A method to incorporate prior information into score test for genetic association studies
© Zakharov et al.; licensee BioMed Central Ltd. 2014
Received: 3 July 2012
Accepted: 17 January 2014
Published: 22 January 2014
The interest of the scientific community in investigating the impact of rare variants on complex traits has stimulated the development of novel statistical methodologies for association studies. The fact that many of the recently proposed methods for association studies suffer from low power to identify a genetic association motivates the incorporation of prior knowledge into statistical tests.
In this article we propose a methodology to incorporate prior information into the region-based score test. Within our framework prior information is used to partition variants within a region into several groups, following which asymptotically independent group statistics are constructed and then combined into a global test statistic. Under the null hypothesis the distribution of our test statistic has lower degrees of freedom compared with those of the region-based score statistic. Theoretical power comparison, population genetics simulations and results from analysis of the GAW17 sequencing data set suggest that under some scenarios our method may perform as well as or outperform the score test and other competing methods.
An approach which uses prior information to improve the power of the region-based score test is proposed. Theoretical power comparison, population genetics simulations and the results of GAW17 data analysis showed that for some scenarios power of our method is on the level with or higher than those of the score test and other methods.
KeywordsPrior information Association analysis Score test Rare variants
In spite of the success of genome-wide association studies (GWAS) in identifying hundreds of common single nucleotide polymorphisms (SNPs) associated with diseases and complex traits (http://www.genome.gov/gwastudies/), in many cases the proportion of heritability explained by these discovered SNPs is low [1–3]. One of the potential explanations of this observation is that rare variants (usually defined as those with minor allele frequency below 1%), which are absent from conventional GWAS studies, are responsible for the missing heritability. Indeed, there is strong evidence that rare variants are associated with some complex traits [4–8].
When performing rare variants association analysis researchers face significant methodological challenges. The single SNP approach, which is popular in GWA studies, is underpowered when applied in rare variants association studies with moderate sample size due to low allele count for each individual rare variant. To overcome this problem, testing multiple rare variants within a region has been recommended [9, 10], and a number of region-based rare variants tests have been proposed [11–15]. However, these novel methodologies may still be underpowered for association studies with moderate sample size . This motivates the development of statistical methods that utilize prior information with the purpose of improving power. Currently, much biological information is publicly available, such as the prediction of degree of deleteriousness of non-synonymous variants (PolyPhen http://genetics.bwh.harvard.edu/pph2/, SIFT http://sift.bii.a-star.edu.sg/), SNP prioritization (FastSNP http://fastsnp.ibms.sinica.edu.tw), functional SNP annotation (SNPnexus http://www.snp-nexus.org/) etc. Although several methods that use prior information have been proposed [17–20], further research is needed to utilize prior knowledge more efficiently  and to expand statistical tools available for researchers.
In this article we propose a method that incorporates prior information into the region-based score test. Within our framework, prior information is used to partition SNPs within a genomic region of interest into groups. Then within each group asymptotically independent SNP scores are combined into a one degree of freedom (d.f.) chi-squared statistics. These group statistics are then used to construct a global test for a region. The proposed methodology has several distinct advantages. First of all, the degrees of freedom of our test statistic equals to the number of groups, which may be much less than the number of SNPs within a region. Secondly, under partitioning that approximately separates associated variants from neutral ones (“informative” partitioning) the proposed method efficiently handles noise introduced by neutral variants. We have evaluated the performance of our method on theoretical power comparison, population genetics simulations and analysis of the GAW17 (http://www.gaworkshop.org/gaw17/) real sequencing data set. The results showed that under some scenarios the proposed methodology performed as well as or outperformed the score test and other competing methods.
Theoretical power comparison
To investigate the impact of the number of groups K on power the following model was considered (Panel 3 of Figure 1): L=50, L1=5, and K=8,16,24,32. As can be seen, the power difference is below zero for large values of K and above zero for small values of K. This may be explained by the fact that the test statistic T1 (4) is distributed as chi-squared random variable with K d.f. Thus, for the fixed NCP r and increasing value of K the power of the proposed method monotonically decreases.
Alongside the models considered above, it is important to include a scenario when causal variants are split between several groups. Panel 4 of Figure 1 depicts the power difference under the following model: L = 54, K = 6, and the number of causal groups m = 2,3,4,5. Also, for simplicity of presentation it was assumed that each group contained equal number of variants (54/6 = 9), and NCP r was split equally between all causal groups. For this scenario the calculation of theoretical distribution of T1 (4) was done using 500,000 simulations of test statistics under the alternative hypothesis, as equation (6) cannot be used here. As can be seen from Panel 4 of Figure 1, our methodology could gain power if associated SNPs were split between several groups. When the number of causal groups became large, our method was slightly worse than the score test. It should be noted that for all models the maximum power gain of our method (and equivalently the maximum power loss) was achieved when the power of the score test was around 50%. Conclusions similar to those above can be derived from Additional file 1 which shows the same scenarios for the fixed type-1 error rate set at the genome-wide level assuming 35,000 genes (α = 0.05/35000).
Population genetics simulations results
Using population genetics simulations we compared our method with the score test and other proposed approaches, namely, weighted selective collapsing strategy (WSCS) , variable threshold (VT) , weighted sum of squared scores test (SSUw) , and optimal sequence kernel association test (SKAT-O) with linear kernel and beta weights . Since these tests utilize only genotype and phenotype information, we applied our method with MAF (minor allele frequency) partitioning: variants within a region are divided into two groups, namely, those with observed MAF above and below 1%. To estimate the empirical type-1 error we ran the analysis of simulated data under the null model of no association (Additional file 2). As can be seen from Additional file 2, the type-1 error was well controlled for all the tests.
GAW17 analysis results
In addition to theoretical power comparison and population genetics simulations we also used the GAW17 data set to compare our test with other methods. The GAW17 data set was designed to mimic a real exome sequencing study of a complex disease. Full description of the method used to generate the GAW17 data can be found in Almasy et al. . Briefly, the whole-exome sequencing data from 1000 Genomes Project (http://www.1000genomes.org) was the basis for simulation of 200 replicates of dichotomous phenotype and 200 replicates of 3 quantitative traits (endophenotypes) in 697 unrelated individuals from six populations (Chinese, Japanese, Yoruba, Luhya, Utah residents with Northen and Western European ancestry – CEPH, and Tuscan). Causal variants were chosen to be common and rare non-synonymous SNPs concentrated in genes from specific pathways. Also, dichotomous phenotype and quantitative traits were impacted by covariates such as smoking status, gender and age.
where q1,q2 and d are regression residuals. These residuals were tested for an association with adjusted genotype . Statistical significance for each method was assessed using 1000 permutations. The power was calculated as a proportion of phenotype replicates significant at the type-1 error rate of 0.05.
GAW17 analysis results: comparison with the score test
We considered three partitionings for our method: MAF partitioning (two groups: variants with MAF above and below 1%), functional partitioning (two groups: non-synonymous variants; synonymous and unknown variants), and combined partitioning (four groups defined by MAF and functionality). To assess the empirical type-1 error rate of our method and those of the score test we ran the analysis with randomly permuted residuals q1,q2 and d from (1). Additional file 3 shows the empirical type-1 error rates for Q1 and Q2 traits (Panel 1) and dichotomous phenotype (Panel 2). It should be noted that the estimate of type-1 error rate is distributed as an observed probability of success for a binomial random variable with the sample size of 200 (number of phenotype replicates) and the probability of success 0.05 (theoretical type-1 error rate). The double-sided 99% confidence interval for the estimate of type-1 error rate is 0.015–0.095. As can be seen from Additional file 3, the type-1 error was well controlled.
Suggested reasons for the difference in power between the score test and our approach for some genes
Suggested reason for the difference in performance
Genes for which our method outperformed the score test
MAF and functionality
All common non-synonymous SNPs are causal.
Association of the three common non-causal SNPs with Q1*.
The only common SNP is causal.
The only common SNP is causal.
MAF and functionality
All common non-synonymous SNPs are causal.
MAF and functionality
All common non-synonymous SNPs are causal.
Genes for which the score test outperformed our method
KDR, BCHE, SOS2
No clear separation of associated variants from neutral ones using any of the considered partitionings.
GAW17 analysis results: comparison with other tests
We contrasted the performance of our method (MAF partitioning) with those of WSCS , VT , SSUw  and SKAT-O with linear kernel and beta weights . Also, we included some of the published results obtained by the participants of the GAW17 workshop: the performance of Multivariate Distance Matrix Regression (MDMR) and Mantel tests on Q2 causal genes , the performance of aggregated U-test (AggregateU) and CMC method (QuTie) on Q1 causal genes . It should be noted that the adjustment for covariates and population stratification for SKAT-O test was done via the function “SKAT_Null_Model” of the R (http://cran.r-project.org) package “SKAT”. Also, in our implementation of the WSCS method rare variant weights were proportional to the absolute value of correlation between those rare variants and phenotype, since the original weights described by Dai et al.  were not applicable due to the adjusted phenotype being quantitative. Additional file 4 shows the empirical type-1 error estimates for WSCS, VT, SSUw and SKAT-O tests. Given that the 99% confidence interval for the empirical estimate of type-1 error rate is 0.015–0.095 as described previously, there was no evidence for type-1 error inflation.
Panel 3 of Figure 3 depicts the analysis results for Q1 causal genes. As can be seen, our method was among the top three most powerful approaches for ANRT, ELAVL4 and HIF1A genes. Panel 1 of Additional file 5 shows the results for Q2 causal genes. Our method was the top performer for VNN3 gene, and was among the top three performing approaches for LPL and VNN1 genes. Panel 2 of Additional file 5 depicts the results of an association analysis of all causal genes with dichotomous phenotype. As can be seen, SKAT-O achieved the most notable power gains for PIK3C3 and PTK2B genes. This emphasizes that SKAT-O may significantly outperform other tests under some phenotype models. However, this method also may significantly underperform other tests for some phenotype models, for example, ELAVL4 gene with Q1 trait and VNN1 gene with Q2 trait. Our proposed method was among the top three performing methods for SOS2, PTK2B, PRKCA genes and some other genes for which the power of all the methods was low.
In this article we have described a methodology that incorporates prior information into a region-based score test with the purpose of improving power to identify an association. Prior information, such as observed MAF, functional annotation, predicted deleteriousness of non-synonymous variants, may be used to partition variants within a region of interest. Then, this partitioning is utilized to construct a test statistic whose distribution under the null hypothesis has lower degrees of freedom compared with those of the score test. Based on the theoretical power comparison, population genetics simulations and GAW17 sequencing data analysis, we have shown that under some scenarios our method may perform as well as or outperform other methods.
Our suggested partitioning is splitting by functionality and if rare variants are present – by MAF with an arbitrary threshold, e.g., 1%. One of the major justifications for considering MAF in a partitioning design is that MAF may distinguish between different evolutionary forces acting on causal variants. Evolutionary theory predicts that variants that confer susceptibility to a disease which in turn reduces fitness are expected to have low MAF due to purifying selection. Empirically, using whole-exome sequencing data, it was found that non-synonymous substitutions are more significantly skewed towards low frequencies compared with synonymous variants. This finding “almost certainly reflects the operation of purifying selection”  acting on many genes across genome. Thus, if a causal gene is under strong purifying selection incorporating MAF into partitioning design is likely to be beneficial. On the other hand, a causal variant could have risen to higher frequency due to other forces such as balancing selection, mutation-selection balance, antagonistic pleiotropy, etc. . Thus, for example, if a causal variant has been under a strong balancing selection it is likely to be common; and incorporating MAF into partitioning design may lead to power improvement. Partitioning by functionality may be beneficial for gene-based analysis in cases when an association signal comes from one or several functional groups of variants. For example, if highly deleterious non-synonymous variants within exons of a gene are causal, then partitioning by functionality is likely to improve power. Although the misclassification rate for prediction tools may be high (e.g., PolyPhen-2 achieved 92% power to detect truly damaging variants at 20% type-1 error rate over HumDiv data ), our method may still benefit from grouping variants by functional significance, since this is an effective way to give more emphasis on variants that are more likely to be causal . The benefit of using functional information was demonstrated in several simulations and candidate gene sequencing studies [18, 34, 35].
There are a few limitations of the proposed methodology. First, it is unknown in advance which prior information is relevant for a given genomic region. Prior information that does not help to uncover groups of associated variants is likely to have negative impact on statistical power. Second, the test statistic (3) is, in general, not invariant with respect to the ordering of SNPs. This can be seen, for example, when the covariance matrix V is invariant with respect to permutation of elements of the score vector U (e.g., all the cross-SNP covariances are the same, and diagonal variances are equal). So, for any SNP ordering the Cholesky decomposition A is the same; hence, the test statistic is, in general, dependent on ordering. To clarify the ambiguity the ordering of SNPs according to a position on a chromosome may be assumed.
A novel statistical approach that incorporates prior information in a region-based score test with the purpose of improving power has been proposed. Theoretical power comparison, population genetics simulations and the results of the GAW17 data set analysis showed that under some scenarios our method may perform as well as or outperform other methods.
where is the vector of scores, is L × L the score covariance matrix assumed to be non-singular, and is the L × L genotype covariance matrix. Under the null hypothesis (none of the variants is associated with phenotype) T0 asymptotically follows chi-squared distribution with rank(V) degrees of freedom (d.f.). Under the alternative hypothesis (at least one variant is associated with phenotype) the asymptotic distribution of (2) can be approximated by a non-central chi-squared distribution.
Following is the description of our proposed method. First, let us transform the coordinates of score vector U to be asymptotically independent under the null hypothesis. If we denote C = (A T )−1, where matrix A is a Cholesky decomposition of V (V = A T A), the vector S = CU, under the null hypothesis, is asymptotically distributed as a standard multivariate normal random vector (since Cov(CU) = CCov(U)C T = (A T )− 1VA− 1 = (A T )− 1(A T A)A− 1 = I). It should be noted that under the alternative hypothesis S is approximately distributed as a multivariate normal random vector with nonzero mean and unit covariance matrix under the assumption of V being a reasonable approximation for the covariance matrix of U. In general, any other decomposition of the covariance matrix V of the form V = A T A can be used. We apply Cholesky decomposition because it is fast to compute even for big covariance matrices.
Under the null hypothesis T1 is distributed as a chi-squared random variable with K d.f. Large value of T1 indicates the deviation from the null hypothesis. Transformation (3) is used to decrease the degrees of freedom of the final test, as some of the groups G k , k = 1, …, K may contain only neutral variants. If kth group does not contain associated SNPs, group G k contributes only 1 d.f. to the final test, whereas it adds L k d.f. to the score test. Thus, under “informative” partitioning our proposed method may gain power compared with the score test.
There are few notes that should be taken into account when applying our method. First, the theoretical approximation for distribution of t k may not hold under moderate sample sizes. Hence, in all our simulations and real data application we used the empirical CDF of t k , denoted as F k , obtained via 1000 permutations under the null hypothesis. Instead of in (3) we calculated a Gaussian kernel CDF estimate F k (t k ) with a multi-stage plug-in bandwidth of Polansky and Baker . R (http://cran.r-project.org) package ‘kerdiest’ contains all the necessary algorithms. Second, the covariance matrix V may happen to be computationally or exactly singular when, for example, considering a region with high LD between common SNPs. In general, to avoid singularity issues we apply Monroe-Penrose pseudoinverse in (2) instead of V−1. This pseudoinverse is uniquely defined and equals the standard inverse matrix when V is non-singular. From the same considerations instead of usual Cholesky factoring we used generalized Cholesky decomposition, which is well defined for square symmetric positive semi-definite matrices. If V is non-singular, the generalized Cholesky decomposition equals the usual Cholesky factoring. Also, it should be noted that in general our test statistic is not invariant with respect to the ordering of variants within a region. To clarify the ambiguity we assume the ordering of SNPs according to a position on a chromosome (see the “Discussion” section).
Theoretical power calculations
Under the alternative hypothesis non-centrality parameter of T0 and those of t1 are equal, since both statistics include all the associated SNPs within a region. Thus, the distribution of H1 is , and the integrand in the equation (7) can be calculated as . This allows us to compute the theoretical power of our method.
Population genetics simulations
Boyko et al.  found a simple two-epoch expansion model to be one of the best-fit models for population genetics simulations of African-American genotype. Also, the authors found strong evidence of selective effects acting on new amino acid replacing mutations and inferred the distribution of those selective effects. The scaled population selection coefficient (defined as a selection coefficient multiplied by twice the effective population size) was assumed to follow a negative gamma distribution with the following parameters: shape 0.184 and scale 8200. Mutation rate was set at 1.8 × 10−8 per nucleotide per generation. The selective disadvantage was additive. Using the forward simulator SFS_CODE (http://sfscode.sourceforge.net) with the parameters described above, we generated 100 haplotype pools for a 3 kb coding genomic region. Each haplotype pool contained 4000 haplotypes. In our simulations we considered only non-synonymous variants. To generate a data replicate we first selected a haplotype pool at random. Within that pool we sampled a pair of haplotypes at random and added those haplotypes to form a multi-site genotype of an “individual”. Dichotomous phenotype was assigned based on this multi-site genotype using a linear logistic model through which we controlled the odds ratios of causal variants. The probability of a disease conditional on the wild type genotype was set at 1% for all phenotype models.
We considered four phenotype scenarios. For the first model called “Rare” randomly chosen 50% of rare variants (defined as those with MAF<1%) within a haplotype pool were assigned to be causal with uniform odds ratio of 3. For the “Low Frequency” scenario one randomly chosen low frequency SNP (MAF between 1% and 5% in a haplotype pool) was causal with odds ratio of 2.5. If there was no SNP with MAF between 1% and 5% in a haplotype pool, we selected a SNP with the lowest MAF above 5%. For the “Common” scenario one common SNP with MAF>5% in a haplotype pool was causal and had the odds ratio of 1.5. Finally, the “Interaction” scenario models the hypothesized interaction of common and rare variants in RET gene associated with Hirschsprung’s disease [40, 41]. The simulation framework for this scenario was previously described by Liu and Leal . Briefly, 50% of rare variants were assigned to be causal. Each causal rare minor allele increased the odds of a disease by 6 times if and only if it was present on the same haplotype as a minor allele of a common SNP randomly chosen beforehand.
Sample sizes were the following: 500 cases and 500 controls for “Rare”, “Low frequency” and “Common” scenarios; 1000 cases and 1000 controls for “Interaction” model. In total 1000 data replicates were generated for each scenario. Power was estimated as a proportion of data replicates significant at a fixed type-1 error of 0.05. For all the statistical tests 1000 permutations were applied to estimate a significance level.
We would like to thank the GAW17 workshop organizers for their permission to use their data in our research. Preparation of the Genetic Analysis Workshop 17 Simulated Exome Data Set was supported by the GAW grant, R01 GM031575, and in part by NIH R01 MH059490. The workshop used sequencing data from the 1000 Genomes Project (http://www.1000genomes.org).
This work was supported by the Agency for Science, Technology and Research (A*STAR, Singapore). The first author is a recipient of the Singapore International Graduate Award.
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