1 Introduction
Genetic variation is what makes us all unique. It refers to the diversity in the DNA sequence in human genomes and it may affect how an individual develops a disease or and responds to drugs, vaccines, pathogens, and etc [5, 2]. The most common type of genetic variation is a singlenucleotide polymorphism (SNP)—i.e., a difference in a single nucleotide in the deoxyribonucleic acid (DNA) [13]. In the past decade, genomewide association studies (GWA studies or GWAS), which aim at revealing the relationships between genetic variants such as SNPs and individual traits, have attracted much attention achieved considerable success [14, 25, 28].
Traditional GWA studies are based on statistical tests. Genetic risk factors are determined by their statistical significance, where a general procedure is to perform a statistical test between each individual SNP and the phenotype under investigation [29, 8, 7]. For example, via metaanalyses, 11 new susceptibility SNPs for Alzheimer’s disease (AD) have been identified [15]; 10 loci that may influence allergic sensitization have been detected [3]. However, such kind of approaches has several limitations. First, it ignores the aggregate effects of multiple SNPs, for example, the epistatic interactions between loci [34, 17]. Second, independent SNP–phenotype testing disregards the SNPs’ structural correlations associated with population genetics (i.e., linkage disequilibrium, LD) and biological relations (e.g. functional relationships between genes) [20].
Later, increasing attention has been focused on Lasso (least absolute shrinkage and selection operator [24]), as an alternative tool for identifying risk SNPs in GWAS [31, 26]. Lasso is a multivariate method that models multiple SNPs simultaneously and highly precarious SNPs (that related to the phenotype under investigation) can be identified through the nonzero components of the model. For example, a previous whole genome association study [31] shows Lasso together with stability selection [19] is promising in detecting risk SNPs associated with Alzheimer’s disease (AD) . However, there are two major drawbacks of Lasso: 1) it tends to arbitrary select only one from a set of highly correlated features [10]; 2) it considers all features equally without any further structural assumptions. To address the above issues, utilizing structured sparse models together with different biological priors has aroused growing concern in GWAS, as incorporating such assumptions is favorable for model construction and interpretation [32]. There are several attempts, for example, group Lasso [18], tree Lasso [26], and absolute fused Lasso [30].
It is worth mentioning that all those aforementioned approaches are based on the nucleotidelevel biological assumptions (e.g. LD or the consistency of successive SNPs). However, in realworld, at nucleotidelevel, neither structural associations, nor functional relationships, nor interaction mechanisms, have been wellstudied to date. On the other hand, studies of biological mechanisms are more rigorous and legitimate at genelevel. For example, GeneMANIA [27] is a powerful tool for revealing genelevel biological networks. It integrates a large set of functional association data, including protein and genetic interactions, pathways, coexpression, colocalization and protein domain similarity. As a consequence, it is potentially beneficial to utilize such genelevel priors in nucleotidelevel GWAS studies.
In this paper, we propose a novel twolevel structured sparse model, called Sparse Group Lasso with Grouplevel Graph structure (SGLGG), which a is promising method for identifying significant SNP–phenotype associations. As its name indicates, SGLGG can be considered as a fusion model of a sparse group Lasso [33, 9] and a grouplevel graph Lasso (a.k.a., graphguided fused Lasso [6]). Essentially, our proposed model involves two levels of predictors—i.e., the nucleotidelevel predictors and the genelevel predictors. And consequently in a GWA study, SGLGG will penalize the following three respects:

the genelevel sparsity;

the graph structure among genelevel predictors;

the nucleotidelevel sparsity.
As a result, SGLGG tends to select only a set of causal SNPs within a gene group and limited gene groups among the entire sequence. Meanwhile, it is capable of taking advantages of biological priors (i.e., gene networks) during the genelevel selection. With the graph constraint, highly relevant genes are likely to be chosen simultaneously, and thus SNPs from different gene scopes are potentially able to connect. SGLGG is hard to solve due to its complex sparseinducing regularizers. To this end, we first transfer the edge constraints among the graph into the matrix form, and then, employ the ADMM (alternating direction method of multipliers [4]) algorithm for optimizing. Experiments have been conducted on the Alzheimer’s Disease Neuroimaging Initiative (ADNI) whole genome sequence (WGS) data and neuroimage data, for both regression tasks and variable selection tasks. Preliminary results show that SGLGG is competitive to the stateofthearts sparse models in predicting ADrelated imaging phenotypes. In addition, stability selection results demonstrate that SGLGG is promising for identifying risk SNPs associated with Alzheimer’s disease.
2 Our Model: SGLGG
Essentially, we consider a linear prediction model. Given a centered data matrix with observations and features, and a corresponding response . Suppose that predictors can be divided into nonoverlapping groups, with the number of lowlevel predictors in group . Accordingly, we denote be the lowlevel predictors and be the grouplevel predictors, respectively. Then, the lowlevel predictor can be represented as . We further denote , where is the Hadamard product operator, is a designed mapping matrix^{1}^{1}1 is a binary matrix, an element if in group ., and is the th element of . The grouplevel graph^{2}^{2}2In this study, we only consider the situation of undirected graph among grouplevel features. information is described by , where is a set of nodes, and is the set of edges. In addition, let
denote the weight vector corresponding to the grouplevel predictors, and
denote the weight of the edge between node and . Hence, in this paper, we consider the following optimization problem:(1) 
where
is a convex empirical loss function (
e.g. the least squares) and the error is calculated based on —a combination of predictors and via ; ; and represent a general monotonically increasing function weight function that enforces a fusion effect between coefficients and .In Eq. (1), the first constraint can be considered as a grouplevel sparsity constraint, the second constraint introduces the grouplevel graph structure via the fused Lasso, and the third constraint penalizes the lowlevel sparsity. Hereby, we call Problem (1), the Sparse Group Lasso with Grouplevel Graph structure (SGLGG) problem. More specifically, in a GAW study, represents the nucleotidelevel predictor, and accordingly, can be considered as the genelevel predictor. Therefore, an ideal solution to Eq. (1) will lead to the following scenarios: 1) only limited gene groups will be selected among the entire sequence; 2) the group selection is guided by the genelevel biological priors—i.e., relevant genes are more likely to be chosen simultaneously; and 3) only a subset of SNPs will be selected within a selected gene. In other words, the genelevel and nucleotidelevel constraints ensure that the most relevant gene groups and SNPs within a gene will be chosen by the model. Meanwhile, the group selection will be affected by the genelevel priors—i.e., some intergene SNP–SNP connections could be revealed by the graph constraint.
Furthermore, the grpah constriant in Eq. (1) can be reformualted into a matrix form. Denote be the sparse matrix constructed from the edge set , where if there is a edge between and . Furthermore, for discussion convenience, we ignore the weight vectors in Eq. (1), then SGLGG problem can be simplified as the following matrix form:
(2) 
3 ADMM for Solving SGLGG
3.1 ADMM basic
Due to the complex sparseinducing regularizers, unconstrianted optimzation problem like (1) are sometimes hard to solve directly. Instead, it is possbile to reformulate the original unconstrianted problem to an equivalent constrained problem. In the sequel, such a problem can be addressed using constrained optimization methods such as the augmented Lagrangian method.
Hereby, we employ the alternating direction method of multipliers (ADMM) [4, 21] algorithm to solve Problem (1). ADMM is a variant of the augmented Lagrangian method. It utilizes dual decomposition and partial updates for the dual variables. Without loss of generality, we consider the following constraint optimization problem:
(3)  
where and are convex, , , , , and . With ADMM, we first reformulate the above problem (3) as:
(4) 
with being the augmented Lagrangian multiplier, and being the nonnegative dual update step length. ADMM solves this problem by iteratively minimizing over , and , one at a time, until convergence. Consequently, the update rule for ADMM is given by
3.2 ADMM for solving SGLGG problem
Suppose be the least squares loss, then the SGLGG problem presented in (2) can be rewritten as the following constrained form:
(5)  
s.t. 
where are slack variables. We employ ADMM to solve Problem (5). The augmented Lagrangian is
(6)  
where are augmented Lagrangian multipliers. Accordingly, in the th iteration, the update rules are as follows:

Update : can be updated by minimizing with fixed:
where , and is an operation that transforms a vector into a square diagonal matrix. The above optimization problem is quadratic, and thus the optimal solution can be obtained by solving the following linear system:
(7) where
It is trivial to show that is symmetric positive definite (SPD), and thus Eq. (7) can be solved efficiently via the conjugate gradient method [11] .

Update : can be updated by minimizing with fixed:
where . Similar to the update rule of , the above optimization problem is quadratic, and thus the optimal solution can be obtained by solving the following linear system:
(8) where
Similarly, since is SPD, Eq. (8) can be solved efficiently via the conjugate gradient method.

Update : Similarly, can be obtained by solving the following problem:
The above optimization problem has a closedfirm solution, known as the softthresholding:
(9) where the softthresholding operator is defined as:

Update : Similarly, can be obtained by solving the following problem:
The closedform solution of the above problem can be obtained by:
(10) 
Update : Similarly, can be obtained by solving the following problem:
The closedform solution of the above problem can be obtained by:
(11) 
Update : In the th iteration, are updated by:
(12) (13) (14)
We summarize the ADMM algorithm for solving the SGLGG Problem (2) in Algorithm 1. Generally, ADMM breaks the original complex optimization problem into a series of smaller subproblems, each of which is then easier to handle. In addition, it is worth mentioning that in practice, it is important to normalize according to its group size.
4 Experiments
To evaluate the performance of the proposed SGLGG approach in GWAS, we conducted a series of experiments on the Alzheimer’s Disease Neuroimaging Initiative (ADNI) whole genome sequence (WGS) data and neuroimage data. Particularly, we focus on two learning tasks: 1) predicting ADrelated imaging phenotypes (based on SNPs data); and 2) identifying risk SNPs w.r.t. AD imaging phenotypes.
4.1 Data processing
4.1.1 ADNI WGS data and neuroimaging data
In this study, we adopt the ADNI WGS data set and MRI data for GWAS. More specifically, the following procedures have been employed for processing SNPs data. First, we employ PLINK [22] together with a series of standard quality control constraints for SNPs data preprocessing. Particularly, a SNP will be removed if its minor allele frequency (MAF) , or missingness , or deviations from HardyWeinberg Equilibrium . In the sequel, we adopt MaCH [16]
for genotype imputation. MaCH is a Markov chain based haplotyper that is capable of resolving long haplotypes or inferring missing genotypes. Eventually, we apply several filters on the imputed data set, including: RSQ (estimated
, specific to each SNP) , FREQ1 (frequency for reference Allele 1) and FREQ1 . As a consequence, the entire genome data contains 1,319 subjects with 6,566,154 SNPs, in which 155,357 SNPs are from Chromosome 19. For subjects composition, there are 327 healthy controls (HC), 249 AD patients, 41 participants with mild cognitive impairment (MCI), 220 early MCI (EMCI) patients, 419 late MCI (LMCI) patients, and 63 patients with significant memory concerns (SMC).Volumes of some major influenced brain regions that are related to Alzheimer’s disease, including the hippocampus (HIPP) and the entorhinal cortex (EC), have been chosen as the neuroimaging phenotypes in this study. Those volumes were extracted from subject’s T1 MRI data using Freesurfer [23],
4.1.2 Candidate AD genes
Hereby, we focus on Alzheimer’s disease genetic risk factors (at both genelevel and nucleotidelevel) on the 19th chromosome of the human genome. Particularly, at genelevel, ten candidate genes are preselected as high ADrisk according to AlzGene [1], including LDLR, GAPDHS, BCAM, PVRL2, TOMM40, APOE, APOC1, APOC4, EXOC3L2, and CD33. Positions of those preselected genes are shown in Figure 1.
The above ten genes have been considered as the most strongly associated genes with AD on Chromosome 19 (Chr.19). In AlzGene, top associated genes are ranked based on genetic variants with the best overall HuGENet/Venice grades [12]. Specifically, for genes with identical grades, the ranking is based on their pvalues; for genes with identical grades & pvalues, the ranking is based on their effect sizes. Basic information on those ADrisk genes is available in Table 1 (top part).
4.1.3 Gene networks
To retrieve genelevel biological priors—i.e., gene networks, we utilized GeneMANIA [27] in our study. Essentially, GeneMANIA is a powerful tool to extract gene networks based on a set of input genes. The network is retrieved from a large set of functional association data, including gene coexpression & colocalization, proteinprotein interaction, genetic interaction, shared protein domains, pathway, and etc. GeneMANIA stands for the Multiple Association Network Integration Algorithm
. It consists of a linear regressionbased algorithm for calculating the functional association network and a label propagation algorithm for predicting gene functions hereafter. In our study, we employ the following two methods to extract gene networks.

Gene network within 10 preselected ADrisk genes in Chr.19.
Ten aforementioned ADrisk genes on Chromosome 19 are utilized as the input genes for GeneMANIA. For network exploration, we only focus on connections within those ten preselected genes. In addition, we adopt the biological processbased method for gene ontology weighting. A visualization of this gene networks is shown in Figure 2 (left). 
Extended gene network based on 10 selected Chr19 ADrelated genes.
Similar to 1, but we allow to introduce ten additional genes for network exploration. This results in totally 20 genes in the graph. A visualization of such a network is shown in Figure 2 (right). Note that, additional genes are selected based on their relations with input genes and thus those genes are not necessary located on Chromosome 19. Additional information of those selected genes is available in Table 1 (bottom part).
Symbol  Assembly  Chr  Location  # of loci^{3}^{3}3This is the number of available loci in our experimental dataset.  
AD Candidate Genes 
LDLR  GRCh37.p13  19  11200037..11244506  135 
GAPDHS  GRCh37.p13  19  36024314..36036221  22  
BCAM  GRCh37.p13  19  45312316..45324678  15  
PVRL2  GRCh37.p13  19  45349393..45392485  164  
TOMM40  GRCh37.p13  19  45394477..45406946  38  
APOE  GRCh37.p13  19  45409039..45412650  5  
APOC1  GRCh37.p13  19  45417577..45422606  14  
APOC4  GRCh37.p13  19  45445495..45448753  7  
EXOC3L2  GRCh37.p13  19  45715879..45737469  88  
CD33  GRCh37.p13  19  51728335..51743274  16  
Associated Genes 
LDLRAP1  GRCh37.p13  1  25870071..25895377  28 
PVRL3  GRCh37.p13  3  110790606..110913017  73  
APOA5  GRCh37.p13  11  116660086..116663136  7  
APOA1  GRCh37.p13  11  116706467..116708338  5  
CRTAM  GRCh37.p13  11  122709255..122743347  75  
GAPDH  GRCh37.p13  12  6643585..6647537  10  
LIPC  GRCh37.p13  15  58702953..58861073  481  
CD226  GRCh37.p13  18  67530192..67624412  149  
APOC2  GRCh37.p13  19  45449239..45452822  17  
SOD1  GRCh37.p13  21  33031935..33041244  15 
Later, the experimental data sets were generated through those two aforementioned methods. More specifically, we first construct a smaller SNPs data set that consists of SNPs from 10 preselected ADrisk genes on Chromosome 19. As a result, such a data set contains 1,381 subjects and 504 SNPs. Next, we generate a larger SNPs data set based on an extended gene network obtained through GeneMANIA—i.e., SNPs from 10 additional genes (as shown in Table 1) are also involved, according to genelevel associations. Accordingly, the larger SNPs data set contains 1,364 SNPs in total from 20 candidate genes.
4.2 Learning task I — Predicting ADrelated phenotypes
In the first series of experiments, we evaluate our proposed SGLGG model in a set of regression tasks—i.e., predicting Alzheimer’s diseaserelated imaging phenotypes. More specifically, SGLGG is compared with a suite of wellknown commonlyused (structured) sparse methods, including Lasso, the fused Lasso (FL) and sparse group Lasso (SGL). For SGL and SGLGG, SNPs in the same gene naturally fall into a group. In addition, we compare SGLGG with the absolute fused Lasso (AFL) [30]—a novel learning model that penalizes SNPs successive similarities. Four imaging phenotypes including volumes of the left entorhinal cortex (LEH), left hippocampus (LHP), right entorhinal cortex (REH), and right hippocampus (RHP), are used as the responses in this study.
Experiments have been conducted on the two SNPs data sets described in Section 4.1.3. We adopt fivefold crossvalidation for each learning task and each sparse model. Comparisons of predictive performance in terms of mean squared error (MSE) of 10 replications are shown in Figure 3 through box plots. In Figure 3, each color represents a modeling method. Labels of the axis are named as follows: the first few letters represent a modeling method, the middle three letters indicate the learning task, and the last number (10 or 20) indicate the data set involved.
From Figure 3, we can observe that our proposed SGLGG model is very competitive compared with other (structured) sparse models. With complex sparseinducing regularizers and complex biopriors, SGLGG can still provide favorable predictive performance in most of the cases. Meanwhile, such a model has better interpretability than traditional ones, as it incorporated extensive prior knowledge during model learning. Therefore, it is potentially beneficial to address realworld GWA studies through the SGLGG model.
4.3 Learning task II — Identifying ADrisk SNPs
One of the benefits of adopting a sparse model for GWAS is that the most relevant genetic factors can be identified through the nonzero components from the model. Hereby, in the following series of experiments, we compare the variable selection (i.e., SNPs selection) results of different structured sparse methods through stability selection [19]. More specifically, experiments were conducted on the smaller SNPs data set mentioned in Sec 4.1.3. We perform 100 simulations for each learning target. Within each simulation, we first randomly subsample half of the subjects and then perform a modeling method 100 times with different regularization parameters (or pairs of parameters). The model selection results are visualized in Figure LABEL:fig:adni_fs_comp. Detailed SNPs selection results are available in Appendix 1. In Figure LABEL:fig:adni_fs_comp, top 50 selected SNPs are marked for each method; each color refers to a modeling method; the axis is a compact illustration of gene/SNPs location on Chromosome 19; green bars together with the axis indicate the negative logarithmic of Pvalues of SNPs associated with each learning task.
From Figure LABEL:fig:adni_fs_comp, we have the following observations:

SNPs selected by Lasso and SGL are spread over a large region in the feature sets (i.e., across different genes). However, most SNPs selected by FL, AFL, and our proposed SGLGG model are clustered in a few small regions.

SNPs groups identified by SGLGG are different from FL or AFL, where the proposed method tends to select more SNPs within a gene but fewer number of genes in total.

Statistical significance in terms of Pvalue of an SNP selected by SGLGG, may not necessarily be small^{4}^{4}4A smaller Pvalue implies higher statistical significance. Since we use the negative logarithm of Pvalues in Figure LABEL:fig:adni_fs_comp, statistically significant SNPs will have higher green bars. (see the bottom two subfigures in Figure LABEL:fig:adni_fs_comp).
The above observations imply that our proposed SGLGG model sparse selection on both nucleotidelevel and genelevel. Within a gene, only the most relevant SNPs will be chosen. The group selection is benefited from genelevel biological prior knowledge—i.e., gene network. Thus, potential intergene SNP–SNP connections could be established by SGLGG. In other words, SGLGG is a promising method and has good prospects in revealing the causal SNPs that associated with a phenotype under investigation.
5 Conclusion
In this paper, we proposed a novel twolevel structured sparse model—SGLGG—for genomewide association studies. Essentially, it can be considered as a sparse group Lasso together with a grouplevel graphguided fused Lasso. Specifically, SGLGG induces sparsities in both nucleotidelevel and genelevel. That is, only the most causal SNPs will be selected within a gene group and only a part of relevant genes will be chosen on the genome. Another benefit of SGLGG is that it also takes advantages of genelevel biological priors during the model construction. Consequently, genelevel biopriors such as protein–protein interactions and pathways can be utilized to explore intergene SNP–SNP connections. To address SGLGG model, we propose an ADMMbased optimization algorithm. Our experiments on the Alzheimer’s disease genome sequence data and neuroimaging data show that SGLGG is very competitive in predict ADrelated phenotypes, compared with other stateofthearts sparse learning models. Furthermore, stability selection results demonstrate that SGLGG is a promising model for identifying ADrisk SNPs. With the help of genelevel biological priors, SGLGG has good prospects for revealing SNP–SNP interactions among different genes.
Acknowledgement
This work was supported in part by NIH BD2K (Big Data to Knowledge) grants to the KnowENG Center, based at UIUC, and the ENIGMA Center for Worldwide Medicine, Imaging & Genomics, based at USC.
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