analyses.

Table 4
NSABP gene-level scan results.

KEGG Set Analyses

In addition to the recursively defined GO gene-sets, we used the KEGG pathways to define gene-sets. Here, there were 221 gene-sets, with the number of genes per set ranging 1-1088, and an average of 70 genes per set. Unlike the GO-set analyses, we did not restrict the number of genes per set for the KEGG sets, primarily because there is no DAG structure among the KEGG sets. Furthermore, with many fewer gene-sets, the time to compute the analyses was reasonable, despite the larger number of genes per set. Result from both the MA.27 and NSABP studies are presented in Table 5. No KEGG sets were near statistical significance, and the top-ranking sets do not suggest reasonable biological hypotheses for these sets.

Table 5
Results from KEGG gene-set analyses.

The results presented above for both MA.27 and NSABP were not adjusted for population stratification by regression-based eigenvector adjustment. These adjusted analyses were run, but they did not differ substantially from those presented, so conclusions would not change if eigenvectors were used for adjustment.

Evaluation of Type-I Error Rate

To evaluate the Type-I error rate for the GO scanning method, we used the NSABP data for bootstrap sampling. For each of 1,000 simulations, we sampled with replacement 1,766 random subjects. We then randomly assigned case-control status (594 cases and 1,172 controls, as for the observed data) and computed the scores for all observed SNPs. The SNP scores were analyzed by gene-sets for the GO biological process domain. The Type-I error rate for the smallest p-value was 0.042, which illustrates that the global FWER is well controlled at the nominal level of 0.05. The type-I error rates for the next 4 smallest p-values were 0.016, 0.010, 0.009, 0.009. These monotonically decreasing error rates result from the way the step-down p-values were created.


Our software, called gene_set_scan, can perform gene-level analyses, gene-set analyses determined by the GO structure, or gene-set analyses determined by KEGG pathways. Although a number of input files are required, only one file must be created by a user. This file contains score statistics for each SNP and each subject. The other supplied files describe how SNPs map to genes, and how genes map to gene-sets (determined by GO data or KEGG data). Types of analyses are controlled by input options. The output from gene_set_scan is structured as HTML files, which provide statistical results with links to web-based annotation (e.g., NCBI GENE information, Amigo for GO terms, or Genome in Japan for KEGG terms). The output is also provided as comma-delimited, for input to other software (e.g., R statistical software for plotting or additional analyses).

The gene_set_scan software is written in C++, and uses the BOOST graphical library for computations on the GO DAG structure. All computations were accomplished on a Beowulf-style Linux cluster. This cluster has a master node that is a dual Athlon 2800+ MP with 2 GB memory. The 20 compute nodes are a mixture of Athlon 2400+ XP CPU with 1 GB memory (PC133 or DDR). The approximate times to compute the analyses are given in Table 6. A key factor that determines the time for computations is the number of genes in a set, due to the calculation of covariance terms among gene-scores within the same set. The GO analyses were restricted to no more than 30 genes in a set. The longer compute times for KEGG analyses were because of the large sizes of some gene-sets, with an average set size of 70 genes. The GO analyses would take longer if the maximum set size were not restricted to 30 genes per set. For examples, when the maximum set size was allowed to be as large as 1,000, the compute times were 26h:36m for NSABP and 14h:50m for MA.27, still quite achievable.

Table 6
Time to Compute Gene and Gene-set Analyses*


We developed a general approach to scan GWAS SNP data for gene-set analyses, primarily based on score statistics for individual SNPs. Although our presentation was for a binary trait modeled with logistic regression, our approach can be used for any type of trait that can be modeled by a generalized linear model. This strategy allows adjustment for covariates, and is computationally efficient. It also capitalizes on the well known statistical advantages of score statistics, such as most power for “local” alternative hypotheses — relatively small SNP effects on a trait. When creating gene-scores, we averaged the square of SNP-level z-scores, so that positive and negative effects of SNPs on a trait would not cancel. This allowed us to then combine gene-level scores into a gene-set score, using known properties of quadratic forms, allowing us to directly account for the covariances among gene-scores within a gene-set. A further computational advantage of our method is use of Lin's approach to compute p-values, based on asymptotic multivariate normal distribution of score statistics, and step-down adjusted p-values.

Some advantages of our proposed quadratic statistics to score genes can be gleaned from both theoretical and simulation studies. By moving from the assumption that the vector β of regression coefficients for p SNPs in a gene is a fixed effect to the assumption of a random effect, as in a mixed model, the null hypothesis Ho : β = 0 can be formulated as testing a variance component, Ho : ν = 0, where the covariance matrix of β is νK, with K a known matrix. Goeman et al. 42 show that the quadratic statistic is the score statistic for testing Ho : ν = 0, when K is the identity matrix. This quadratic statistic has optimal power for alternative hypotheses that are “local” to the null hypothesis. They further showed that the quadratic statistic has greater power than an F-test for linear regression of a trait on all covariates (e.g., all SNPs in a gene) whenever there are a limited number of latent variables, such that the latent variables are captured by a few of the large variance principal components of the SNP correlation matrix. This would occur, for example, when there are multiple tag-SNPs for a gene, but much fewer unmeasured causal variants. One could apply principal components to the SNPs in a gene to choose a few principal components to be used in a regression model F-test, as many have suggested 43-45. But, as Goeman et al. emphasize, this is not necessary when using the locally most powerful quadratic statistic that already has greater power than the F-test. Another surprising result by Goeman et al. is that the quadratic statistic has greater power than using the most extreme statistic (e.g., smallest p-value) even when there is only a single non-zero regression coefficient. Once again, these conclusions hinge on the correlation of the tag-SNPs with the latent casual variants and the causal variants having small effects on the trait. As we emphasized in the development of our methods, and Appendix B, this mixed model formulation provides a powerful framework to develop other types of gene-scores. This is based on the connection between generalized linear mixed models and kernel-machine learning 30, primarily driven by choice of the kernel matrix.

A number of simulation-based studies compared different multi-marker statistics to detect associations between a gene and disease 46-49. These include Fisher's method for combining P-values, the minimum P-value of all SNPs in a gene, a Fourier-transform-based approach 50, regression on principal components, and the quadratic score statistic of Goeman et al. 42. The power advantage of the competing methods depends on the underlying genetic mechanisms, but the general conclusions were that the variance components score statistic and principal components methods provided similar power, often greater than competing methods. The greatest power occurred when there were multiple tag-SNPs associated with disease, and when the minor allele frequencies of the tag-SNPs were not too small (eg., MAF > 10%).

There is some similarity in our gene-scoring approach with that proposed by Luo et al, 51, who also used linear combinations of SNP scores to score a gene, and used the SNP correlation matrix to account for their correlations. A critical distinction, however, is that we used a linear combination of the squared statistics, so that positive and negative associations would not cancel. Luo et al. were vague about whether their methods of converting p-values to normal distribution z-scores assumed 1-sided or 2-sided p-values. It is clear that their approach works for signed statistics (1-sided), but it is not valid for when using absolute values of the statistics (2-sided), because in this case, the moments of a truncated multivariate normal would be required 32. Our gene-scoring approach is also similar in spirit to that implemented in the VEGAS software, except that we use the GWAS data to estimate the correlation of SNPs, in contrast to VEGAS that uses a reference population to estimate the correlation structure, and then uses this correlation matrix to simulate from a multivariate normal distribution to determine the null distribution of gene-scores 52.

Although our approach requires genotype data used for GWAS analyses, it overcomes the size-bias problem when using the smallest p-value among all SNPs in a gene. Many reports on gene-set analyses recognize the potential for gene size and SNP correlation structure to bias the scoring of genes by extreme p-values, yet few have found adequate solutions. A clever approach proposed by Segre et al. 53 is to convert the extreme p-value to a z-score, and then attempt to adjust for gene characteristics correlated with the z-score. Using linear regression, they found the following gene characteristics to be associated with extreme p-values: gene size (in kilobases, kb), number of SNPs per kb, number of independent SNPs per kb, number of recombinant hot-spots per kb, and linkage disequilibrium units per kb. These characteristics, however, were based on reference data, not the GWAS data that generated the p-values. All of these factors are likely correlated among themselves, and are merely indirect reflections of the underlying correlation structure of the SNPs in a gene. In contrast, our approach uses the observed correlation structure of the GWAS data to directly account for the SNP correlation structure, both within and between genes.

We defined recursive gene-sets that take advantage of the DAG structure of GO. This can be advantageous if in fact the associations of genes with a trait are well represented by the DAG structure. A strength of GO is the large amount of information, with over 16 million annotations. Yet, a limitation is that over 95% of the annotations are computationally derived, not experimentally validated 54. This is achieved by taking experimentally derived biological knowledge from a limited number of model organisms to infer knowledge about similar gene products in other organisms using a phylogenetic framework 55. As of November 1, 2010, the GO web pages show that 50% of the 198,207 annotations for humans have been electronically inferred. Hence, errors are likely to exist about how genes map to the GO, and the DAG structure of the GO. For the GWAS of musculoskeletal adverse events, the strongest single-SNP test was near the TCL1A gene. Further functional studies in cell lines revealed a totally new and unexpected function for TCL1A— estrogen dependent regulation of cytokine expression. This illustrates the limitations of databases if our prior knowledge is incomplete.

One way to begin modeling incomplete prior knowledge is by using ideas from fuzzy-set theory 56, whereby errors and ambiguity could be addressed in the analyses by weighting GO terms according to the accuracy of their information or according to the probability of function (either at the gene level or the GO term level). Alternatively, the DAG structure could be enhanced by using edge weights, to differentially weight relationships when creating gene-sets. However, at this point in time, it is difficult to develop a biologically sensible weighting scheme. Perhaps clever use of gene expression and proteomic experiments to model the DAG would help, but this is beyond the scope of this current work.

As discussed by Holmans 22, there are multiple sources for pathway information to create gene-sets, and unresolved issues relate to which pathways to use (quality vs. quantity of information). One approach is to try different pathway resources, but it can be difficult to decide which gives the most reasonable results in light of the large number of genes and gene-sets. Another might be to use multiple sources of information to create larger numbers of gene-sets, recognizing that genes are likely to overlap for the more well-established sets. Our approach would allow for these types of complex sets, accounting for the within-set correlations of genes by our analytic derivations, while allowing for between-set correlations in the way that we compute p-values (i.e., simulation-based p-values allow for between-set correlations that are not conservative like Bonferroni-adjusted p-values).

Application of our methods to two different GWAS provided guidance on the potential strengths and weaknesses of our gene-set analyses. Although no gene-set results were close to statistical significance, hints of genes and gene-sets worthy to evaluate in functional studies were suggested. Whether there are true effects of gene-sets in either of these two data sets is unknown, and so simulations would be useful to evaluate the power of our approach. However, empirical power studies would be best when comparing the relative power of a number of competing analysis strategies across a variety of assumed genetic effects. This is beyond the scope of this initial work, but we outline our future directions. First, we described ways to score genes and score gene-sets, based on averaging, centering about the null mean, and then scaling by the null standard error. If we replace averaging with a general function, but still center about the null mean of the general function, and scale by the standard error of the general function, we can extend our methods yet still account for size biases. We no longer have multivariate normal theory to guide us on how to compute the null mean and standard error, so we would need to use simulations to compute them. This approach could allow a general function to incorporate thresholding of SNPs based on their SNP-level z-scores, effectively filtering out a large number of random noise variables. Whether this approach is more powerful than the averaging approach we described in this paper will require simulations. In fact, a wide range of general scoring functions could be considered, requiring a wide range of generic scenarios to evaluate which scoring functions are best for different genetic mechanisms.

Besides considering different functions to score genes and gene-sets, an alternative strategy is to reduce the number of statistical tests to improve power. One strategy, similar to Goeman's focus-level, is to scan all sets defined by the GO DAG to find the most extreme statistic. If that set is statistically significant, then look at the statistics of its neighbors (parents and children). If any of those are significant, then look at their neighbors, and so forth. This approach reduces the multiple testing burden, but at the price of reduced power if gene sets unrelated to each other contain causal genes. A second strategy would be to begin scanning only the gene-sets defined by the leaves of the GO DAG; if any are significant, then scan their parents, and so forth. This bottom-up approach might be more powerful when the causal genes map to the most specific GO terms, otherwise it could miss gene-sets that are at higher levels in the DAG.

Finally, our applications illustrated that when a single gene with strong associations with a trait is grouped in a set with many other null genes, the signal for the strong gene can be lost. This is a consequence of using score statistics, all computed under the null hypothesis of no associations. An alternative approach would be use of different kernels to score genes (see Appendix B). Another viable strategy is to use penalized models to select the strongest genes and gene-sets. We are actively developing such models for the DAG structure of the GO, and other gene-set structures, using lasso-type penalized generalized linear models. These penalized models will be compared with our score-statistic methods presented in this paper, in terms of their practical utility in large GWAS data, and their relative power evaluated with simulations.


This research was supported by the U.S. Public Health Service, National Institutes of Health, contract grant number GM065450.


This research was supported by the U.S. Public Health Service, National Institutes of Health (NIH), contract grant number GM065450 (DJS, JPS). The NSABP trials were supported by NIH contract grant numbers U10-CA-37377 and U10-CA-69974. The MA.27 trail was supported by NIH contract grant numbers U01GM61388, U01GM63173, P50CA116201, U10CA77202, and the National Cancer Institute of Canada contract grant number CCS 015469.The administrative coordination and data analyses were supported by NIH contract number U19 GM6138 (Mayo PGRN). The genotyping was supported the Biobank Japan Project funded by the Ministry of Education, Culture, Sports, Science and Technology, Japan. Genotyping was performed at the RIKEN Center for Genomic Medicine, Japan.

Appendix A. Mapping SNPs, Genes, and Gene-Sets

To map SNPs to genes, and genes to the Gene Ontology (GO), we used publically available maps for both SNPs and known genes, and the GO data files. These files and steps are described below.

Converting SNP maps from NCBI human genome build 36.3 to build 37.1

Because the HapMap SNP maps are based on NCBI human genome build 36.3, but the gene maps are based on build 37.1, we needed to first convert the HapMap SNP map to build 37.1. For build 36.3, there are 4,098,136 SNPs in HapMap, which includes most, if not all, SNPs on panels that are commercially available. To convert HapMap from build 36.3 to build 37.1 positions, we wrote a perl script,, that updates the positions in the PLINK formatted SNP map file using SNP positions for build 37.1 from the NCBI site. The files needed for this step are publicly available at these sites:

PLINK file with build 36.3 positions

hapmap_r23a.bim is within

NCBI file with build 37.1 positions

Of the HapMap SNPs, a total of 11,949 SNPs were dropped when converting from build 36.3 to build 37.1, leaving 4,086,187 SNPs. The counts and reasons for the excluded SNPs are given in Table A1.

Table A1
SNPs excluded when converting from build 36.3 to build 37.1

Mapping SNPs to Genes

To map the HapMap SNPs to genes, we used the updated HapMap genome build 37.1 map file described above, and the “Seq_Gene” file from NCBI for gene start/stop positions, available at the site Our Perl script,, creates a text file with columns Gene_ID, rs_ID, and position, where position is the position of the SNP relative to the gene. Position has the value zero if the SNP is within the gene's start/stop position. Otherwise, position is an integer for the number of base pairs outside of the gene, with a negative value if the SNP occurs before the gene start position, and a positive value if the SNP occurs after the gene stop position. Our default base pair maximum absolute distance for SNPs outside a gene is 50,000 bp. Our software, however, allows the user to choose shorter distances for analyses. If distances larger than 50,000 bp are required, one would need to set the maximum distance in the script, and rerun it to create a new file.

To process the “Seq_Gene” file, we used the following steps.

  1. Only match SNPs to entries in Seq_Gene labeled as “GENE” in the “Primary Assembly” group.
  2. Some genes (e.g. GeneID:100126533) are on multiple chromosomes. Our Perl script includes all of them by creating multiple lines in the output file, one for each chromosome. This means that a gene can be paired with different SNPs that occur on different chromosomes.
  3. The “X” chromosome in Seq_Gene file is labeled “23” in the SNP map file, and chromosome “Y” in seq_gene is labeled “24” in the SNP map file; our script accounts for this.
  4. Some genes are not localized to a particular position on a chromosome, indicated with a different format in Seq_Gene — our script excludes these genes.

Mapping Genes to GO Terms

To map genes to terms in the GO, we use the file gene2go, available from the NCBI ftp site: The gene2go file contains human and other organisms, so we subset to humans, based on taxonomy identifier for humans, “9606”.

Creating and Processing the GO DAG Structure File

To create the directed acyclic graph (DAG) structure to represent the GO terms (vertices) and their connecting edges, we use the GO OBO format file available from

We wrote a Perl script,, that parses the OBO file to create a comma-separated file that contains the three GO namespaces (Biological Process, Molecular Function, and Cellular Component), a child-term, an edge relationship, and a parent-term, where the parent-term is more general than the child-term. The output file format is namespace, child-term, relationship, parent-term.

The edge relationship is one of the following: “is_a”, “part_of”, “has_part”, “regulates”, “negatively_regulates” or “positively_regulates”. This file is read into our software and processed as follows. The edge relationships point from child to parent, except for the “has_part” relationship, which points from parent to child, so the child and parent are swapped for this type of relationship. We used the Boost Graph Library, BGL, 57 to create a DAG from an input file. For this, direction goes from left vertex to right vertex. So, when reading the above file format, the child-term is read as left vertex and parent-term as right vertex. This would make the direction to go from child to parent. However, we prefer to have directions to go from parent to child, so our software puts the parent-term as the left vertex and the child-term as the right vertex. By reversing direction, we consider the most general GO terms as the “roots” and the most specific GO terms as the “leaves”. This allows us to use the depth-first-search algorithm of the BGL to traverse the DAG structure, flagging leaf vertices when first visited, and then recursively flagging parent vertices after their child vertices are flagged. This flagging operation allows us to determine the order to visit vertices of the DAG so that we can then create gene-sets in a recursive manner. That is, a recursive gene-set for a specific GO term is the set of genes that map to the specific GO term, and all the genes that map to GO terms that descend from the specific GO term. At the highest level, say a single DAG root, the gene-set would be all genes that map to the entire GO structure. At the lowest level, say a leaf, the gene-set would be only those genes that map to the leaf.

At this point, we have not included weights for the edge relationships. Including weights would be simple, by adding weights to the input records, and making use of the edge weights in the BGL algorithms. However, at this point it is not clear how best to specify weights, so this feature will be a topic of future research.

Merging Observed SNP Scores with Above Mappings

For a SNP to be used in analyses, it must exist in the SNP-score file, and also exist in the gene-SNP map file described above. For a gene to be used in analyses, it must exist in the gene-SNP file, and also exist in the GO-gene mapping file. Hence, for a SNP to be included in a gene-set analysis, the SNP must exist in the SNP-score file, it must map to a gene, and the gene it maps to must map to the GO structure.

Appendix B. Scoring Genes by Kernel Matrices

Based on work by Lin 58, Wu et al. 30 derived powerful variance component score tests for binary traits, called logistic kernel-machine tests, that have the quadratic form Q = (y – [p with hat]o)′K(y – [p with hat]o), where y is a vector of length n for binary indicators of disease status, [p with hat]o, also a vector of length n, contains the logistic regression fitted values based on an intercept, and possibly adjusting covariates, and the n x n symmetric matrix K is a positive semidefinite kernel matrix. If the n x m matrix X contains the SNP allele dosage scores, then the linear kernel is K = XX′. Substituting this linear kernel into the expression for Q, reduces to Q = U′U, where U is a vector containing the U-scores. If the U-scores are standardized by their standard errors, Q = Z'Z, which is the quadratic form we use to score genes in the main text. Wu et al. evaluated the power of this linear kernel, along with a radial-basis Gaussian kernel and an identity-by-state (IBS) kernel. Although the linear kernel had good power, they found that the IBS kernel had greater power when there was interaction among SNPs, on the logit scale. This suggests that a variety of kernels might be considered when scoring genes. Here, we illustrate the form of a general kernel gene-score, following results in the Appendix A of Wu et al. 30.

Suppose that kernel matrix Ki is used for gene i to compute the gene score Qi = (y – [p with hat]o)′Ki(y – [p with hat]o). For example, if IBS were used, then Ki would be an n x n matrix of IBS scores based on SNPs in the ith gene. Then, the null expected value of Qi is E[Qi] = tr(KiVo), where Vo is the null covariance matrix for the residual vector (y[p with hat]o), which can be estimated by Vo = DoDoA(A′DoA)–1A′Do. The matrix A is the design matrix for the intercept and any adjusting covariates, and Do is a diagonal matrix with binomial variance terms [p with hat]oi(1 – [p with hat]oi). Furthermore, the covariance of Qi and Qj is Cov(Qi,Qi) = 2tr(KiVoKjVo). Knowing these moments, we can follow our procedure of computing weighted averages of gene-scores to compute a set-score, and then create a normalized set-score statistic by subtracting the null value of the expected set-score and dividing by the standard error of the set-score. This illustrates that different kernels could be used for different genes, although it might be most sensible to use consistent kernel definitions, such as linear kernels or IBS kernels across all genes. Note that if the design matrix A has only an intercept, then [p with hat]oi is constant over all subjects (the fraction of cases in the sample), and equation M33, where I is the identity matrix and J is a matrix of all 1's.


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