Gene–environment interaction analysis via deep learning

Abstract

Gene–environment (G–E) interaction analysis plays an important role in studying complex diseases. Extensive methodological research has been conducted on G–E interaction analysis, and the existing methods are mostly based on regression techniques. In many fields including biomedicine and omics, it has been increasingly recognized that deep learning may outperform regression with its unique flexibility (e.g., in accommodating unspecified nonlinear effects) and superior prediction performance. However, there has been a lack of development in deep learning for G–E interaction analysis. In this article, we fill this important knowledge gap and develop a new analysis approach based on deep neural network in conjunction with penalization. The proposed approach can simultaneously conduct model estimation and selection (of important main G effects and G–E interactions), while uniquely respecting the “main effects, interactions” variable selection hierarchy. Simulation shows that it has superior prediction and feature selection performance. The analysis of data on lung adenocarcinoma and skin cutaneous melanoma overall survival further establishes its practical utility. Overall, this study can advance G–E interaction analysis by delivering a powerful new analysis approach based on modern deep learning.

1 ∣. INTRODUCTION

For complex diseases, there is extensive evidence that gene–environment (G–E) interactions can play an important role beyond the main G and E effects. G–E interaction analysis can assist in better understanding disease biology, identifying targets for therapeutic developments, and building more accurate disease models. The literature on G–E interaction analysis is too vast to be reviewed here. For representative studies, we refer to Hunter (2005), McAllister et al. (2017), and others. A long array of statistical methods has been developed for G–E interaction analysis, taking different perspectives. Some of the existing methods examine marginal statistical significance (S. Zhang et al., 2020), while others conduct joint modeling. In this study, we focus on the latter, which may better reflect disease biology and can be technically more challenging (X. Liu et al., 2020).

Most of the existing G–E interaction analyses are built on regression techniques (M. Wu et al., 2020). In a “standard” analysis, a (semi)parametric model, for example, the Cox model, is assumed to link a response variable with main G effects, main E effects, and G–E interactions. G–E interaction analysis can be seriously challenged by high data dimensionality, the presence of extensive “noises,” and other factors. Accordingly, a myriad of dimension reduction and variable selection methods have been developed. For recent reviews, we refer to M. Wu and Ma (2019) and Zhou et al. (2021). Unlike some other analyses, a unique challenge is brought by the “main effects, interactions” hierarchy (Bien et al., 2013), under which, if an interaction is identified (as associated with the response), the corresponding main G effect needs to be automatically identified—this leads to additional constraints to estimation. For estimation, identification (of important interactions and main effects), and respect for the hierarchy, penalization has emerged as an especially appealing technique (C. Wu, Jiang, et al., 2018). One example is Zemlianskaia et al. (2022), which introduces a selection method based on Lasso with hierarchical constraints. This method can be generalized to more than one E variable; however, the number of E variables needs to be small for the efficiency of the proposed block coordinate descent algorithm.

In the past decade, it has been increasingly recognized that deep learning may outperform regression, especially in the prediction and accommodation of unspecified nonlinear effects (Kang et al., 2022). “Standard” deep neural network (DNN) and its “siblings,” such as recurrent neural network and convolutional neural network, have been extensively adopted in many fields such as engineering, finance, and biomedicine (Mobadersany et al., 2018). In quite a few recent studies, deep learning has demonstrated highly appealing performance with genetic and other omics data (Ching et al., 2018). However, our literature review suggests that there has been a lack of deep learning study tailored to G–E interaction analysis. Additionally, most of the existing deep learning techniques are for categorical and continuous responses, and, relatively, developments for censored survival response have been limited. In deep learning-based omics data analysis, two unique challenges have been identified: high dimensionality of input variables (which may lead to overfitting and ill-behaved estimates) and excessive “noises” (which demand distinction and removal of noises). One possible solution is to first statistically or biologically screen omics measurements and reduce dimensionality to a much lower level, which may have a risk of losing important input variables in screening. As a more effective alternative, regularization in neural network construction has been proposed (Kukačka et al., 2017). As in regression, penalization has again emerged as a preferred tool, and two penalization strategies have been commonly adopted (Li et al., 2016). The first strategy is to apply sparsity penalties (in particular Lasso) to the input layer. Then, variable selection can be conducted at the input layer, and only a subset of selected variables enters the neural network construction (Li et al., 2018). The second strategy is to apply penalization to the hidden layers. For example, the ridge penalty has been commonly applied to ensure that the hidden layer weights are sensible (e.g., not extremely large) (Cai et al., 2011). Lasso and other sparsity penalties have been applied to generate sparse and more interpretable architectures (Lemhadri et al., 2021). Here, it is noted that the aforementioned techniques have not been applied to G–E interaction analysis. It is conjectured that, as in regression analysis, directly applying them may lead to a violation of the “main effects, interactions” hierarchy.

Motivated by the significance of G–E interaction analysis and superior performance of deep learning in other omics data analysis contexts, the goal of this study is to develop a deep learning-based approach tailored to G–E interaction analysis. This study may advance the existing literature in multiple important aspects. First, advancing from those based on regression techniques (J. Liu et al., 2013), it can provide a deep learning-based alternative approach, which can share superior prediction performance and flexibility in accommodating unspecified nonlinear effects with the existing deep learning methods. Second, unlike the existing deep learning methods, the proposed method can automatically respect the “main effects, interactions” hierarchy. As such, the analysis results can have similar high interpretability as from regression analysis. Third, this study can further enrich the family of deep learning tools for censored survival data, which has been underdeveloped. Last but not least, it can provide a practically useful tool for analyzing The Cancer Genome Atlas (TCGA) and other cancer omics studies.

2 ∣. METHODS

2.1 ∣. Network architecture

The overall architecture of the proposed neural network is graphically presented in the left panel of Figure 1. To simplify visualization, we present only two hidden layers. For comparison, in the right panel, we present the architecture of a fully connected neural network that has no special consideration for G–E interactions—it treats all input variables in the same manner. It is easy to see that the two architectures are significantly different. Similar to some of the published ones (Li et al., 2016), the proposed neural network has a sparse layer, one or multiple hidden layers, and a Cox layer. The sparse layer is a one-to-one mapping of the input variables. As described in detail below, we apply penalization to select relevant G effects and G–E interactions, and only the selected input variables are involved in the subsequent network. As described above, a unique challenge here is the “main effects, interactions” hierarchy, which demands a penalty more sophisticated than, for example, Lasso. In most of the existing G–E studies, E variables have a much lower dimension and are usually manually selected. As such, as in the literature (J. Liu et al., 2013), there is no selection of E variables. There can be one or multiple hidden layers, which conduct affine transformations and introduce nonlinearity via activation functions. For the first hidden layer, input includes the selected main G effects, selected G–E interactions, and all E effects. For the subsequent hidden layers, input comes from the previously hidden layer. There are multiple choices for the activation function. In our numerical study, we adopt ReLU, which can be easily optimized and have a lower probability of gradient vanishment. The last is the Cox layer. As described below, the adopted loss function has been motivated by the partial likelihood function of the Cox model. This layer has only one output node, which has been referred to as the prognostic index in the literature (Ching et al., 2018). This output corresponds to the linear risk score of the Cox model.

FIGURE 1.

Left: the proposed neural network architecture. Right: for comparison, the fully connected neural network architecture.

2.2 ∣. Formulation

2.2.1 ∣. Loss function

For many existing neural networks, loss functions have been motivated by their regression counterparts. In this study, we analyze censored survival data, for which the most popular regression model is the Cox (Cox & Oakes, 1984). In several deep learning studies with censored survival responses (Katzman et al., 2018; Kvamme et al., 2019; Lee et al., 2018), loss functions are based on the partial likelihood function of the Cox model. We adopt the same strategy here. Briefly, assume $n$ i.i.d. observations. For subject $i$, we use $W_i$, to generically denote input variables. Denote $Y_i$ and $C_i$ as the survival and censoring times, respectively. We observe $T_i=\min (Y_i,C_i)$ and the event indicator ${\delta }_i=I(Y_i\le C_i)$. As in many regression studies, independent censoring, under which the hazard of failure at time $T$ is equal to that given that censoring has not yet occurred, is assumed. Then, the loss function is

$$l(\theta )=-\sum _{t:{\delta }_i=1}\left(h_{\theta }(W_i)-\log \sum _{j\in \mathcal{R}(T_i)}{e}^{h_{\theta }(W_j)}\right),$$

where $\theta$ generically denotes the neural network weights, $\mathcal{R}(T_i)=\{i^{\prime }:T_{i^{\prime }}\ge T_i\}$ is the at-risk set at time $T_i$, and $h_{\theta }$ is the prognostic index. In the Cox model, $h_{\theta }$ is the linear combination of explanatory variables. Here, it is the forward propagation of explanatory variables.

2.2.2 ∣. Penalized estimation

Denote $Z$ as the length-$p$ vector of G variables and $X$ as the length-$q$ vector of E variables. Then, the input variables corresponding to the $j$th G variable are ${\mathit{W}}_j={(Z_j,X_1{Z}_j,\dots ,X_q{Z}_j)}^T$. The overall input variable vector is $\mathit{W}={\left({\mathit{W}}_1^T,\dots ,{\mathit{W}}_p^T\right)}^T$, which has dimensionality $pq+p+q$. For the sparse layer, denote ${\mathit{b}}_j={({\gamma }_j,{\beta }_{1j},\dots ,{\beta }_{qj})}^T={(b_{j1},\dots ,b_{j(q+1)})}^T$ as the coefficient (weight) vector corresponding to the $j$th G variable—noting that it includes the main effect ${\gamma }_j$ as well as the interactions ${\beta }_{1j},\dots ,{\beta }_{qj}$. Further, it is noted that, with the nonlinear nature of the neural network, these coefficients/weights do not have the same implications as their counterparts in regression models. Denote $K$ as the number of fully connected layers. For the $k$th layer, denote ${\omega }_k$ as the weight matrix. Overall, we propose the objective function:

$$\begin{gathered} L(\theta )=l(\theta )+\sum _{j=1}^p\rho (\Vert {\mathit{b}}_j\Vert ;{\lambda }_1,s) \\ +\sum _{j=1}^p\sum _{k=1}^q\rho (\mid {\beta }_{kj}\mid ;{\lambda }_2,s)+\lambda \left(\sum _{k=1}^K\Vert {\omega }_k{\Vert }_F^2\right). \end{gathered}$$

Here, $\Vert \cdot \Vert$ is the $L_2$ norm, and $\rho (t;\lambda ,s)=\lambda {\int }_0^{\mid t\mid }{(1-\frac{x}{\lambda s})}_{+}dx$ is the minimax concave penalty (MCP). ${\lambda }_1$, ${\lambda }_2>0$ are tuning parameters. Following J. Liu et al. (2013), we set ${\lambda }_1=\sqrt{q+1}{\lambda }_2$, which can reduce the number of tunings and computational cost. $s$ is the regularization parameter of MCP. Following the practice in regression (C.-H. Zhang, 2010), we set $s=3$. ${\Vert \cdot \Vert }_{\mathrm{F}}$ denotes the Frobenius norm of a matrix. $\lambda >0$ is another tuning parameter. With the sparse penalization, some weights of the sparse layer can be estimated as exactly zero. The main G effects and interactions that have nonzero estimates in the sparse layer will be identified as associated with the survival response. All of the main E effects will have nonzero estimates and automatically be identified as associated with the response. As the proposed approach is built on deep learning and MCP and tailored to G–E interaction analysis, below we refer it to as DeepGE-MCP.

In the proposed objective function, the first two penalty terms achieve sparse group penalization (J. Liu et al., 2013). It conducts the selection of important G effects and G–E interactions. More importantly, it can ensure that if a ${\beta }_{kj}$ is nonzero, then ${\gamma }_j$ is nonzero—that is, the hierarchy is respected. This strategy has been adopted in multiple regression-based G–E interaction analyses, but not under the deep learning framework. Another advancement from the existing neural networks is that we adopt MCP as opposed to Lasso. Many regression studies have demonstrated the superior performance of MCP over Lasso (C.-H. Zhang, 2010); however, MCP has not been well adopted in deep learning. For the hidden layers, we apply the ridge penalization, which has been adopted in several recent deep learning studies to control network complexity and avoid the swelling of hidden layer parameters.

2.3 ∣. Algorithm

To optimize the MCP-based penalties, we adopt the local quadratic algorithm (LQA) technique (Fan & Li, 2001). The overall computational algorithm is iterative. In what follows, the superscript “($m$)” denotes the $m$th iteration. The first two penalty terms can be approximated by

$$\begin{gathered} \sum _{j=1}^p\frac{{\rho }_{{\lambda }_1}^{\prime }\left(\Vert {\mathit{b}}_j^{(m)}\Vert \right)}{2\Vert {\mathit{b}}_j^{(m)}\Vert }\frac{\mid {\gamma }_j^{(m)}\mid }{\Vert {\mathit{b}}_j^{(m)}\Vert }{\gamma }_j^2 \\ +\sum _{j=1}^p\sum _{k=1}^q\left[\frac{{\rho }_{{\lambda }_1}^{\prime }\left(\Vert {\mathit{b}}_j^{(m)}\Vert \right)}{2\Vert {\mathit{b}}_j^{(m)}\Vert }\frac{\mid {\beta }_{kj}^{(m)}\mid }{\Vert {\mathit{b}}_j^{(m)}\Vert }+\frac{{\rho }_{{\lambda }_2}^{\prime }\left(\mid {\beta }_{kj}^{(m)}\mid \right)}{2\mid {\beta }_{kj}^{(m)}\mid }\right]{\beta }_{kj}^2, \end{gathered}$$ (1)

where ${\rho }_{\lambda }^{\prime }(t)={(\lambda -\frac{t}{s})}_{+}$ is the first order derivative of the MCP penalty. This is also referred to as the MCP-LQA penalty.

Algorithm 1 Training of DeepGE-MCP.

$$\begin{array}{cc}\hfill & \mathrm{Input}:\hfill \\ \hfill & \text{Training data}\{({\mathit{W}}_1,T_1,{\delta }_1),\dots ,({\mathit{W}}_n,T_n,{\delta }_n)\};\hfill \\ \hfill & \text{Learning rate of the sparse layer}{\alpha }_1;\hfill \\ \hfill & \text{Learning rate of the fully connected layers}{\alpha }_2;\hfill \\ \hfill & \text{Tuning and regularization parameters of the MCP}\hfill \\ \hfill & \text{penalty}\{{\lambda }_1,{\lambda }_2,s\};\hfill \\ \hfill & \text{Tuning parameter of the fully connected layers}\lambda ;\hfill \\ \hfill & \mathrm{Input}:\hfill \\ \hfill & \text{Estimates for the sparse layer}\mathit{b}={\left({\mathit{b}}_1^T,\dots ,{\mathit{b}}_p^T\right)}^T\hfill \\ \hfill & \text{with}{\mathit{b}}_j({\gamma }_j,{\beta }_{1j},\dots ,{\beta }_{qj}{)}^T;\hfill \\ \hfill & \text{Estimates for the}k\text{th fully connected}\hfill \\ \hfill & \text{layer}\{{\omega }_k\},k=1,\dots ,K;\hfill \\ \hfill & 1\text{Initialize:}\hfill \\ \hfill & {\mathit{b}}^{(0)}\sim \text{Uniform}(0,1),{\omega }_k^{(0)}\sim \text{Uniform}(-\sqrt{\frac{1}{l_k}},\sqrt{\frac{1}{l_k}}),k=1,,\hfill \\ \hfill & \dots ,K-1,{\omega }_K^{(0)}\sim \text{Uniform}\hfill \\ \hfill & (-0.001,0.001)\hfill \\ \hfill & \text{and}{l}_k\text{is the number of input variables for the}k\text{th fully}\hfill \\ \hfill & \text{connected layer};\hfill \\ \hfill & 2\text{Initialize}m=0;\hfill \\ \hfill & 3\mathbf{repeat}\hfill \\ \hfill & 4\text{Given the current estimate}{\mathit{b}}^{(m)},\text{update the MCP}-\text{LQA}\hfill \\ \hfill & \text{penalty}(1);\hfill \\ \hfill & 5\text{Update}loss=l(\theta )+\text{MCP}-\text{LQA penalty}+\lambda \sum \begin{array}{c}K\hfill \\ k=1\hfill \end{array}{\Vert {\omega }_k^{(m)}\Vert }_F^2;\hfill \\ \hfill & 6\text{Conduct back propagation, and obtain the gradients}\frac{\partial los{s}^F}{\partial {\mathit{b}}_j^{(m)}}\hfill \\ \hfill & \text{and}\frac{\partial loss}{\partial {\omega }_k^{(m)}};\hfill \\ \hfill & 7\mathbf{for}j=1,\dots ,p\mathbf{do}\hfill \\ \hfill & 8\text{Update estimates}{\gamma }_j^{(m+1)}={\gamma }_j^{(m)}-{\alpha }_1\frac{\partial loss}{\partial {\gamma }_j^{(m)}};\hfill \\ \hfill & 9\mathbf{for}k=1,\dots ,q\mathbf{do}\hfill \\ \hfill & 10\text{Update estimates}{\beta }_{kj}^{(m+1)}={\beta }_{kj}^{(m)}-{\alpha }_1\frac{\partial loss}{\partial {\beta }_{kj}^{(m)}};\hfill \\ \hfill & 11\mathbf{end}\hfill \\ \hfill & 12\mathbf{end}\hfill \\ \hfill & 13\mathbf{for}k=1,\dots ,K\mathbf{do}\hfill \\ \hfill & 14\text{Update}{\omega }_k^{(m+1)}={\omega }_k^{(m)}-{\alpha }_2\frac{\partial loss}{\partial {\omega }_k^{(m)}};\hfill \\ \hfill & 15\mathbf{end}\hfill \\ \hfill & 16\text{Update}m=m+1;\hfill \\ \hfill & 17\mathbf{until}convergenceormreachesitsmaximum;\hfill \end{array}$$

The proposed algorithm is summarized in Algorithm 1. It is a “combination” of the existing algorithms for DNNs (Rumelhart et al., 1986) (for which backpropagation is a standard tool) and penalized estimation (Fan & Li, 2001). There are a few points worth noting. First, in penalization regression, the initial values are often set as zero. With DNNs, nonzero initial values are needed (Thimm & Fiesler, 1995). Second, although in principle the optimizer of the proposed objective function can have exactly zero components, with the LQA approximation and nonzero initial values, the estimates may not be exactly zero. This has been noted in the penalized deep learning literature (Li et al., 2016). Following published studies (Li et al., 2016), we set estimates below a predefined threshold to be zero. We note that this does not destroy the variable selection hierarchy. Third, in some deep learning studies, the dropout technique has been adopted to regularize estimation with hidden layers. In our approach, we apply ridge penalization, which may achieve a similar function. When needed, dropout can be added in a standard manner. Fourth, we conclude convergence if the absolute difference of estimates from two consecutive iterations is smaller than a prespecified threshold. Convergence is satisfactorily achieved in all of our numerical studies. The maximum value for $m$ can be imposed for safeguarding. Last, for tuning parameter selection, we adopt a cross-validation-type approach. In particular, we consider a sequence of tuning parameter combinations. For each combination, we conduct neural network construction using a training data set and then evaluate it on an independent testing data set. The combination with the smallest testing data loss is selected as optimal.

3 ∣. SIMULATION

Many published deep learning studies have only conducted an analysis of real data. Without knowing the underlying data generation mechanisms, it is challenging to objectively evaluate performance. Here, we conduct a simulation, with which we can objectively assess the performance of the proposed method and gauge against relevant alternatives.

3.1 ∣. Settings

We set $q=5$ and first generate $q$ variables from a multivariate normal distribution with marginal means 0 and variances 1. Then, we generate the E variables by dichotomizing two of those normally distributed variables at 0, which leads to three continuous and two binary E variables. For the G variables, we set $p=500$ and generate a multivariate normal distribution with marginal means of 0 and variances of 1. For the correlation structures of both the G and E variables, we consider the autoregressive (AR) and banded structures. Under the AR structure, the correlation between the $j$th and $k$th variables is ${\sigma }^{\mid j-k\mid }$, with $\sigma =0.25$ (AR1), 0.5 (AR2), and 0.75 (AR3). Under the first banded structure (Band1), the $j$th and $k$th variables have a correlation coefficient of 0.33 if $\mid j-k\mid =1$, and 0 otherwise. Under the second scenario (Band2), the $j$th and $k$th variables have a correlation of 0.6 if $\mid j-k\mid =1$ if $\mid j-k\mid =2$, and 0 otherwise. There are 30 nonzero main G effects and 30 nonzero G–E interactions. The “main effects, interactions” hierarchy is satisfied. All E variables have nonzero effects. Overall, there are 3005 “candidate” effects, among which 65 are nonzero and associated with the response. More details are described below.

The censoring time $C$ is independently generated from an exponential distribution with mean $U^{\ast }\exp \{h_{\theta }(\cdot )\}$, where $U$ is generated from Uniform (1, 3). Overall, the censoring rate is about 30%. The event time is generated from a Cox model with a constant baseline hazard function. For the risk score, we consider the following four cases:

• Linear case, with the risk score taking the form:

  $$\begin{gathered} h_{\theta }(Z,X)=\sum _{j=1}^{30}{\gamma }_j{Z}_j+\sum _{\{j,k\}\in \mathcal{J}\times \mathcal{K}}{\beta }_{jk}{Z}_j{X}_k \\ +\sum _{k=1}^5{\alpha }_k{X}_k, \end{gathered}$$

  where ${\gamma }_j,{\beta }_{jk},{\alpha }_k\sim \mathit{Uniform}(0.5,0.8)$. That is, the first 30 G effects and 5 E effects are important. $\mathcal{J}=\{j\mid 1,\dots ,30\}$, $\mathcal{K}=\{k\mid 1,\dots ,5\}$, and we randomly select 30 $\{j,k\}$’s from $\mathcal{J}\times \mathcal{K}$, which ensures the variable selection hierarchy.
• Nonlinear case 1, with the risk score taking the form:

  $$\begin{gathered} h_{\theta }(Z,X)=\sum _{j=1}^{30}\sin (Z_j)+\sum _{\{j,k\}\in \mathcal{J}\times \mathcal{K}}\sin (Z_j{X}_k) \\ +\sum _{k=1}^5\sin (X_k). \end{gathered}$$

  There are also 30 main G effects, 5 main E effects, and 30 G-E interactions. The nonlinear effects take a simple additive form.
• Nonlinear case 2, with the risk score taking the form:

  $$\begin{gathered} h_{\theta }(Z,X)=\sum _{j=1}^{30}\sigma (Z_j)+\sum _{\{j,k\}\in \mathcal{J}\times \mathcal{K}}\tanh (Z_j{X}_k) \\ +\sum _{k=1}^5\sigma (X_k), \end{gathered}$$

  where $\sigma (x)=1∕(1+\exp (-x))$ and $\tanh (x)=(\exp (x)-\exp (-x))∕(\exp (x)+\exp (-x))$. Different from the nonlinear case 1, here the main effects have a smaller impact than the interactions. This setting may favor methods that respect the variable selection hierarchy, as the identification of weak main effects can be “assisted” by that of strong interactions.
• Nonlinear case 3, which is a single index model, and the risk score takes the form:

  $$\begin{gathered} h_{\theta }(Z,X)=\log [1+\text{exp}(\sum _{j=1}^{30}{\gamma }_j{Z}_j+\sum _{\{j,k\}\in \mathcal{J}\times \mathcal{K}}{\beta }_{jk}{Z}_j\phantom{)}\phantom{]} \\ \phantom{[}\phantom{(}{X}_k+\sum _{k=1}^5{\alpha }_k{X}_k)]. \end{gathered}$$

For sample size, we consider 500, 1000 and 1500, which is sensible for genetic studies.

3.2 ∣. Alternatives

To better position the proposed approach, we consider the following alternatives: (a) DeepGE-Lasso, which is similar to the proposed approach, with the MCP replaced by Lasso. In regression, MCP has been shown to outperform Lasso. Here, we examine their relative performance with a neural network; (b) FNN, which is a fully connected neural network. Its architecture is shown in the right panel of Figure 1. Its biggest difference from the proposed approach is the lack of sparsity; (c) MA, under which we conduct the marginal regression analysis of one gene, all E variables, and their interactions at a time. This marginal analysis conducts selection based on significance ($p$ value), is perhaps still the most popular approach, and serves as a benchmark here; (d) CoxMCP, which conducts regression analysis under the Cox model with MCP for regularized estimation and selection; and (e) CoxLasso, which is similar to CoxMCP but applies Lasso as opposed to MCP. CoxMCP and CoxLasso represent the popular penalization-based regression methods, and having both of them can make the comparison “more complete”.

3.3 ∣. Metrics

For evaluating prediction accuracy, we generate independent testing data under the same settings for each simulated replicate. Models are generated using training data and used to make predictions for testing data. We consider C-index. It ranges between 0.5 and 1, and a larger value indicates better prediction performance. For evaluating feature selection accuracy, we consider the number of true positives (TPs) and the number of false positives (FPs). These metrics have been commonly adopted in the literature. Here, we note that the proposed and other neural network analyses cannot generate estimates of regression coefficients. As such, it is not possible to evaluate estimation accuracy.

3.4 ∣. Results

Results based on 100 replicates are summarized in Tables 1 and 2 (main text) and Tables A1-A7. It is observed that the proposed approach has highly competitive performance across the whole spectrum of simulation. First consider the linear case, which should favor regression analysis. As a representative example, consider the AR2 correlation setting and $n=1000$. The C-index values are 0.920 (DeepGE-MCP), 0.907 (DeepGE-Lasso), 0.667 (FNN), 0.891 (CoxMCP), and 0.821 (CoxLasso). Here, it is noted that marginal analysis cannot generate a predictive model. In general, it is observed that the proposed approach outperforms DeepGE-Lasso by a small to moderate margin, which can be at least partly attributed to the superiority of MCP over Lasso. FNN is dense and includes both signals and noises, which leads to inferior performance. The comparison between CoxMCP and CoxLasso also “reconfirms” the advantages of MCP over Lasso. Overall, CoxMCP and CoxLasso have reasonable performance, which, however, is inferior to the proposed approach. We acknowledge that there are possibly better penalized regression methods, which, however, usually are computationally more expensive. The proposed approach is also observed to have competitive variable selection performance. The TP values for the main effects are 29.3 (DeepGE-MCP), 28.6 (DeepGE-Lasso), 0.3 (MA), 30.0 (CoxMCP), and 28.0 (CoxLasso). It is noted that FNN cannot conduct variable selection. With the correlations among covariates, the MA approach is completely unacceptable. For interactions, the TP values are 27.9 (DeepGE-MCP), 26.8 (DeepGE-Lasso), 0.0 (MA), 30.0 (CoxMCP), and 25.7 (CoxLasso). Different from the satisfactory TP identification, it is observed that DeepGE-Lasso, CoxMCP, and CoxLasso can be inferior with many FPs.

TABLE 1.

Simulation (linear case): mean (SD) based on 100 replicates.

Correlation	n	Method	C-index	Main effects		Interactions
				M:TP	M:FP	I:TP	I:FP
AR1	500	DeepGE-MCP	0.704 (0.044)	14.9 (3.5)	9.1 (4.3)	8.1 (2.7)	12.8 (9.1)
		DeepGE-Lasso	0.678 (0.043)	17.1 (3.2)	35.2 (19.0)	10.9 (2.6)	195.0 (109.7)
		FNN	0.565 (0.064)	–	–	–	–
		MA	–	0.7 (1.5)	0.0 (0.1)	0.1 (0.4)	0.2 (0.6)
		CoxMCP	0.619 (0.097)	9.1 (3.2)	1.2 (1.1)	4.4 (2.2)	7.1 (3.4)
		CoxLasso	0.702 (0.106)	23.5 (6.9)	30.3 (21.6)	18.8 (8.3)	128.7 (80.9)
	1000	DeepGE-MCP	0.900 (0.019)	29.7 (0.6)	4.2 (4.2)	27.8 (1.9)	6.9 (5.8)
		DeepGE-Lasso	0.881 (0.030)	29.0 (1.5)	39.0 (31.5)	27.0 (2.8)	201.0 (165.6)
		FNN	0.614 (0.045)	–	–	–	–
		MA	–	2.7 (4.2)	0.0 (0.1)	0.6 (1.3)	0.4 (1.3)
		CoxMCP	0.883 (0.075)	30.0 (0.0)	3.0 (1.7)	30.0 (0.0)	13.6 (4.2)
		CoxLasso	0.819 (0.072)	29.9 (0.5)	45.3 (34.8)	29.4 (1.6)	198.3 (140.8)
	1500	DeepGE-MCP	0.922 (0.008)	30.0 (0.0)	4.2 (10.6)	30.0 (0.2)	6.3 (11.8)
		DeepGE-Lasso	0.922 (0.007)	30.0 (0.1)	14.1 (23.7)	30.0 (0.2)	63.4 (100.9)
		FNN	0.634 (0.053)	–	–	–	–
		MA	–	3.0 (5.1)	0.1 (0.3)	0.7 (1.5)	0.5 (1.1)
		CoxMCP	0.878 (0.059)	30.0 (0.0)	0.6 (0.8)	30.0 (0.0)	2.9 (2.3)
		CoxLasso	0.834 (0.065)	29.8 (1.1)	19.8 (26.9)	29.5 (2.5)	95.1 (122.1)
AR2	500	DeepGE-MCP	0.806 (0.035)	18.6 (3.1)	5.5 (3.5)	10.7 (2.8)	9.6 (6.8)
		DeepGE-Lasso	0.779 (0.038)	20.0 (2.7)	24.4 (13.2)	13.0 (2.5)	173.4 (99.5)
		FNN	0.604 (0.072)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.1)	0.0 (0.1)	0.1 (0.2)
		CoxMCP	0.679 (0.114)	10.1 (1.3)	0.7 (0.9)	5.5 (2.2)	5.4 (2.9)
		CoxLasso	0.728 (0.132)	21.7 (8.1)	13.9 (17.3)	15.7 (9.1)	63.4 (68.3)
	1000	DeepGE-MCP	0.920 (0.014)	29.3 (1.4)	4.3 (4.5)	27.9 (1.9)	9.0 (7.3)
		DeepGE-Lasso	0.907 (0.017)	28.6 (1.7)	31.1 (27.7)	26.8 (2.4)	157.8 (143.3)
		FNN	0.667 (0.048)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.1)	0.0 (0.1)	0.1 (0.1)
		CoxMCP	0.891 (0.133)	30.0 (0.0)	3.0 (1.9)	30.0 (0.1)	11.6 (4.5)
		CoxLasso	0.821 (0.123)	28.0 (4.7)	17.7 (26.5)	25.7 (7.3)	81.7 (103.4)
	1500	DeepGE-MCP	0.934 (0.007)	30.0 (0.1)	3.4 (4.1)	30.0 (0.3)	7.4 (5.9)
		DeepGE-Lasso	0.933 (0.009)	29.8 (0.5)	13.5 (21.9)	29.6 (0.9)	70.0 (102.0)
		FNN	0.695 (0.046)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.2)	0.0 (0.1)	0.1 (0.1)
		CoxMCP	0.856 (0.146)	29.9 (1.2)	0.5 (0.9)	30.0 (0.5)	2.1 (2.3)
		CoxLasso	0.825 (0.141)	29.3 (2.5)	12.8 (27.5)	27.8 (5.1)	62.1 (116.0)
AR3	500	DeepGE-MCP	0.893 (0.021)	21.0 (3.0)	2.9 (2.9)	12.9 (3.1)	11.3 (5.2)
		DeepGE-Lasso	0.883 (0.023)	20.0 (2.1)	10.4 (6.6)	13.4 (2.9)	82.6 (62.8)
		FNN	0.657 (0.083)	–	–	–	–
		MA	–	0.0 (0.0)	0.0 (0.0)	0.0 (0.0)	0.0 (0.2)
		CoxMCP	0.674 (0.170)	7.5 (1.2)	0.7 (0.9)	5.4 (2.1)	6.5 (3.2)
		CoxLasso	0.714 (0.092)	15.4 (7.5)	2.2 (7.4)	9.0 (7.1)	15.0 (29.0)
	1000	DeepGE-MCP	0.933 (0.010)	28.4 (1.2)	4.6 (3.9)	24.3 (2.5)	17.1 (6.4)
		DeepGE-Lasso	0.923 (0.012)	25.4 (2.4)	13.0 (15.1)	21.0 (2.9)	83.5 (93.7)
		FNN	0.634 (0.123)	–	–	–	–
		MA	–	0.0 (0.1)	0.0 (0.1)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.771 (0.186)	15.7 (2.6)	3.7 (2.3)	15.8 (3.4)	17.9 (6.1)
		CoxLasso	0.711 (0.199)	19.5 (8.1)	1.6 (5.3)	12.7 (9.4)	16.1 (27.5)
	1500	DeepGE-MCP	0.946 (0.007)	29.6 (0.7)	3.7 (2.8)	27.8 (1.5)	15.7 (7.4)
		DeepGE-Lasso	0.940 (0.008)	28.6 (1.3)	5.1 (12.6)	26.2 (2.3)	32.4 (64.2)
		FNN	0.665 (0.132)	–	–	–	–
		MA	–	0.1 (0.4)	0.0 (0.0)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.811 (0.190)	29.7 (1.3)	8.2 (3.3)	27.8 (2.1)	28.0 (8.4)
		CoxLasso	0.735 (0.185)	23.7 (7.3)	2.7 (13.2)	18.4 (9.7)	21.5 (53.6)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions. Abbreviations: AR, autoregressive; FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE 2.

Simulation (linear case, cont.): mean (SD) based on 100 replicates.

Correlation	n	Method	C-index	Main effects		Interactions
				M:TP	M:FP	I:TP	I:FP
Band1	500	DeepGE-MCP	0.701 (0.044)	13.0 (3.5)	4.0 (3.2)	9.0 (3.0)	6.7 (4.8)
		DeepGE-Lasso	0.673 (0.055)	13.7 (5.0)	53.2 (77.6)	10.0 (6.2)	400.2 (540.3)
		FNN	0.570 (0.059)	–	–	–	–
		MA	–	0.6 (1.2)	0.1 (0.4)	0.1 (0.4)	0.2 (0.4)
		CoxMCP	0.650 (0.142)	9.2 (3.0)	1.3 (1.4)	5.1 (2.5)	7.6 (3.1)
		CoxLasso	0.714 (0.092)	23.5 (6.7)	33.2 (21.2)	19.9 (8.0)	137.2 (80.0)
	1000	DeepGE-MCP	0.892 (0.028)	28.8 (1.8)	10.0 (17.0)	27.1 (3.0)	16.9 (30.2)
		DeepGE-Lasso	0.723 (0.124)	22.7 (6.8)	149.2 (173.6)	22.1 (7.6)	975.5 (978.2)
		FNN	0.615 (0.047)	–	–	–	–
		MA	–	1.5 (3.6)	0.0 (0.2)	0.2 (0.7)	0.2 (0.7)
		CoxMCP	0.880 (0.082)	30.0 (0.0)	2.7 (1.7)	30.0 (0.0)	12.5 (3.9)
		CoxLasso	0.830 (0.079)	29.2 (3.2)	42.6 (31.3)	28.7 (4.3)	195.0 (131.0)
	1500	DeepGE-MCP	0.924 (0.008)	30.0 (0.1)	4.2 (4.9)	30.0 (0.2)	8.1 (7.6)
		DeepGE-Lasso	0.907 (0.048)	29.4 (1.6)	23.9 (62.6)	29.0 (2.6)	136.2 (349.3)
		FNN	0.600 (0.074)	–	–	–	–
		MA	–	3.0 (5.9)	0.1 (0.3)	0.5 (1.4)	0.3 (0.9)
		CoxMCP	0.885 (0.073)	30.0 (0.0)	0.7 (0.8)	30.0 (0.0)	2.9 (1.9)
		CoxLasso	0.834 (0.065)	29.7 (2.8)	19.9 (28.7)	29.6 (3.0)	92.1 (119.8)
Band2	500	DeepGE-MCP	0.830 (0.034)	18.7 (3.2)	5.2 (5.7)	12.3 (3.6)	10.3 (7.3)
		DeepGE-Lasso	0.804 (0.036)	19.0 (2.5)	20.2 (8.9)	12.4 (2.6)	159.4 (85.8)
		FNN	0.597 (0.069)	–	–	–	–
		MA	–	0.2 (0.6)	0.0 (0.0)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.699 (0.112)	9.3 (1.1)	0.9 (1.0)	6.1 (2.5)	6.8 (3.5)
		CoxLasso	0.728 (0.122)	21.7 (7.4)	20.6 (21.1)	17.2 (9.0)	83.7 (76.5)
	1000	DeepGE-MCP	0.913 (0.015)	27.9 (1.8)	6.4 (8.1)	25.8 (3.4)	14.8 (13.6)
		DeepGE-Lasso	0.806 (0.113)	19.3 (6.5)	53.3 (114.1)	18.1 (8.2)	533.1 (786.4)
		FNN	0.659 (0.048)	–	–	–	–
		MA	–	0.2 (0.8)	0.0 (0.0)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.818 (0.122)	16.4 (5.1)	4.1 (3.1)	22.6 (5.4)	18.5 (8.2)
		CoxLasso	0.802 (0.126)	27.8 (3.7)	15.2 (24.6)	25.1 (6.2)	73.8 (95.1)
	1500	DeepGE-MCP	0.930 (0.010)	29.1 (1.1)	4.9 (9.3)	28.7 (2.4)	12.6 (16.0)
		DeepGE-Lasso	0.905 (0.044)	25.1 (4.9)	14.4 (44.1)	24.8 (6.5)	147.3 (259.9)
		FNN	0.673 (0.065)	–	–	–	–
		MA	–	0.1 (0.6)	0.1 (0.5)	0.0 (0.0)	0.1 (0.0)
		CoxMCP	0.858 (0.131)	27.4 (3.0)	6.7 (3.7)	29.9 (0.5)	24.6 (10.3)
		CoxLasso	0.821 (0.130)	28.8 (3.9)	11.7 (21.4)	27.3 (5.7)	58.5 (86.1)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

For the nonlinear cases, the proposed approach remains highly competitive. Its superiority may get more prominent because of the nonlinearity. Consider for example nonlinear case 3, AR3 correlation, and $n=1000$. The superiority of the proposed approach in terms of C-index is similar to the above, with values 0.780 (DeepGE-MCP), 0.614 (DeepGE-Lasso), 0.669 (FNN), 0.693 (CoxMCP), and 0.658 (CoxLasso). In variable selection, CoxLasso misses many more TPs than the proposed. In particular, it has M:TP 11.3 (compared to 24.0 of DeepGE-MCP) and I:TP 4.4 (compared to 13.7 of DeepGE-MCP). CoxMCP has performance similar to CoxLasso, for example, with M:TP 7.0 and I:TP 4.6. DeepGE-Lasso may identify more TPs, however, at the price of dramatically more FPs.

To examine whether the proposed approach is still advantageous with more severe censoring, we consider nonlinear case 3 and AR correlation. The censoring rate is set as 50%, which better mimics the data analysis below. The results are presented in Table A7. The performance of all the methods deteriorates, which is as expected. However, the proposed approach is still observed to substantially outperform the alternatives.

4 ∣. DATA ANALYSIS

4.1 ∣. Analysis of lung adenocarcinoma (LUAD) data

Lung cancer has the highest incidence and mortality rate among all cancers in the world. LUAD is the most common histological subtype of non-small cell lung cancer (NSCLC), accounting for about 40% of lung malignant tumors. Here, we analyze data collected by TCGA, which is jointly organized by the National Cancer Institute and National Human Genome Research Institute and has published data on 36 cancer types. For G variables, we consider gene expressions, which were measured using the IlluminaHiseq RNAseq V2 platform. In the original data, there are 20,531 gene expression measures. To improve performance, as in many published studies, we first conduct marginal screening and reduce dimensionality to 1000 (noting that the proposed approach can be directly applied to a higher data dimensionality). For E variables, we take a “looser” definition and consider demographic and clinical variables, whose interactions with G measurements have been strongly advocated in recent literature (McAllister et al., 2017). In particular, we consider age, gender, smoking pack years (smoking), and American Joint Committee on Cancer (AJCC) tumor pathologic stage (stage), all of which have been suggested as playing important roles in lung cancer prognosis (Network, 2014). Among them, age and smoking are continuous, and the others are categorical. As in the literature (M. Wu, Huang, et al., 2018), we focus on samples of the White. After removing samples with missingness, there are 262 samples. Among them, 93 died during follow-up, and the survival times range from 0.13 to 104.11 months, with a median of 19.94 months. The 169 censored subjects have the observed times ranging from 0.43 to 238.11 months, with a median of 21.39 months.

With the proposed approach, a total of 45 main G effects and 34 G–E interactions are identified. The identification results are provided in Table 3. It is noted that, different from regression analysis, there is not an explicit estimator for each identification. A quick literature search suggests that the identified genes may have significant implications. For example, gene S100G belongs to the S100 protein family and is involved in cancer cell invasion and metastasis. It has been correlated with worse overall survival in patients without a smoking history. Cav3.1 overexpression is associated with negative characteristics and prognosis in NSCLC. It is also a potential biomarker for poorer outcomes. The overexpression of ACY1 can significantly increase cell growth, migration, invasion, and tumorigenesis, and studies have shown that it may promote the progression of NSCLC in a PTEN-dependent manner by activating the PI3K/AKT signaling pathway. Researchers have found that circ-BANP is overexpressed in lung cancer. The higher the expression of circ-BANP, the lower the survival rate. Silencing circ-BANP can significantly inhibit the proliferation, migration, and invasion of lung cancer cells in vitro and inhibit the proliferation of tumors in vivo. Low $C3$ expression has been related to poor prognosis in NSCLC, and a significant correlation has been found between $C3$ and age. CCT6A is upregulated in LUAD tissues, and CCT6A overexpression is associated with poor recurrence-free survival and overall survival in LUAD. Published literature has shown that NTSR1 overexpression is able to predict the poor 5-year overall survival of patients with stage I LUAD treated by surgery alone.

TABLE 3.

LUAD data analysis: identification results using the proposed method.

Main G	Interactions
	Age	Gender	Stage	Smoking
AACSP1				*
ABAT
ACY1	*
AVPR2
BANP				*
BLID		*
C17orf58	*
C1QTNF9B		*		*
C3
CAPN15				*
CAV3
CCDC64				*
CCT6A	*
CLUL1		*
CSAG1				*
DCLRE1B		*
DIO3		*
FAM71E1
GPT
HAL			*
HECTD3
HEPN1
HLA.DRB5
HOMEZ	*
ISCU		*
LOC646999		*
LOC727896	*
LRRTM2
MC5R	*
MGC4856		*
MRPS17	*
NCOR1P1				*
NTSR1			*
OR2F2		*	*
PRR21	*
PTCHD4
S100G	*		*
SLC6A18
SRRM5		*
SYS1
TMEM180				*
TSC22D2	*
UMPS		*
WFIKKN2
WNT9B		*

Note:

^*denotes an identified interaction.

Abbreviations: G, gene; LUAD, lung adenocarcinoma.

Beyond the proposed approach, we also apply the alternatives considered in the simulation. The comparison results are shown in Table 4. Beyond the number of findings/overlapping, we also use the RV coefficient to measure overlapping information, which may better accommodate genes with correlated expressions/similar functions but different names. RV coefficient ranges between 0 and 1, with a larger value indicating a stronger overlap in information. It is noted again that FNN cannot conduct any selection. Table 4 suggests that, in terms of the names of main G effects and G-E interactions, different approaches lead to significantly different findings. Based on the RV coefficient, it is concluded that different approaches lead to moderately overlapping findings. With the differences observed in the simulation, this observation is reasonable. To get more insights into the proposed and alternative approaches, we resort to a resampling approach. In particular, we randomly select 80% of the subjects to form the training data, and the rest 20% forms the testing data. We analyze the training data, record selection results, and make predictions for the testing data. To avoid an extreme split, this process is repeated 100 times. With survival data, we use the C-index to measure prediction performance. The mean C-index values for the sparse approaches are 0.578 (proposed), 0.503 (DeepGE-Lasso), 0.533 (CoxMCP), and 0.533 (CoxLasso)—here it is noted that MA cannot generate predictive models. As a “byproduct,” we also obtain the observed occurence index (OOI) value for each effect, which is the percentage of times that an effect is identified across the splittings. The five main G effects with the highest OOI values (i.e., the most stable findings) are S100G (OOI = 0.74), MMP27 (0.67), HEPN1 (0.65), MC5R (0.59), and SCGB1A1 (0.48). The five interactions with the highest OOI values are S100G with age (OOI = 0.37), gender (0.28), and stage (0.28), TNNI2 with smoking (0.31), and SYT8 with smoking (0.32). More detailed results are available from the authors. Overall, it is observed that the proposed approach leads to sensible identifications, improved prediction, and reasonable stability.

TABLE 4.

LUAD data analysis using the proposed and alternative methods: number of overlapping identifications (RV coefficient) for the main G effects and G–E interactions.

Main G	DeepGE-MCP	DeepGE-Lasso	FNN	MA	CoxMCP	CoxLasso
DeepGE-MCP	45	10 (0.480)	–	0 (0.108)	2 (0.409)	12 (0.492)
DeepGE-Lasso		234	–	2 (0.476)	8 (0.686)	21 (0.564)
FNN			–	–	–	–
MA				9	3 (0.291)	2 (0.282)
CoxMCP					39	19 (0.638)
CoxLasso						75
Interactions	DeepGE-MCP	DeepGE-Lasso	FNN	MA	CoxMCP	CoxLasso
DeepGE-MCP	34	2 (0.320)	–	1 (0.198)	0 (0.285)	2 (0.343)
DeepGE-Lasso		199	–	2 (0.451)	1 (0.458)	7 (0.723)
FNN			–	–	–	–
MA				24	1 (0.271)	2 (0.412)
CoxMCP					13	9 (0.271)
CoxLasso						68

Abbreviations: E, environment; G, gene; LUAD, lung adenocarcinoma.

4.2 ∣. Analysis of skin cutaneous melanoma (SKCM) data

Skin cutaneous melanoma (SKCM) accounts for about 72% of skin cancer deaths and is one of the most malignant and invasive cancers. As in the above section, we also analyze the TCGA data. For G variables, we also consider mRNA gene expressions. With the same processing as above, we reduce the dimensionality of gene expressions from 20, 531 to 1000. For E variables, we consider age, gender, AJCC tumor pathologic stage (stage), Clark level, and Breslow thickness at diagnosis. Among them, age and Breslow thickness are continuous, and the others are categorical. We also focus on samples of the White. After processing, there are 291 samples, among whom 140 are censored. For the 151 subjects with observed death, the times range from 2.6 to 339.88 months, with a median of 36.24 months. For the censored subjects, the observed times range from 0.36 to 241.20 months, with a median of 50.69 months.

With the proposed approach, a total of 53 main G effects and 41 interactions are identified. The detailed identification results are provided in Table 5. Literature search also suggests that the identification results are sensible. For example, PAEP has been identified in previous statistical analyses. For females, PAEP is mainly expressed in secretory/decidualized endometrial glands after progesterone exposure. The somatic copy number deletions that occurred in gene CALML6 may affect the melanogenesis pathways and then pose an impact on SKCM. Researchers have found that a lower methylation value of BANP is associated with improved SKCM survival. Arg280His polymorphism of gene XRCC1 has been found to be associated with an approximately 3.5-fold increase in skin cancer risk in homozygote codominant and recessive models. The genes encoding HLA proteins are related to some tumors and they may be responsible for one of the mechanisms involved in the development of melanoma and nonmelanoma skin cancers. KLKs have been used as prognostic/diagnostic markers for many cancer types and participate in skin immune homeostasis. Dysregulation of the KLK cascade directly leads to skin diseases with severe inflammatory aspects. Among all KLKs, KLK14 seems to be the most specific to skin. Its content is lower in other tissues but relatively higher in the skin. It can affect the migration of keratinocytes and also process prefilament proteins. The fusion of the NKAIN3 and TOX genes has been associated with SKCM.

TABLE 5.

SKCM data analysis: identification results using the proposed method.

Main G	Interactions
	Age	Gender	Stage	Clark level	Breslow thickness
ANKRD28			*
APOL1				*
B4GALNT3			*
BANP			*
BLCAP		*
CACNA2D2		*
CALML6				*
CHAT				*
CLDN20		*
CREM			*
CTAGE4
CTAGE9		*
CYTL1
EYA3		*
FAM15B	*
FAM181B				*
FCRLA			*
FXYD1		*
GP2		*
H1F0
H2AX
HIST1H2AJ					*
HLA.DMA
HLA.DQA2
IFNA1					*
IGSF21
KLK14				*
L3HYPDH
LINC00339
MAPKAP1
MGC13038		*
MTMR11				*
NDUFB2		*
NKAIN3	*
OR2A10
OR2J2	*
PAEP		*		*
PCGF5
PKD1L3
RORB	*
SNORA14B	*
SOGA3	*		*	*
SPARCL1
SPG7
SULT1E1	*		*	*
TEX35		*
TMEM169		*
TMEM217
TMEM35				*
TMTC1		*	*
TSC1		*
WASHC2A
XRCC1

Note:

^*denotes an identified interaction.

Abbreviation: SKCM, skin cutaneous melanoma.

The comparison results are summarized in Table 6. The observations on overlapping findings are similar to the above. In the random splitting-based evaluation, the average prediction C-index values are 0.612 (proposed), 0.552 (DeepGE-Lasso), 0.544 (CoxMCP), and 0.605 (Cox-Lasso) for the sparse approaches. In the stability evaluation, the five main G effects with the highest OOI values are RORB (OOI = 0.36), SCARNA4 (0.32), PSG7 (0.29), KRTAP19.7 (0.24), and HIST1H2AD (0.24). The five interactions with the highest OOI values are RORB with age (0.20), RORB with gender (0.21), SCARNA4 with stage (0.18), PSG7 with gender (0.17), and HIST1H2AD with gender (0.19). The overall findings are similar to the above.

TABLE 6.

SKCM data analysis using the proposed and alternative methods: number of overlapping identifications (RV coefficient) for the main G effects and G–E interactions.

Main G	DeepGE-MCP	DeepGE-Lasso	FNN	MA	CoxMCP	CoxLasso
DeepGE-MCP	53	14 (0.493)	-	3 (0.431)	3 (0.414)	19 (0.596)
DeepGE-Lasso		314	-	8 (0.727)	18 (0.701)	25 (0.683)
FNN			-	-	-	-
MA				23	1 (0.466)	3 (0.509)
CoxMCP					56	30 (0.791)
CoxLasso						77
Interactions	DeepGE-MCP	DeepGE-Lasso	FNN	MA	CoxMCP	CoxLasso
DeepGE-MCP	41	2 (0.613)	-	0 (0.034)	0 (0.057)	0 (0.254)
DeepGE-Lasso		232	-	0 (0.019)	1 (0.121)	4 (0.368)
FNN			–	–	–	–
MA				1	0 (0.001)	0 (0.039)
CoxMCP					15	5 (0.391)
CoxLasso						44

Abbreviations: E, environment; G, gene; SKCM, skin cutaneous melanoma.

5 ∣. DISCUSSION

The significance of G–E interaction analysis and the power of neural networks have been well established in recent literature. In this study, we have significantly advanced G–E interaction analysis by conducting neural network-based analysis. The proposed approach has taken advantage of both neural networks (e.g., power in accommodating unspecified nonlinear effects, novel architecture, etc.) and regression analysis (e.g., sparsity, interpretability, and respect for the variable selection hierarchy). It has an intuitive formulation and can be effectively realized. Another significant difference in this study is that we have conducted extensive simulations, which have convincingly shown the superiority of the proposed approach. Finally, in the analysis of TCGA data on LUAD and SKCM, biologically sensible findings that are different from the alternatives have been made. The proposed analysis also has sensible prediction and stability performance.

Deep learning for genetic interaction analysis is a largely unexplored area. It will be of interest to couple other neural network architectures with our proposed strategy (in particular, sparsity and respect of the variable selection strategy). It will also be of interest to explore other ways of achieving sparsity and selection hierarchy. In a handful of studies, it has been inexplicitly argued that, with nonlinear activations, even a “standard” neural network can be viewed as having some capacity of accommodating interactions. We note that this aspect has not been thoroughly studied and that the proposed analysis can provide a more explicit way of interaction analysis. In this study, we have focused on survival data. Extending to continuous, categorical, and other data types can be straightforward. Additionally, G–E interaction analysis and deep learning are both fast-moving. It will be of interest to conduct more extensive comparisons—noting that the adopted alternatives can be the most relevant. Last but not least, we have focused on methodological and numerical developments. In future research, it can be worthwhile to examine theoretical properties, for example, under what conditions the proposed approach can have variable selection consistency.

APPENDIX A

TABLE A1.

Simulation (nonlinear case 1): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
AR1	500	DeepGE-MCP	0.610 (0.046)	6.1 (2.6)	15.0 (6.3)	2.0 (1.6)	20.2 (8.7)
		DeepGE-Lasso	0.617 (0.043)	13.6 (2.9)	46.4 (9.7)	6.7 (2.4)	236.8 (37.3)
		FNN	0.594 (0.046)	–	–	–	–
		MA	–	1.8 (2.3)	0.0 (0.3)	0.3 (0.7)	0.3 (0.7)
		CoxMCP	0.628 (0.056)	8.9 (3.1)	1.5 (1.5)	2.3 (1.7)	7.5 (3.6)
		CoxLasso	0.647 (0.078)	16.3 (8.5)	18.9 (21.6)	7.3 (6.0)	78.4 (82.6)
	1000	DeepGE-MCP	0.732 (0.063)	20.0 (3.9)	29.7 (78.9)	15.9 (4.5)	163.9 (581.2)
		DeepGE-Lasso	0.719 (0.059)	24.3 (4.5)	44.6 (98.2)	14.8 (5.8)	303.0 (615.9)
		FNN	0.619 (0.026)	–	–	–	–
		MA	–	6.3 (5.4)	0.1 (0.4)	0.6 (1.1)	0.5 (0.9)
		CoxMCP	0.691 (0.049)	21.3 (2.0)	2.2 (1.5)	9.5 (2.7)	13.7 (4.7)
		CoxLasso	0.712 (0.055)	26.1 (4.4)	17.7 (25.7)	15.2 (6.7)	85.1 (112.5)
	1500	DeepGE-MCP	0.779 (0.063)	24.9 (2.9)	25.6 (69.8)	22.4 (3.1)	139.9 (532.4)
		DeepGE-Lasso	0.746 (0.091)	26.3 (4.7)	54.3 (124.5)	17.3 (6.1)	386.1 (776.0)
		FNN	0.630 (0.021)	–	–	–	–
		MA	–	8.9 (8.0)	0.2 (0.4)	1.2 (1.6)	0.8 (1.4)
		CoxMCP	0.729 (0.048)	27.6 (1.7)	3.4 (2.0)	19.4 (3.5)	21.1 (7.8)
		CoxLasso	0.729 (0.050)	28.3 (4.0)	11.7 (23.1)	18.5 (6.9)	57.2 (95.3)
AR2	500	DeepGE-MCP	0.736 (0.039)	13.0 (2.9)	5.0 (3.5)	6.4 (2.2)	9.0 (5.2)
		DeepGE-Lasso	0.738 (0.039)	21.9 (2.4)	28.6 (5.6)	10.9 (2.2)	109.9 (15.5)
		FNN	0.631 (0.042)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.1)	0.0 (0.1)	0.1 (0.2)
		CoxMCP	0.685 (0.095)	10.4 (1.6)	1.0 (1.1)	3.2 (1.7)	5.4 (2.7)
		CoxLasso	0.703 (0.094)	19.2 (6.5)	12.6 (17.9)	8.0 (5.7)	52.5 (65.8)
	1000	DeepGE-MCP	0.796 (0.067)	22.0 (3.0)	19.2 (63.5)	16.2 (3.7)	111.7 (479.4)
		DeepGE-Lasso	0.773 (0.073)	26.9 (1.8)	45.5 (94.0)	17.5 (4.1)	220.0 (494.0)
		FNN	0.653 (0.032)	–	–	–	–
		MA	–	2.1 (4.0)	0.0 (0.2)	0.0 (0.2)	0.2 (0.5)
		CoxMCP	0.700 (0.073)	14.5 (1.4)	1.3 (1.4)	8.4 (2.6)	7.9 (5.4)
		CoxLasso	0.733 (0.071)	25.3 (4.7)	8.6 (22.9)	11.7 (7.4)	40.4 (86.5)
	1500	DeepGE-MCP	0.826 (0.058)	26.5 (2.3)	14.1 (45.1)	22.0 (2.8)	65.9 (343.0)
		DeepGE-Lasso	0.819 (0.051)	27.5 (4.2)	15.9 (69.0)	17.5 (4.9)	136.8 (500.1)
		FNN	0.677 (0.026)	–	–	–	–
		MA	–	3.2 (5.3)	0.0 (0.1)	0.1 (0.4)	0.3 (0.8)
		CoxMCP	0.718 (0.074)	17.5 (1.7)	2.4 (2.3)	14.7 (3.1)	14.2 (9.0)
		CoxLasso	0.724 (0.077)	26.9 (4.7)	8.0 (23.6)	14.7 (8.1)	40.9 (107.4)
AR3	500	DeepGE-MCP	0.831 (0.028)	15.0 (3.0)	3.6 (3.3)	6.7 (2.4)	9.4 (4.9)
		DeepGE-Lasso	0.812 (0.029)	20.7 (2.3)	24.0 (5.6)	10.0 (2.5)	84.5 (12.7)
		FNN	0.689 (0.084)	–	–	–	–
		MA	–	0.2 (0.9)	0.0 (0.1)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.706 (0.363)	6.6 (0.9)	0.6 (0.9)	3.0 (1.5)	4.6 (3.0)
		CoxLasso	0.672 (0.130)	14.6 (6.6)	3.5 (9.4)	5.0 (4.7)	17.2 (37.4)
	1000	DeepGE-MCP	0.861 (0.016)	21.3 (3.1)	4.7 (4.7)	12.7 (2.6)	13.8 (7.8)
		DeepGE-Lasso	0.844 (0.019)	24.1 (2.0)	12.1 (5.7)	13.3 (2.7)	53.1 (21.6)
		FNN	0.732 (0.029)	–	–	–	–
		MA	–	0.6 (1.8)	0.0 (0.1)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.705 (0.112)	9.0 (1.4)	0.9 (1.2)	6.1 (1.8)	8.2 (4.8)
		CoxLasso	0.710 (0.139)	20.7 (5.9)	3.0 (15.5)	8.6 (6.2)	18.2 (51.2)
	1500	DeepGE-MCP	0.871 (0.013)	26.3 (2.2)	4.4 (4.0)	16.3 (3.0)	17.0 (8.1)
		DeepGE-Lasso	0.862 (0.035)	25.8 (2.8)	6.4 (45.0)	13.8 (3.5)	38.5 (233.1)
		FNN	0.720 (0.025)	–	–	–	–
		MA	–	1.3 (3.2)	0.0 (0.1)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.708 (0.109)	10.3 (1.4)	1.0 (1.2)	7.7 (2.1)	9.6 (5.9)
		CoxLasso	0.707 (0.119)	23.9 (4.9)	4.1 (14.7)	11.8 (7.1)	24.8 (60.2)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: AR, autoregressive; FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A2.

Simulation (nonlinear case 1, continuous): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
Band1	500	DeepGE-MCP	0.645 (0.047)	8.6 (3.2)	5.4 (4.7)	4.4 (2.1)	7.9 (5.8)
		DeepGE-Lasso	0.572 (0.060)	20.4 (9.0)	265.6 (187.3)	20.5 (10.2)	1592.8 (943.1)
		FNN	0.523 (0.056)	–	–	–	–
		MA	–	0.6 (1.4)	0.0 (0.2)	0.1 (0.2)	0.1 (0.3)
		CoxMCP	0.629 (0.075)	8.6 (2.4)	1.6 (1.3)	2.6 (1.7)	7.8 (3.4)
		CoxLasso	0.668 (0.066)	13.4 (9.2)	14.7 (20.1)	7.0 (6.3)	57.0 (72.0)
	1000	DeepGE-MCP	0.750 (0.040)	21.2 (4.1)	15.1 (34.8)	15.3 (3.5)	45.0 (243.9)
		DeepGE-Lasso	0.533 (0.055)	26.9 (4.5)	381.5 (115.0)	27.1 (4.9)	2167.9 (501.4)
		FNN	0.520 (0.050)	–	–	–	–
		MA	–	5.0 (5.6)	0.1 (0.3)	0.4 (0.8)	0.6 (1.1)
		CoxMCP	0.670 (0.046)	17.4 (1.4)	2.2 (1.4)	9.1 (2.7)	12.5 (5.3)
		CoxLasso	0.709 (0.075)	23.7 (5.7)	11.7 (22.9)	11.4 (6.7)	52.4 (91.5)
	1500	DeepGE-MCP	0.775 (0.078)	25.8 (3.1)	32.8 (85.9)	22.0 (3.3)	193.1 (623.8)
		DeepGE-Lasso	0.523 (0.051)	28.8 (2.0)	437.0 (44.8)	28.8 (1.2)	2340.3 (78.2)
		FNN	0.566 (0.067)	–	–	–	–
		MA	–	8.4 (8.0)	0.1 (0.3)	1.1 (1.5)	0.5 (0.9)
		CoxMCP	0.713 (0.066)	21.2 (2.2)	3,4 (2.4)	17.6 (3.3)	19.5 (7.8)
		CoxLasso	0.718 (0.050)	27.6 (3.9)	5.3 (9.0)	16.0 (6.4)	29.3 (43.6)
Band2	500	DeepGE-MCP	0.749 (0.044)	12.8 (3.5)	5.2 (5.6)	5.6 (2.6)	8.7 (7.2)
		DeepGE-Lasso	0.595 (0.060)	18.2 (8.1)	217.5 (172.8)	24.1 (8.2)	1937.3 (696.0)
		FNN	0.533 (0.069)	–	–	–	–
		MA	–	0.7 (1.5)	0.1 (0.3)	0.0 (0.2)	0.3 (0.8)
		CoxMCP	0.676 (0.081)	9.0 (1.0)	1.0 1.0)	2.8 (1.6)	5.6 (3.4)
		CoxLasso	0.697 (0.085)	16.8 (6.4)	11.1 (16.9)	6.7 (5.5)	45.3 (62.8)
	1000	DeepGE-MCP	0.817 (0.022)	20.8 (3.4)	5.2 (5.7)	13.6 (2.9)	13.3 (10.2)
		DeepGE-Lasso	0.570 (0.059)	26.0 (5.0)	368.4 (112.5)	28.5 (1.4)	2330.9 (96.1)
		FNN	0.525 (0.062)	–	–	–	–
		MA	–	1.4 (3.1)	0.1 (0.2)	0.0 (0.2)	0.2 (0.6)
		CoxMCP	0.710 (0.076)	10.6 (0.9)	1.7 (1.7)	7.8 (2.0)	10.4 (5.2)
		CoxLasso	0.714 (0.076)	22.4 (4.7)	9.0 (25.8)	10.0 (6.8)	39.2 (100.2)
	1500	DeepGE-MCP	0.825 (0.054)	23.2 (2.8)	8.5 (50.0)	16.8 (3.3)	57.1 (344.1)
		DeepGE-Lasso	0.522 (0.067)	28.9 (1.3)	441.0 (19.4)	28.8 (1.1)	2362.5 (14.5)
		FNN	0.557 (0.080)	–	–	–	–
		MA	–	2.2 (4.2)	0.0 (0.1)	0.1 (0.3)	0.2 (0.7)
		CoxMCP	0.737 (0.087)	11.4 (0.8)	1.9 (1.9)	12.0 (2.6)	15.0 (6.6)
		CoxLasso	0.723 (0.095)	24.8 (5.1)	3.9 (10.7)	12.0 (7.4)	21.9 (44.5)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A3.

Simulation (nonlinear case 2): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
AR1	500	DeepGE-MCP	0.675 (0.062)	8.8 (2.8)	6.2 (6.6)	12.8 (4.2)	7.5 (5.4)
		DeepGE-Lasso	0.663 (0.056)	2.9 (5.7)	44.7 (86.1)	20.7 (3.9)	164.1 (384.5)
		FNN	0.569 (0.042)	–	–	–	–
		MA	–	0.0 (0.0)	0.2 (0.4)	5.4 (2.9)	1.3 (1.7)
		CoxMCP	0.681 (0.049)	0.1 (0.2)	1.3 (1.2)	12.8 (3.9)	8.2 (3.6)
		CoxLasso	0.717 (0.051)	1.3 (1.5)	19.9 (21.5)	22.2 (6.5)	85.3 (77.9)
	1000	DeepGE-MCP	0.783 (0.062)	18.5 (2.7)	31.5 (63.9)	28.0 (2.4)	124.9 (477.9)
		DeepGE-Lasso	0.730 (0.109)	7.2 (10.9)	113.7 (169.4)	27.6 (3.9)	309.7 (684.2)
		FNN	0.582 (0.034)	–	–	–	–
		MA	–	0.0 (0.1)	0.4 (0.6)	17.4 (3.1)	2.9 (2.3)
		CoxMCP	0.822 (0.026)	0.2 (0.4)	2.6 (2.0)	29.3 (1.2)	15.6 (7.0)
		CoxLasso	0.802 (0.024)	0.5 (0.8)	7.2 (4.0)	29.3 (1.0)	36.8 (18.4)
	1500	DeepGE-MCP	0.826 (0.017)	19.9 (1.7)	9.3 (9.2)	29.7 (0.7)	15.2 (13.2)
		DeepGE-Lasso	0.724 (0.141)	20.3 (12.0)	315.5 (184.6)	27.7 (3.4)	270.4 (589.4)
		FNN	0.605 (0.034)	–	–	–	–
		MA	–	0.1 (0.2)	0.5 (0.8)	24.6 (2.0)	4.5 (2.6)
		CoxMCP	0.836 (0.016)	0.2 (0.4)	1.4 (1.5)	30.0 (0.1)	11.8 (6.7)
		CoxLasso	0.825 (0.016)	0.2 (0.5)	3.4 (4.4)	30.0 (0.1)	19.4 (20.1)
AR2	500	DeepGE-MCP	0.763 (0.047)	12.7 (3.3)	6.0 (10.0)	16.8 (4.0)	10.9 (13.1)
		DeepGE-Lasso	0.737 (0.054)	3.0 (7.0)	51.8 (107.1)	21.7 (4.5)	116.5 (275.7)
		FNN	0.616 (0.045)	–	–	–	–
		MA	–	0.0 (0.2)	0.2 (0.5)	4.0 (2.7)	1.1 (1.7)
		CoxMCP	0.733 (0.040)	0.1 (0.2)	0.8 (1.0)	12.5 (2.8)	6.6 (3.1)
		CoxLasso	0.777 (0.040)	0.5 (1.2)	9.3 (11.7)	23.5 (4.0)	47.8 (46.7)
	1000	DeepGE-MCP	0.830 (0.041)	19.7 (2.4)	15.7 (14.6)	28.3 (1.5)	26.5 (18.8)
		DeepGE-Lasso	0.680 (0.159)	16.3 (13.9)	254.9 (217.7)	24.9 (6.3)	673.2 (974.6)
		FNN	0.652 (0.032)	–	–	–	–
		MA	–	0.0 (0.1)	0.4 (0.7)	11.1 (4.6)	3.7 (2.8)
		CoxMCP	0.821 (0.035)	0.2 (0.4)	2.0 (1.8)	25.3 (3.5)	13.5 (7.2)
		CoxLasso	0.829 (0.031)	0.2 (0.6)	3.7 (8.8)	29.3 (1.2)	27.1 (33.3)
	1500	DeepGE-MCP	0.851 (0.016)	20.1 (2.0)	8.2 (9.5)	29.8 (0.5)	16.7 (13.1)
		DeepGE-Lasso	0.725 (0.170)	13.8 (14.2)	215.5 (221.3)	29.3 (1.1)	835.0 (1115.7)
		FNN	0.657 (0.038)	–	–	–	–
		MA	–	0.0 (0.2)	0.2 (0.5)	4.0 (2.7)	1.1 (1.7)
		CoxMCP	0.839 (0.031)	0.2 (0.5)	2.0 (2.0)	28.8 (2.0)	15.0 (7.6)
		CoxLasso	0.834 (0.026)	0.1 (0.3)	1.35 (3.4)	29.9 (0.4)	15.3 (17.8)
AR3	500	DeepGE-MCP	0.853 (0.023)	16.3 (2.8)	5.2 (7.6)	17.3 (3.2)	17.8 (10.9)
		DeepGE-Lasso	0.812 (0.029)	20.7 (2.2)	24.0 (5.6)	10.0 (2.5)	84.5 (12.7)
		FNN	0.721 (0.055)	–	–	–	–
		MA	–	0.0 (0.2)	0.1 (0.4)	0.9 (1.7)	0.9 (1.8)
		CoxMCP	0.776 (0.052)	0.0 (0.2)	0.6 (1.1)	7.1 (2.1)	7.4 (3.7)
		CoxLasso	0.816 (0.047)	0.2 (0.8)	2.3 (5.6)	20.0 (3.7)	25.8 (20.3)
	1000	DeepGE-MCP	0.882 (0.016)	20.3 (2.6)	11.8 (12.6)	25.6 (2.3)	28.5 (17.4)
		DeepGE-Lasso	0.848 (0.101)	4.7 (9.7)	73.9 (153.8)	25.3 (3.5)	168.3 (481.8)
		FNN	0.766 (0.039)	–	–	–	–
		MA	–	0.0 (0.1)	0.1 (0.5)	2.4 (2.9)	2.6 (3.5)
		CoxMCP	0.805 (0.044)	0.1 (0.3)	1.5 (2.2)	12.5 (2.9)	11.8 (6.2)
		CoxLasso	0.820 (0.044)	0.1 (0.5)	1.4 (4.5)	25.9 (3.2)	25.6 (18.4)
	1500	DeepGE-MCP	0.892 (0.012)	21.3 (2.6)	7.0 (8.5)	28.4 (1.4)	24.1 (11.9)
		DeepGE-Lasso	0.821 (0.063)	2.0 (4.9)	29.6 (74.2)	20.1 (3.0)	116.7 (325.1)
		FNN	0.787 (0.040)	–	–	–	–
		MA	–	0.0 (0.0)	0.1 (0.5)	4.0 (4.3)	4.6 (6.3)
		CoxMCP	0.808 (0.050)	0.2 (0.5)	1.8 (2.3)	15.5 (2.7)	15.6 (7.4)
		CoxLasso	0.823 (0.037)	0.0 (0.1)	0.9 (3.1)	28.0 (2.1)	25.0 (17.4)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: AR, autoregressive; FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A4.

Simulation (nonlinear case 2, continuous): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
Band1	500	DeepGE-MCP	0.645 (0.055)	8.6 (3.3)	4.8 (3.5)	4.6 (2.2)	7.5 (5.0)
		DeepGE-Lasso	0.515 (0.050)	27.7 (1.7)	433.1 (23.1)	19.9 (5.6)	930.2 (662.9)
		FNN	0.527 (0.054)	–	–	–	–
		MA	–	0.0 (0.1)	0.2 (0.4)	4.7 (2.5)	1.0 (1.4)
		CoxMCP	0.684 (0.050)	0.1 (0.3)	1.3 (1.2)	13.0 (3.5)	8.7 (3.7)
		CoxLasso	0.721 (0.051)	1.6 (2.0)	19.9 (20.0)	22.3 (6.0)	84.4 (72.6)
	1000	DeepGE-MCP	0.775 (0.073)	16.7 (3.7)	32.1 (64.1)	26.0 (3.8)	124.0 (478.4)
		DeepGE-Lasso	0.516 (0.040)	28.8 (1.2)	448.1 (5.8)	26.3 (3.3)	1185.3 (648.9)
		FNN	0.582 (0.041)	–	–	–	–
		MA	–	0.0 (0.2)	0.4 (0.8)	15.4 (3.2)	2.8 (2.2)
		CoxMCP	0.822 (0.028)	0.2 (0.4)	2.8 (2.2)	29.0 (1.7)	17.1 (6.6)
		CoxLasso	0.805 (0.022)	0.5 (0.8)	7.3 (5.3)	29.2 (1.5)	39.4 (19.5)
	1500	DeepGE-MCP	0.823 (0.037)	18.6 (2.1)	2.2 (3.0)	28.4 (1.3)	3.8 (3.3)
		DeepGE-Lasso	0.552 (0.114)	28.1 (4.3)	429.9 (65.6)	27.8 (3.6)	1030.3 (1158.3)
		FNN	0.583 (0.054)	–	–	–	–
		MA	–	0.0 (0.2)	0.3 (0.5)	22.1 (2.2)	4.3 (1.9)
		CoxMCP	0.836 (0.017)	0.1 (0.3)	1.6 (1.5)	29.9 (0.4)	12.2 (5.7)
		CoxLasso	0.827 (0.014)	0.4 (0.6)	3.7 (3.1)	29.9 (0.2)	23.0 (12.7)
Band2	500	DeepGE-MCP	0.752 (0.045)	13.1 (3.2)	4.6 (3.5)	5.6 (2.4)	8.3 (5.1)
		DeepGE-Lasso	0.506 (0.049)	28.6 (1.1)	450.2 (4.8)	27.2 (42.4)	2119.1 (229.1)
		FNN	0.570 (0.095)	–	–	–	–
		MA	–	0.0 (0.0)	0.1 (0.3)	3.0 (2.2)	1.2 (1.5)
		CoxMCP	0.746 (0.039)	0.0 (0.2)	0.7 (1.0)	9.8 (2.3)	6.8 (3.2)
		CoxLasso	0.781 (0.038)	0.3 (0.7)	7.1 (12.5)	20.1 (4.8)	38.9 (42.5)
	1000	DeepGE-MCP	0.833 (0.022)	17.5 (3.1)	13.0 (16.2)	25.0 (3.2)	22.6 (21.3)
		DeepGE-Lasso	0.517 (0.039)	28.7 (1.2)	451.3 (3.8)	26.7 (4.7)	1959.8 (644.3)
		FNN	0.637 (0.066)	–	–	–	–
		MA	–	0.0 (0.2)	0.4 (0.8)	8.8 (3.4)	4.1 (3.0)
		CoxMCP	0.813 (0.045)	0.1 (0.3)	2.2 (2.3)	19.2 (3.9)	13.5 (7.0)
		CoxLasso	0.829 (0.027)	0.3 (0.5)	3.3 (3.6)	28.4 (1.5)	27.6 (14.1)
	1500	DeepGE-MCP	0.843 (0.036)	18.1 (2.4)	4.6 (36.5)	26.5 (2.0)	31.6 (244.9)
		DeepGE-Lasso	0.665 (0.166)	25.1 (7.8)	395.4 (120.6)	24.9 (4.1)	345.9 (819.9)
		FNN	0.668 (0.054)	–	–	–	–
		MA	–	0.0 (0.1)	0.4 (0.7)	13.6 (3.9)	7.3 (4.6)
		CoxMCP	0.828 (0.034)	0.2 (0.6)	2.3 (2.1)	23.9 (3.5)	17.2 (9.8)
		CoxLasso	0.835 (0.025)	0.1 (0.3)	1.7 (2.6)	29.5 (0.8)	20.6 (11.6)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A5.

Simulation (nonlinear case 3): mean (sd) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
AR1	500	DeepGE-MCP	0.578 (0.079)	8.2 (6.2)	48.7 (110.6)	6.3 (8.5)	288.1 (765.8)
		DeepGE-Lasso	0.525 (0.061)	24.7 (7.1)	368.7 (120.9)	24.0 (7.4)	1931.6 (631.2)
		FNN	0.552 (0.071)	–	–	–	–
		MA	–	0.6 (1.4)	0.0 (0.1)	0.1 (0.3)	0.2 (0.6)
		CoxMCP	0.600 (0.094)	5.4 (2.8)	1.1 (1.1)	2.5 (1.7)	6.3 (3.9)
		CoxLasso	0.592 (0.099)	8.2 (6.5)	7.6 (13.9)	4.5 (4.4)	33.9 (53.2)
	1000	DeepGE-MCP	0.647 (0.036)	12.4 (3.3)	17.8 (27.4)	8.2 (3.5)	46.8 (223.6)
		DeepGE-Lasso	0.529 (0.057)	22.7 (5.2)	244.3 (155.5)	23.2 (6.4)	1593.0 (740.9)
		FNN	0.576 (0.032)	–	–	–	–
		MA	–	2.5 (3.9)	0.0 (0.0)	0.5 (1.2)	0.4 (0.7)
		CoxMCP	0.690 (0.104)	16.6 (2.3)	2.2 (1.7)	11.0 (02.4)	13.1 (4.6)
		CoxLasso	0.677 (0.105)	18.3 (6.3)	11.5 (6.0)	31.8 (65.2)
	1500	DeepGE-MCP	0.724 (0.077)	22.7 (3.9)	31.1 (94.7)	21.9 (4.6)	181.2 (622.2)
		DeepGE-Lasso	0.502 (0.044)	29.1 (1.0)	455.8 (8.7)	29.0 (1.0)	2392.9 (11.1)
		FNN	0.592 (0.032)	–	–	–	–
		MA	–	2.9 (5.0)	0.1 (0.3)	0.7 (1.5)	0.4 (1.0)
		CoxMCP	0.738 (0.074)	23.1 (2.3)	3.2 (2.1)	20.3 (2.6)	17.7 (6.1)
		CoxLasso	0.725 (0.080)	24.1 (6.1)	4.9 (7.4)	18.5 (7.3)	27.7 (37.8)
AR2	500	DeepGE-MCP	0.641 (0.086)	12.7 (5.7)	64.7 (129.3)	8.9 (9.4)	406.6 (896.7)
		DeepGE-Lasso	0.530 (0.072)	23.8 (5.8)	334.1 (113.6)	25.2 (6.4)	2035.2 (531.8)
		FNN	0.582 (0.046)	–	–	–	–
		MA	–	0.2 (0.9)	0.0 (0.2)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.650 (0.135)	8.0 (1.5)	1.0 (1.1)	3.0 (1.7)	6.4 (2.9)
		CoxLasso	0.644 (0.122)	9.7 (6.3)	4.9 (11.0)	4.1 (4.2)	22.5 (44.4)
	1000	DeepGE-MCP	0.720 (0.076)	20.2 (4.0)	38.5 (116.2)	13.1 (5.7)	206.6 (665.8)
		DeepGE-Lasso	0.560 (0.061)	22.9 (6.2)	250.1 (166.3)	24.3 (5.8)	1816.6 (657.5)
		FNN	0.652 (0.032)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.1)	0.0 (0.1)	0.1 (0.2)
		CoxMCP	0.729 (0.122)	12.5 (1.2)	1.2 (1.1)	9.1 (2.2)	8.6 (4.7)
		CoxLasso	0.697 (0.150)	17.5 (7.3)	2.8 (7.8)	9.1 (6.9)	15.8 (34.7)
	1500	DeepGE-MCP	0.772 (0.020)	24.5 (2.3)	2.0 (1.8)	20.4 (3.3)	8.3 (4.1)
		DeepGE-Lasso	0.521 (0.060)	24.4 (4.7)	292.3 (125.6)	26.2 (4.0)	2018.5 (386.5)
		FNN	0.623 (0.039)	–	–	–	–
		MA	–	0.3 (1.2)	0.0 (0.2)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.740 (0.123)	14.1 (1.4)	1.6 (1.6)	13.7 (2.2)	11.5 (6.1)
		CoxLasso	0.718 (0.161)	21.8 (6.5)	2.6 (9.7)	13.4 (7.0)	15.6 (42.1)
AR3	500	DeepGE-MCP	0.740 (0.053)	13.3 (2.9)	6.1 (34.9)	5.7 (3.3)	34.0 (244.1)
		DeepGE-Lasso	0.569 (0.070)	18.8 (6.1)	215.9 (123.3)	24.0 (5.9)	1928.1 (487.9)
		FNN	0.625 (0.048)	–	–	–	–
		MA	–	0.0 (0.0)	0.0 (0.0)	0.0 (0.0)	0.0 (0.2)
		CoxMCP	0.634 (0.189)	5.4 (0.9)	0.5 (0.8)	2.6 (1.2)	4.6 (2.5)
		CoxLasso	0.625 (0.190)	8.9 (5.6)	2.2 (6.8)	3.4 (3.4)	12.1 (26.3)
	1000	DeepGE-MCP	0.780 (0.034)	24.0 (2.5)	4.6 (11.0)	13.7 (3.0)	16.8 (7.7)
		DeepGE-Lasso	0.614 (0.045)	15.6 (7.3)	135.1 (146.9)	21.5 (9.2)	1733.7 (804.9)
		FNN	0.669 (0.034)	–	–	–	–
		MA	–	0.0 (0.1)	0.0 (0.1)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.693 (0.167)	7.0 (1.0)	1.0 (1.1)	4.6 (1.6)	8.9 (4.5)
		CoxLasso	0.658 (0.197)	11.3 (6.6)	1.5 (12.4)	4.4 (4.1)	9.3 (43.3)
	1500	DeepGE-MCP	0.785 (0.020)	20.4 (2.7)	0.2 (0.5)	15.3 (3.0)	7.8 (3.5)
		DeepGE-Lasso	0.602 (0.053)	18.3 (7.1)	171.3 (154.3)	26.1 (5.4)	2052.7 (519.3)
		FNN	0.700 (0.031)	–	–	–	–
		MA	–	0.1 (0.4)	0.0 (0.0)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.706 (0.165)	7.9 (1.0)	1.4 (2.0)	6.0 (1.8)	10.5 (4.8)
		CoxLasso	0.684 (0.158)	14.7 (7.0)	3.2 (21.0)	6.9 (5.6)	18.2 (80.9)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: AR, autoregressive; FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A6.

Simulation (nonlinear case 3, continuous): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
Band1	500	DeepGE-MCP	0.575 (0.072)	5.1 (3.1)	8.3 (39.7)	2.9 (3.2)	29.3 (243.0)
		DeepGE-Lasso	0.566 (0.053)	22.0 (7.0)	298.9 (131.0)	23.5 (5.9)	1854.0 (513.0)
		FNN	0.524 (0.057)	–	–	–	–
		MA	–	0.7 (1.7)	0.1 (0.4)	0.1 (0.2)	0.2 (0.4)
		CoxMCP	0.602 (0.106)	5.8 (2.6)	1.4 (1.4)	2.8 (1.8)	7.5 (4.1)
		CoxLasso	0.607 (0.107)	9.3 (6.1)	8.5 (14.7)	5.5 (4.2)	38.1 (56.0)
	1000	DeepGE-MCP	0.685 (0.043)	17.6 (4.5)	12.1 (9.8)	12.1 (3.6)	17.1 (13.6)
		DeepGE-Lasso	0.537 (0.049)	15.4 (3.3)	21.1 (41.0)	12.5 (5.2)	359.6 (478.5)
		FNN	0.591 (0.035)	–	–	–	–
		MA	–	1.4 (3.4)	0.0 (0.2)	0.2 (0.5)	0.2 (0.7)
		CoxMCP	0.684 (0.078)	15.0 (1.8)	2.2 (1.7)	10.8 (2.6)	13.2 (4.5)
		CoxLasso	0.697 (0.080)	18.2 (6.5)	5.6 (11.2)	12.1 (5.9)	29.7 (48.2)
	1500	DeepGE-MCP	0.744 (0.019)	24.3 (2.6)	10.5 (5.7)	20.0 (2.6)	15.8 (7.1)
		DeepGE-Lasso	0.510 (0.050)	19.5 (5.6)	116.6 (157.3)	20.9 (6.6)	1209.2 (831.1)
		FNN	0.604 (0.041)	–	–	–	–
		MA	–	3.0 (5.8)	0.1 (0.3)	0.5 (1.6)	0.3 (0.9)
		CoxMCP	0.737 (0.073)	18.2 (1.3)	2.7 (2.0)	18.4 (2.8)	16.3 (7.3)
		CoxLasso	0.731 (0.072)	23.8 (5.3)	6.3 (20.2)	18.3 (6.9)	32.4 (82.5)
Band2	500	DeepGE-MCP	0.687 (0.044)	9.6 (3.1)	4.3 (3.4)	4.0 (2.1)	7.5 (5.4)
		DeepGE-Lasso	0.599 (0.045)	20.1 (6.3)	252.7 (130.3)	23.5 (5.4)	1836.9 (489.5)
		FNN	0.534 (0.065)	–	–	–	–
		MA	–	0.3 (1.1)	0.0 (0.0)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.661 (0.119)	7.2 (1.6)	1.0 (1.0)	3.3 (1.5)	6.2 (2.8)
		CoxLasso	0.641 (0.111)	9.1 (6.4)	3.3 (7.4)	4.1 (4.0)	15.5 (28.8)
	1000	DeepGE-MCP	0.742 (0.043)	17.7 (2.9)	3.4 (6.6)	10.6 (3.4)	44.8 (286.2)
		DeepGE-Lasso	0.572 (0.062)	20.5 (5.2)	193.0 (126.7)	23.6 (5.9)	1779.9 (566.0)
		FNN	0.618 (0.046)	–	–	–	–
		MA	–	0.2 (0.8)	0.0 (0.0)	0.0 (0.1)	0.1 (0.3)
		CoxMCP	0.723 (0.118)	9.8 (0.9)	1.5 (1.5)	8.1 (2.0)	10.9 (5.1)
		CoxLasso	0.727 (0.126)	17.9 (5.3)	2.7 (13.1)	9.6 (4.9)	17.0 (51.0)
	1500	DeepGE-MCP	0.773 (0.019)	25.1 (2.2)	6.7 (5.0)	17.5 (3.3)	15.8 (8.6)
		DeepGE-Lasso	0.587 (0.087)	15.1 (3.1)	8.2 (44.9)	12.2 (6.9)	479.9 (678.0)
		FNN	0.644 (0.048)	–	–	–	–
		MA	–	0.1 (0.6)	0.1 (0.5)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.738 (0.120)	10.6 (0.7)	2.2 (2.0)	10.9 (2.2	14.7 (6.8)
		CoxLasso	0.742 (0.123)	20.6 (6.7)	2.0 (4.7)	12.7 (6.7)	15.6 (25.0)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

TABLE A7.

Simulation (nonlinear case 3, AR correlation, censoring rate 0.5): mean (SD) based on 100 replicates.

				Main effects		Interactions
Correlation	n	Method	C-index	M:TP	M:FP	I:TP	I:FP
AR1	500	DeepGE-MCP	0.574 (0.063)	6.0 (5.3)	34.9 (80.2)	2.8 (3.5)	92.8 (300.4)
		DeepGE-Lasso	0.530 (0.066)	24.8 (5.2)	356.7 (93.0)	24.9 (4.4)	1984.8 (355.3)
		FNN	0.558 (0.053)	–	–	–	–
		MA	–	0.3 (0.8)	0.0 (0.2)	0.1 (0.3)	0.2 (0.5)
		CoxMCP	0.561 (0.092)	3.4 (2.6)	1.1 (1.2)	1.5 (1.3)	5.8 (4.7)
		CoxLasso	0.558 (0.080)	8.8 (7.2)	17.7 (22.3)	5.1 (4.9)	70.0 (82.1)
	1000	DeepGE-MCP	0.624 (0.039)	8.7 (3.2)	9.3 (15.1)	5.6 (3.3)	62.2 (305.2)
		DeepGE-Lasso	0.569 (0.054)	23.0 (6.0)	291.4 (136.4)	21.8 (8.3)	1594.1 (761.7)
		FNN	0.581 (0.033)	–	–	–	–
		MA	–	1.1 (2.4)	0.0 (0.1)	0.2 (0.5)	0.2 (0.7)
		CoxMCP	0.649 (0.095)	13.7 (2.7)	2.5 (2.0)	8.1 (2.5)	14.1 (5.1)
		CoxLasso	0.655 (0.087)	17.0 (8.1)	18.0 (28.4)	11.3 (7.0)	81.5 (121.0)
	1500	DeepGE-MCP	0.673 (0.053)	12.3 (4.6)	14.2 (62.5)	10.4 (5.1)	80.3 (421.1)
		DeepGE-Lasso	0.532 (0.055)	22.8 (3.6)	157.2 (120.2)	22.1 (5.4)	1250.0 (683.1)
		FNN	0.596 (0.046)	–	–	–	–
		MA	–	1.3 (3.1)	0.0 (0.2)	0.3 (1.0)	0.2 (0.5)
		CoxMCP	0.712 (0.068)	20.1 (2.3)	3.5 (2.2)	15.9 (2.9)	19.4 (6.0)
		CoxLasso	0.708 (0.077)	22.0 (7.1)	10.4 (21.8)	16.5 (7.3)	56.0 (99.8)
AR2	500	DeepGE-MCP	0.637 (0.060)	7.7 (3.4)	9.3 (38.6)	3.8 (5.3)	102.4 (470.1)
		DeepGE-Lasso	0.503 (0.069)	23.2 (6.9)	339.6 (125.2)	26.2 (4.8)	2126.2 (401.0)
		FNN	0.584 (0.049)	–	–	–	–
		MA	–	0.0 (0.2)	0.0 (0.2)	0.0 (0.0)	0.2 (0.6)
		CoxMCP	0.633 (0.133)	6.6 (2.0)	1.0 (1.2)	2.3 (1.3)	6.6 (3.3)
		CoxLasso	0.606 (0.133)	9.2 (7.0)	11.1 (18.7)	4.2 (4.2)	43.6 (66.5)
	1000	DeepGE-MCP	0.703 (0.038)	13.3 (3.8)	1.8 (2.0)	7.0 (2.7)	5.7 (4.8)
		DeepGE-Lasso	0.530 (0.068)	26.5 (3.8)	385.4 (70.1)	27.7 (1.8)	2237.0 (105.2)
		FNN	0.599 (0.034)	–	–	–	–
		MA	–	0.1 (0.3)	0.0 (0.1)	0.0 (0.0)	0.1 (0.3)
		CoxMCP	0.695 (0.130)	11.5 (1.5)	1.3 (1.3)	7.6 (2.1)	9.5 (4.3)
		CoxLasso	0.695 (0.139)	16.4 (7.8)	6.9 (16.1)	9.7 (6.4)	34.1 (65.8)
	1500	DeepGE-MCP	0.752 (0.026)	20.1 (3.0)	3.4 (2.9)	13.2 (2.8)	8.5 (4.0)
		DeepGE-Lasso	0.552 (0.067)	23.3 (4.2)	186.3 (153.7)	23.2 (5.8)	1487.4 (750.7)
		FNN	0.626 (0.060)	–	–	–	–
		MA	–	0.2 (0.7)	0.0 (0.1)	0.0 (0.0)	0.1 (0.4)
		CoxMCP	0.717 (0.133)	13.0 (1.3)	1.6 (1.5)	11.1 (2.3)	12.5 (6.3)
		CoxLasso	0.701 (0.161)	19.0 (8.4)	5.4 (16.4)	11.6 (7.8)	29.1 (74.2)
AR3	500	DeepGE-MCP	0.732 (0.045)	13.5 (2.9)	5.3 (4.1)	4.5 (2.3)	11.2 (6.0)
		DeepGE-Lasso	0.528 (0.081)	18.5 (7.6)	230.3 (145.4)	25.6 (5.3)	2062.4 (452.8)
		FNN	0.602 (0.055)	–	–	–	–
		MA	–	0.0 (0.2)	0.0 (0.1)	0.0 (0.0)	0.1 (0.5)
		CoxMCP	0.625 (0.154)	4.9 (0.9)	0.6 (0.9)	2.0 (1.2)	5.1 (2.2)
		CoxLasso	0.590 (0.152)	7.8 (5.9)	5.4 (14.0)	2.9 (3.2)	20.9 (44.3)
	1000	DeepGE-MCP	0.766 (0.026)	13.6 (2.2)	0.3 (0.6)	7.4 (2.5)	6.3 (2.6)
		DeepGE-Lasso	0.570 (0.071)	22.0 (6.1)	285.6 (120.9)	25.5 (5.4)	2068.8 (457.3)
		FNN	0.657 (0.041)
		MA		0.0 (0.0)	0.0 (0.1)	0.0 (0.0)	0.0 (0.1)
		CoxMCP	0.655 (0.205)	6.3 (0.9)	0.8 (1.0)	3.9 (1.4)	7.6 (4.0)
		CoxLasso	0.640 (0.166)	12.2 (6.9)	6.6 (20.7)	5.4 (4.9)	29.2 (73.6)
	1500	DeepGE-MCP	0.788 (0.022)	22.9 (2.4)	1.7 (1.5)	14.5 (2.9)	13.5 (4.2)
		DeepGE-Lasso	0.596 (0.054)	24.1 (4.3)	292.2 (99.7)	26.7 (3.3)	2090.0 (258.5)
		FNN	0.675 (0.058)
		MA		0.0 (0.1)	0.1 (0.3)	0.0 (0.0)	0.1 (0.3)
		CoxMCP	0.668 (0.159)	7.4 (1.0)	1.2 (1.5)	5.2 (1.7)	10.0 (4.8)
		CoxLasso	0.654 (0.192)	13.5 (8.0)	1.4 (7.9)	6.4 (5.3)	12.1 (33.0)

Note: C-index and TP/FP: true/false positives for main effects and interactions separately. There are 30 nonzero main effects and 30 nonzero interactions.

Abbreviations: AR, autoregressive; FNN, fully-connected neural network; MA, marginal analysis; MCP, minimax concave penalty.

DATA AVAILABILITY STATEMENT

The TCGA data that support the findings of this study are available from TCGA Program. Data are generated by The TCGA Research Network at https://www.cancer.gov/about-nci/organization/ccg/research/structural-genomics/tcga.