Volume 3 Supplement 7
Genetic Analysis Workshop 16
Application of Bayesian classification with singular value decomposition method in genomewide association studies
 Soonil Kwon^{1},
 Jinrui Cui^{1},
 Shannon L Rhodes^{1},
 Donald Tsiang^{1},
 Jerome I Rotter^{1} and
 Xiuqing Guo^{1}Email author
DOI: 10.1186/175365613S7S9
© Kwon et al; licensee BioMed Central Ltd. 2009
Published: 15 December 2009
Abstract
To analyze multiple singlenucleotide polymorphisms simultaneously when the number of markers is much larger than the number of studied individuals, as is the situation we have in genomewide association studies (GWAS), we developed the iterative Bayesian variable selection method and successfully applied it to the simulated rheumatoid arthritis data provided by the Genetic Analysis Workshop 15 (GAW15). One drawback for applying our iterative Bayesian variable selection method is the relatively long running time required for evaluation of GWAS data. To improve computing speed, we recently developed a Bayesian classification with singular value decomposition (BCSVD) method. We have applied the BCSVD method here to the rheumatoid arthritis data distributed by GAW16 Problem 1 and demonstrated that the BCSVD method works well for analyzing GWAS data.
Background
Genomewide association studies (GWAS) evaluate genetic variants throughout the entire genome with the goal of identifying susceptibility genes for diseases or conditions of interest. While a large number (m) of singlenucleotide polymorphisms (SNPs) are usually evaluated in a GWAS, sample size (n) is often limited due to substantial costs of recruitment and phenotype measurements. The fact that m>>n makes it unrealistic to analyze all SNPs simultaneously using traditional statistical methods, such as multiple linear regression analysis. It is therefore common for the analyses to be conducted one SNP at a time in GWAS. This means that 300,000 to 1,000,000 tests will be carried out for each GWAS study against each phenotype of interest. Such a large number of tests lead inevitably to a considerable problem with false positive results. To address this multiple testing issue in GWAS, two potential solutions have been investigated in recent years: evaluate false positive rates, e.g., Benjamini and Hochberg's false discovery rate and the qvalue [1, 2], or develop novel statistical methods for analyzing datasets where m>>n [3, 4]. As one of the first proposed methods for analyzing multiple SNPs simultaneously when m>>n, we introduced the iterative Bayesian variable selection (IBVS) method [3]. Although sufficient to produce accurate and reliable results, the IBVS method has one barrier to efficient use: a relative long run time for GWAS data sets. To further improve the running speed, Kwon and Guo developed a Bayesian classification with singular value decomposition (BCSVD) method [4]. As a comparison, we applied the BCSVD method to the same subsamples of the simulated rheumatoid arthritis (RA) data provided by Genetic Analysis Workshop (GAW) 15 Problem 3 and successfully identified the common genetic variants associated with RA status. Using exactly the same computer and the same data, we found that the runtime for BCSVD is less than half compared to that required for the IBVS method [4]. We applied the BCSVD method here to the GWAS data for RA sample provided in GAW16 Problem 1.
Methods
The BCSVD method
where L = FD and . Expressed as a linear combination of the original parameters (β), we call γ a superfactor vector. The joint distribution of γ and z can be expressed as the product of the prior distribution of γ and the likelihood function of z given γ, i.e., p(γ, z) ∝ p(γ)p(zγ). The joint posterior distribution of γ and z given y can be written by multiplying p(γ, z) with the likelihood function of y given γ and z. By integrating out z and γ, respectively, from p(γ, zy), we can have the posterior distributions of z and γ, respectively. With these posterior distributions, we can fit the model using Markov chain Monte Carlo with Gibbs sampler. The 95% credibility interval was used to check for the convergence of sampler. To transform γ back to β, we used the most general solution form for the linear equation (γ = A'β) and achieved the unique solution for β by choosing the generalized inverse of A' as A [5]. The test statistic for association was generated by permutation. Let (i = 1, ⋯, p) be the estimate of i^{th} SNP effect from the raw data and be the estimate of i^{th} SNP effect from the j^{th} shuffled. Let us define as the difference between and . Then, under the null hypothesis (H_{0}: β_{ i }= 0), the statistic Λ_{ i }follows the standard normal distribution when k is large: , where is the sample mean of values, j = 1, ⋯, k, and se( ) is the standard error of . With the statistic Λ_{ i }(i = 1, ⋯, p), we provided the pvalue to reject the null hypothesis.
Association analysis
The evaluation of the BCSVD method for association analysis was performed in two steps. As the first step, we performed a genomewide single SNP association analysis using the logistic regression model option in PLINK. The PLINK analysis results served for two purposes, one was for the comparison with the results from BCSVD method, and the other was for the selection of genomic regions. Even though the BCSVD method can be applied to the whole genomewide association data, the requirement on computer memory is still a limiting factor. We therefore focused on chromosome regions selected through PLINK analysis results in our BCSVD analysis.
Study sample
We used the wholegenome association data of the North American Rheumatoid Arthritis Consortium (NARAC) in GAW16 Problem 1. There were 2,062 subjects in the study, including 868 cases and 1,194 controls. Quality control on genotype data was performed with PLINK software. We eliminated 133,616 SNPs that failed the following quality control criteria: pvalue < 10^{5} for HardyWeinberg equilibrium (HWE) test, minor allele frequency <1%, or missing data >10%. As a result, 411,464 SNPs were included in the PLINK association analysis.
Imputation
For BCSVD analysis, we first selected chromosomes that have SNPs with pvalue < 10^{7}. The best SNP on each chromosome that has the smallest pvalue was identified. SNPs within 2 Mb upstream and downstream of the best SNP were then selected. Software MACH 1.0 [6] and HapMap phase 3 genotype data for the HapMap CEU sample were used for genotype imputation. Imputed SNPs with a squared correlation with true genotypes (r^{2}) < 0.3 were excluded.
BCSVD study sample
Analyzing all selected SNPs simultaneously for 2,062 samples requires tremendous computer memory that our current computers cannot yet handle. We therefore generated two data sets based on the imputed data: one had 1,000 subject (500 cases and 500 controls) randomly selected from 868 cases and 1,194 controls; the other had 200 subjects (100 cases and 100 controls) randomly selected from the above selected 1,000 subjects.
Results
Step 1. Single SNP association from GWAS
Step 2. Evaluating multiple SNPs simultaneously with the BCSVD method
Conclusion
The BCSVD method was applied to RA casecontrol data from Problem 1 of GAW16 for 8 selected regions. When we evaluated the association between RA affection status and all SNPs in selected regions simultaneously using BCSVD, significant associations were detected for all the 8 chromosomal regions, and the highest peak was observed on chromosome 6, which were consistent with the PLINK results. Even though the magnitude of significance [log_{10}(pvalue)] appeared smaller than those from PLINK, we have to keep in mind that we used only datasets with 200 and 1,000 samples, respectively, in the BCSVD analysis, compared to the 2,062 samples in PLINK. More importantly, we have successfully avoided multiple testing issues because we performed only one test by evaluating all SNPs simultaneously. Similar results were observed in the datasets with 200 samples and 1,000 samples. We therefore conclude that the BCSVD method is a practical method for identifying genetic determinants in GWAS when sample size is much smaller than number of markers (m>>n). The BCSVD method has been implemented in our BAMGAS (Bayesian analysis methods for genetic association studies) program. While we are still working on a webbased userfriendly version, an executive version of the software is available from the authors.
List of abbreviations used
 BCSVD:

Bayesian classification with singular value decomposition
 GAW:

Genetic Analysis Workshop
 GWAS:

Genomewide association studies
 HWE:

HardyWeinberg equilibrium
 IBVS:

Iterative Bayesian variable selection
 NARAC:

North American Rheumatoid Arthritis Consortium
 RA:

Rheumatoid arthritis
 SNP:

Singlenucleotide polymorphism
 SVD:

singular value decomposition
Declarations
Acknowledgements
This study was supported partially by grants DK046763, GM008243, HL088457, and the CedarsSinai Board of Governors Chair in Medical Genetics.
This article has been published as part of BMC Proceedings Volume 3 Supplement 7, 2009: Genetic Analysis Workshop 16. The full contents of the supplement are available online at http://www.biomedcentral.com/17536561/3?issue=S7.
Authors’ Affiliations
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