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A Penalized Regression Approach for Integrative Analysis in Gen ome- Wide Association Studies | OMICS International
ISSN: 2155-6180
Journal of Biometrics & Biostatistics

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A Penalized Regression Approach for Integrative Analysis in Gen ome- Wide Association Studies

Liu J1*, Wang F2, Gao X3, Zhang H4, Wan X5 and Yang C6

1Centre of Quantitative Medicine, Duke-NUS Graduate Medical School, Singapore

2Department of Biostatistics and Epidemiology, University of Pennsylvania, USA

3Department of Ophthalmology and Visual Science, University of Illinois, Chicago, USA

4Department of Psychiatry, Yale University, USA

5Department of Computer Science, Hong Kong Baptist University, Hong Kong

6Department of Mathematics, Hong Kong Baptist University, Hong Kong

*Corresponding Author:
Liu J
Centre of Quantitative Medicine
Duke-NUS Graduate Medical School, Singapore
Tel: +65 6516 7666
E-mail: [email protected]

Received date: February 13, 2015; Accepted date: May 22, 2015; Published date: May 29, 2015

Citation: Liu J, Wang F, Gao X, Zhang H, Wan X, et al. (2015) A Penalized Regression Approach for Integrative Analysis in Genome-Wide Association Studies. J Biomet Biostat 6: 228. doi: 10.4172/2155-6180.1000228

Copyright: © 2015 Liu J, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are are credited.

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Over one thousand genome-wide association studies (GWAS) have been conducted in the past decade. Increasing biological evidence suggests the polygenic genetic architecture of complex traits: a complex trait is affected by many risk variants with small or moderate effects jointly. Meanwhile, recent progress in GWAS suggests that complex human traits may share common genetic bases, which is known as “pleiotropy”. To further improve statistical power of detecting risk genetic variants in GWAS, we propose a penalized regression method to analyze the GWAS dataset of primary interest by incorporating information from other related GWAS. The proposed method does not require the individual-level of genotype and phenotype data from other related GWAS, making it useful when only summary statistics are available. The key idea of the proposed approach is that related traits may share common genetic basis. Specifically, we propose a linear model for integrative analysis of multiple GWAS, in which risk genetic variants can be detected via identification of nonzero coefficients. Due to the pleiotropy effect, there exist genetic variants affecting multiple traits, which correspond to a consistent nonzero pattern of coefficients across multiple GWAS. To achieve this, we use a group Lasso penalty to identify this nonzero pattern in our model, and then develop an efficient algorithm based on the proximal gradient method. Simulation studies showed that the proposed approach had satisfactory performance. We applied the proposed method to analyze a body mass index (BMI) GWAS dataset from a European American (EA) population and achieved improvement over single GWAS analysis.


Integrative analysis of GWAS; Penalized methods; Scaled group Lasso


Genome-wide association studies (GWAS) provide an unprecedented opportunity for identifying disease-associated genetic variants. Although disease associated SNPs at genome-wide significance level (e.g. P-value<5 × 10-8) were identified for some diseases [1-4], those identified SNPs only explained a small fraction of genetic contributions to the etiology of the diseases. This phenomenon is referred to as “missing heritability”. Rather than only using genomewide significant SNPs, Yang et al. [5] showed that 45% of the variance for human height can be explained by using all of the genotyped common SNPs. This result suggests that most of the “missing heritability” is not missing but remains hidden in the genome: due to the limited sample size, many individual effects of genetic markers are too weak to pass the genome-wide significance level, and thus those risk genetic variants remain undiscovered. So far, people have found similar genetic architectures for many other complex diseases [4], such as psychiatric disorders [6,7], i.e., the phenotype is affected by many genetic variants with small or moderate effects, which are referred to as “polygenicity”. The polygenicity of complex diseases is further supported by recent GWAS with larger sample sizes, in which more associated common SNPs with moderate effects have been identified (e.g., GWAS data from 34,840 patients and 114,981 healthy people are analyzed to understand the genetic architecture of type 2 diabetes [8]). However, large sample recruitment may be expensive and time-consuming.

For single GWAS analysis, many existing statistical methods have been proposed [9,10]. Among them, penalized regression methods [11-14] have drawn particular attention in GWAS. However, due to limited sample size of a single GWAS and polygenicity of a complex trait, many existing methods do not have enough power to uncover the remaining risk genetic variants. Recently, increasing evidence suggests that complex traits may share common genetic bases, which is known as “pleiotropy” [15-17]. A systematic investigation of pleiotropy [18] suggests that 16.9% of genes and 4.6% of SNPs have been reported to show pleiotropic effects. Therefore, it is possible to further improve statistical power in GWAS data analysis by integrating multiple GWAS. The difficulties of integrative analysis of GWAS mainly come from two aspects. First, a direct pool of samples from multiple GWAS is questionable due to heterogeneity in different studies. Second, some existing methods (e.g. [19]) require the availability of all genotype data from multiple GWAS, which could be practically difficult due to the privacy restrictions on sharing individual-level data.

In this work, we aim at improving statistical power of identifying associated markers for the given GWAS data by integrating information from other GWAS, where only the summary statistics rather than the geneotype data of some GWAS are needed. We propose a penalized method for integrating multiple GWAS (pIGWAS). The key idea of the proposed approach is that genetically related traits can share common genetic bases [18,20], which enables us to borrow information from some related GWAS when analyzing the trait of primary interest. Specifically, we propose a novel loss function that combines multiple GWAS together and use a group-Lasso penalty to integrate information from different GWAS. We further derive a gradient-based algorithm to efficiently optimize the model parameters. Based on both simulation study and real data analysis, we showed that the proposed method had advantage over single-GWAS analysis.

Material and Method


Suppose we have q GWAS, in which we have complete data for GWAS 1, including its genotype data and phenotype data, and only have summary statistics (marginal regression coefficients) for the rest q-1 of GWAS. There are p SNPs shared by all q GWAS. Let image and image be the phenotype vector and genotype matrix of GWAS 1, respectively, where n1 is the sample size of GWAS 1. Let image be the vector of marginal regression coefficients from kth GWAS, k=2, . . . , q.

Consider a linear model between the phenotype y1 and genotype X1 of GWAS 1,

y1 =X1b1+e1,


Where image is the coefficient vector, image is the error term image denotes the noise variance. For the rest of GWAS with only summary statistics, we assume that image is an estimate of the true effect size image with noise image i.e.,



where image denotes the noise variance for the k-th GWAS, k=2, . . . , q. Here we use a simple example to illustrate the key idea. Suppose the vectors of the true effect sizes from three GWAS are given as follows:




The joint analysis of b1, b2 and b3 can improve the statistical power of identifying risk variants as the same loci consistently produce non-zero effect sizes among different studies. Therefore, we propose to consider all the effect sizes of the same variant as a group. For this toy example, we have imageas the vector of the true effect sizes of the first group, imagefor the second group, and Bj for the j-th group, j=1, .. . , p. For convenience, we denote imageas the effect size matrix and image is the j-th column.

To integrate information from multiple GWAS, we propose the following optimization problem


Where γ is the regularization parameter controlling the sparsity of B, ||.|| denote the l2-norm of a vector. The proposed optimization is closely related to the scaled Lasso problem [21]. Here we emphasize on integration of information from multiple GWAS and use the group penalty to achieve this goal.


Now we present our algorithm for parameter estimation in the above model. Noticing that objective function (1) is jointly convex in (B, σ1, . . . , σk), it is very convenient for us to use an alternating strategy in optimization. For fixed values of σk, k=1, . . . , q, we optimize (1) with respect to B. Then we update σk (k=1, . . . , q) using the current fitted B. The details of the algorithm are given below.

Fixing image the optimization problem becomes

image              (2)

Since imageis non-differentiable, we adopt the proximal gradient method [22].


image and image. The proximal gradient algorithm solves optimization problem (2) iteratively using the proximal operator of g(B):

image          (3)

where the superscript m indicates the mth iteration, τ is the Lipschitz constant of f (B) and


Is the gradient of f (B) evaluated at B(m-1). Note that imageis q × p matrix which does not involve the optimization variable B. Let its image denote the j-th column of image. Then optimization problem (3) can be rewritten into p separate optimization problems and be solved analytically:

image       (4)

To further accelerate the convergence of the above proximal gradient algorithm, we use the accelerated proximal gradient algorithm (APG) [22], where two points {B(m-1), B(m-2)} are employed to find the optimal solution for a fixed value of tuning parameter γ [23]. The detail of the APG algorithm is given in Algorithm 1.

Algorithm 1: Accelerated proximal gradient algorithm (APG)

Given a value of the tuning parameter γ, we first initialize Lipschitz constant imageand t1=1, where image is the maximum eigenvalue of matrix image

for m ≥ 1 do

1. image

2. image

3. image

4. image


Let image be the solution of Algorithm 1. Fixing B at image, we can take the derivative of (1) with respect to σk, k=1, . . . , q and set them to zeros, yielding the following updating equations:

image              (5)

image              (6)

Based on the above alternative optimization strategy, we now summarize the overall working algorithm in Algorithm 2.

Algorithm 2: Working Algorithm to Solve (1)

We first initialize image using the null model. for l ≥ 1 do

1. Using Algorithm 1 with image to optimize (2) that results image

2. With image we can update σk, k=1, . . . , q using (5).


For the tuning parameter γ, we searched for optimal settings using a five-fold cross validation to search the best γ in image , where E=0.05 in our experiment, and γmax is the minimum γ such that all the elements in B are estimated to be zero. A sequence of 100 γ values is generated equally in the log-space of image . The optimal γ is chosen according to the criteria that the minimum prediction error in primary GWAS is selected.

Simulation Study

We conducted a simulation study to evaluate the performance of the proposed method. For comparison, we also considered scaled Lasso on one GWAS with genotype data. We simulated two sets of genotype data, one for GWAS with genotype and one for summary statistics of GWAS. In the simulation study, we set n1=500 with n2=500 or n2=2000 for the sample sizes of two GWAS, while we set the number of SNPs to be p=5000 or p=10000. We considered the auto-regressive correlation (AR) and block AR. For AR, SNP j and k have correlation coefficient ρ|j-k|. For block AR, we set block size to be 20 equally distributed over all SNPs and the correlation coefficient for SNP j and k within a block is set to ρ|j-k| and 0 otherwise. We considered three scenarios with ρ=0.2, 0.5 and 0.8, corresponding to weak, moderate, and strong correlations, respectively. SNPs in the simulation study were generated with a two-stage procedure [11]. First, we drew the predictor vector xi from a p- dimensional multivariate normal distribution under different correlation structure. Then, the genotype of the ith SNP was set to be 0, 1, or 2 according to whether xij<-c,-c ≤ xij ≤ c, or xij>c. The cutoff point c was the first quartile of a standard normal distribution. In this simulation, we considered that the first ten SNPs were associated with the trait in both datasets and regression coefficients were generated under normal distribution in the way that signal-to-noise ratio was controlled at 1:1 (corresponding to heritability=50%). To be specific, we first generated genotype data using the way described above. Then we normalized the genotype such that image and image. Finally, we generated the quantitative trait using the linear model,

yk=Xk bk+ek, k=1, . . . , q,

where ek was the error term under normal distribution with mean zero. In both correlation structures, there were ten trait-associated markers and in block AR, each of the ten blocks contained one trait-associated marker. For the second dataset, we applied the single-marker analysis and obtained the summary statistics for the integration, in which we only used this partial information without knowledge of genotype data.

In total, there were twenty-four scenarios with different combinations of correlation structures (AR and block AR), the sample size of the second GWAS n2 and the total number of SNPs p for comparing the proposed method with the scaled Lasso. We used area under the curve (AUC) to show the selection performance. We also used the square of correlation coefficients (r2) of observed values and predictive values, based on cross-validation. The results are shown in Figure 1. As indicated by AUC and r2, the proposed method has better selection and prediction performance than scaled Lasso. The selection and prediction performance of the proposed method can further improve as the sample size of the second GWAS n2 increase from 500 to 2000, which indicates the proposed pIGWAS method is able to effectively integrate additional information.


Figure 1: Boxplots of areas under the curve (AUC) and r2 under different combinations of n2, p and correlation structure (AR, Block AR).

Real Data Analysis

We applied our pIGWAS method to the quantitative trait-body mass index (BMI). We primarily used European American (EA) samples from two GWAS-Study of Addiction: Genetics and Environment (SAGE) and the Collaborative Study on the Genetics of Alco- holism (COGA). The summary statistics of height were downloaded from the web- site of Genetic Investigation of Anthropometric Traits (GIANT) consortium ( php/GIANT_consortium_data_files) [24]. After quality control, there were 656,848 SNPs with minor allele frequency (MAF) ≥ 0.01 and pvalue ≥ 0.001 for Hardy-Weinberg equilibrium test in both GWAS data. For summary statistics from GIANT, we used SNPs with no missing values in MAF. Overall, there were 619,651 SNPs used in all chromosomes satisfying the pruning criteria in genotype data and existing non-missing values of MAF in summary statistics. The Manhattan plots for log10 p-value and image using conventional marginal analysis for GWAS data are given in Figure 2. Obviously, there is not a rich set of findings using SAGE and COGA EA samples for phenotype BMI (the upper panel of Figure 2) compared to the meta-analysis conducted in [25] (the lower panel of Figure 2).


Figure 2: Manhattan plots of SAGE-COGA and GIANT. Upper panel: Manhattan plots of the summary statistics (coefficient β and p-value) from SAGE-COGA. Lower panel: Man- hattan plots of the summary statistics (coefficient β and p-value) from GAINT

First, we performed pIGWAS using combined GWAS data of SAGE and COGA with summary statistics from GIANT. Specifically, we used the EA samples containing genotype data with corresponding summary statistics implemented with the proposed method. The analysis was conducted chromosome by chromosome. We also evaluated the relative stability of the selected SNPs using random sampling [26]. Specifically, we randomly sampled 80% of the subjects and applied pIGWAS to identify associated SNPs. This process was repeated 100 times. For each SNP, we were able to calculate the proportion of times that the SNP was associated with the trait out of 100 samplings, i.e., the observed occurrence index (OOI). For comparison, we conducted single data analysis of GWAS with EA samples using scaled Lasso, and evaluated its relative stability of the selected markers using the OOI. The associated markers identified by integrative and single data analysis were listed in Table 1. The average OOI of SNPs selected by pIGWAS is 0.649 while that of SNPs selected by scaled Lasso is 0.648, suggesting a limited improvement of pIGWAS over scaled Lasso on the COGA-SAGE data set. There may be two reasons for this. First, the GWAS signals of BMI from the COGA-SAGE may be too weak to be distinguished from noise (Figure 1). Second, the pleiotropic effects between BMI and height may not be strong enough to boost the power of pIGWAS. It is expected that pIGWAS could achieve a better performance in presence of well-powered GWAS signals and pleiotropy information.

SNP Chr Position Genea Band pIGWAS Scaled Lasso
rs11588887 1 157983786 CRP q23.2 1.39 0.99 3.49 0.99
rs1254207 1 234434850 GPR137B q42.3 -0.36 0.84 -0.78 0.75
rs2477586 1 234451564 ERO1LB q42.3 -0.32 0.81 -0.61 0.73
rs1860875 7 21078168 RPL23P8 p15.3 0.23 0.70
rs2390470 7 21088622 RPL23P8 p15.3 -0.06 0.34
rs2192300 7 121194239 PTPRZ1 q31.32 -0.14 0.81
rs1054611 12 10061428 CLEC12B p13.2 0.10 0.78 0.09 0.66
rs4565970 12 81715248 TMTC2 q21.31 0.03 0.56 0.01 0.52
rs12370680 12 83436479 MIR548T q21.31 0.25 0.87 0.35 0.83
rs4393415 12 102464392 STAB2 q23.3 0.10 0.72 0.13 0.60
rs4405407 12 102466587 STAB2 q23.3 0.04 0.60 0.03 0.40
rs1336850 13 22680577 SGCG q12.12 -1.19 0.86
rs622227 13 27937214 FLT1 q12.3 1.00 0.78
rs1058214 13 38885772 LHFP q13.3 1.32 0.80
rs7323630 13 39747343 LINC00548 q14.11 -0.16 0.59
rs4473069 13 43009971 ENOX1 q14.11 -0.03 0.40
rs2786712 13 44192569 LINC00330 q14.11 -0.18 0.56
rs1330476 13 81854308 SLITRK1 q31.1 -0.04 0.61
rs9531358 13 81964898 SLITRK1 q31.1 -0.14 0.89 -3.55 1.00
rs2777825 13 83099526 SLITRK1 q31.1 -1.20 0.83
rs9531489 13 83152986 SLITRK1 q31.1 -0.57 0.62
rs9546479 13 83154395 SLITRK1 q31.1 -0.03 0.33
rs9319013 13 83522913 MIR548F1 q31.1 -0.29 0.55
rs723576 13 95002092 CLDN10 q32.1 -0.11 0.47
rs1547166 13 95071328 DZIP1 q32.1 -0.37 0.46
rs7338545 13 95073552 DZIP1 q32.1 -0.16 0.39
rs8018440 14 32981820 NPAS3 q13.1 0.25 0.71 0.03 0.52
rs4903707 14 39909619 FBXO33 q21.1 0.57 0.78 0.38 0.73
rs9944120 14 40082198 FBXO33 q21.1 0.04 0.51
rs7149526 14 80098349 CEP128 q31.1 0.00 0.47
rs1951980 14 95322780 TCL1A q32.13 -1.47 1.00 -1.54 0.97
rs1345300 16 8671929 ABAT p13.2 -0.10 0.68
rs2283557 16 24273067 CACNG3 p12.1 -0.01 0.63
rs4784651 16 54869275 GO1 q13 0.15 0.42
rs9922112 16 54872188 GO1 q13 0.10 0.39
rs2587878 16 54872860 GO1 q13 -0.16 0.46 -0.02 0.48
rs8047093 16 59619195 CDH8 q21 0.09 0.64
rs8044561 16 70108642 CHST4 q22.3 0.18 0.28
rs310334 16 70131048 CHST40.11769305 q22.3 0.12 0.29
rs2432524 16 70141834 CHST4 q22.3 1.36 0.92 1.66 0.97
rs8056272 16 72267976 LOC100506172 q22.3 0.04 0.50
rs12928065 16 77094975 WWOX q23.1 -0.13 0.61 -0.02 0.52
rs933374 17 13742206 CDRT15P1 p12 0.04 0.69 0.16 0.65
rs9889937 17 18512091 FOXO3B p11.2 0.00 0.58 0.01 0.58
rs4792855 17 40815480 ARHGAP27 q21.31 -0.18 0.87 -0.48 0.87
rs17673185 17 48822712 MIR548AJ2 q22 -0.02 0.57 -0.12 0.55
rs7265169 20 312747 TRIB3 p13 0.09 0.63
rs459012 20 410008 CSNK2A1 p13 0.25 0.59
rs6053384 20 5354093 LINC00658 p12.3 0.83 0.82 0.75 0.80
rs1555669 20 12598312 SPTLC3 p12.1 0.01 0.41
rs6074541 20 12926517 SPTLC3 p12.1 -0.01 0.46
rs6081333 20 18660916 DTD1 p11.23 -0.38 0.72 -0.16 0.57
rs2067845 20 19446645 SLC24A3 p11.23 0.88 0.91 0.83 0.81
rs6035387 20 19524045 SLC24A3 p11.23 0.22 0.46 0.03 0.42
rs6515030 20 19529688 SLC24A3 p11.23 0.39 0.67 0.23 0.51
rs942990 20 19533661 SLC24A3 p11.23 0.07 0.32
rs199575 20 19902601 RIN2 p11.23 -0.11 0.73
rs56916 20 19936892 RIN2 p11.23 1.15 0.90 1.26 0.82
rs199572 20 19940313 RIN2 p11.23 0.17 0.42
rs200175 20 19949483 A20 p11.23 0.45 0.54 0.21 0.35
rs6050359 20 25070945 LOC284798 p11.21 0.33 0.50 0.10 0.41
rs6050372 20 25081225 LOC284798 p11.21 0.77 0.81 0.84 0.88
rs6050418 20 25118643 LOC284798 p11.21 0.48 0.62 0.36 0.63
rs3787076 20 25143018 ENTPD6 p11.21 0.37 0.68 0.07 0.57
rs2073077 20 25143913 ENTPD6 p11.21 0.13 0.54
rs6022419 20 36083479 TTI1 q11.23 -0.08 0.67
rs6030352 20 40658434 PTPRT q12 0.66 0.87 0.52 0.71
rs1010310 20 44268451 CDH22 q13.12 -0.02 0.48
rs846743 20 48768050 PARD6B q13.13 -0.13 0.59
rs6021702 20 50145309 ZFP64 q13.2 -0.13 0.64
rs7268780 20 56735571 STX16-NPEPL1 q13.32 0.03 0.65
rs2823209 21 15586648 NRIP1 q21.1 0.24 0.66 0.22 0.54
rs2823216 21 15591805 NRIP1 q21.1 0.34 0.74 0.30 0.60
rs463370 21 30177240 GRIK1 q21.3 1.40 0.95 1.51 1.00

Table 1: SNPs selected incorporating summary statistics from public available source by using pIGWAS and SNPs selected using scaled Lasso for GWAS with genotype.


GWAS suffer from low statistical power due to the individual weak effects of genetic variants. In this study, we proposed a statistical approach to jointly analyzing primary GWAS data with summary statistics together from other source. The key idea of the proposed approach lies on the existence of pleiotropic effects of genetic variants, which allows us to borrow information from genetically related GWAS. Specifically, we proposed a novel penalized regression that combines multiple GWAS together. The computationally efficient algorithm is derived for optimizing the model parameters. Based on both simulation study and real data analysis, we demonstrated the advantages of the proposed method over single-GWAS analysis.


This study was supported by National Institutes of Health grants R01EY022651, Hong Kong Research grant HKBU12202114, Hong Kong Baptist University FRG2/13-14/005 and Duke- NUS Graduate Medical School WBS: R-913-200-098-263.


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