Identification of Gene-environment Interactions with Marginal Penalization

Abstract

Gene-environment (G-E) interaction analysis has been extensively conducted for complex diseases. In marginal analysis, the common practice is to conduct likelihood-based (and other “standard”) estimation with each marginal model, and then select significant G-E interactions and main effects based on p-values and multiple comparisons adjustment. One limitation of this approach is that the identification results often do not respect the “main effects, interactions” hierarchy, which has been stressed in recent G-E interaction analyses. There is some recent effort tackling this problem, however, with very complex formulations. Another limitation of the common practice is that it may not perform well when regularization is needed for example because of “non-normal” distributions. In this article, we propose a marginal penalization approach which adopts a novel penalty to directly tackle the aforementioned problems. The proposed approach has a framework more coherent with that of the recently developed joint analysis methods and an intuitive formulation, and can be effectively realized. In simulation, it outperforms the popular significance-based analysis and simple penalization-based alternatives. Promising findings are made in the analysis of a SNP and a gene expression data.

1. Introduction

For many complex diseases such as cancer and cardiovascular diseases, gene-environment (G-E) interactions have critical implications for etiology, prognosis, response to treatment, and biomarkers (Hunter, 2005; Thomas, 2010; Simonds et al., 2016). Marginal G-E interaction analysis, which examines one or a small number of genes at a time, has been routinely conducted (Wu et al., 2012; Hutter et al., 2013; Sun et al., 2018). Denote Y as a disease outcome/phenotype of interest, Z = (Z₁, Z₂,…, Z_p) as p genes, and X = (X₁,…, X_q) as q environmental risk factors. Here we use the generic terminology “gene”, which can be gene expression (GE), single nucleotide polymorphism (SNP), or other omics measurements. For environmental (E) factors, as in quite a few other studies, we take a looser definition to also include clinical and other risk factors. In the literature, the most popular approach proceeds as follows: (i) For j = 1,…,p, fit the regression model $Y\sim \varphi ({\sum }_{k=1}^q{\alpha }_k{X}_k+{\beta }_j{Z}_j+{\sum }_{k=1}^q{\gamma }_{j,k}{X}_k{Z}_j)$, where the form of model ϕ is assumed to be known, α_k’s, β_j, and γ_j,k’s are the unknown regression coefficient. As usually the number of environmental risk factors q is small, this is a “standard” estimation problem, and likelihood-based and other simple estimation techniques are adopted. Denote p_j as the p-value of ${\widehat{\beta }}_j$, the estimate of β_j, and p_j,k as the p-value of ${\widehat{\gamma }}_{j,k}$, the estimate of γ_j,k; (ii) With {p_j : j = 1,…,p} and {p_j,k : j = 1,… ,p, k = 1,…, q}, conduct multiple comparisons adjustment with the Bonferroni or FDR approach, and identify significant effects. Some studies identify significant interactions and main effects together (Kraft et al., 2007; Dai et al., 2012), while others conduct separate identification. It is noted that there have been variations of this approach. For example, some studies use objective functions other than likelihood functions (Ritchie et al., 2001).

Despite great successes, the above approach has limitations. The most important is that the “main effects, interactions” hierarchy may be violated. That is, an interaction term may have a significant p-value and concluded as significantly associated with the response, but the corresponding main effects are not. In G-E interaction analysis, usually E factors are manually selected based on prior knowledge (and hence likely to be important) and have a low dimensionality. As such, the main interest and hierarchy are on G-E interactions and main G effects. In joint G-E interaction analysis, which analyzes a large number of G factors in a single model, the statistical and biological importance of this hierarchy has been stressed (Bien et al., 2013; Hao and Zhang, 2017), and a long array of methods have been developed to respect this hierarchy. In marginal analysis, comparatively, there has been much less attention to this hierarchy. One foundational work is Bien et al. (2015), which establishes the essential importance of the “main effects, interactions” hierarchy in marginal analysis. It is argued that a significant G-E interaction (identified as associated with the response) would demand the significance of the corresponding main G effect (hence also identified as response-associated). It is also pointed out that it is insufficient to simply include the main G effect in modeling if it is not significant. This work is in the hypothesis testing framework, and, despite great successes, suffers from very complex formulation and computation. Another related strategy is to conduct two-step identification (Hao and Zhang, 2017), which identifies significant main G effects first and then searches for significant interactions only among those identified in the first step. This can ensure that a significant interaction always has a corresponding significant main G effect. However, Hao and Zhang (2017) points out that this approach demands assumptions. In addition, Bien et al. (2015) suggests that this may miss important interactions if the corresponding main effects are not sufficiently significant. The second limitation is that simple estimations may not behave well enough. Consider, for example, linear regression models with random errors deviating from normal. One potential solution is to adopt objective functions with certain robustness properties (Shi et al., 2014; Li et al., 2013). But often such objective functions are not easy to compute and conduct inference with.

In this article, our goal is to develop a new marginal G-E interaction analysis method, which may differ from the existing ones in multiple aspects. Advancing from the “straightforward” significance-based analysis, it respects the “main effects, interactions” hierarchy. That is, if a G-E interaction is identified as important for the response variable, then the corresponding main G effect is automatically identified. To achieve this, we propose using penalization for G-E interaction identification and estimation, which differs fundamentally from the significance-based approaches. Although sharing the same philosophy of respecting the hierarchy, the proposed method differs from Bien et al. (2015) by adopting a penalized estimation/selection, not hypothesis testing, strategy. In addition, implementation of the proposed method is much simpler than Bien et al. (2015). It also differs from the two-step approach by searching for important interactions and main effects simultaneously, which may improve performance by “borrowing information” between the two identifications. In addition, penalized estimation can also be advantageous over “standard” estimation (Gauderman et al., 2013). It is noted that it is not “abnormal” to employ penalization in low-dimensional estimation. In particular, the method in Bien et al. (2015) is also solved via penalized estimation. With the hierarchy and regularized estimation, the proposed approach has an analysis framework more coherent with that of the joint analysis (Liu et al., 2013; Wu et al., 2018; Wu et al., 2017; Zhao et al., 2018; Wang et al., 2019). This coherence facilitates methodological developments as well as biological interpretations. We fully acknowledge the connections between the proposed analysis strategy and technique with those of the joint genetic interaction analysis. Marginal and joint analysis cannot replace each other. Considering the importance of marginal analysis and lack of approaches that possess the aforementioned desirable properties, this study is warranted and will provide a practically useful new way of conducting marginal G-E interaction analysis.

2. Methods

For the jth gene Z_j(j = 1,…, p), consider the regression model

$$\begin{gathered} Y~\varphi ({\alpha }_{j,0}+\sum _{k=1}^q{X}_k{\alpha }_{j,k}+Z_j{\beta }_j+\sum _{k=1}^q{X}_k{Z}_i{\gamma }_{j,k}) \\ =\varphi ({\alpha }_{j,0}+\mathit{X}{\alpha }_j+{\mathit{W}}_j{\mathit{b}}_j), \end{gathered}$$ (1)

where the form of function ϕ is known, α_0,j is the unknown intercept to accommodate models such as linear and logistic, α_j = (α_j,1,…, α_j,q)′ is the vector of unknown regression coefficients representing the main E effects, W_j ≜ (Z_j, Z_jX₁,…, Z_jX_q) accommodates the main G effect and G-E interactions, and b_j = (β_j, γ_j,1,…, γ_j,q)′ ≜ (b_j,1, b_j,2,…, b_j,q+1)′ is the vector of unknown coefficients for W_j. It is noted that the proposed analysis can accommodate a wide variety of response variables (such as continuous, categorical, and censored survival) and models (such as linear, logistic, and Cox) as well as G (for example, gene expression and SNP) and E (environmental, clinical, etc.) measurements.

Denote l(α_0,j, α_j, b_j) as the negative log-likelihood function. Other objective functions, for example estimating equation-based, can be analyzed in the same manner. We introduce a penalized objective function

$$Q({\alpha }_{0,j},{\alpha }_j,{\mathit{b}}_j)=l({\alpha }_{0,j},{\alpha }_j,{\mathit{b}}_j)+P_{\lambda }({\mathbf{b}}_j),$$ (2)

where P is the penalty function and λ is the vector of data-dependent tuning parameters. Minimizing the objective function (2) with respect to the parameters α_0,j, α_j and b_j, we get the estimate

$$({\widehat{\alpha }}_{0,j},{\widehat{\alpha }}_j,{\widehat{\mathit{b}}}_j)=\text{argmin}Q({\alpha }_{0,j},{\alpha }_j,{\mathit{b}}_j).$$ (3)

For the penalty function, we adopt the sparse group MCP (sgMCP) with

$$P_{\lambda }({\mathit{b}}_j)=\rho (\Vert {\mathit{b}}_j\Vert ;\sqrt{q+1}{\lambda }_1,\xi )+\sum _{k=2}^{q+1}\rho (\mid b_{j,k}\mid ;{\lambda }_2,\xi ),$$ (4)

where $\rho (t;\lambda ,\xi )=\lambda {\int }_0^t(1-\frac{x}{\lambda \xi })+dx$ is the MCP penalty (Zhang et al., 2010), λ₁, λ₂ > 0 are data-dependent tuning parameters, and ξ is the regularization parameter.

Penalized estimation is applied to all p genes. To ensure that all genes are analyzed on the same ground, the same tuning/regularization parameters are applied in all estimations. This is also coherent with that in joint analysis. Different from joint analysis, each gene has its own model and likelihood function. Interactions and main G effects corresponding to the nonzero components of ${\widehat{\mathit{b}}}_j$ ’s are identified as important and associated with the response. This is an estimation-based identification strategy and has notable differences from the significance-based (including that in Bien et al., 2015).

Rationale

Similar to other marginal analysis, one gene is considered at a time, and a marginal regression model is assumed. We adopt the penalized estimation and identification strategy, which has been extensively adopted in joint analysis (Liu et al., 2013; Kim et al., 2017) but only a few marginal analysis (Bien et al. 2015). The sparse group MCP automatically ensures that if b_j,k ≠ 0 for any k ≥ 2 (that is, an interaction term), then b_j,1 ≠ 0 (that is, the main G effect). The MCP-based penalty is adopted because of its satisfactory performance demonstrated in many publications. There are multiple strong reasons for adopting the sparse group MCP. It can conduct the selection of both interactions and main effects and be advantageous over, for example, applying penalization to interactions only (which cannot discriminate important main G effects against noises). Even when only interactions are of interest, by reinforcing the hierarchy, the proposed penalization can “borrow information” from main effects in the estimation/identification of interactions. Consider for example when a main G effect is zero (not identified). Then this suggests that the corresponding interactions should not be identified. That is, there is additional information for interaction identification. This may make the proposed analysis advantageous over two-step analysis, which does not use information on interactions in the first step of main effect analysis. Adopting this penalty also makes the frameworks of marginal and joint analyses (Liu et al., 2013 and others) more coherent. A “byproduct” of penalization is its regularization property. Consider, for example, linear regression with random errors deviating from normal. It has been well recognized that in this case, even with a low dimensionality, regularization may be needed to generate reliable estimation. This can be achieved with the proposed penalization.

As in other penalization problems, the identification of important effects is achieved by examining the estimates. With a sequence of tuning parameters, a sequence of identification results can be generated, which is “equivalent” to varying the p-value cutoff in significance-based approaches. When it is desirable to generate one fixed set of identification results, we propose using the extended BIC (EBIC; Chen and Chen, 2008) criterion, which has been extensively adopted in high-dimensional penalization problems. We note that high dimensional inference under penalization is still a highly challenging problem, and the present problem has remarkable differences from the joint penalized estimation. Our preliminary exploration does not suggest an easy way of making rigorous inference. As such, as a limitation of the proposed approach, there is not a simple significance interpretation associated with the proposed identification.

Computation

It is first noted that, as in other marginal analysis, computation can be conducted in a highly parallel manner. Only the tuning parameter selection demands pooling all genes together, which is needed in other marginal analyses too.

First consider a continuous response and linear models. To simplify notation, we suppress the dependence and hence subscript on j. For subject i(= 1,…,n), let Y_i be the response of interest, X_i· = (X_i1,…, X_iq) be the q-dimensional vector of E measurements, and Z_i. be the jth G measurement, consider the model

$$Y_i={\alpha }_0+{\mathit{X}}_i.\alpha +{\mathit{W}}_i.\mathit{b}+{\epsilon }_i,$$ (5)

where ε_i is the random error, W_i· = (Z_i·, Z_i·X_i1,…, Z_i·X_iq)′. With normalization, α₀ = 0. Consider the objective function

$$Q(\alpha ,\mathit{b})=\frac{1}{2n}\Vert \mathit{Y}-\mathit{X}\alpha -\mathbf{W}\mathit{b}{\Vert }_2^2+P(\mathit{b};\xi ,{\lambda }_1,{\lambda }_2),$$ (6)

where Y is the vector composed of Y_i’s, and X and W are the matrices composed of X_i·’s and W_i·’s. With fixed tuning parameters, we consider the following iterative algorithm: (i) Initialize $\widehat{\mathit{b}}=0$; (ii) With the estimate of b fixed at $\widehat{\mathit{b}}$, compute $\widehat{\mathit{\alpha }}=({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}{\mathit{X}}^{\prime }(\mathit{Y}-\mathit{W}\widehat{\mathit{b}})$; (iii) Compute $\widehat{\mathit{b}}$ as the minimizer of the objective function with α fixed at $\widehat{\mathit{\alpha }}$; (iv) Iterate Step (ii) -(iii) until convergence. In this computation, the key is Step (iii), and the details of this step are provided in Appendix. Different from joint analysis, here each penalized estimation is a low dimensional problem. Convergence properties of this iterative algorithm can be established following the literature (Liu et al., 2013). Convergence is achieved in all of our numerical studies.

The proposed approach and algorithm are computationally highly affordable. For a simulated dataset with n = 500, 4 E variables, 1,000 genes, and 400 pairs of (λ₁, λ₂) values, computation can be accomplished within 10 seconds using a regular laptop without parallel computing. It is expected that computer time can be much reduced with parallelization.

The proposed approach is applicable to a wide variety of data types and models. In Appendix, we provide details on accommodating censored survival data under the accelerated failure time (AFT) model, which is examined in our data analysis. Consider for example a binary response and logistic regression models, as analyzed below in simulation. In this case, we iterate between a Taylor expansion that keeps the quadratic term and the above computational algorithm. This strategy has also been adopted in the literature (Breheny and Huang, 2011).

3. Simulation

For each subject, we first simulate q = 4 normally distributed E factors with marginal mean 0 and variance 1. The correlation between the jth and kth E factors is ρ^∣j–k∣ with ρ = 0.2. We then dichotomize one E variable at 0, which leads to three continuous and one binary E variables for analysis. For the G variables, we consider two simulation strategies. First we sample SNP measurements from the GENEVA diabetes data (details below). To maintain the correlation structure, we sample consecutive SNPs. This way, realistic SNP distributions and correlations can be achieved. In addition, we also simulate from parametric distributions, as has been done in a large number of published studies. In simulating continuous distributions to mimic gene expression data, we generate multivariate normal distributions with marginal mean 0 and variance 1. For the correlation of G factors, we consider two structures. The first is the AR (auto-regressive) structure, under which genes j and k have correlation ρ^∣j–k∣ with ρ = 0.2. The second is the Band (banded) structure, under which genes j and k have correlation 0.6 if ∣j – k∣ = 1, 0.33 if ∣j – k∣ = 2, and 0 otherwise (we have also examined negative correlations and made similar findings). To mimic SNP data, we further dichotomize the continuous G variables simulated above, setting MAF (minor allele frequency) as 0.3. We consider the number of G factors p = 500 and 1,000. Among the p main G effects and p × q G-E interactions, 16 and 12 are set as associated with the response. In addition, to examine whether performance of the proposed approach scales well, we also consider the scenario with 8 important main G effects and 6 interactions. The main effects and interactions are set such that the hierarchy is respected. All E factors have important main effects. The nonzero coefficients of important effects are randomly generated from a uniform distribution Unif[0.2, 0.8] (we have also examined negative coefficients and made similar findings). We simulate continuous responses under linear regression models. Three scenarios for the random error distributions are considered: (a) standard normal distribution (Norm), (b) 0.8N(0,1)+0.2LN(0,1) distribution (MNL), and (c) t distribution with degree of freedom 2 (t). In addition, we also simulate binary responses under logistic regression models with intercept set as 0. Sample size is set as n = 100 or 500.

Besides the sgMCP penalization approach, we also analyze simulated data using (a) Sig, which is the benchmark analysis, ranks interactions based on their p-values, and selects the important ones. A parallel analysis is also conducted on the main effects. This approach does not automatically respect the “main effects, interactions” hierarchy. Approaches such as the one in Bien et al. (2015) are significance-based and respect the hierarchy. However, they are very complex to realize and have not been extensively adopted; (b) Lasso, which, for each marginal model, applies the popular Lasso penalization to the main G effect and interactions. To ensure that all G factors are analyzed on the same ground, all marginal models share the same tuning. Important interactions and main G effects are identified as those with nonzero estimates; (c) MCP, which is the same as (b) but with the MCP penalty; (d) Lassointer, which differs from (b) by applying Lasso to interactions only. This approach includes all main G effects in the model and cannot discriminate important G effects from noises; and (e) MCPinter, which is the same as (d) but with the MCP penalty. Here, comparing with Sig can directly establish the merit of the penalization strategy; comparing with Lasso and MCP can directly establish the merit of respecting the “main effects, interactions” hierarchy; and comparing with Lassointer and MCPinter can establish the benefit of jointly selecting important main G effects and interactions. We note that in the majority of existing studies, identification of important main G effects is also conducted.

It is realized that the tuning parameters of different approaches may have different implications (in Sig, the p-value cutoff is viewed as the tuning parameter). In addition, the penalization approaches have different numbers of tunings. To eliminate the impact of tuning parameter selection as much as possible, we consider a sequence of tunings, compute TP (true positive) and FP (false positive) values at each tuning, and use the AUC (area under curve) under the ROC (receiver operating characteristic) framework for comparing the identification accuracy across approaches. This evaluation approach has been adopted in quite a few recent studies (Shi et al., 2014; Meinshausen and Bühlmann, 2010). In addition, we also consider the partial AUC (pAUC) (Walter, 2015), denoted by AUC_rs where r and s mark the range of the FP value. We specifically consider (r = 0, s = 0.5). Another measure we consider is Top40, defined as the number of TPs when 40 interactions (or main effects) are identified.

Representative results using SNP measurements extracted from the diabetes data based on 200 replicates are provided in Figures 1 and 2. The rest of the results are in Appendix, where for example “C-Norm” stands for continuous response and normal error, and “B-Logistic” stands for binary response and logistic model. Note that for the ROC based evaluation, the AUC (pAUC) × 100 values are presented. It is observed that across all the simulated scenarios, the proposed approach has competitive performance. For example in Figure 1 (with detailed numerical results in Table A1), consider the scenario with continuous response and C-MNL error distribution. For the identification of interactions, the mean AUC values are 69.9 (Sig), 70.9 (Lasso), 69.4 (MCP), 72.6 (Lassointer), 72.1 (MCPinter), and 78.7 (sgMCP). Taking into variation, we can see that the improvement of sgMCP is significant. Similar advantageous performance is observed with the pAUC and Top40 measures. For the identification of main effects, the mean AUC values are 68.4 (Sig), 68.6 (Lasso), 68.3 (MCP), and 72.7 (sgMCP). Note that Lassointer and MCPinter cannot discriminate important main G effects from noises. Similar advantageous performance is observed with the pAUC and Top40 measures. As shown in the “B-Logistic” results, the proposed approach is also advantageous with binary responses. Consider for example Figure 2 with detailed numerical results in Table A2. For the identification of interactions, the mean AUC values are 58.4 (Sig), 40.6 (Lasso), 39.7 (MCP), 44.5 (Lassointer), 44.3 (MCPinter), and 65.2 (sgMCP). For the identification of main effects, the mean AUC values are 52.2 (Sig), 27.9 (Lasso), 30.1 (MCP), and 61.3 (sgMCP). Similar advantageous performance is observed with pAUC and Top40.

Figure 1:

Simulation results based on 200 replication. n = 100, p = 500 and SNP data from real data, which has 16 main effects and 12 interactions. Upper/lower panels: main effects/interactions.

Figure 2:

Simulation results based on 200 replication. n = 500, p = 500 and SNP data from real data, which has 16 main effects and 12 interactions. Upper/lower panels: main effects/interactions.

We further examine performance of the proposed and alternative approaches with selected tunings. Representative results are provided in Table A4 (Appendix). For the Sig approach, as it is found that the numbers of TP are small at target FDR=0.1, we also consider the looser criterion with target FDR=0.5. The FP and TP values suggest competitive performance of the proposed approach with EBIC selected tunings. For example, with C-Norm and GE data, the (TP, FP) values are (4.29, 0.72) for Sig_0.1 (Sig with FDR=0.1), (9.78, 16.67) for Sig_0.5, (15.52, 15.7) for Lasso, (15.13, 31.64) for MCP, and (16.67, 13.25) for the proposed sgMCP. As Lassointer and MCPinter cannot select important main effects, results are not presented. When taking both TP and FP into consideration, the proposed approach is clearly advantageous.

We have also examined a few other simulation settings and observed similar patterns (results omitted). It is noted that overall the identification performance of all approaches may be not as “exciting” as in some publications. This is caused by the higher complexity of our simulated data, including weak signals, correlations among variables, etc. Comparing with alternatives, especially the significance-based method, on exactly the same ground can convincingly establish the merit of the proposed approach. It is also noted that the simulated data have dimensions lower than that in a whole genome study. Marginal analysis analyzes one gene at a time, and the summary performance is not as strongly affected by the number of genes. In addition, in practical data analysis, screening is often conducted prior to modeling to reduce dimensionality. As such, the simulated settings can be sufficient.

4. Data analysis

4.1. GENEVA diabetes data

The Health Professionals Follow-up Study (HPFS) was launched in 1986 and organized by the National Institutes of Health (NIH) as part of the Gene Environment Association Studies (GENEVA). We analyze the GENEVA type 2 diabetes data, where a major goal is to identify genetic factors that are associated with type 2 diabetes phenotypes, biomarkers, and others. In our study, data are downloaded from dbGaP (accession number phs000091.v2.p1). Here we focus on the analysis of BMI (body mass index), which is continuously distributed. BMI is the principal measure of adiposity which plays an important role in cardiovascular diseases and metabolic disorders, such as hypertension and diabetes. Following recently published studies, we take a “loose” definition of E factors. Specifically, E factors considered include age, family history of diabetes among first degree relatives (famdb), total physical activity (act), trans fat intake (trans), cereal fiber intake (ceraf), and heme iron intake (heme), all of which have been suggested as potentially associated with BMI. For G factors, we analyze SNPs on chromosome 4, which plays an important role in many disorders, such as Huntington’s disease, Parkinson’s disease, and others. Preprocessing similar to that in Wu et al. (2014) is conducted, which includes subject matching, standard quality control for SNPs, and missing data imputation. The working dataset contains 2,558 subjects with 40,568 SNPs. As the number of SNPs that are associated with BMI and related disorders and diseases is not expected to be large, we further conduct a marginal screening based on SNPs’ marginal associations with BMI. This screening may help generate more reliable findings, although we note that this step is not essential. We select the region of 10,000 consecutive SNPs with the smallest sum of p-values for downstream analysis. Here we select a region as opposed to individual SNPs to keep the physical adjacency (and hence correlation) structure.

With a continuous outcome variable, we adopt the linear regression models. With the EBIC selected tunings, the proposed approach identifies 63 main SNP effects and 115 G-E interactions. The detailed estimation results are provided in Table 1, where we also present genes that the SNPs belong to or are closest to. Note that the 0.0000’s represent estimates with small magnitudes but are actually not zero. Interactions with all six E factors are observed. Similar findings have also been made in the literature. It is clearly observed that the “main effects, interactions” hierarchy is respected. Previous literature suggests that some genes have direct effects on BMI. For example, FAM13A expression is abundant in human adipose tissues. It is shown that the expression of FAM13A is associated with adipose morphology, specifically adipocyte cell numbers (hyperplasia). FAM13A is also identified to promote fatty acid oxidation (FAO) possibly by interacting with and activating Sirtuin 1, which contributes to body fat distribution and is strongly associated with waist to hip ratio. The enzyme encoded by gene GK2 has a notable effect on the regulation of glycerol uptake and metabolism, which is associated with obesity. PPA2 may have a function in feeding behavior via controlling the phosphate level of the cell, and PPA2 is a negative regulator of the insulin metabolic signaling pathway and may contribute to abnormal BMI. The cellular m6A methylation status plays a role in the regulation of fat mass and obesity, and YTHDF1 is one of m6A readers/effector proteins. In addition, some genes can be related to BMI by affecting other diseases. A large number of studies support the hypothesis that many serious diseases may increase the risk of underweight. MIR1269A is involved in multiple cellular programs, including proliferation, differentiation, apoptosis, and differentiation, which are some of the most important processes primarily altered during the development and progression of complex diseases, including breast cancer, endometrial cancer, lung squamous cell cancer, and melanoma. The loss of heterozygosity in NAA11 is shown to correlate with a poor prognosis in hepatocellular carcinoma patients. The lack of NAA11 expression also contributes to human cancerous tissues. NAA11 is found to co-express with NAA10, which is associated with different types of cancer, in several human cell lines. Subjects with these cancers are likely to have low BMI as they suffer from pain, depression, and surgery. Obesity in men, particularly when central, is associated with lower total testosterone, free testosterone and sex hormone-binding globulin. Those hormones are related to the high expression of C4orf22 in testis, trachea, lung, fetal lung and epididymis. Furthermore, an even higher expression of C4orf22 is shown in the condition of soft tissue tumor and muscle tissue tumor, which may cause underweight.

Table 1:

Analysis of the GENEVA diabetes data using the proposed approach: identified main effects and interactions.

SNP	Gene*	main	age	famdb	act	trans	ceraf	heme
rs17090278	RP11-593F5.2	−33.9906	0.4713	0.0118	0.0000	0.0450	0.0477
rs17090286	RP11-593F5.2	−34.0006	0.4726	0.0107	0.0000	0.0381	0.0411
rs13122165	RP11-593F5.2	−12.2239	0.1648
rs17828144	RP11-593F5.2	−23.5662	0.3333
rs17085296	RP11-63H19.1	−28.9288	0.3775	−0.1094	−0.0027	−0.1187	0.0572	1.3963
rs1430504	RP11-707A18.1	−7.3531	0.0897	−0.0043	0.0002	−0.0400	0.0222	0.5266
rs6551878	RP11-707A18.1	−29.5479	0.3729	−0.0271	0.0004	−0.0992	0.0962	1.4930
rs6823601	RP11-707A18.1	−29.5479	0.3729	−0.0271	0.0004	−0.0992	0.0962	1.4930
rs13107026	MIR1269A	0.0391
rs1397755	MIR1269A	0.0925
rs13151560	MIR1269A	0.0459
rs1858306	MIR1269A	0.0202
rs10016795	MIR1269A	0.0072
rs12331987	MIR1269A	0.0504
rs10000219	MIR1269A	0.0492
rs4860208	RPS23P3	0.0844
rs1511286	RPS23P3	0.0871
rs2136822	RPS23P3	0.0032	0.0000
rs11936928	RPS23P3	0.0321	0.0001	−0.0002	0.0000	−0.0024	0.0007	0.0272
rs6838523	RPS23P3	0.0289	0.0000	−0.0002	0.0000	−0.0023	0.0007	0.0264
rs17088752	UBA6-AS1	30.1278	−0.4304
rs17088764	UBA6-AS1	0.0052	0.0000	−0.0001
rs353169	UBA6-AS1	0.0052	0.0000	−0.0001
rs10033058	YTHDC1	0.0079
rs2293595	YTHDC1	0.0172
rs17089267	YTHDC1	0.0182
rs11249477	CSN1S2AP	−0.0962	0.0000	0.0001	0.0000	0.0011	0.0010
rs1399247	CSN1S2AP	−0.0260
rs1717600	CSN1S2AP	−0.0362
rs10003790	DCK	13.0274	−0.1394	−0.0888	−0.0017	0.1351	−0.0814	−1.7329
rs10012631	DCK	9.6975	−0.1012	−0.0677	−0.0013	0.1095	−0.0644	−1.3904
rs9790462	DCK	0.0170	−0.0001	0.0000	0.0000
rs12649753	RN7SL218P	0.0127	0.0000	0.0000	0.0000	−0.0001	0.0001	0.0023
rs7681755	LINC01088	−0.0727
rs11731223	NAA11	−0.0097
rs17003746	GK2	−0.0105
rs17003749	GK2	−0.0051
rs10004901	C4orf22	−0.0343	0.0000	−0.0002	0.0000	0.0010	0.0014
rs1391262	RP11-689K5.3	−0.0340
rs35036928	RP11-689K5.3	−0.0752
rs4693369	RP11-689K5.3	−0.0603
rs7672440	RP11-689K5.3	−0.0739
rs676592	RP11-689K5.3	−0.1460
rs1993798	RP11-689K5.3	10.0827	−0.1347	−0.0039	0.0001	−0.0138	−0.0119
rs2868257	RP11-689K5.3	6.1677	−0.0820
rs6535281	RP11-689K5.3	7.4534	−0.0984	−0.0035	0.0001	−0.0124	−0.0107
rs612318	RP11-689K5.3	−0.0542
rs1824657	RP11-689K5.3	0.0547	−0.0001	−0.0001	0.0000	−0.0003	−0.0003
rs11722328	RP11-689K5.3	0.0751	0.0000	0.0000	0.0000	−0.0045
rs2199487	RP11-689K5.3	0.0752	0.0000	0.0000	0.0000	−0.0047
rs392112	RP11-218C23.1	−0.0232
rs434193	RP11-218C23.1	−0.0219
rs6842681	RP11-218C23.1	−0.0107
rs416035	RP11-218C23.1	−0.0274
rs432755	RP11-218C23.1	−0.0139
rs375432	RP11-218C23.1	−0.0186
rs407430	RP11-218C23.1	−0.0168
rs400023	RP11-218C23.1	−0.0247
rs585787	RP11-218C23.1	−0.0033
rs3775373	FAM13A	−0.0088	0.0000	0.0016
rs2726516	PPA2	0.0071	0.0000	−0.0020
rs2636739	PPA2	0.0085	0.0000	−0.0025
rs2686293	RP13-612N21.1	−0.0751

^*Genes that SNPs belong to or are the closest to.

The stability of findings is evaluated using a resampling approach (Huang and Ma, 2010). Specifically, we randomly sample the subjects and apply the proposed approach. With 500 resamplings, we compute the OOI (observed occurrence index) values, which are the probabilities that interactions and main effects are identified. The OOI values are presented in Table A16 in Appendix. It is observed that in general the findings have high to moderate OOI values. In comparison, we also compute the OOI values for interactions and main effects that are not identified by the proposed approach, and obtain the 95% confidence interval as [0.234, 0.269]. The clear separation of the OOI values between the effects identified and not identified provides support to the validity of our analysis.

Data analysis is also conducted the alternative Sig, Lasso, and MCP approaches. As a large number of the SNPs are expected to be noises, and with inferior performance observed in simulation, Lassointer and MCPinter are not applied. Summary comparison results are provided in Table 3. Detailed estimation results using the alternatives are available from the authors. It is observed that different approaches make quite different discoveries. More specifically, the Sig approach identifies main effects and interactions with almost no overlap with the proposed approach. The MCP and Lasso approaches generate quite similar results, which have moderate overlaps with sgMCP. The observed differences are not surprising with the differences observed in simulation and increased complexity of practical data.

Table 3:

Data analysis: numbers of main G effects and interactions identified by different approaches and their overlaps.

GENEVA	Main effects				Interactions
	Sig	MCP	Lasso	sgMCP	Sig	MCP	Lasso	sgMCP
Sig	51	0	0	3	57	0	0	0
MCP		24	19	19		103	77	11
Lasso			20	19			83	11
sgMCP				63				115
SKCM	Main effects				Interactions
	Sig	MCP	Lasso	sgMCP	Sig	MCP	Lasso	sgMCP
Sig	27	0	0	0	173	0	0	0
MCP		63	63	33		60	59	5
Lasso			66	36			66	5
sgMCP				46				110

4.2. TCGA skin cutaneous melanoma data

The Cancer Genome Atlas (TCGA), as a hallmark cancer genetic program organized by the National Cancer Institute (NCI), has published high quality genetic, epigenetic, transcriptomic, and proteomic data. In this study, we consider skin cutaneous melanoma (SKCM) and download the processed level 3 data from TCGA Provisional using the R package cgdsr. The outcome of interest is (censored) overall survival, whose importance has been established in multiple publications. For G factors, we consider mRNA gene expressions. The analyzed E factors include Age, AJCC nodes pathologic stage (PN), Gender, Breslow’s depth, and Clark level, all of which have been extensively studied in the literature. In TCGA, gene expression measurements are z-scores, which have been lowess-normalized, log-transformed and median-centered, and quantify the relative expressions of tumor samples with respect to normals. Data are available on 298 subjects and 18,934 gene expressions. Among the subjects, 152 died during follow-up. Marginal screening is also conducted, and the 10,000 genes with the strongest marginal associations with survival are selected for G-E interaction analysis.

With a censored survival outcome, we adopt the AFT model with the Kaplan-Meier weighted estimation approach (Stute, 1996), which adds a nonzero weight to each event and a zero weight to each censored subject. Details on estimation under the AFT model are provided in Appendix. It can be seen that the algorithm developed for linear models with uncensored data can be applied here with minor modifications. The proposed approach identifies 46 main G effects and 110 G-E interactions. The detailed estimation results are provided in Table 2. Interactions with all five E factors are identified, and the hierarchy is clearly observed. Published literature suggests that the identification results are biologically sensible. Some genes are found to be directly related to melanoma. Melanoma cell response to a hypoxic environment is transcriptionally regulated by HIF1A, and HIF1A has a higher expression in malignant melanoma, which suggests that it may play an important role in the generation and development of malignant melanoma. The expression level of CUL2 can be used to discriminate between two classes of uveal melanomas. RASSF8 expression is low in metastatic melanoma cells and decreases with melanoma progression. RASSF8 can induce apoptosis in melanoma cells by activating the P53-P21 pathway, and in vivo studies demonstrate that inhibiting RASSF8 increases the tumorigenic properties of human melanoma xenografts. EWSR1 encodes an RNA-binding protein and is involved in the recurrent translocations associated with a number of sarcomas including melanoma. KIF5B contributes to the outward transport of melanosomes and regulates tyrosinase expression level. In addition, KIF5B regulates the intracellular distribution of melanosomes, and the distribution of KIF5B is consistent with that of melanosomes. Targeting KIF5B can reduce melanosome transport and promote melanogenesis. The APC gene promoter is methylated in 60% of primary cutaneous melanomas and 90% of metastases. Reducing APC expression to a certain level may contribute to the development of malignant melanoma. The p53-inducible RRM2B (also named p53R2) gene plays a crucial role in DNA repair and synthesis after DNA damage. Its expression and activity are associated with the resistance of human cancer cells to anticancer agents. RRM2B expression is strongly associated with the progression of melanoma, and the proliferation of melanoma cells is inhibited by RRM2B silencing. Some of the identified genes may be indirectly related to melanoma by affecting other important factors. For example, the increased expression level of gene CASC4 is associated with HER-2/neu proto-oncogene overexpression. CCPG1 may be involved in cell cycle regulation. EEA1 is involved in phagocytic processes and regulates membrane fusion. The product of gene LRP12 is a transmembrane protein that is differentially expressed in many cancer cells. Translocation of GOLGA5 has been found in tumor tissues.

Table 2:

Analysis of the TCGA SKCM data using the proposed approach: identified main effects and interactions.

Gene		Age	PN	Gender	Breslow’s depth	Clark level
ZNF25	7.0336	−0.0876	−0.1225	−0.7268	0.0240
ZNF37A	0.0445	−0.0003	−0.0122	−0.0118	0.0028	0.0090
PJA2	0.0178
KIF5B	0.1047	−0.0001	−0.0141	−0.0153	0.0021
NUDT4	0.0259
KTN1	0.0961	0.0001	−0.0080	−0.0014	0.0006
LRP12	0.0477
TRIP11	0.0428	0.0000	−0.0059	−0.0021	0.0007
EEA1	0.0544
ARHGAP12	6.5295	−0.0752	−0.0825	−1.0930	0.0243
APC	0.0050	0.0000	0.0001	−0.0005
FNDC3A	0.0005	0.0000	0.0000
EIF5	4.7374	−0.0737	−0.1937	0.2275	0.0030	0.6600
DGLUCY	0.0597	0.0000	0.0037	−0.0113
ARHGAP5	0.1040
RRM2B	0.0062	0.0000	−0.0010
UHRF1BP1L	0.0209	0.0000	−0.0005	−0.0002	0.0000
SEL1L	0.0548
VPS13C	0.0275	0.0000	−0.0005
PDP2	0.0014	0.0000	0.0000	0.0000	0.0000
GOLGA5	5.7936	−0.0713	0.0098	−0.3226
HSPA13	0.0019	0.0000	−0.0002	0.0001	0.0001
RASSF8	0.0457
EFR3A	0.0016	0.0000	−0.0001	0.0000	0.0000
PCNX1	0.0194	0.0000	−0.0022	0.0004	0.0004
PPM1A	0.0583	0.0000	−0.0002	0.0000	0.0001
DNAL1	0.0276	−0.0001	−0.0030	−0.0084	0.0008
CUL2	0.0023	0.0000	−0.0005	−0.0002	0.0001
ZMYND11	0.1752	−0.0004	0.0007	−0.0525	0.0003
MPP5	0.0176	0.0000	−0.0003	0.0000	0.0000
CCPG1	0.0332
HIF1A	0.0020
PNMA1	0.0271	−0.0001	0.0006	−0.0060
EXOC5	0.0098	0.0000	−0.0011	0.0000	0.0002
GSKIP	3.0672	−0.0386	−0.0428	−0.0055	0.0056
ARL1	0.0159
EWSR1	−0.0036	0.0000	0.0005
CASC4	0.0143
RCC2	−0.0335
ARHGAP21	0.0534
JKAMP	0.0901
ACBD5	0.0463	−0.0001	−0.0066	−0.0148	0.0016
SETD3	0.0063	0.0000	−0.0007	−0.0009	0.0001
EHMT1	−0.0031
RAB18	0.0127	0.0000	−0.0005	−0.0003	0.0001
CREG1	0.0006

Identification stability is evaluated in the same manner as for the diabetes dataset. The OOI values for the identified main effects and interactions are provided in Table A17. For those effects not identified by the proposed approach, the 95% confidence interval of the OOI values is [0.139 0.180]. The patterns are similar as for the previous data, with the effects identified by the proposed approach having high to moderate OOIs, and with a clear separation of OOI values between the identified and non-identified effects.

Data is also analyzed using the alternatives. The summary comparison results are provided in Table 3, and detailed estimation results using the alternatives are available from authors. It is observed that Sig generates findings with no overlap with the penalization approaches, Lasso and MCP generate highly similar findings, and the proposed approach generates moderate overlapping in main G effect identification with Lasso and MCP but small overlapping in interaction identification.

5. Conclusion

The importance of G-E interaction analysis and demand for new and more effective methods have been well established in the literature. This study has developed a new marginal G-E interaction analysis method, which adopts penalization to respect the “main effects, interactions” hierarchy and achieve regularized estimation. It is advantageous over approaches that do not respect the hierarchy and easier to implement than Bien et al. (2015) and others. It also has an analysis framework more coherent with that of penalized joint interaction analysis, which has been popular in the past a few years. In simulation, it demonstrates notable superiority over the competitors. And in data analysis, different findings, with sound biological implications and satisfactory stability, are made.

Marginal and joint G-E interaction analyses have different objectives. However, it is still of great interest and importance to have coherent analysis frameworks and to “borrow strength” across analysis. This study may mark an important step towards that. There have been great developments in joint G-E interaction analysis in recent literature. It can be of interest to continue “migrating” some of these methods to marginal analysis. The MCP-based penalization adopted in this study can be potentially replaced by other regularization methods that also respect the hierarchy. It can also be of interest to extend our simulation to other types of response, for example count data. Although bioinformatics and statistical evaluations have been conducted with the data analysis results, it is crucial to further validate the findings.

Appendix

Details for Step (iii) (update $\widehat{\mathit{b}}$)

Again, to simplify notation, we suppress the dependence and hence subscript on j. The update of $\widehat{\mathit{b}}$ proceeds as follows. With a fixed $\widehat{\alpha }$, consider the problem with loss function

$$L(\mathit{b})=\frac{1}{2n}\Vert \mathit{r}-\mathbf{W}\mathit{b}{\Vert }^2+\rho (\Vert \mathit{b}\Vert ;\sqrt{q+1}{\lambda }_1,\xi )+\sum _{k=2}^{q+1}\rho (\mid b_{jk}\mid ;{\lambda }_2,\xi ),$$ (A.1)

where W is a n × (q + 1) matrix and $\mathit{r}=\mathit{Y}-\mathbf{X}\widehat{\mathit{\alpha }}$. Assume that W has been orthogonalized with $\frac{1}{n}{W}^{\prime }W=I_{q+1}$. In practice, this can be achieved via Cholesky decomposition and proper transformations.

By setting the first order derivative of (A.1) to zero, we have

$$\frac{\partial L(\mathit{b})}{\partial \mathit{b}}=-\frac{1}{n}{\mathbf{W}}^{\prime }\mathit{r}+\mathit{b}+\frac{\mathit{b}}{\Vert \mathit{b}\Vert }\{\begin{array}{c}\sqrt{q+1}{\lambda }_1-\frac{\Vert \mathit{b}\Vert }{\xi },\mathrm{if}\Vert \mathit{b}\Vert \le \xi \sqrt{q+1}{\lambda }_1\hfill \\ 0,\mathrm{if}\Vert \mathit{b}\Vert >\xi \sqrt{q+1}{\lambda }_1\hfill \end{array}\phantom{\}}+\mathit{h}=0,$$ (A.2)

where

$$\mathit{h}={\left(0,sgn(b_2)\{\begin{array}{c}{\lambda }_2-\frac{\mid b_2\mid }{\xi },\mathrm{if}\mid b_2\mid \le \xi {\lambda }_2\mid \hfill \\ 0,\mathrm{if}\mid b_2\mid >\xi {\lambda }_2\hfill \end{array}\phantom{\}},\dots ,sgn(b_{q+1})\{\begin{array}{c}{\lambda }_2-\frac{\mid b_{q+1}\mid }{\xi },\mathrm{if}\mid b_{q+1}\mid \le \xi {\lambda }_2\hfill \\ 0,\mathrm{if}\mid b_{q+1}\mid >\xi {\lambda }_2\hfill \end{array}\phantom{\}}\right)}^{\prime },$$ (A.3)

and sgn(·) is the sign function. We can rewrite as

$$-\mu +g(\mathit{b})\mathit{b}+\mathit{h}=0,$$ (A.4)

where $\mu =-\frac{1}{n}{\mathbf{W}}^{\prime }\mathit{r}$ and $g(\mathbf{b})=1+\frac{\mathit{b}}{\Vert \mathit{b}\Vert }\{\begin{array}{c}\sqrt{q+1}{\lambda }_1-\frac{\Vert \mathit{b}\Vert }{\xi },\mathrm{if}\Vert \mathit{b}\Vert \le \xi \sqrt{q+1}{\lambda }_1\hfill \\ 0,\mathrm{if}\Vert \mathit{b}\Vert >\xi \sqrt{q+1}{\lambda }_1\hfill \end{array}\phantom{\}}$. Denote μ_k as the kth element of μ. To solve this equation, we first fix g(b) at the current estimate $\widehat{\mathit{b}}$, and denote $\widehat{g}=g(\widehat{\mathit{b}})$. Then we can update the solution to (A.4) as

$$\widehat{g}{b}_1={\mu }_1,\widehat{g}{b}_k=\{\begin{array}{c}\frac{S({\mu }_k,{\lambda }_2)}{1-\frac{1}{\xi \widehat{g}}},\mathrm{if}\mid {\mu }_k\mid \le \xi {\lambda }_2\widehat{g}\hfill \\ {\mu }_k,\mathrm{if}\mid {\mu }_k\mid >\xi {\lambda }_2\widehat{g},\hfill \end{array}\phantom{\}}$$ (A.5)

where $S(\mu ,\lambda )=(1-\frac{1}{\mu })+\mu$. Further, we set

$$\widehat{\nu }=\mathit{b}\widehat{g}.$$ (A.6)

Taking $\widehat{v}$ back into its definition in (A.5), we have

$$\mathit{b}+\frac{\mathit{b}}{\Vert \mathit{b}\Vert }\{\begin{array}{c}\sqrt{q+1}{\lambda }_1-\frac{\Vert b\Vert }{\xi },\mathrm{if}\Vert \mathit{b}\Vert \le \xi \sqrt{q+1}{\lambda }_1\hfill \\ 0,\mathrm{if}\Vert \mathit{b}\Vert >\xi \sqrt{q+1}{\lambda }_1\hfill \end{array}\phantom{\}}=\widehat{\nu }.$$ (A.7)

Solving the above equation, we can obtain the final estimate as

$$\widehat{\mathit{b}}=\{\begin{array}{c}\frac{\xi }{\xi -1}S(\widehat{\nu },\sqrt{q+1}{\lambda }_1),\mathrm{if}\Vert \widehat{\nu }\Vert \le \xi \sqrt{q+1}{\lambda }_1\hfill \\ \widehat{\nu },\mathrm{if}\Vert \widehat{\nu }\Vert >\xi \sqrt{q+1}{\lambda }_1\hfill \end{array}\phantom{\}}.$$ (A.8)

Estimation under the AFT model

For subject i, denote T_i as the survival time of interest. For T_i, consider the accelerated failure time (AFT) model

$$\log (T_i)={\alpha }_0+{\mathit{X}}_i.\alpha +{\mathit{W}}_i.\mathit{b}+{ϵ}_i.$$ (A.9)

Denote C_i as the censoring time for subject i. Under right censoring, we observe Y_i = log(min(T_i, C_i)) and δ_i = I(T_i ≤ C_i). Assume that data {(Y_i, X_i·, W_i·), i = 1,…, n} have been sorted according to Y_i from the smallest to the largest. Consider the following weighted least squared loss function

$$\frac{1}{2n}\sum _{i=1}^n{w}_i[Y_i-{\alpha }_0-{\mathit{X}}_i.\alpha +{\mathit{W}}_i.\mathit{b}{]}^2,$$ (A.10)

where w_i’s are the Kaplan-Meier weights defined as

$$w_1=\frac{{\delta }_1}{n},w_i=\frac{{\delta }_i}{n-i+1}\prod _{i=1}^{i-1}{\left(\frac{n-l}{n-l+1}\right)}^{{\delta }_l},i=2,\dots ,n.$$

With a slight abuse of notation, consider

$$Y_i=\sqrt{w_i}(Y_i-\overline{Y}),{\mathit{X}}_i.=\sqrt{w_i}({\mathit{X}}_i.-\overline{\mathit{X}}),{\mathit{W}}_i.=\sqrt{w_i}({\mathit{W}}_i.-\overline{\mathit{W}}),$$ (A.11)

where $\overline{Y}={\sum }_{i=1}^n{w}_i{Y}_i∕ws$, $\overline{\mathit{X}}={\sum }_{i=1}^n{w}_i{\mathit{X}}_{i.}∕ws$, $\overline{\mathit{W}}={\sum }_{i=1}^n{w}_i{\mathit{W}}_{i.}∕ws$, and $ws={\sum }_{i=1}^n{w}_i$. Then loss function (A.10) can be written as

$$\frac{1}{2n}\Vert \mathit{Y}-\mathit{X}\alpha -\mathbf{W}\mathit{b}{\Vert }_2^2,$$

where notations have similar definitions as in the main text.

Table A1:

Simulation results mean(sd) based on 200 replication. n = 100, p = 500 and SNP data from real data, which has 16 main effects and 12 interactions.

		Main			Interaction
		AUC	pAUC	Top40	AUC	pAUC	Top40
C-Norm	Sig	71.6(0.06)	53.8(0.07)	5.3(1.37)	75.6(0.07)	59.2(0.1)	2.4(1.12)
	Lasso	71.8(0.05)	52.7(0.07)	5.5(1.35)	76.6(0.07)	60.3(0.09)	2.7(1.18)
	MCP	71.4(0.05)	52.4(0.07)	5.4(1.32)	74.5(0.07)	56.8(0.09)	2.5(1.13)
	Lassointer	0(0)	0(0)	1.3(0)	70.2(0.08)	51.3(0.1)	1.9(1.07)
	MCPinter	0(0)	0(0)	1.3(0)	68.1(0.07)	48.6(0.1)	1.8(0.88)
	sgMCP	76.1(0.05)	55.4(0.07)	5.7(1.53)	84.9(0.05)	66.9(0.11)	3.6(1.62)
C-MNL	Sig	68.4(0.06)	49.1(0.08)	4.3(1.34)	69.9(0.08)	51.3(0.11)	1.6(0.95)
	Lasso	68.6(0.07)	48.1(0.09)	4.5(1.31)	70.9(0.07)	52(0.1)	1.9(1.09)
	MCP	68.3(0.06)	48(0.08)	4.5(1.33)	69.4(0.08)	49.8(0.1)	1.8(1.02)
	Lassointer	0(0)	0(0)	1.3(0)	72.6(0.07)	54.2(0.1)	2(1.07)
	MCPinter	0(0)	0(0)	1.3(0)	72.1(0.07)	53.6(0.09)	2(1.09)
	sgMCP	72.7(0.06)	49.9(0.09)	4.7(1.48)	78.7(0.08)	56.8(0.12)	2.2(1.34)
C-t	Sig	67.6(0.07)	48.5(0.09)	4.6(1.44)	61.8(0.1)	40.1(0.12)	1(0.82)
	Lasso	67.4(0.07)	46.5(0.09)	4.5(1.51)	62.3(0.09)	40.1(0.11)	1.1(0.92)
	MCP	67.2(0.08)	46.4(0.1)	4.5(1.53)	61.3(0.1)	38.9(0.11)	1(0.77)
	Lassointer	0(0)	0(0)	1.3(0)	64.4(0.1)	43.6(0.12)	1.4(1.12)
	MCPinter	0(0)	0(0)	1.3(0)	63.4(0.1)	42.2(0.12)	1.3(1.01)
	sgMCP	71(0.07)	48.7(0.1)	4.7(1.49)	71.5(0.08)	47.1(0.12)	1.8(1.15)
B-Logistic	Sig	49.4(0.07)	24.2(0.08)	1.2(0.94)	49(0.1)	23.5(0.1)	0.3(0.54)
	Lasso	18.9(0.13)	20.3(0.1)	1.4(1.06)	25.8(0.13)	24.3(0.1)	0.3(0.45)
	MCP	19(0.13)	20.4(0.1)	1.4(1.12)	16.6(0.13)	18.8(0.11)	0.3(0.49)
	Lassointer	0(0)	0(0)	1.3(0)	39.4(0.1)	23.3(0.09)	0.2(0.46)
	MCPinter	0(0)	0(0)	1.3(0)	37.2(0.1)	23.5(0.09)	0.2(0.45)
	sgMCP	54.5(0.07)	26.1(0.08)	1.6(1.15)	59.4(0.12)	33.4(0.14)	0.5(0.88)

Table A2:

Simulation results mean(sd) based on 200 replication. n = 500, p = 500 and SNP data from real data, which has 16 main effects and 12 interactions.

		Main			Interaction
		AUC	pAUC	Top40	AUC	pAUC	Top40
C-Norm	Sig	95.8(0.02)	90.9(0.03)	12.4(1.16)	93.5(0.04)	87.1(0.05)	6.2(1.05)
	Lasso	95(0.02)	87.7(0.04)	12.1(1.23)	92.4(0.04)	85.6(0.06)	6.5(1.24)
	MCP	95(0.02)	88.3(0.04)	12.1(1.22)	91.9(0.04)	84.6(0.06)	6.4(1.21)
	Lassointer	0(0)	0(0)	1.3(0)	93(0.04)	87.4(0.06)	7.7(1.24)
	MCPinter	0(0)	0(0)	1.3(0)	92.1(0.04)	85.3(0.06)	7.3(1.27)
	sgMCP	96.5(0.01)	90.8(0.04)	12.5(1.6)	97.3(0.01)	92.5(0.03)	6.2(1.21)
C-MNL	Sig	87.5(0.03)	77.9(0.05)	10.1(1.22)	93.8(0.04)	87.7(0.06)	6.4(1.06)
	Lasso	86.9(0.03)	75.7(0.05)	9.8(1.22)	93.4(0.03)	86.9(0.05)	7(1.38)
	MCP	86.9(0.04)	75.7(0.05)	9.7(1.24)	92.6(0.04)	85.6(0.06)	6.6(1.31)
	Lassointer	0(0)	0(0)	1.3(0)	89.6(0.04)	80.9(0.06)	6.1(0.96)
	MCPinter	0(0)	0(0)	1.3(0)	89.1(0.04)	80(0.06)	5.9(1)
	sgMCP	90.8(0.03)	79.2(0.06)	10.4(1.52)	96.7(0.02)	86.3(0.11)	7.6(1.3)
C-t	Sig	84.1(0.09)	72.2(0.14)	8.6(2.33)	82(0.09)	69(0.14)	3.6(1.57)
	Lasso	83.7(0.09)	69.9(0.13)	8.3(2.2)	82.5(0.09)	69.2(0.14)	3.9(1.73)
	MCP	83.6(0.09)	69.7(0.13)	8.3(2.21)	81.4(0.09)	68.1(0.13)	3.9(1.71)
	Lassointer	0(0)	0(0)	1.3(0)	84.3(0.08)	72.4(0.12)	4.7(1.66)
	MCPinter	0(0)	0(0)	1.3(0)	83.5(0.08)	71.1(0.12)	4.3(1.65)
	sgMCP	86.9(0.08)	71.8(0.14)	8.8(2.71)	90.6(0.08)	77.1(0.14)	4.5(1.69)
B-Logistic	Sig	52.2(0.07)	27.5(0.08)	1.6(1.25)	58.4(0.08)	34.9(0.11)	0.7(0.65)
	Lasso	27.9(0.1)	33.7(0.1)	2.9(1.23)	40.6(0.09)	38.5(0.1)	1.1(0.9)
	MCP	30.1(0.09)	32(0.08)	2.8(1.26)	39.7(0.09)	33.6(0.1)	0.9(0.77)
	Lassointer	0(0)	0(0)	1.3(0)	44.5(0.1)	38.7(0.1)	0.8(0.8)
	MCPinter	0(0)	0(0)	1.3(0)	44.3(0.09)	37.9(0.11)	0.8(0.8)
	sgMCP	61.3(0.07)	35.2(0.08)	2.7(1.38)	65.2(0.12)	44.5(0.15)	1.5(1.49)

Table A3:

Simulation results mean(sd) based on 200 replication. p = 500 and GE data with AR Correlation, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	65.5(0.08)	44.4(0.11)	3.8(1.68)	59.2(0.07)	37(0.08)	0.8(0.7)
		Lasso	75.4(0.07)	61(0.09)	6.5(1.4)	68.5(0.07)	51.8(0.09)	2.9(1.01)
		MCP	69(0.08)	51.6(0.11)	5.4(1.82)	61.4(0.07)	42.1(0.08)	2.4(0.94)
		Lassointer	0(0)	0(0)	1.3(0)	67.2(0.07)	47(0.09)	1.8(0.89)
		MCPinter	0(0)	0(0)	1.3(0)	66.6(0.07)	46.2(0.09)	1.7(0.89)
		sgMCP	80.2(0.05)	60.4(0.08)	6.4(1.21)	84.1(0.06)	63(0.11)	4.4(1.57)
	C-MNL	Sig	63.5(0.07)	42.5(0.1)	3.7(1.51)	57.4(0.09)	33.4(0.11)	0.5(0.52)
		Lasso	72.6(0.07)	56.4(0.1)	6.1(1.71)	66.4(0.08)	50.1(0.1)	2.2(1.09)
		MCP	66.4(0.07)	48.4(0.09)	5.3(1.5)	59.1(0.08)	37.9(0.11)	1.9(1.04)
		Lassointer	0(0)	0(0)	1.3(0)	65.2(0.06)	44.8(0.08)	1.7(0.77)
		MCPinter	0(0)	0(0)	1.3(0)	64.4(0.06)	44.1(0.08)	1.7(0.78)
		sgMCP	80(0.06)	60.3(0.1)	6.4(1.76)	81.2(0.07)	58.7(0.13)	3.1(1.28)
	C-t	Sig	60.7(0.08)	37.8(0.1)	3.1(1.58)	53.8(0.09)	31.4(0.11)	0.8(0.77)
		Lasso	67.8(0.08)	49.9(0.11)	4.7(1.8)	61(0.09)	42.6(0.11)	1.8(1.1)
		MCP	64.6(0.08)	45.9(0.11)	4.5(1.59)	54.9(0.07)	34.7(0.08)	1.6(0.94)
		Lassointer	0(0)	0(0)	1.3(0)	63.2(0.08)	42.2(0.11)	1.2(0.92)
		MCPinter	0(0)	0(0)	1.3(0)	62.4(0.09)	40.9(0.11)	1.2(0.93)
		sgMCP	75(0.07)	51.6(0.1)	4.8(1.86)	78.2(0.11)	57.6(0.15)	2.6(1.85)
	B-Logistic	Sig	58.6(0.07)	35.4(0.09)	2.5(1.29)	52.3(0.09)	27.7(0.11)	0.4(0.53)
		Lasso	64.8(0.07)	47.7(0.09)	4.6(1.32)	54.6(0.09)	36.6(0.11)	1.1(0.92)
		MCP	61(0.08)	44.3(0.1)	4.5(1.33)	50.2(0.09)	30.1(0.11)	1.1(0.96)
		Lassointer	0(0)	0(0)	1.3(0)	62(0.07)	42.2(0.09)	1.1(0.82)
		MCPinter	0(0)	0(0)	1.3(0)	61.3(0.07)	41.6(0.09)	1.1(0.8)
		sgMCP	72.6(0.06)	48.2(0.09)	4.4(1.32)	69.2(0.09)	42.1(0.12)	1.6(1.16)
n=500	C-Norm	Sig	92.9(0.04)	86.8(0.05)	11.6(1.27)	79.1(0.05)	64(0.08)	3(0.9)
		Lasso	96(0.02)	93.1(0.03)	13.3(1.15)	88.9(0.06)	83.2(0.07)	7.1(1.36)
		MCP	92.2(0.04)	85.5(0.05)	11.3(1.59)	76.4(0.05)	60.5(0.06)	4.7(0.98)
		Lassointer	0(0)	0(0)	1.3(0)	82.8(0.05)	70.3(0.07)	5.4(0.85)
		MCPinter	0(0)	0(0)	1.3(0)	82.4(0.05)	69.7(0.07)	5.4(0.86)
		sgMCP	98.4(0.01)	93(0.04)	13.1(1.72)	99(0.01)	85.9(0.13)	9.6(1.68)
	C-MNL	Sig	87.4(0.04)	78.1(0.06)	9.9(1.73)	80.8(0.07)	66.4(0.1)	3.4(1.2)
		Lasso	93.1(0.03)	88.9(0.04)	12.4(1.42)	89.6(0.06)	83.7(0.09)	6.8(1.5)
		MCP	87.8(0.05)	79.4(0.07)	10.4(1.73)	81.3(0.07)	69.2(0.1)	5.6(1.5)
		Lassointer	0(0)	0(0)	1.3(0)	87.3(0.05)	78.1(0.07)	6.7(0.92)
		MCPinter	0(0)	0(0)	1.3(0)	86.7(0.05)	76.7(0.07)	6.5(0.89)
		sgMCP	96.3(0.02)	89.8(0.06)	12.6(1.72)	98.2(0.01)	91.9(0.07)	8.9(1.52)
	C-t	Sig	80.3(0.07)	66.9(0.1)	7.3(2)	74.8(0.07)	58.1(0.09)	2.2(1.22)
		Lasso	88.3(0.05)	79.7(0.08)	10.5(2.09)	85.7(0.07)	78.4(0.1)	6.3(1.73)
		MCP	83.3(0.06)	72.8(0.09)	9.4(2.06)	74.8(0.06)	59.3(0.08)	4.2(1.13)
		Lassointer	0(0)	0(0)	1.3(0)	77.6(0.07)	61.6(0.09)	3.6(0.96)
		MCPinter	0(0)	0(0)	1.3(0)	77.4(0.07)	61.2(0.09)	3.6(0.94)
		sgMCP	95.2(0.04)	86.2(0.09)	11.2(2.09)	97(0.04)	82.9(0.16)	8(1.98)
	B-Logistic	Sig	79.9(0.06)	65.6(0.08)	7.2(1.67)	64.8(0.08)	43.5(0.1)	1.1(0.86)
		Lasso	81.8(0.05)	79.1(0.06)	10.4(1.45)	69.5(0.08)	62.9(0.11)	4.7(1.33)
		MCP	77.1(0.06)	70.5(0.07)	9.3(1.79)	59.7(0.07)	47.3(0.09)	3.4(1.07)
		Lassointer	0(0)	0(0)	1.3(0)	73.8(0.07)	62(0.09)	3.5(1.09)
		MCPinter	0(0)	0(0)	1.3(0)	73.5(0.07)	61.4(0.09)	3.5(1.1)
		sgMCP	93.3(0.03)	82.9(0.06)	10.2(1.41)	94.8(0.03)	79.2(0.13)	7.3(1.54)

Table A4:

Simulation: mean (sd) of TP and FP values for main effects and interactions combined. p = 500, n = 500, and AR Correlation. 16 main effects and 12 interactions.

		SNP		GE
		FP	TP	FP	TP
C-Norm	Sig0.1	0.56(0.96)	3.96(2.62)	0.72(1.07)	4.29(2.18)
	Sig0.5	9.69(12.94)	9.65(3.09)	16.67(22.64)	9.78(3.03)
	Lasso	18.69(16.79)	18.3(2.56)	15.7(6.92)	15.52(2.04)
	MCP	126.95(57.53)	20.41(2.33)	31.64(20.47)	15.13(2.39)
	sgMCP	14.47(4.36)	19.83(2.45)	13.25(3.39)	16.67(2.61)
C-MNL	Sig0.1	0.46(0.9)	1.44(1.62)	0.63(1.35)	1.91(1.66)
	Sig0.5	10.21(19.34)	4.57(3.22)	18.63(31.17)	6.23(3.55)
	Lasso	19.6(19.82)	14.47(3.16)	16.22(8.89)	17.99(2.22)
	MCP	120.57(71.23)	15.25(3.19)	28.51(23.34)	11.63(2.45)
	sgMCP	12.26(3.78)	15.85(3.26)	13.97(3.63)	17.82(2.76)
C-t	Sig0.1	4.79(18.64)	0.35(0.66)	5.75(31.64)	1.29(1.54)
	Sig0.5	45.85(140.55)	1.74(2.43)	35.3(96.29)	4.62(3.28)
	Lasso	11.88(16.93)	8.08(3.28)	17.64(14.04)	11.17(3.66)
	MCP	19.89(38.82)	8.3(3.33)	23.47(24.5)	8.52(2.77)
	sgMCP	14.66(26.34)	9.6(3.81)	12.53(14.92)	12.73(3.5)
B-Logistic	Sig0.1	0.12 (0.44)	0 (0)	0.4 (1.03)	1.38 (1.52)
	Sig0.5	8.16 (22.92)	0.2 (0.76)	11 (8.39)	6.34 (2.88)
	Lasso	5.68(3.53)	1.44(1.34)	7.56(2.65)	9.56(2.98)
	MCP	4.92 (3.28)	1.38 (1.01)	6 (2.57)	8.94 (2.15)
	sgMCP	6.58 (3.87)	1.22 (1)	9.3 (3.7)	11.62 (3.02)

Table A5:

Simulation results mean(sd) based on 200 replication. p = 500 and GE data with Band structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	70.4(0.06)	51.6(0.09)	5(1.56)	57.2(0.09)	33.7(0.11)	0.6(0.66)
		Lasso	81.6(0.06)	70.4(0.07)	8.5(1.28)	65.7(0.09)	48.9(0.11)	2.4(1.21)
		MCP	74.4(0.06)	58.7(0.09)	7.1(1.58)	57.7(0.1)	36.6(0.12)	2(1.37)
		Lassointer	0(0)	0(0)	1.3(0)	72.2(0.07)	54(0.09)	2.2(1.04)
		MCPinter	0(0)	0(0)	1.3(0)	71.7(0.07)	52.6(0.09)	2.3(1.04)
		sgMCP	84.8(0.05)	69.3(0.07)	8.2(1.38)	84.5(0.07)	64.9(0.13)	3.5(1.12)
	C-MNL	Sig	63.8(0.09)	42.6(0.11)	3.5(1.56)	57.7(0.11)	35.6(0.12)	0.8(0.74)
		Lasso	72.6(0.07)	57.2(0.09)	6.1(1.55)	65.5(0.09)	47.5(0.1)	1.8(1.02)
		MCP	66.9(0.09)	48.8(0.11)	5.3(1.67)	59.4(0.1)	39.2(0.11)	1.6(1.04)
		Lassointer	0(0)	0(0)	1.3(0)	69.5(0.07)	49.7(0.08)	1.7(0.95)
		MCPinter	0(0)	0(0)	1.3(0)	68.9(0.06)	48.7(0.08)	1.7(0.93)
		sgMCP	80.1(0.05)	60(0.07)	6(1.42)	78.4(0.08)	56.3(0.12)	2.5(1.1)
	C-t	Sig	58(0.09)	35.3(0.13)	2.7(1.82)	53.1(0.09)	29.5(0.1)	0.5(0.53)
		Lasso	66.5(0.09)	47.6(0.11)	4.7(1.7)	62.8(0.09)	45.1(0.12)	1.9(1.18)
		MCP	62.3(0.09)	42.8(0.11)	4.3(1.68)	55.7(0.09)	35.6(0.09)	1.5(0.9)
		Lassointer	0(0)	0(0)	1.3(0)	64.9(0.09)	43.9(0.11)	1.6(1.08)
		MCPinter	0(0)	0(0)	1.3(0)	64(0.09)	43(0.11)	1.5(1.05)
		sgMCP	73.7(0.07)	50.9(0.11)	5(1.66)	78.7(0.08)	57.8(0.13)	2.6(1.67)
	B-Logistic	Sig	58.1(0.06)	35.4(0.07)	2.7(1.02)	54.4(0.07)	29.8(0.09)	0.4(0.48)
		Lasso	64.9(0.06)	48.9(0.07)	4.8(1.3)	54(0.06)	34.3(0.08)	0.9(0.69)
		MCP	60.5(0.06)	44.5(0.08)	4.6(1.38)	52.9(0.07)	32.3(0.08)	0.8(0.66)
		Lassointer	0(0)	0(0)	1.3(0)	55.9(0.07)	33.1(0.09)	1(0.76)
		MCPinter	0(0)	0(0)	1.3(0)	54.9(0.07)	32.4(0.09)	0.9(0.74)
		sgMCP	73.5(0.05)	50(0.07)	4.4(1.38)	69.4(0.09)	40.2(0.11)	1.2(0.97)
n=500	C-Norm	Sig	86.7(0.05)	76.6(0.06)	9.5(1.51)	82.8(0.05)	69.9(0.07)	3.8(1.08)
		Lasso	89.7(0.04)	82.5(0.05)	11.1(1.21)	91.8(0.04)	87(0.06)	7.8(1.36)
		MCP	84.9(0.05)	74.5(0.07)	9.5(1.72)	80.3(0.05)	66.8(0.06)	5(1.01)
		Lassointer	0(0)	0(0)	1.3(0)	92.4(0.04)	85.1(0.05)	6.7(1.19)
		MCPinter	0(0)	0(0)	1.3(0)	91.9(0.04)	84.5(0.05)	6.7(1.18)
		sgMCP	95.3(0.02)	86.4(0.06)	12.4(1.68)	97.3(0.02)	88.9(0.07)	9(1.27)
	C-MNL	Sig	85.7(0.05)	74.8(0.07)	9(1.45)	73.9(0.08)	57.4(0.09)	2.5(1)
		Lasso	90.2(0.04)	83.3(0.05)	10.9(1.19)	84.4(0.08)	77.3(0.1)	6.4(1.54)
		MCP	84.6(0.05)	72.8(0.07)	8.5(1.5)	74.7(0.08)	61.4(0.09)	5.2(1.26)
		Lassointer	0(0)	0(0)	1.3(0)	92.8(0.04)	85.5(0.06)	6.7(1.21)
		MCPinter	0(0)	0(0)	1.3(0)	91.9(0.04)	83.7(0.07)	6.4(1.16)
		sgMCP	94.9(0.02)	85.7(0.06)	12.2(1.26)	95.5(0.03)	81.7(0.11)	7.3(1.58)
	C-t	Sig	76.7(0.11)	61.5(0.14)	6.7(2.37)	69.1(0.08)	51.2(0.11)	2(1.13)
		Lasso	83.2(0.13)	73.2(0.16)	9.2(2.78)	76.7(0.1)	65.3(0.15)	4.8(1.76)
		MCP	77.6(0.14)	65.1(0.17)	8.2(2.55)	68.6(0.08)	51.6(0.12)	3.5(0.97)
		Lassointer	0(0)	0(0)	1.3(0)	84.5(0.07)	72(0.11)	4.6(1.64)
		MCPinter	0(0)	0(0)	1.3(0)	83.7(0.07)	70.9(0.11)	4.4(1.62)
		sgMCP	90(0.11)	77.5(0.16)	9.3(2.76)	93.3(0.11)	77.1(0.21)	6.7(2.41)
	B-Logistic	Sig	76.7(0.06)	61.3(0.08)	6.5(1.44)	70.6(0.07)	51.8(0.09)	2.1(1.04)
		Lasso	78.9(0.04)	73.8(0.06)	9.2(1.45)	77(0.07)	74(0.08)	5.3(1.32)
		MCP	74.4(0.05)	66.9(0.08)	8.4(1.66)	67.4(0.07)	58.2(0.1)	4.1(1.23)
		Lassointer	0(0)	0(0)	1.3(0)	66.3(0.06)	51.5(0.08)	2.7(0.89)
		MCPinter	0(0)	0(0)	1.3(0)	65.4(0.06)	50.3(0.08)	2.7(0.88)
		sgMCP	92(0.03)	80.2(0.07)	9.8(1.25)	91.6(0.13)	76.8(0.13)	5.9(1.31)

Table A6:

Simulation results mean(sd) based on 200 replication. p = 500 and SNP data with AR structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	69.6(0.08)	50(0.11)	4.6(1.47)	57.6(0.07)	34.6(0.09)	0.9(0.75)
		Lasso	73.5(0.06)	55.7(0.09)	4.8(1.74)	76.9(0.08)	68.3(0.09)	3.8(1.06)
		MCP	72(0.08)	56.1(0.1)	6.4(1.82)	59.5(0.08)	39.8(0.1)	2.5(1.36)
		Lassointer	0(0)	0(0)	1.3(0)	67.4(0.08)	47.4(0.1)	2.1(1.02)
		MCPinter	0(0)	0(0)	1.3(0)	66.2(0.07)	45.5(0.09)	1.8(0.86)
		sgMCP	85.1(0.04)	66.9(0.08)	8.3(1.44)	85.3(0.05)	66.9(0.09)	4(1.21)
	C-MNL	Sig	59.4(0.08)	37.4(0.1)	3(1.35)	55.2(0.09)	31.1(0.11)	0.6(0.59)
		Lasso	62.3(0.08)	40.5(0.1)	3.1(1.5)	70.6(0.1)	58.4(0.11)	2.9(1.35)
		MCP	60.7(0.08)	40.2(0.1)	3.2(1.36)	58.8(0.09)	39.6(0.09)	2(1.07)
		Lassointer	0(0)	0(0)	1.3(0)	67.9(0.09)	47.9(0.11)	1.8(1.19)
		MCPinter	0(0)	0(0)	1.3(0)	66.4(0.09)	45.8(0.11)	1.8(1.1)
		sgMCP	75.3(0.08)	53.9(0.11)	5.4(1.8)	79.8(0.08)	56.9(0.14)	2.5(1.41)
	C-t	Sig	57.4(0.09)	33.8(0.1)	2.6(1.38)	55.1(0.1)	30.5(0.12)	0.4(0.74)
		Lasso	59.2(0.12)	37.7(0.11)	3(1.38)	66.1(0.11)	49.8(0.15)	1.9(1.38)
		MCP	56.1(0.11)	33.8(0.1)	3.1(1.52)	58.2(0.1)	37.1(0.12)	1.6(1.19)
		Lassointer	0(0)	0(0)	1.3(0)	58(0.1)	34.2(0.12)	0.9(0.87)
		MCPinter	0(0)	0(0)	1.3(0)	57.3(0.1)	32.9(0.12)	0.9(0.9)
		sgMCP	71.5(0.13)	48.2(0.14)	4.7(2.21)	72.1(0.15)	46.7(0.18)	1.9(1.45)
	B-Logistic	Sig	49.6(0.07)	23.9(0.08)	1.2(0.85)	45.9(0.08)	22.3(0.09)	0.3(0.47)
		Lasso	43(0.12)	33.5(0.1)	2.2(1.38)	29.4(0.11)	26.7(0.09)	0.4(0.59)
		MCP	39(0.12)	27.3(0.09)	1.9(1.19)	23.1(0.1)	22.4(0.07)	0.5(0.59)
		Lassointer	0(0)	0(0)	1.3(0)	44.7(0.09)	26.8(0.09)	0.3(0.5)
		MCPinter	0(0)	0(0)	1.3(0)	41.8(0.1)	26.9(0.08)	0.3(0.51)
		sgMCP	56(0.08)	27(0.1)	1.8(1.31)	56.7(0.1)	28.8(0.12)	0.3(0.61)
n=500	C-Norm	Sig	90.5(0.05)	83(0.06)	11(1.48)	79.5(0.07)	65.5(0.09)	3.3(0.95)
		Lasso	90.5(0.04)	84.4(0.05)	11.4(1)	86.8(0.06)	81.6(0.07)	7.5(1.06)
		MCP	88.8(0.05)	81.4(0.07)	10.9(1.92)	76.5(0.07)	63.8(0.09)	4.1(1.09)
		Lassointer	0(0)	0(0)	1.3(0)	88.4(0.05)	79.1(0.07)	5.8(1.16)
		MCPinter	0(0)	0(0)	1.3(0)	87.8(0.05)	77.9(0.07)	5.6(1.06)
		sgMCP	98.8(0.01)	95.3(0.03)	14(1.07)	97.8(0.01)	91.5(0.09)	8.4(0.95)
	C-MNL	Sig	81.5(0.07)	68.4(0.1)	8.1(1.94)	77.8(0.08)	63(0.11)	3.3(1.3)
		Lasso	85.1(0.05)	75.5(0.08)	9.3(1.54)	88.3(0.07)	84.4(0.09)	7.6(1.58)
		MCP	79.7(0.07)	66.6(0.11)	7.3(2.06)	73.1(0.07)	56.8(0.1)	3(1.05)
		Lassointer	0(0)	0(0)	1.3(0)	78.5(0.07)	65.1(0.09)	4.3(0.99)
		MCPinter	0(0)	0(0)	1.3(0)	78.4(0.07)	65.2(0.09)	4.3(0.99)
		sgMCP	93.8(0.03)	83.8(0.07)	11.7(1.39)	98(0.02)	83.8(0.15)	8.5(1.65)
	C-t	Sig	69.3(0.09)	50.5(0.12)	4.8(1.87)	70.1(0.1)	50.4(0.13)	1.5(1.06)
		Lasso	73.6(0.07)	57(0.1)	5.8(2.05)	84.1(0.07)	76.8(0.1)	6.3(1.64)
		MCP	70.1(0.08)	52.8(0.11)	5(2.04)	68.6(0.08)	48.8(0.1)	2.7(0.9)
		Lassointer	0(0)	0(0)	1.3(0)	78.6(0.08)	64(0.11)	3.7(1.4)
		MCPinter	0(0)	0(0)	1.3(0)	77.7(0.08)	62.4(0.12)	3.6(1.32)
		sgMCP	86.9(0.07)	72.9(0.11)	9(2.17)	89(0.07)	75.7(0.11)	4.8(2.12)
	B-Logistic	Sig	56.2(0.1)	32.5(0.1)	2.3(1.52)	56.9(0.08)	34.4(0.1)	0.6(0.66)
		Lasso	46.5(0.07)	40.9(0.08)	3.2(1.28)	43.3(0.08)	42(0.1)	0.8(0.79)
		MCP	43.4(0.08)	36.2(0.09)	3.2(1.24)	39(0.08)	34.1(0.08)	0.7(0.66)
		Lassointer	0(0)	0(0)	1.3(0)	49.4(0.08)	39.5(0.11)	0.9(0.81)
		MCPinter	0(0)	0(0)	1.3(0)	50.3(0.08)	40.8(0.11)	0.9(0.79)
		sgMCP	60.9(0.06)	33.9(0.08)	2.6(1.35)	60.1(0.09)	31.5(0.11)	0.5(0.89)

Table A7:

Simulation results mean(sd) based on 200 replication. p = 500 and SNP data with Band structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	65.6(0.07)	46.3(0.08)	4.3(1.25)	63.8(0.08)	41.9(0.1)	1.1(0.77)
		Lasso	69.1(0.07)	50.3(0.09)	4.5(1.27)	71.8(0.08)	57.9(0.11)	3(1.4)
		MCP	62.8(0.07)	41.6(0.1)	4.1(1.69)	59.4(0.08)	36.1(0.09)	1.5(0.87)
		Lassointer	0(0)	0(0)	1.3(0)	65.8(0.08)	45.5(0.11)	1.8(0.99)
		MCPinter	0(0)	0(0)	1.3(0)	64.8(0.08)	43.6(0.11)	1.7(0.93)
		sgMCP	81.9(0.05)	63.3(0.09)	7.2(1.34)	82.4(0.06)	63.2(0.12)	3.1(1.26)
	C-MNL	Sig	61.2(0.08)	39.5(0.09)	3.1(1.43)	56.1(0.08)	33(0.1)	0.6(0.67)
		Lasso	66.7(0.08)	47.5(0.11)	4.3(1.77)	67.6(0.09)	53.2(0.11)	2.2(1.24)
		MCP	63.5(0.08)	43.1(0.11)	3.8(1.72)	56.8(0.08)	35.6(0.1)	1.6(0.89)
		Lassointer	0(0)	0(0)	1.3(0)	69(0.09)	49.2(0.12)	1.8(1.07)
		MCPinter	0(0)	0(0)	1.3(0)	68.2(0.1)	48.2(0.12)	1.8(1.08)
		sgMCP	76.8(0.06)	55.7(0.09)	6(1.7)	74.3(0.07)	49.6(0.11)	2(1.01)
	C-t	Sig	55.6(0.07)	32.4(0.1)	2.6(1.39)	56.5(0.11)	32.9(0.12)	0.5(0.63)
		Lasso	59.2(0.08)	36.6(0.1)	2.9(1.48)	67.1(0.1)	52.7(0.12)	2.5(1.04)
		MCP	57.6(0.08)	35.5(0.1)	3.1(1.41)	57.5(0.09)	35.7(0.09)	1.8(1.04)
		Lassointer	0(0)	0(0)	1.3(0)	64.7(0.09)	43.6(0.12)	1.2(0.91)
		MCPinter	0(0)	0(0)	1.3(0)	64.4(0.09)	42.7(0.11)	1.2(0.95)
		sgMCP	69.1(0.07)	44.9(0.09)	4.1(1.41)	75.2(0.1)	52(0.15)	2(1.46)
	B-Logistic	Sig	47.5(0.08)	22.6(0.09)	1.1(0.96)	49.4(0.08)	25(0.1)	0.3(0.42)
		Lasso	43.4(0.11)	30.9(0.09)	2.3(1.15)	31.1(0.12)	27.6(0.09)	0.6(0.71)
		MCP	42.1(0.11)	27.8(0.08)	2.1(1.29)	25.1(0.12)	24(0.1)	0.3(0.53)
		Lassointer	0(0)	0(0)	1.3(0)	41(0.09)	25.8(0.09)	0.3(0.51)
		MCPinter	0(0)	0(0)	1.3(0)	36.2(0.1)	26(0.09)	0.3(0.52)
		sgMCP	57.1(0.08)	28.4(0.09)	1.8(1.23)	59.5(0.1)	30.6(0.12)	0.7(0.78)
n=500	C-Norm	Sig	89.8(0.05)	82(0.07)	10.8(1.34)	80(0.08)	65.8(0.11)	3.2(1.4)
		Lasso	91.6(0.04)	85.9(0.05)	11.1(1.36)	88.6(0.06)	84.1(0.07)	7(1.41)
		MCP	85.2(0.05)	74.6(0.08)	7.8(1.83)	71.2(0.08)	52.2(0.13)	2.8(1.06)
		Lassointer	0(0)	0(0)	1.3(0)	90.3(0.04)	83(0.06)	6.7(1.04)
		MCPinter	0(0)	0(0)	1.3(0)	90.2(0.04)	81.9(0.06)	6.8(1.05)
		sgMCP	97(0.01)	90.1(0.04)	13.4(1.11)	96.6(0.02)	85.2(0.08)	6.9(1.04)
	C-MNL	Sig	79.5(0.07)	65.8(0.1)	7.5(1.88)	76(0.07)	60.3(0.08)	2.9(1.15)
		Lasso	83.6(0.05)	72.3(0.07)	8.3(1.4)	87.2(0.07)	83.6(0.07)	7.7(1.39)
		MCP	77.8(0.06)	63.9(0.08)	7.5(1.8)	72.7(0.06)	58.1(0.07)	3.5(1.24)
		Lassointer	0(0)	0(0)	1.3(0)	94.3(0.04)	88.2(0.06)	7.4(1.32)
		MCPinter	0(0)	0(0)	1.3(0)	93.8(0.04)	87.7(0.06)	7.2(1.33)
		sgMCP	92.5(0.03)	80.7(0.07)	11.2(1.59)	97.9(0.02)	85.6(0.15)	7.9(1.89)
	C-t	Sig	72.3(0.13)	55.5(0.15)	5.6(2.22)	66.2(0.1)	44(0.12)	1.2(0.86)
		Lasso	75.6(0.13)	61.2(0.15)	6.7(2.19)	82.7(0.09)	75.5(0.12)	6.1(1.85)
		MCP	71.7(0.11)	55(0.14)	5.8(2.16)	67.7(0.09)	49.9(0.11)	4(1.25)
		Lassointer	0(0)	0(0)	1.3(0)	81.1(0.08)	67.8(0.12)	4.2(1.6)
		MCPinter	0(0)	0(0)	1.3(0)	80.5(0.08)	66.8(0.12)	4.1(1.57)
		sgMCP	88.4(0.08)	75.7(0.15)	9.5(2.86)	93.9(0.06)	82.6(0.15)	6(3)
	B-Logistic	Sig	53(0.08)	28.5(0.09)	1.6(1.08)	55.6(0.09)	31.2(0.11)	0.5(0.56)
		Lasso	43.8(0.06)	41.8(0.07)	3.7(1.28)	39.5(0.09)	35.4(0.11)	0.7(0.76)
		MCP	40.4(0.05)	37(0.05)	3.3(1.12)	35.6(0.08)	28.5(0.1)	0.8(0.67)
		Lassointer	0(0)	0(0)	1.3(0)	45.6(0.1)	36(0.11)	0.7(0.77)
		MCPinter	0(0)	0(0)	1.3(0)	46.9(0.09)	37.2(0.11)	0.8(0.78)
		sgMCP	60.1(0.06)	32.4(0.08)	2.6(1.22)	55.2(0.1)	26(0.11)	0.5(0.72)

Table A8:

Simulation results mean(sd) based on 200 replication. p = 500 and GE data with AR structure, which has 8 main effects and 6 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	82.2(0.09)	69.8(0.12)	4.2(1.2)	62.8(0.13)	42(0.15)	0.5(0.5)
		Lasso	91(0.06)	84.3(0.09)	5.6(1.13)	77(0.1)	66.3(0.13)	2.3(0.87)
		MCP	83.1(0.09)	71.2(0.13)	4.5(1.4)	63.1(0.12)	45.5(0.15)	1.7(0.97)
		Lassointer	0(0)	0(0)	0.6(0)	80.7(0.08)	68.6(0.1)	2.4(0.76)
		MCPinter	0(0)	0(0)	0.6(0)	79.8(0.08)	67.4(0.1)	2.4(0.73)
		sgMCP	93.6(0.04)	84.6(0.08)	5.4(1.09)	97.5(0.02)	89.8(0.07)	3.7(0.75)
	C-MNL	Sig	70.6(0.11)	52.3(0.15)	2.6(1.27)	64.5(0.11)	43.7(0.15)	0.5(0.59)
		Lasso	81.2(0.09)	69.8(0.12)	4.3(1.21)	76.3(0.1)	63.3(0.13)	1.8(1.06)
		MCP	73.6(0.09)	58.6(0.13)	3.7(1.16)	68.7(0.11)	52.6(0.16)	1.7(1.05)
		Lassointer	0(0)	0(0)	0.6(0)	82.7(0.08)	70.2(0.1)	2.4(0.76)
		MCPinter	0(0)	0(0)	0.6(0)	81.5(0.08)	68(0.11)	2.2(0.76)
		sgMCP	90(0.06)	74.8(0.12)	4.7(1.23)	90.5(0.06)	73.8(0.16)	2.5(1.02)
	C-t	Sig	62.8(0.1)	42.5(0.14)	1.8(1.03)	63.2(0.1)	41.3(0.13)	0.4(0.42)
		Lasso	69.6(0.11)	52.4(0.14)	2.5(1.29)	73.2(0.11)	58.2(0.14)	1.6(0.9)
		MCP	65.2(0.1)	47.2(0.14)	2.6(1.24)	63.7(0.1)	43.9(0.1)	1.2(0.74)
		Lassointer	0(0)	0(0)	0.6(0)	57.6(0.12)	34(0.14)	0.4(0.49)
		MCPinter	0(0)	0(0)	0.6(0)	57.4(0.12)	34(0.14)	0.4(0.49)
		sgMCP	78.2(0.1)	58(0.13)	3.1(1.19)	88.6(0.07)	72.2(0.15)	2.3(1.37)
	B-Logistic	Sig	64.6(0.09)	43.3(0.12)	1.7(1.02)	56.5(0.11)	32.1(0.14)	0.2(0.31)
		Lasso	75.5(0.08)	64.8(0.1)	3.9(1)	63.5(0.12)	48.6(0.16)	0.9(0.76)
		MCP	71.5(0.08)	60(0.1)	3.7(0.94)	55.4(0.11)	36.4(0.14)	0.7(0.66)
		Lassointer	0(0)	0(0)	0.6(0)	59.1(0.11)	38.4(0.14)	0.5(0.55)
		MCPinter	0(0)	0(0)	0.6(0)	58.5(0.12)	37.9(0.14)	0.5(0.54)
		sgMCP	83.5(0.07)	64.4(0.11)	3.5(1.15)	82.4(0.08)	61.1(0.15)	1.4(0.99)
n=500	C-Norm	Sig	96.1(0.04)	92.1(0.07)	6.6(0.85)	96.1(0.04)	91.8(0.05)	3.7(0.58)
		Lasso	96.8(0.03)	95.1(0.04)	6.8(0.75)	97.1(0.03)	96.1(0.04)	4.8(0.66)
		MCP	94.2(0.04)	89.7(0.07)	6.3(0.88)	93.2(0.04)	88(0.07)	3.8(0.9)
		Lassointer	0(0)	0(0)	0.6(0)	97.2(0.02)	93.2(0.05)	4.2(0.74)
		MCPinter	0(0)	0(0)	0.6(0)	97.1(0.02)	93.1(0.05)	4.3(0.75)
		sgMCP	99.2(0.01)	94.6(0.04)	6.6(0.98)	99.5(0)	94.1(0.04)	5.1(0.64)
	C-MNL	Sig	93.3(0.05)	87.5(0.07)	6.2(0.8)	87.5(0.09)	76.9(0.13)	2.3(0.99)
		Lasso	95.3(0.04)	92(0.06)	6.7(0.69)	95.3(0.04)	92.3(0.05)	4.1(0.68)
		MCP	91.6(0.04)	85.2(0.06)	5.7(0.91)	87.6(0.09)	78.8(0.12)	3.6(0.86)
		Lassointer	0(0)	0(0)	0.6(0)	92(0.06)	85.6(0.08)	3.6(0.79)
		MCPinter	0(0)	0(0)	0.6(0)	91.3(0.06)	84.6(0.08)	3.5(0.72)
		sgMCP	98.5(0.02)	93(0.05)	6.7(0.8)	99.6(0)	93.3(0.09)	5.4(0.66)
	C-t	Sig	71.6(0.11)	53.9(0.15)	2.9(1.38)	78.9(0.11)	64.1(0.16)	1.6(0.95)
		Lasso	80.1(0.11)	68.3(0.14)	4.2(1.27)	87.9(0.1)	80.4(0.14)	3.3(1.15)
		MCP	73.5(0.11)	59.6(0.14)	3.6(1.14)	78.7(0.1)	64.2(0.15)	2.2(1.02)
		Lassointer	0(0)	0(0)	0.6(0)	81.8(0.09)	67.4(0.13)	2(0.87)
		MCPinter	0(0)	0(0)	0.6(0)	81.1(0.09)	66.6(0.13)	2(0.87)
		sgMCP	91.1(0.09)	78.8(0.15)	4.9(1.55)	93.7(0.07)	84.3(0.13)	3.3(1.25)
	B-Logistic	Sig	83.6(0.08)	72.5(0.11)	4.5(1.19)	69.8(0.11)	51.4(0.15)	1(0.58)
		Lasso	84.5(0.06)	83.8(0.09)	5.8(0.91)	73.5(0.1)	70.1(0.14)	2.7(0.94)
		MCP	81.4(0.07)	78.3(0.1)	5.6(0.91)	61.2(0.1)	48.6(0.13)	1.7(0.66)
		Lassointer	0(0)	0(0)	0.6(0)	88.8(0.06)	85.2(0.09)	3.6(0.83)
		MCPinter	0(0)	0(0)	0.6(0)	88.6(0.06)	85.1(0.09)	3.6(0.84)
		sgMCP	97.7(0.03)	91.7(0.05)	6.2(0.79)	97.3(0.03)	84.8(0.12)	4.5(0.84)

Table A9:

Simulation results mean(sd) based on 200 replication. p = 500 and GE data with Band structure, which has 8 main effects and 6 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	69.6(0.08)	50.3(0.12)	2.3(1.28)	79.1(0.12)	64.3(0.16)	1.2(0.75)
		Lasso	77.1(0.09)	63.3(0.11)	3.5(1.04)	91.6(0.05)	86.1(0.07)	3.3(0.78)
		MCP	67.8(0.09)	49.5(0.12)	2.6(1.2)	82.7(0.09)	73.3(0.11)	3.1(0.87)
		Lassointer	0(0)	0(0)	0.6(0)	72.6(0.1)	54.5(0.14)	1.2(0.77)
		MCPinter	0(0)	0(0)	0.6(0)	72.1(0.1)	53.4(0.13)	1.2(0.78)
		sgMCP	89.6(0.05)	76.2(0.09)	4.8(0.94)	95.2(0.04)	85.3(0.08)	3.5(0.74)
	C-MNL	Sig	69.6(0.1)	50.8(0.13)	2.4(0.97)	62.5(0.1)	39.3(0.14)	0.4(0.57)
		Lasso	77.2(0.1)	63.4(0.13)	3.5(1.29)	74.5(0.12)	61(0.17)	1.7(0.89)
		MCP	72.7(0.11)	55.7(0.15)	3.2(1.3)	63.3(0.1)	44.1(0.13)	1.3(0.68)
		Lassointer	0(0)	0(0)	0.6(0)	82.7(0.09)	68.9(0.12)	2(0.77)
		MCPinter	0(0)	0(0)	0.6(0)	81.9(0.09)	67.8(0.12)	2(0.76)
		sgMCP	85.8(0.07)	70.3(0.12)	4(1.12)	89.2(0.09)	74.4(0.14)	2.3(1.13)
	C-t	Sig	63.1(0.11)	41.6(0.15)	1.8(1.29)	57(0.13)	35(0.14)	0.4(0.55)
		Lasso	69.3(0.12)	50.9(0.16)	2.7(1.36)	65.2(0.13)	49.2(0.15)	1.1(0.82)
		MCP	64.2(0.11)	43.8(0.15)	2.3(1.42)	60(0.13)	41.1(0.16)	1.1(0.8)
		Lassointer	0(0)	0(0)	0.6(0)	74(0.11)	56.5(0.15)	1.3(0.86)
		MCPinter	0(0)	0(0)	0.6(0)	73.9(0.11)	56.2(0.15)	1.3(0.86)
		sgMCP	78.4(0.09)	57.8(0.14)	3.3(1.32)	79.8(0.12)	59.1(0.19)	1.3(1.02)
	B-Logistic	Sig	62.2(0.08)	39.9(0.11)	1.7(0.97)	53.5(0.11)	28.2(0.12)	0.1(0.34)
		Lasso	71.2(0.09)	57.9(0.13)	3.2(1.15)	54.9(0.13)	35.5(0.14)	0.4(0.46)
		MCP	66(0.1)	52(0.14)	3(1.1)	52.6(0.11)	32.2(0.13)	0.3(0.42)
		Lassointer	0(0)	0(0)	0.6(0)	64.1(0.1)	44.3(0.14)	0.7(0.63)
		MCPinter	0(0)	0(0)	0.6(0)	63.3(0.1)	43.7(0.13)	0.7(0.62)
		sgMCP	79.2(0.08)	58.7(0.13)	3.1(1.04)	72.2(0.09)	44.5(0.11)	0.8(0.61)
n=500	C-Norm	Sig	95.9(0.04)	92(0.06)	6.6(0.82)	85.8(0.09)	77.2(0.11)	2.9(0.59)
		Lasso	95.7(0.04)	93(0.05)	6.9(0.77)	87.2(0.1)	82.2(0.11)	3.8(0.76)
		MCP	94(0.04)	89.5(0.07)	6.4(0.96)	80.7(0.1)	70(0.12)	2.2(0.52)
		Lassointer	0(0)	0(0)	0.6(0)	97.7(0.02)	93.8(0.04)	4.5(0.61)
		MCPinter	0(0)	0(0)	0.6(0)	97.7(0.02)	93.8(0.04)	4.6(0.61)
		sgMCP	97.5(0.02)	92.8(0.05)	6.9(0.79)	99(0.01)	93.9(0.06)	5(0.56)
	C-MNL	Sig	89.9(0.06)	82.6(0.08)	5.5(1.03)	84.5(0.07)	71.9(0.12)	2.1(0.75)
		Lasso	93.5(0.04)	89.1(0.06)	6.4(0.8)	89.1(0.08)	82.4(0.1)	3.5(0.84)
		MCP	87.5(0.06)	78.5(0.09)	5.3(0.98)	83.3(0.08)	70.7(0.13)	2.5(0.77)
		Lassointer	0(0)	0(0)	0.6(0)	97.3(0.02)	93.6(0.04)	4.3(0.62)
		MCPinter	0(0)	0(0)	0.6(0)	97.2(0.02)	93(0.04)	4.3(0.63)
		sgMCP	98.8(0.02)	94.9(0.03)	7.1(0.68)	98.9(0.01)	89.6(0.16)	4.7(0.8)
	C-t	Sig	79.2(0.12)	64.7(0.18)	3.6(1.6)	72.5(0.11)	55.1(0.16)	1.1(0.88)
		Lasso	87.6(0.1)	78.8(0.14)	5.2(1.65)	82.3(0.13)	74.6(0.16)	2.9(1.27)
		MCP	81(0.11)	68.9(0.15)	4.3(1.48)	75.2(0.11)	62.1(0.14)	2.4(0.98)
		Lassointer	0(0)	0(0)	0.6(0)	80.9(0.1)	66.4(0.13)	2.1(0.71)
		MCPinter	0(0)	0(0)	0.6(0)	80.4(0.1)	66.2(0.13)	2.1(0.71)
		sgMCP	93.7(0.06)	84.1(0.13)	5.5(1.41)	94(0.07)	83.5(0.14)	3.3(1.63)
	B-Logistic	Sig	72.2(0.09)	55.1(0.13)	2.9(1.11)	79.9(0.09)	65.3(0.12)	1.6(0.76)
		Lasso	73.7(0.09)	65.8(0.12)	3.9(1.04)	77.8(0.09)	73.9(0.11)	2.5(0.96)
		MCP	70.2(0.09)	61.3(0.12)	3.5(1.11)	74.7(0.09)	69.3(0.12)	2.3(0.86)
		Lassointer	0(0)	0(0)	0.6(0)	89.1(0.05)	85.4(0.07)	3.5(0.79)
		MCPinter	0(0)	0(0)	0.6(0)	88.9(0.05)	85.5(0.07)	3.5(0.8)
		sgMCP	90.5(0.05)	78.1(0.08)	5.2(0.93)	90.5(0.07)	77.9(0.12)	2.8(1.09)

Table A10:

Simulation results mean(sd) based on 200 replication. p = 500 and SNP data with AR structure, which has 8 main effects and 6 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	75.9(0.1)	60.3(0.13)	3.3(1.15)	72.5(0.12)	54.5(0.16)	0.9(0.72)
		Lasso	78.9(0.08)	64.7(0.11)	3.4(1.11)	86.1(0.1)	79.5(0.12)	3.2(0.8)
		MCP	72.2(0.1)	53.8(0.13)	2.3(1.24)	67.9(0.11)	50.7(0.12)	2.1(0.68)
		Lassointer	0(0)	0(0)	0.6(0)	79.4(0.08)	65.7(0.11)	2(0.89)
		MCPinter	0(0)	0(0)	0.6(0)	78.7(0.09)	64.6(0.11)	2(0.86)
		sgMCP	87(0.06)	72.5(0.1)	4.4(0.89)	93.1(0.05)	83.6(0.09)	3.6(0.88)
	C-MNL	Sig	63(0.11)	41.7(0.13)	1.9(1.08)	56.1(0.11)	32(0.14)	0.3(0.43)
		Lasso	67.3(0.11)	47.8(0.15)	2.1(1.15)	68.5(0.14)	55.1(0.18)	1.6(0.94)
		MCP	62.7(0.11)	42.1(0.15)	1.9(1.17)	57.2(0.12)	37.8(0.15)	1.2(0.89)
		Lassointer	0(0)	0(0)	0.6(0)	67.3(0.11)	48.4(0.14)	0.9(0.72)
		MCPinter	0(0)	0(0)	0.6(0)	66.1(0.11)	46.7(0.15)	0.9(0.73)
		sgMCP	81.3(0.08)	62.9(0.13)	3.7(1.21)	83.5(0.11)	65.1(0.18)	1.9(1.24)
	C-t	Sig	57.3(0.1)	32.4(0.12)	1.2(0.98)	60.4(0.14)	37.4(0.17)	0.3(0.5)
		Lasso	59.5(0.12)	35.8(0.15)	1.4(1.12)	67.1(0.17)	51.1(0.21)	1.2(0.96)
		MCP	58.1(0.11)	35.8(0.16)	1.6(1.37)	59.7(0.13)	38.7(0.14)	0.7(0.61)
		Lassointer	0(0)	0(0)	0.6(0)	62.5(0.12)	40.9(0.14)	0.6(0.65)
		MCPinter	0(0)	0(0)	0.6(0)	61.7(0.12)	39.8(0.14)	0.6(0.64)
		sgMCP	71.3(0.11)	47.4(0.16)	2.2(1.29)	72.3(0.14)	48(0.2)	0.8(0.7)
	B-Logistic	Sig	52.6(0.1)	26.9(0.12)	0.5(0.67)	51.8(0.13)	24.1(0.13)	0.1(0.23)
		Lasso	52.1(0.1)	34.9(0.12)	1.4(0.74)	46.2(0.13)	32.8(0.15)	0.3(0.56)
		MCP	51(0.1)	32.2(0.11)	1.3(0.71)	38.5(0.13)	22.6(0.13)	0.2(0.39)
		Lassointer	0(0)	0(0)	0.6(0)	41.1(0.13)	23.1(0.14)	0.1(0.29)
		MCPinter	0(0)	0(0)	0.6(0)	38.9(0.13)	23(0.13)	0.1(0.29)
		sgMCP	60(0.09)	32.7(0.11)	1.1(0.72)	57(0.14)	30.2(0.17)	0.3(0.55)
n=500	C-Norm	Sig	96.7(0.03)	93.6(0.05)	6.8(0.75)	90.7(0.08)	83(0.12)	2.8(0.9)
		Lasso	96(0.02)	94.3(0.04)	6.7(0.65)	94.8(0.04)	94.6(0.05)	4.9(0.57)
		MCP	95.4(0.03)	92.9(0.04)	6.9(0.64)	80.7(0.09)	68.3(0.13)	2(1.12)
		Lassointer	0(0)	0(0)	0.6(0)	92(0.05)	85.7(0.07)	3.7(0.75)
		MCPinter	0(0)	0(0)	0.6(0)	91.9(0.05)	85.7(0.07)	3.7(0.72)
		sgMCP	99.3(0.01)	96.2(0.03)	7(0.67)	99.8(0)	96.6(0.03)	5.4(0.71)
	C-MNL	Sig	88.4(0.07)	79.1(0.11)	5(1.14)	73.9(0.11)	58.4(0.14)	1.3(0.74)
		Lasso	91.1(0.06)	85.4(0.09)	5.8(1.22)	81.5(0.12)	74.9(0.14)	3.4(0.95)
		MCP	83.4(0.08)	71.1(0.13)	4.1(1.4)	68.7(0.1)	52.8(0.12)	1.3(0.73)
		Lassointer	0(0)	0(0)	0.6(0)	93.3(0.05)	86.9(0.08)	3.5(0.78)
		MCPinter	0(0)	0(0)	0.6(0)	92.7(0.05)	86(0.08)	3.3(0.79)
		sgMCP	97.6(0.02)	90.9(0.05)	6.5(0.9)	98.9(0.02)	91.6(0.09)	4.7(1.1)
	C-t	Sig	78.5(0.1)	64.4(0.13)	3.5(1.33)	71.3(0.09)	51.7(0.14)	1(0.66)
		Lasso	80.4(0.1)	68.5(0.13)	4(1.17)	85(0.08)	78.7(0.11)	3.3(0.92)
		MCP	73(0.1)	54(0.15)	2.7(1.2)	70.2(0.1)	51.7(0.14)	1.7(0.78)
		Lassointer	0(0)	0(0)	0.6(0)	83(0.08)	70.7(0.12)	2.5(0.9)
		MCPinter	0(0)	0(0)	0.6(0)	82.1(0.09)	69(0.13)	2.4(0.95)
		sgMCP	91.2(0.07)	80.9(0.11)	5.3(1.37)	92.3(0.05)	78.4(0.1)	3.4(1.36)
	B-Logistic	Sig	66.2(0.12)	45.6(0.16)	2.1(1.35)	63.9(0.1)	41.9(0.14)	0.7(0.66)
		Lasso	63.8(0.09)	57.8(0.1)	3.3(0.99)	59(0.12)	59.7(0.16)	1.6(0.69)
		MCP	60.9(0.1)	53.9(0.13)	3.1(1.14)	48.5(0.09)	40.6(0.12)	1(0.63)
		Lassointer	0(0)	0(0)	0.6(0)	62.9(0.1)	52.2(0.13)	1.2(0.75)
		MCPinter	0(0)	0(0)	0.6(0)	62.9(0.1)	52.4(0.14)	1.2(0.74)
		sgMCP	78(0.08)	58.4(0.12)	3.4(1.2)	79.1(0.1)	59.1(0.15)	1.3(0.91)

Table A11:

Simulation results mean(sd) based on 200 replication. p = 500 and SNP data with Band structure, which has 8 main effects and 6 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	75.9(0.09)	60.3(0.12)	3.1(1.01)	74.4(0.12)	57.2(0.16)	1.2(0.82)
		Lasso	79(0.07)	65.9(0.09)	3.9(0.93)	84.4(0.09)	77.1(0.12)	2.7(0.99)
		MCP	75.3(0.09)	60.6(0.12)	3.7(1.12)	67.7(0.12)	47.6(0.17)	1.3(0.87)
		Lassointer	0(0)	0(0)	0.6(0)	80.6(0.1)	65.3(0.13)	1.4(0.76)
		MCPinter	0(0)	0(0)	0.6(0)	79.4(0.09)	63.3(0.13)	1.4(0.77)
		sgMCP	90.1(0.05)	77.7(0.08)	5.2(1.09)	94.4(0.04)	83.1(0.1)	2.9(1.09)
	C-MNL	Sig	67.1(0.09)	47.1(0.12)	2.2(1.2)	58.6(0.11)	35.3(0.13)	0.4(0.49)
		Lasso	71.5(0.09)	53.7(0.13)	2.4(1.16)	77.8(0.11)	68.5(0.14)	2.3(0.9)
		MCP	65.6(0.09)	44.9(0.12)	2.1(1.08)	62.2(0.1)	44.9(0.12)	1.5(0.83)
		Lassointer	0(0)	0(0)	0.6(0)	65.1(0.12)	45.1(0.15)	0.8(0.74)
		MCPinter	0(0)	0(0)	0.6(0)	64.4(0.12)	44.2(0.15)	0.8(0.74)
		sgMCP	83.4(0.07)	67.3(0.12)	4(1.18)	88.9(0.07)	73.8(0.14)	2.5(1.09)
	C-t	Sig	62.9(0.12)	41.4(0.16)	1.9(1.25)	57.9(0.14)	34.8(0.15)	0.2(0.36)
		Lasso	63.7(0.11)	42.5(0.14)	2(1.07)	69.7(0.13)	53.8(0.16)	1.2(0.97)
		MCP	60.2(0.12)	37.8(0.16)	1.9(1.22)	60.4(0.13)	40.6(0.13)	0.9(0.63)
		Lassointer	0(0)	0(0)	0.6(0)	64.4(0.13)	43.6(0.16)	0.7(0.68)
		MCPinter	0(0)	0(0)	0.6(0)	63.4(0.13)	42.6(0.16)	0.7(0.68)
		sgMCP	78.2(0.1)	58.3(0.16)	2.9(1.41)	79(0.1)	58(0.15)	1.2(1)
	B-Logistic	Sig	50.9(0.12)	26.9(0.13)	0.7(0.75)	45.4(0.13)	22.4(0.12)	0.1(0.36)
		Lasso	55.8(0.09)	38.3(0.11)	1.5(0.98)	48.2(0.11)	34.1(0.12)	0.3(0.41)
		MCP	53(0.1)	35(0.11)	1.3(0.94)	40(0.12)	24.7(0.13)	0.2(0.38)
		Lassointer	0(0)	0(0)	0.6(0)	49.7(0.12)	29.5(0.14)	0.2(0.4)
		MCPinter	0(0)	0(0)	0.6(0)	48(0.12)	29.6(0.14)	0.2(0.41)
		sgMCP	60.8(0.1)	34.3(0.11)	1.1(0.86)	60.7(0.13)	31.6(0.16)	0.2(0.33)
n=500	C-Norm	Sig	95.3(0.04)	90.4(0.07)	6.5(0.8)	94.2(0.05)	88.4(0.08)	3.4(0.74)
		Lasso	95.3(0.02)	93.9(0.04)	6.6(0.73)	94.2(0.04)	94.2(0.05)	4.8(0.73)
		MCP	89.7(0.03)	82.4(0.06)	4.7(1.04)	86.8(0.05)	80(0.08)	2.1(0.65)
		Lassointer	0(0)	0(0)	0.6(0)	92.3(0.06)	86.2(0.08)	3.5(0.9)
		MCPinter	0(0)	0(0)	0.6(0)	91.5(0.06)	84.2(0.08)	3.3(0.8)
		sgMCP	98.9(0.01)	93.9(0.04)	7.2(0.73)	99.7(0)	94.4(0.04)	5.5(0.45)
	C-MNL	Sig	91.7(0.07)	85.3(0.11)	5.9(1.2)	80(0.1)	68.2(0.12)	2.1(0.76)
		Lasso	93(0.06)	87.6(0.09)	5.9(1)	83.4(0.09)	77.4(0.11)	3.3(0.9)
		MCP	85.2(0.09)	72.6(0.14)	4(1.75)	73.5(0.1)	58.6(0.13)	1.4(0.86)
		Lassointer	0(0)	0(0)	0.6(0)	95.4(0.04)	91.3(0.06)	4.1(0.9)
		MCPinter	0(0)	0(0)	0.6(0)	95.4(0.04)	91.1(0.07)	4.1(0.89)
		sgMCP	97(0.03)	90.6(0.05)	6.7(0.69)	99.5(0)	95.9(0.03)	5(1.1)
	C-t	Sig	82.5(0.1)	70.1(0.15)	4.1(1.47)	68.1(0.13)	50(0.16)	0.8(0.71)
		Lasso	87.2(0.08)	78(0.13)	4.9(1.32)	83.6(0.11)	78.5(0.14)	3.6(0.99)
		MCP	79(0.1)	65.3(0.14)	3.7(1.38)	70.4(0.12)	56.2(0.15)	3(1.04)
		Lassointer	0(0)	0(0)	0.6(0)	69.9(0.12)	52.1(0.14)	1.5(0.72)
		MCPinter	0(0)	0(0)	0.6(0)	69.6(0.12)	51.7(0.14)	1.5(0.7)
		sgMCP	96(0.08)	89.2(0.12)	6.3(1.4)	95.2(0.08)	84(0.14)	3.3(1.28)
	B-Logistic	Sig	64.9(0.09)	43.6(0.11)	1.8(0.98)	56.8(0.12)	36.1(0.12)	0.3(0.47)
		Lasso	66.6(0.07)	60.5(0.09)	3.1(0.86)	52(0.09)	46.7(0.11)	0.2(0.44)
		MCP	61.6(0.09)	55.6(0.11)	3(1.03)	44.6(0.1)	35.1(0.13)	0.3(0.48)
		Lassointer	0(0)	0(0)	0.6(0)	55.2(0.12)	41.7(0.15)	0.6(0.56)
		MCPinter	0(0)	0(0)	0.6(0)	55.2(0.12)	41(0.15)	0.6(0.57)
		sgMCP	73(0.07)	50.7(0.11)	2.5(0.95)	72.9(0.08)	49.6(0.13)	1.1(0.67)

Table A12:

Simulation results mean(sd) based on 200 replication. p = 1,000 and GE data with AR structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	65.5(0.08)	44.9(0.1)	2.8(1.34)	60.4(0.09)	37.4(0.12)	0.6(0.61)
		Lasso	77.3(0.06)	63.6(0.08)	5.4(1.46)	70(0.08)	55.2(0.1)	2.4(1.03)
		MCP	69.7(0.08)	53.5(0.1)	4.8(1.37)	62.9(0.09)	43(0.1)	2.1(1)
		Lassointer	0(0)	0(0)	0.6(0)	69.8(0.08)	51.1(0.1)	1.5(0.9)
		MCPinter	0(0)	0(0)	0.6(0)	68.9(0.08)	49.8(0.1)	1.5(0.9)
		sgMCP	82.5(0.05)	63.8(0.07)	5.2(1.56)	85.1(0.07)	67(0.13)	3.8(1.24)
	C-MNL	Sig	63.7(0.08)	43(0.1)	2.3(1.13)	58.7(0.08)	35.8(0.08)	0.5(0.44)
		Lasso	72.7(0.06)	56.1(0.08)	4.6(1.65)	64(0.09)	45.6(0.11)	1.7(0.89)
		MCP	67.4(0.06)	49.7(0.08)	4.2(1.65)	59.6(0.08)	38(0.1)	1.4(0.76)
		Lassointer	0(0)	0(0)	0.6(0)	72.7(0.07)	54.4(0.09)	1.8(0.93)
		MCPinter	0(0)	0(0)	0.6(0)	72.1(0.07)	53.7(0.09)	1.8(0.92)
		sgMCP	79.6(0.05)	58.7(0.08)	4.7(1.9)	80.4(0.06)	59(0.1)	2.4(1.17)
	C-t	Sig	59.4(0.09)	36.6(0.11)	1.6(1.2)	55.9(0.08)	31.9(0.1)	0.3(0.45)
		Lasso	65.1(0.08)	46.8(0.1)	3.4(1.55)	62(0.08)	42.8(0.1)	1.1(0.72)
		MCP	62.2(0.07)	41.9(0.08)	2.9(1.47)	57.7(0.08)	36.1(0.1)	1(0.78)
		Lassointer	0(0)	0(0)	0.6(0)	64.8(0.09)	42.9(0.11)	0.8(0.76)
		MCPinter	0(0)	0(0)	0.6(0)	64.1(0.09)	41.8(0.11)	0.8(0.76)
		sgMCP	74.1(0.07)	51.2(0.11)	3.5(1.71)	73.8(0.06)	51.9(0.09)	1.1(0.8)
	B-Logistic	Sig	59.1(0.07)	36(0.09)	1.7(1.07)	53(0.1)	29.3(0.1)	0.2(0.46)
		Lasso	63.1(0.07)	46.6(0.08)	3.2(1.36)	55.2(0.09)	38.2(0.11)	0.8(0.69)
		MCP	60.8(0.07)	43.8(0.08)	3(1.27)	50.8(0.1)	31.9(0.11)	0.8(0.72)
		Lassointer	0(0)	0(0)	0.6(0)	54.8(0.08)	33.2(0.09)	0.8(0.69)
		MCPinter	0(0)	0(0)	0.6(0)	54.6(0.08)	33(0.09)	0.8(0.69)
		sgMCP	72.1(0.05)	47.5(0.08)	2.7(1.2)	71.9(0.08)	44.8(0.11)	1.2(1.19)
n=500	C-Norm	Sig	89.9(0.05)	82(0.07)	8.7(1.31)	82.5(0.06)	69.7(0.08)	2.8(0.94)
		Lasso	93.8(0.03)	88.6(0.05)	11.1(1.42)	91.6(0.05)	87.5(0.06)	7.2(1.25)
		MCP	90(0.04)	82.6(0.06)	9.7(1.7)	82.1(0.06)	71(0.09)	5.4(1.39)
		Lassointer	0(0)	0(0)	0.6(0)	85.6(0.06)	74.9(0.08)	4.5(1.1)
		MCPinter	0(0)	0(0)	0.6(0)	84.8(0.06)	73.6(0.08)	4.4(1.06)
		sgMCP	97.4(0.02)	90.1(0.06)	11(2.05)	98(0.02)	89.2(0.11)	8.5(1.56)
	C-MNL	Sig	87.2(0.05)	78.5(0.07)	8.2(1.54)	72.2(0.06)	55.5(0.08)	1.7(0.54)
		Lasso	92.4(0.03)	87.8(0.05)	11.4(1.43)	83.2(0.07)	75.4(0.08)	5.8(1.17)
		MCP	87.8(0.04)	79.8(0.06)	9.9(1.64)	72.9(0.05)	59.3(0.07)	4.2(1.17)
		Lassointer	0(0)	0(0)	0.6(0)	91.3(0.04)	83.3(0.06)	6.2(1.13)
		MCPinter	0(0)	0(0)	0.6(0)	90.6(0.04)	82.4(0.06)	6.1(1.08)
		sgMCP	96.2(0.02)	89.1(0.05)	11.5(1.51)	98.5(0.02)	84.9(0.15)	8(1.77)
	C-t	Sig	77.9(0.09)	64.5(0.13)	5.5(2.04)	70.8(0.09)	52.2(0.13)	1.3(0.86)
		Lasso	87(0.09)	78.7(0.12)	8.9(2.49)	84.2(0.08)	76.3(0.11)	5.6(1.53)
		MCP	80.7(0.09)	69.8(0.14)	7.3(2.34)	73.4(0.07)	59.1(0.09)	4.2(1.53)
		Lassointer	0(0)	0(0)	0.6(0)	83.9(0.07)	71.1(0.1)	3.8(1.09)
		MCPinter	0(0)	0(0)	0.6(0)	83.8(0.07)	70.9(0.1)	3.8(1.08)
		sgMCP	92(0.06)	79.5(0.11)	9.1(2.68)	96.4(0.06)	88.6(0.12)	7.5(2.78)
	B-Logistic	Sig	78.2(0.06)	63.2(0.08)	5.1(1.56)	69.9(0.08)	51.7(0.11)	1.2(0.68)
		Lasso	79.6(0.05)	74.9(0.06)	8.1(1.48)	70.6(0.06)	64.5(0.08)	3.4(0.98)
		MCP	75.7(0.05)	68(0.07)	7.1(1.54)	64.7(0.08)	54.1(0.09)	2.7(0.99)
		Lassointer	0(0)	0(0)	0.6(0)	76.5(0.06)	66.5(0.08)	3.4(0.97)
		MCPinter	0(0)	0(0)	0.6(0)	76(0.06)	65.7(0.08)	3.4(0.95)
		sgMCP	90.6(0.04)	78.4(0.07)	8.2(1.6)	89.3(0.07)	71.3(0.15)	5.3(1.18)

Table A13:

Simulation results mean(sd) based on 200 replication. p = 1,000 and GE data with Band structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	65.2(0.08)	45.3(0.1)	2.6(1.35)	61.5(0.09)	39.7(0.1)	0.6(0.58)
		Lasso	74.9(0.06)	59.7(0.08)	4.9(1.14)	69.5(0.08)	52.6(0.09)	2(0.79)
		MCP	69.2(0.08)	52.2(0.1)	4.6(1.39)	62.3(0.08)	41.6(0.09)	1.6(0.67)
		Lassointer	0(0)	0(0)	0.6(0)	68.6(0.07)	48.9(0.08)	1.6(0.94)
		MCPinter	0(0)	0(0)	0.6(0)	67.7(0.07)	47.8(0.08)	1.6(0.92)
		sgMCP	82.1(0.05)	63.6(0.09)	4.8(1.24)	81.4(0.06)	62.2(0.1)	2.8(1.02)
	C-MNL	Sig	62.7(0.09)	40.6(0.11)	1.9(1.2)	57.8(0.08)	34.9(0.1)	0.4(0.38)
		Lasso	70(0.08)	52.8(0.09)	3.6(1.16)	66.6(0.09)	51.3(0.1)	2.1(0.96)
		MCP	64.6(0.09)	44.6(0.11)	3.2(1.3)	58.7(0.08)	38(0.1)	1.4(0.68)
		Lassointer	0(0)	0(0)	0.6(0)	69(0.07)	49.7(0.09)	1.4(0.97)
		MCPinter	0(0)	0(0)	0.6(0)	68(0.07)	48.1(0.09)	1.4(0.97)
		sgMCP	77.6(0.06)	56(0.08)	4.1(1.22)	81.5(0.07)	61(0.12)	3(0.78)
	C-t	Sig	60.5(0.08)	38.7(0.09)	2(1.06)	55.3(0.09)	31(0.1)	0.2(0.31)
		Lasso	69.7(0.07)	51.9(0.1)	3.8(1.31)	59.5(0.1)	39.4(0.12)	1.1(0.92)
		MCP	64.4(0.07)	45(0.1)	3.3(1.36)	56(0.09)	32.8(0.11)	0.9(0.85)
		Lassointer	0(0)	0(0)	0.6(0)	63.6(0.07)	41.5(0.09)	0.8(0.76)
		MCPinter	0(0)	0(0)	0.6(0)	62.8(0.07)	40.3(0.08)	0.8(0.76)
		sgMCP	75.9(0.07)	53.9(0.11)	3.7(1.75)	76.2(0.09)	54.4(0.14)	1.5(1.19)
	B-Logistic	Sig	56.4(0.08)	32.7(0.09)	1.6(1.05)	54.9(0.08)	29.7(0.1)	0.3(0.46)
		Lasso	60.7(0.08)	44(0.1)	2.9(1.15)	58.3(0.08)	41.3(0.09)	0.8(0.67)
		MCP	57.9(0.08)	40.3(0.11)	2.8(1.18)	55.2(0.08)	36.5(0.09)	0.8(0.65)
		Lassointer	0(0)	0(0)	0.6(0)	59.6(0.07)	39.2(0.09)	0.9(0.68)
		MCPinter	0(0)	0(0)	0.6(0)	58.9(0.07)	38.8(0.09)	0.9(0.68)
		sgMCP	70(0.06)	45.6(0.09)	2.8(1.25)	70.4(0.08)	46.4(0.12)	1(0.97)
n=500	C-Norm	Sig	87.4(0.04)	78(0.07)	8.2(1.42)	81(0.07)	68.1(0.1)	2.6(0.82)
		Lasso	92.7(0.03)	87.8(0.04)	11.3(1.26)	89.8(0.04)	84.3(0.06)	6.7(1.06)
		MCP	87.1(0.04)	78.4(0.06)	9.1(1.71)	78.2(0.07)	64.5(0.1)	4.8(1.19)
		Lassointer	0(0)	0(0)	0.6(0)	87.1(0.05)	76.5(0.06)	5(0.91)
		MCPinter	0(0)	0(0)	0.6(0)	86.5(0.05)	75.6(0.06)	5(0.88)
		sgMCP	96.7(0.02)	89.7(0.05)	11.8(1.25)	98.4(0.02)	89.5(0.08)	7.8(1.03)
	C-MNL	Sig	83.6(0.05)	72.4(0.08)	6.7(1.56)	83.1(0.06)	70.1(0.09)	2.6(0.85)
		Lasso	89.7(0.05)	82.3(0.06)	9.6(1.28)	88.9(0.06)	82.1(0.07)	5.8(1.51)
		MCP	85.2(0.06)	75.7(0.08)	8.5(1.7)	81.3(0.06)	67.2(0.09)	4.1(1.08)
		Lassointer	0(0)	0(0)	0.6(0)	82.1(0.06)	68.4(0.09)	3.6(0.94)
		MCPinter	0(0)	0(0)	0.6(0)	81.7(0.06)	67.9(0.09)	3.6(0.95)
		sgMCP	94.8(0.02)	85.7(0.07)	10.4(1.44)	96.2(0.02)	89.8(0.05)	6.7(1.1)
	C-t	Sig	79.3(0.09)	65.6(0.12)	5.6(1.67)	73.7(0.09)	57.3(0.11)	1.6(0.97)
		Lasso	85.7(0.07)	75.4(0.08)	8.1(1.7)	81.9(0.08)	72(0.11)	5(1.49)
		MCP	81.6(0.08)	69.5(0.11)	7.4(1.99)	73.7(0.08)	59.4(0.09)	3.7(1.2)
		Lassointer	0(0)	0(0)	0.6(0)	85.7(0.07)	74.7(0.11)	4.5(1.54)
		MCPinter	0(0)	0(0)	0.6(0)	85(0.07)	73.3(0.11)	4.5(1.55)
		sgMCP	92.3(0.05)	81.1(0.09)	9.2(2.46)	94.6(0.05)	80.8(0.17)	6.4(1.96)
	B-Logistic	Sig	75.6(0.06)	60(0.09)	4.8(0.94)	74.1(0.08)	57.6(0.1)	1.5(0.75)
		Lasso	77.5(0.06)	72.1(0.07)	7.6(1.58)	78.5(0.06)	75.5(0.08)	5(1.26)
		MCP	72.8(0.06)	64.8(0.08)	6.7(1.47)	70.7(0.08)	63.3(0.1)	4.3(1.16)
		Lassointer	0(0)	0(0)	0.6(0)	66.8(0.06)	53(0.08)	2.8(0.8)
		MCPinter	0(0)	0(0)	0.6(0)	66.3(0.06)	52.2(0.08)	2.7(0.81)
		sgMCP	91.7(0.03)	78.2(0.06)	9.5(1.4)	93.9(0.1)	72.9(0.15)	6.2(1.62)

Table A14:

Simulation results mean(sd) based on 200 replication. p = 1,000 and SNP data with AR structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	66.9(0.07)	46.7(0.09)	2.5(1.19)	63.1(0.08)	41(0.1)	0.7(0.62)
		Lasso	70.6(0.06)	52.3(0.08)	3.2(1.46)	75.8(0.07)	64.2(0.08)	2.8(1.01)
		MCP	64.5(0.06)	43.5(0.09)	3.1(1.73)	60.9(0.08)	38.8(0.1)	1.1(0.76)
		Lassointer	0(0)	0(0)	0.6(0)	70.7(0.08)	51.9(0.1)	1.6(0.93)
		MCPinter	0(0)	0(0)	0.6(0)	69.2(0.08)	49.6(0.09)	1.5(0.81)
		sgMCP	83.7(0.04)	65.2(0.08)	5.9(1.24)	84.7(0.06)	65.7(0.11)	2.6(0.9)
	C-MNL	Sig	60.8(0.07)	37.8(0.08)	1.6(0.98)	56.6(0.09)	33.2(0.1)	0.3(0.43)
		Lasso	63.3(0.07)	41.3(0.09)	1.9(1.15)	68.7(0.09)	55.3(0.11)	2.3(1.19)
		MCP	60.6(0.07)	38(0.09)	1.6(1.08)	56.6(0.09)	35.9(0.11)	1.3(0.64)
		Lassointer	0(0)	0(0)	0.6(0)	70.5(0.08)	52.2(0.11)	1.5(0.97)
		MCPinter	0(0)	0(0)	0.6(0)	69.1(0.08)	50.2(0.12)	1.5(0.95)
		sgMCP	73.4(0.06)	50.7(0.09)	3.6(1.33)	79.9(0.07)	60.2(0.13)	2.7(1.65)
	C-t	Sig	57(0.08)	34.3(0.09)	1.6(1.17)	53.5(0.08)	28.1(0.09)	0.3(0.4)
		Lasso	61.6(0.07)	39.3(0.09)	1.8(1.27)	61.9(0.1)	43.4(0.13)	1.1(0.9)
		MCP	58.4(0.07)	36.1(0.1)	1.9(1.37)	55.9(0.09)	33.7(0.12)	1(0.89)
		Lassointer	0(0)	0(0)	0.6(0)	60.5(0.09)	37.8(0.11)	0.6(0.66)
		MCPinter	0(0)	0(0)	0.6(0)	59.8(0.09)	36.6(0.11)	0.6(0.65)
		sgMCP	71.9(0.08)	48.6(0.11)	3.3(1.79)	71.9(0.09)	49.1(0.12)	1.3(1.01)
	B-Logistic	Sig	45.9(0.09)	20.8(0.09)	0.4(0.62)	47.3(0.09)	21.1(0.1)	0.1(0.28)
		Lasso	43.7(0.13)	33.2(0.08)	1.3(0.92)	28.8(0.1)	26.1(0.09)	0.1(0.28)
		MCP	38.1(0.15)	28.3(0.1)	1.1(0.89)	23.3(0.12)	23.5(0.1)	0.2(0.4)
		Lassointer	0(0)	0(0)	0.6(0.06)	39.5(0.1)	23.5(0.09)	0.1(0.22)
		MCPinter	0(0)	0(0)	0.6(0.06)	35.9(0.1)	24.4(0.09)	0.1(0.22)
		sgMCP	54.9(0.06)	26(0.06)	0.9(0.83)	52.5(0.08)	23.2(0.09)	0.1(0.23)
n=500	C-Norm	Sig	89.5(0.04)	82(0.06)	9.1(1.5)	82(0.07)	69.1(0.09)	2.7(0.78)
		Lasso	91.2(0.03)	85.7(0.04)	10.5(1.62)	89.9(0.06)	87.6(0.07)	8.1(1.34)
		MCP	85.7(0.05)	76.3(0.09)	7.2(2.28)	74.4(0.06)	58.7(0.08)	2.2(0.89)
		Lassointer	0(0)	0(0)	0.6(0)	83(0.05)	72.2(0.07)	4.8(1.18)
		MCPinter	0(0)	0(0)	0.6(0)	82.9(0.05)	72.1(0.06)	4.8(1.17)
		sgMCP	97.6(0.02)	90.2(0.04)	12.7(1.34)	98.8(0.01)	86.7(0.12)	9.1(1.67)
	C-MNL	Sig	83.6(0.05)	71.9(0.07)	6.7(1.56)	77.6(0.07)	62.6(0.09)	2.2(0.9)
		Lasso	85.9(0.05)	75.9(0.08)	7.3(1.53)	88.8(0.06)	84.3(0.08)	7(1.23)
		MCP	76.8(0.06)	58.9(0.09)	4.2(1.59)	69.1(0.07)	48.4(0.09)	2(1.02)
		Lassointer	0(0)	0(0)	0.6(0)	82(0.06)	68.9(0.09)	3.1(1.27)
		MCPinter	0(0)	0(0)	0.6(0)	81.5(0.06)	68.3(0.09)	3.1(1.22)
		sgMCP	94.7(0.03)	84.9(0.06)	10.6(1.62)	98.2(0.02)	82.7(0.13)	7.5(2.31)
	C-t	Sig	71.9(0.1)	54.7(0.13)	4.1(1.85)	65.9(0.1)	45.6(0.12)	0.9(0.64)
		Lasso	76.2(0.08)	61.6(0.11)	5(1.77)	79.6(0.08)	71.2(0.11)	4.8(1.55)
		MCP	71(0.09)	53.4(0.13)	3.8(1.85)	64(0.08)	44.5(0.09)	2.4(0.86)
		Lassointer	0(0)	0(0)	0.6(0)	71.1(0.09)	53.8(0.12)	2(1.37)
		MCPinter	0(0)	0(0)	0.6(0)	70.5(0.09)	52.5(0.12)	1.9(1.27)
		sgMCP	89.2(0.06)	76.2(0.11)	8.2(2.41)	90(0.06)	70.6(0.17)	4.4(2.14)
	B-Logistic	Sig	56.3(0.08)	32.3(0.09)	1(0.99)	56.4(0.08)	32.3(0.1)	0.2(0.33)
		Lasso	47.5(0.07)	44.9(0.07)	2.8(1.02)	41.4(0.06)	39.1(0.08)	0.9(0.82)
		MCP	45(0.06)	39.9(0.08)	2.7(1.01)	37.5(0.07)	31.5(0.09)	0.6(0.58)
		Lassointer	0(0)	0(0)	0.6(0)	46.2(0.09)	36(0.11)	0.5(0.6)
		MCPinter	0(0)	0(0)	0.6(0)	47(0.09)	36.9(0.11)	0.5(0.61)
		sgMCP	64.4(0.06)	38.5(0.08)	2.3(1.38)	64.3(0.08)	38.2(0.09)	0.8(0.74)

Table A15:

Simulation results mean(sd) based on 200 replication. p = 1,000 and SNP data with Band structure, which has 16 main effects and 12 interactions.

			Main			Interaction
			AUC	pAUC	Top40	AUC	pAUC	Top40
n=100	C-Norm	Sig	65.1(0.07)	44.6(0.09)	2.6(1.3)	60.5(0.07)	38.5(0.08)	0.5(0.65)
		Lasso	69.3(0.05)	50.1(0.07)	2.5(1.38)	72.5(0.09)	61.2(0.1)	2.6(1.03)
		MCP	62.7(0.06)	41.8(0.09)	2.4(1.4)	61.4(0.08)	42.1(0.1)	1.9(0.89)
		Lassointer	0(0)	0(0)	0.6(0)	67.6(0.07)	47.8(0.09)	1.3(0.88)
		MCPinter	0(0)	0(0)	0.6(0)	66.7(0.07)	46.2(0.09)	1.3(0.88)
		sgMCP	80.4(0.04)	62.8(0.07)	5.5(1.7)	82.9(0.05)	63.8(0.12)	3(1.15)
	C-MNL	Sig	61.2(0.08)	38(0.1)	1.9(1.27)	59.6(0.08)	37.1(0.09)	0.5(0.53)
		Lasso	66.8(0.08)	46.6(0.11)	2.9(1.52)	71.3(0.08)	59.1(0.1)	2(1.05)
		MCP	64.4(0.09)	44.5(0.12)	3.3(1.67)	59.6(0.08)	37.6(0.09)	1.1(0.76)
		Lassointer	0(0)	0(0)	0.6(0)	69.5(0.09)	50.5(0.11)	1.4(1.03)
		MCPinter	0(0)	0(0)	0.6(0)	67.9(0.08)	48.4(0.11)	1.3(0.98)
		sgMCP	75.3(0.07)	53.9(0.1)	4.3(1.63)	76(0.08)	51.5(0.11)	1.6(1.06)
	C-t	Sig	55.8(0.07)	31.5(0.09)	1.4(1.03)	54.7(0.09)	30.3(0.1)	0.3(0.5)
		Lasso	55.6(0.08)	31.4(0.09)	1.4(0.9)	64.8(0.11)	48.2(0.16)	1.3(1.19)
		MCP	54.5(0.08)	30.4(0.1)	1.6(1.18)	55.3(0.08)	32.3(0.1)	1(0.93)
		Lassointer	0(0)	0(0)	0.6(0)	59.9(0.09)	36.9(0.12)	0.5(0.64)
		MCPinter	0(0)	0(0)	0.6(0)	59(0.09)	35.6(0.11)	0.5(0.61)
		sgMCP	67.3(0.07)	42.3(0.1)	2.4(1.18)	71.4(0.08)	45.8(0.13)	0.9(1)
	B-Logistic	Sig	48.5(0.07)	23.6(0.07)	0.6(0.56)	49.5(0.07)	24.8(0.08)	0.1(0.26)
		Lasso	39.2(0.12)	30.4(0.09)	1.2(0.99)	31.4(0.11)	29.5(0.09)	0.3(0.47)
		MCP	38.1(0.14)	27.8(0.09)	1.1(0.97)	25.1(0.12)	25.7(0.09)	0.2(0.43)
		Lassointer	0(0)	0(0)	0.6(0)	44.9(0.08)	25.7(0.08)	0.1(0.3)
		MCPinter	0(0)	0(0)	0.6(0)	43.7(0.09)	25.9(0.08)	0.1(0.3)
		sgMCP	55.3(0.09)	25.9(0.1)	0.9(0.78)	57.5(0.1)	29.2(0.11)	0.2(0.35)
n=500	C-Norm	Sig	91(0.04)	83.9(0.05)	9.3(1.23)	77.8(0.05)	63.6(0.07)	2.4(0.85)
		Lasso	92.3(0.03)	87.4(0.04)	10.1(1.29)	85.5(0.06)	80.7(0.07)	6.5(1.06)
		MCP	86.4(0.04)	75.8(0.07)	6.3(1.75)	73.7(0.06)	59.3(0.07)	2.4(0.6)
		Lassointer	0(0)	0(0)	0.6(0)	76.3(0.07)	61.6(0.08)	3.2(0.85)
		MCPinter	0(0)	0(0)	0.6(0)	75.6(0.07)	60.6(0.08)	3.1(0.8)
		sgMCP	96.2(0.02)	89.7(0.05)	12.2(1)	99.4(0)	90.7(0.1)	9.1(1.39)
	C-MNL	Sig	81.7(0.06)	69.7(0.08)	6.1(1.52)	79.9(0.07)	65.2(0.1)	1.9(1.1)
		Lasso	83.1(0.06)	72.4(0.08)	7.2(1.74)	91.9(0.04)	88.6(0.06)	7.3(1.49)
		MCP	75.7(0.06)	58(0.09)	4.8(1.88)	74.7(0.06)	58(0.09)	2.4(0.63)
		Lassointer	0(0)	0(0)	0.6(0)	87.4(0.05)	78.3(0.07)	5.5(1.31)
		MCPinter	0(0)	0(0)	0.6(0)	87(0.05)	77.9(0.07)	5.6(1.35)
		sgMCP	94.2(0.03)	83.1(0.08)	10.7(1.73)	97.2(0.02)	89.6(0.08)	7.6(1.53)
	C-t	Sig	68.6(0.09)	50.6(0.11)	3.6(1.68)	61.5(0.08)	39(0.11)	0.6(0.57)
		Lasso	71.1(0.09)	54.1(0.12)	4.5(1.61)	73.8(0.1)	62.8(0.14)	3.7(1.6)
		MCP	66.7(0.09)	47.9(0.11)	3.9(1.95)	61.7(0.08)	41.8(0.1)	2.3(1.13)
		Lassointer	0(0)	0(0)	0.6(0)	79.9(0.09)	67(0.12)	3.6(1.66)
		MCPinter	0(0)	0(0)	0.6(0)	79.2(0.09)	65.9(0.13)	3.5(1.58)
		sgMCP	84.4(0.07)	68.9(0.11)	7.3(1.97)	89.6(0.05)	75.9(0.11)	4.8(2.1)
	B-Logistic	Sig	55.3(0.08)	31.2(0.08)	1(0.92)	55.2(0.09)	33.1(0.11)	0.3(0.44)
		Lasso	46.7(0.07)	42.1(0.08)	3.1(1.11)	44(0.07)	43.9(0.09)	1(0.89)
		MCP	45.2(0.08)	38.6(0.1)	2.6(1.05)	36.6(0.08)	30.1(0.08)	0.2(0.51)
		Lassointer	0(0)	0(0)	0.6(0)	47.4(0.08)	36(0.1)	0.5(0.57)
		MCPinter	0(0)	0(0)	0.6(0)	48(0.08)	36.9(0.1)	0.5(0.58)
		sgMCP	65.2(0.07)	39.2(0.09)	2.4(1.19)	64.3(0.1)	38.9(0.12)	0.6(0.75)

Table A16:

Analysis of the GENEVA diabetes data using the proposed approach: observed occurence index.

SNP	Gene*	main	age	famdb	act	trans	ceraf	heme
rs17090278	RP11-593F5.2	0.86	0.85	0.74	0.72	0.69	0.64
rs17090286	RP11-593F5.2	0.87	0.86	0.77	0.72	0.68	0.63
rs13122165	RP11-593F5.2	0.57	0.49
rs17828144	RP11-593F5.2	0.72	0.69
rs17085296	RP11-63H19.1	0.77	0.76	0.75	0.75	0.68	0.63	0.61
rs1430504	RP11-707A18.1	0.75	0.74	0.69	0.68	0.66	0.61	0.58
rs6551878	RP11-707A18.1	0.74	0.74	0.7	0.68	0.68	0.65	0.61
rs6823601	RP11-707A18.1	0.74	0.74	0.7	0.68	0.68	0.65	0.61
rs13107026	MIR1269A	0.64
rs1397755	MIR1269A	0.73
rs13151560	MIR1269A	0.66
rs1858306	MIR1269A	0.59
rs10016795	MIR1269A	0.55
rs12331987	MIR1269A	0.66
rs10000219	MIR1269A	0.64
rs4860208	RPS23P3	0.72
rs1511286	RPS23P3	0.74
rs2136822	RPS23P3	0.55	0.5
rs11936928	RPS23P3	0.78	0.72	0.72	0.69	0.67	0.63	0.61
rs6838523	RPS23P3	0.78	0.73	0.71	0.69	0.66	0.63	0.61
rs17088752	UBA6-AS1	0.77	0.73
rs17088764	UBA6-AS1	0.54	0.49	0.48
rs353169	UBA6-AS1	0.54	0.49	0.48
rs10033058	YTHDC1	0.53
rs2293595	YTHDC1	0.57
rs17089267	YTHDC1	0.57
rs11249477	CSN1S2AP	0.72	0.69	0.67	0.63	0.61	0.52
rs1399247	CSN1S2AP	0.6
rs1717600	CSN1S2AP	0.63
rs10003790	DCK	0.7	0.7	0.69	0.68	0.68	0.66	0.63
rs10012631	DCK	0.68	0.68	0.67	0.67	0.67	0.65	0.63
rs9790462	DCK	0.57	0.53	0.49	0.47
rs12649753	RN7SL218P	0.54	0.46	0.45	0.44	0.42	0.39	0.35
rs7681755	LINC01088	0.67
rs11731223	NAA11	0.52
rs17003746	GK2	0.54
rs17003749	GK2	0.52
rs10004901	C4orf22	0.64	0.61	0.58	0.56	0.55	0.51
rs1391262	RP11-689K5.3	0.58
rs35036928	RP11-689K5.3	0.7
rs4693369	RP11-689K5.3	0.66
rs7672440	RP11-689K5.3	0.69
rs676592	RP11-689K5.3	0.87
rs1993798	RP11-689K5.3	0.68	0.63	0.57	0.51	0.48	0.45
rs2868257	RP11-689K5.3	0.71	0.68
rs6535281	RP11-689K5.3	0.68	0.63	0.57	0.51	0.48	0.44
rs612318	RP11-689K5.3	0.59
rs1824657	RP11-689K5.3	0.63	0.59	0.54	0.48	0.46	0.4
rs11722328	RP11-689K5.3	0.7	0.67	0.65	0.63	0.61
rs2199487	RP11-689K5.3	0.7	0.67	0.65	0.62	0.61
rs392112	RP11-218C23.1	0.62
rs434193	RP11-218C23.1	0.63
rs6842681	RP11-218C23.1	0.61
rs416035	RP11-218C23.1	0.63
rs432755	RP11-218C23.1	0.61
rs375432	RP11-218C23.1	0.63
rs407430	RP11-218C23.1	0.61
rs400023	RP11-218C23.1	0.63
rs585787	RP11-218C23.1	0.59
rs3775373	FAM13A	0.56	0.51	0.5
rs2726516	PPA2	0.54	0.53	0.51
rs2636739	PPA2	0.55	0.54	0.51
rs2686293	RP13-612N21.1	0.72

^*Genes that SNPs belong to or are the closest to.

Table A17:

Analysis of the TCGA SKCM data using the proposed approach: observed occurrence index.

Gene		Age	PN	Gender	Breslow’s depth	Clark level
ZNF25	0.68	0.67	0.67	0.67	0.55
ZNF37A	0.42	0.42	0.42	0.42	0.38	0.2
PJA2	0.41
KIF5B	0.58	0.56	0.56	0.56	0.47
NUDT4	0.49
KTN1	0.68	0.61	0.6	0.56	0.46
LRP12	0.62
TRIP11	0.54	0.47	0.45	0.45	0.43
EEA1	0.57
ARHGAP12	0.7	0.66	0.66	0.64	0.55
APC	0.4	0.35	0.35	0.34
FNDC3A	0.39	0.34	0.34
EIF5	0.48	0.47	0.47	0.47	0.45	0.38
DGLUCY	0.59	0.53	0.52	0.5
ARHGAP5	0.65
RRM2B	0.42	0.39	0.38
UHRF1BP1L	0.45	0.37	0.37	0.31	0.27
SEL1L	0.54
VPS13C	0.45	0.29	0.28
PDP2	0.39	0.32	0.3	0.28	0.22
GOLGA5	0.68	0.65	0.6	0.58
HSPA13	0.4	0.31	0.31	0.29	0.28
RASSF8	0.53
EFR3A	0.35	0.31	0.31	0.3	0.28
PCNX1	0.42	0.36	0.35	0.33	0.28
PPM1A	0.68	0.54	0.49	0.47	0.39
DNAL1	0.4	0.4	0.4	0.4	0.31
CUL2	0.37	0.35	0.35	0.35	0.33
ZMYND11	0.52	0.49	0.49	0.49	0.25
MPP5	0.45	0.34	0.32	0.32	0.25
CCPG1	0.49
HIF1A	0.4
PNMA1	0.43	0.38	0.37	0.37
EXOC5	0.43	0.38	0.38	0.37	0.32
GSKIP	0.42	0.38	0.35	0.34	0.25
ARL1	0.43
EWSR1	0.41	0.3	0.29
CASC4	0.41
RCC2	0.45
ARHGAP21	0.53
JKAMP	0.64
ACBD5	0.46	0.45	0.45	0.45	0.4
SETD3	0.45	0.43	0.41	0.41	0.37
EHMT1	0.32
RAB18	0.41	0.37	0.35	0.35	0.3
CREG1	0.42