Assisted differential network analysis for gene expression data

Abstract

In the analysis of gene expression data, when there are two or more disease conditions/groups (for example, diseased and normal, responder and non-responder, and multiple stages/subtypes), differential analysis has been extensively conducted to identify key differences and has important implications. Network analysis takes a system perspective and can be more informative than that limited to simple statistics such as mean and variance. In differential network analysis, a common practice is to first estimate a gene expression network for each condition/group, and then spectral clustering can be applied to the network difference(s) to identify key genes and biological mechanisms that lead to the differences. Compared to “simple” analysis such as regression, differential network analysis can be more challenging with the significantly larger number of parameters. In this study, taking advantage of the increasing popularity of multidimensional profiling data, we develop an assisted analysis strategy and propose incorporating regulator information to improve the identification of key genes (that lead to the differences in gene expression networks). An effective computational algorithm is developed. Comprehensive simulation is conducted, showing that the proposed approach can outperform the benchmark alternatives in identification accuracy. With the TCGA lung adenocarcinoma (LUAD) data, we analyze the expressions of genes in the KEGG cell cycle pathway, assisted by copy number variation data. The proposed assisted analysis leads to identification results similar to the alternatives but different estimations. Overall, this study can deliver an efficient and cost-effective way of improving differential network analysis.

1. Introduction

Gene expression data has been playing a uniquely important role in biomedical research. The analysis of gene expression data has led to a deeper understanding of disease etiology and actionable targets for the development of treatment/prevention strategies. An important type of analysis is to compare gene expression profiles under different conditions/groups, which can be diseased and normal, responder and non-responder, multiple subtypes, and others. For practical examples of such analysis, we refer to Costa-Silva et al. (2017), Rapaport et al. (2013), and many others.

When comparing gene expressions between conditions, the simplest way is through means (medians, etc.; Costa-Silva et al. 2017). It has been recognized that the first moment does not contain all relevant information. Accordingly, variance (second moment)-based analysis has been conducted (Costa-Silva et al. 2017), motivated by the understanding that a higher variation indicates lower stability, which increases disease susceptibility and severity. Further advancing from such analyses, network-based analysis has been developed (Kolaczyk, 2009), which takes a system perspective and describes properties of not only individual gene expressions but also their interconnections.

Gene expression network analyses can be mainly classified into two categories: unconditional and conditional. In unconditional analysis, the goal is to quantify whether any two gene expressions are independent while “disregarding” the others. A representative example is the WGCNA (Weighted Gene Co-Expression Network Analysis; Horvath, 2011). In the WGCNA and other unconditional analyses, variance-covariance matrix is often an effective starting point. Unconditional gene expression networks can be either directional or undirectional, either weighted or unweighted, and either sparse or dense. In comparison, in conditional analysis, the goal is to quantify whether two gene expressions are independent conditional on the rest of the genes. The most extensively conducted conditional analysis is perhaps the Gaussian Graphical Model (GGM; Friedman et al. 2008), under which it is assumed that gene expressions have a multivariate normal distribution. Here quantifying conditional independence is equivalent to determining whether precision matrix (inverse of variance-covariance matrix) has the corresponding elements being zero. When the normality assumption is violated, alternative approaches have been developed, which replace Pearson’s correlation (in variance-covariance matrix) by robust, for example Kendall’s tau, correlations, and then proceed in the same way as under the GGM. In both unconditional and conditional analysis, when sparsity is desirable – which is usually the case for gene expression analysis, regularization can be applied (Mazumder and Hastie, 2012). It is noted that this may be routinely needed in conditional analysis, which demands the joint estimation of a large number of parameters. For example, for GGM, the graphical Lasso approach (Friedman et al. 2008), which applies Lasso penalization to GGM estimation, has been popular.

In this article, we consider comparing gene expressions between two conditions, and note that the discussions and proposed approach are also applicable to more than two conditions. The first step, very naturally, is to determine whether the gene expression networks under the two conditions are significantly different. A “straightforward” approach is to first take the difference of the two networks, and then compute a certain norm of this difference. Norms considered in the literature include the Frobenius, ℓ_∞, and others (Shojaie 2020). Some studies have derived asymptotic distributions of such norms. An alternative solution is to apply, for example, permutation-based techniques. Once a p-value is computed, the significance of network difference can be inferred.

For many “simple” problems, for example the comparison between disease subtypes or between normal and diseased, significant differences are obviously expected. In this case, the natural next step is to identify which genes contribute to such differences. For both unconditional and conditional analysis, with the difference of networks, a simple approach is to examine which genes correspond to large elements. A more rigorous approach is via spectral clustering (Yu et al. 2017), which amounts to conducting SVD (singular value decomposition) of the difference of networks. In many analyses, sparsity is assumed, under which only a small number of genes contribute to the difference. Then regularization is needed along with SVD to differentiate “signals” from “noises”. A “straightforward” choice is the SSVD (sparse SVD) technique, with which some components of the singular vectors can be estimated as exactly zero (Lee et a. 2010). When the first sparse singular vector is estimated, the nonzero components correspond to the first gene expression cluster that contributes to the difference (Yu et al. 2017). If desirable, the SSVD procedure can be repeated to identify the subsequent gene expression clusters that also contribute to the difference. This analysis pipeline has been developed in the literature (Yu et al. 2017). Here we note that it can be conducted in the same manner for both unconditional and conditional networks.

As well recognized in the literature, network analysis is challenged by the high dimensionality of parameters and limited sample size. This can be especially true in differential analysis, under which at least two networks need to be estimated. Gene expressions are heavily regulated. We note that here we take a loose definition and generically refer to upstream molecular mechanisms that can a ect gene expression levels, including but not limited to copy number variation and other DNA mutations (for example, somatic mutations in cancer studies), methylation, and microRNA, as “regulators”. Under other and simpler contexts, assisted analysis has been developed to take advantage of regulator information and assist the analysis of gene expressions. One example is collaborative regression (Gross and Tibshirani 2015). Here for a response variable, a regression model is built using gene expressions only, and a separate model is built using regulators only. Considering the gene-expression-regulator relationship, this approach promotes the two models to have similar estimates for the response variable. This approach may be limited by not explicitly modeling the regulation relationship. To tackle this problem, the ARMI approach includes the additional gene-expression-regulator modeling step (Chai et al. 2017). Assisted analysis has also been developed for clustering (Li et al. 2018) and other purposes. In a recent study (Yi et al. 2021), for a single sample group, assisted analysis is developed for conditional gene expression network analysis. The aforementioned studies have provided strong evidences on the effectiveness of gene expression analysis assisted by regulators.

In this study, we consider comparing gene expression networks between two conditions and identifying key contributors. This can accommodate multiple application scenarios. For each subject, it is assumed that both gene expression and regulator measurements are available. Here it is noted that, as in the published assisted studies, the proposed analysis does not demand all collected regulators to be relevant, or all relevant regulators to be collected. This study advances from the existing literature in the following important aspects. First, it advances from differential analysis based on mean and variance by studying the interconnections among genes. Second, it advances from the existing differential network analysis by incorporating information in regulators. Third, its data settings are fundamentally different from those in Yi et al. (2021) and some other studies. Specifically, in Yi et al. (2021), the two networks have different formulating components: one without regulators, and the other with regulators. In contrast, in the present analysis, regulator data is used in network construction under both conditions. This study also advances by conducting both conditional and unconditional network analysis. Fourth, this study advances from assisted regression and clustering analysis by conducting more complex network analysis.

2. Methods

2.1. Data and benchmark analysis

Consider the setting with two sample groups. For each subject, data is available on gene expressions and their regulators. Use Y₁ and Y₂ to denote the p gene expressions for the two groups. Further, use X₁ and X₂ to denote the q regulators for the two groups. When there are two or more types of regulators, measurements can be stacked together to create “mega” regulators – this seemingly simple approach has been shown as effective (Zhu et al. 2016). Here we note that “ordinary” data processing, for example normalization, data matching, and accommodation of missing data, is still needed prior to analysis.

For differential network analysis, the first step is to construct networks for the two groups separately. As discussed in the previous section, this can be achieved in multiple ways. In this article, we consider two popular options: (a) For unconditional network analysis, we adopt the correlation matrix to measure gene expression interconnections, where a large value indicates a stronger interconnection. Sparsity can be further imposed to generate simpler network structures, although this is not pursued in our numerical study. (b) For conditional network analysis, we adopt the GGM approach and use the precision matrix to measure gene expression interconnections. We impose Lasso for sparsity, under which a nonzero component of precision matrix indicates an interconnection (Friedman et al. 2008). In what follows, we focus on adjacency matrix, which is a squared matrix. Its elements describe whether an interconnection exists between the corresponding nodes, and the magnitudes can describe the strengths of interconnections (Kolaczyk, 2009). Use p × p matrices G₁ and G₂ to denote the network adjacency matrices. Denote G_diff = G₁ − G₂.

To identify genes that contribute the most to the network differences, the existing “benchmark” first finds the best rank-one approximation to G_diff. This can be achieved via SVD. When sparsity is desired, SSVD, for example as developed in Lee et al. (2010), can be applied. Denote the first component of SVD (or SSVD) as $s_1^G{v}_1{w}_1^{\top }$, where s₁ is the largest singular value, and vectors v₁ and w₁ have unit norms. Examining the large components of v₁ under SVD or its nonzero components under SSVD can lead to the identification of the first gene expression cluster that contributes the most to the differences. When it is desirable to identify additional contributing gene expressions, we take $G_{\mathit{\text{diff}}}=G_{\mathit{\text{diff}}}-s_1^G{v}_1{w}_1^{\top }$, conduct SVD (or SSVD), and examine the first component again.

2.2. Assisted estimation

In the above analysis, information contained in regulators is not used. For contexts other than differential network analysis, published studies (Gross and Tibshrani, 2015; Chai et al. 2017; Li et al. 2018) have shown that incorporating regulator information can improve analysis. Motivated by such successes, here we develop an assisted estimation approach.

With X₁ and X₂, we construct their q × q network adjacency matrices R₁ and R₂ using the same technique as for gene expressions. Denote R_diff = R₁ − R₂. To more effectively identify key contributing gene expressions by taking advantage of information contained in R_diff, we propose the penalized objective function:

$$Q\left(v_1,w_1,s_1^G,u_1,z_1,s_1^R\right)={\Vert G_{\mathit{\text{diff}}}-s^G{v}_1{w}_1^{\top }\Vert }_F^2+{\Vert R_{\mathit{\text{diff}}}-s^R{u}_1{z}_1^{\top }\Vert }_F^2+P_{v_1}+P_{w_1}+P_{u_1}+P_{z_1}-P_{\text{similarity}}.$$ (1)

Here v₁, w₁, and $s_1^G$ have the same definitions as in the last subsection, and u₁, z₁, and $s_1^R$ are defined accordingly for the regulators. Further,

$$\begin{gathered} P_{v_1}=\rho \left(\left|v_1\right|;{\lambda }_1,a\right),P_{w_1}=\rho \left(\left|w_1\right|;{\lambda }_1,a\right),P_{u_1}=\rho \left(\left|u_1\right|;{\lambda }_2,a\right),P_{z_1}=\rho \left(\left|z_1\right|;{\lambda }_2,a\right), \\ {P}_{\text{similarity}}={\lambda }_{3}I{\left(v_1\ne 0\right)}^{\top }\cdot \left|\text{cor}\left(Y_1,X_1\right)+\mathit{\text{cor}}\left(Y_2,X_2\right)\right|\cdot I\left(u_1\ne 0\right). \end{gathered}$$

Here λ₁, λ₂ and λ₃ are data-dependent tuning parameters. To simplify computation, it is possible to set λ₁ = λ₂. $\rho \left(|\cdot |;\lambda ,a\right)=\lambda {\int }_0^{|\cdot |}{\left(1-\frac{x}{\lambda a}\right)}_{+}dx$ is the MCP (minimax concave penalty), and a is its regularization parameter. The indicator function I(·) is defined component-wise. cor(·,·) is the correlation matrix. With the estimate of v₁, gene expressions contributing the most to the differences can be identified as those with nonzero estimates. As a “byproduct”, regulators that contribute the most to the differences of regulator networks can also be identified. When it is desirable to identify additional contributing gene expressions (and regulators), G_diff and R_diff can be updated in the same way as described in the last subsection, and the procedure described above can be repeated.

In (1), the indicator function is discontinuous, which may bring significant computational challenges. To simplify computation, we propose the approximation:

$$I\left(\eta \ne 0\right)\approx \left(1-\text{exp}\left(-\frac{{\eta }^2}{\tau }\right)\right),$$

where τ is a small positive constant. In our numerical study and the published ones, it is found that the value of τ is not critical, as long as it is small enough. We set τ = 0.1 in our numerical study and note that it may be prudent to consider additional values under other circumstances.

Rationale

In (1), if λ₃ = 0, the analysis reduces to the identification of important gene expressions (contributing to the gene expression network differences) and that of important regulators (contributing to the regulator network differences) separately. The key advancement lies in the newly added penalty P_similarity. In this penalty, the matrix |cor(Y₁, X₁) + cor(Y₂, X₂)| provides a simple but effective description of the regulations between gene expressions and regulators. In the literature, more sophisticated regulation measures have been developed (for example, the sparse ones; Shi et al. 2015) and can also be used here. We note that this definition measures “average regulations” and is effective when regulations have the same “directions” in the two sample groups. On the other hand, when regulations have conflicting signs – which has not been commonly observed, it may be ineffective. A quick examination of the signs of cor(Y, X) is recommended prior to analysis. I(v₁ ≠ 0) and I(u₁ ≠ 0) single out the important gene expressions and regulators. Overall, P_similarity promotes the important gene expressions to be regulated by the important regulators, which facilitates biological interpretation. More specifically, the underlying intuition is that a set of important regulators lead to the regulator network differences, and regulate a set of important gene expressions that contribute to the gene expression network differences. This strategy is coherent with that for regression analysis (Chai et al. 2017) and clustering (Li et al. 2018). Approximating the indicator function using a smooth function has been common in the literature, and there are other equally applicable approximations.

A toy example

To gain some intuition into the working characteristics of the proposed approach, we simulate a toy example. The settings are similar to those in the Simulation section but with a lower dimensionality – there are only 20 gene expressions. In Figure 1 (Appendix), for conditional analysis, we show the true network difference (upper-left), estimated network difference (upper-right), matrix that describes the regulations of gene expressions by regulators, as well as the sets of truly important and identified gene expressions. For this specific simulated example, the proposed approach can accurately identify the important genes, while the alternatives miss one. More definitive results are based on larger-scale simulations and presented in the next Section.

Remarks

The proposed analysis has two “layers”: gene expressions and their regulators. Following similar spirit, it may be possible to extend to three or more layers, for example, gene expressions, methylation, and somatic mutations, or proteins, gene expressions, and regulators. In this case, three network differences need to be simultaneously analyzed, and two P_similarity penalties need to be imposed. This may be cumbersome and will not be pursued in this study.

2.3. Computation

We first further consider the approximation:

$$\left(1-\text{exp}\left(-\frac{{\eta }^2}{\tau }\right)\right)\approx \left[1-\text{exp}\left(-\frac{{\tilde{\eta }}^2}{\tau }\right)\right]+\text{exp}\left(-\frac{{\tilde{\eta }}^2}{\tau }\right)\cdot \frac{2\tilde{\eta }}{\tau }\cdot \left(\eta -\tilde{\eta }\right),$$

when η is “not far from” $\tilde{\eta }$. Accordingly, we can approximate the vector:

$$I\left(v_1\ne 0\right)\approx C_1\left({\tilde{v}}_1\right)+C_2\left({\tilde{v}}_1\right)\left(v_1-{\tilde{v}}_1\right),$$

where $C_1\left({\tilde{v}}_1\right)$ is a p×1 vector, and $C_2\left({\tilde{v}}_1\right)$ is a diagonal matrix, both of which are constants depending on ${\tilde{v}}_1$. Similarly, we can approximate I(u₁ ≠ 0) at ${\tilde{u}}_1$ as $I\left(u_1\ne 0\right)\approx C_3\left({\tilde{u}}_1\right)+C_4\left({\tilde{u}}_1\right)\left(u_1-{\tilde{u}}_1\right)$.

Overall, we propose the following algorithm based on the coordinate descent and local linear approximation techniques. It optimizes over one component at a time and consecutively cycles through all components. To simplify notation, we denote T = |cor(Y₁, X₁) + cor(Y₂, X₂)|. The proposed algorithm proceeds as follows.

• Step 1. Initialization. Apply the standard SVD to G_diff and R_diff separately. Let $\left\{{\tilde{s}}^G,{\tilde{v}}_1,{\tilde{w}}_1;{\tilde{s}}^R,{\tilde{u}}_1,{\tilde{z}}_1\right\}$ denote the first SVD triplets. If there is concern on high dimensionality and/or noisy data, the SSVD approach can be used.

• Step 2. Update:

  1. Set

     $$v_{\text{temp}}=\frac{1}{2}\mathit{\text{sign}}\left[2G_{\mathit{\text{diff}}}{\tilde{w}}_1+{\lambda }_3{C}_2\left({\tilde{v}}_1\right)TI\left({\tilde{u}}_1\ne 0\right)\right]\cdot {\left[\left|2G_{\mathit{\text{diff}}}{\tilde{w}}_1+{\lambda }_3{C}_2\left({\tilde{v}}_1\right)TI\left({\tilde{u}}_1\ne 0\right)\right|-\dot{\rho }\left(\left|{\tilde{v}}_1\right|;{\lambda }_1,a\right)\right]}_{+}.$$
     Let $s^G=\sqrt{\Vert v_{\mathit{\text{temp}}}\Vert F\cdot \Vert {\tilde{w}}_1\Vert F}$ and v₁ = v_temp/s^G.
  2. Set $w_{\mathit{\text{temp}}}=\frac{1}{2}\mathit{\text{sign}}\left[G_{\mathit{\text{diff}}}{v}_1\right]\cdot {\left[2G_{\mathit{\text{diff}}}{v}_1-\dot{\rho }\left(\left|{\tilde{w}}_1\right|;{\lambda }_1,a\right)\right]}_{+}$. Let $s^G=\sqrt{\Vert v_1\Vert F\cdot \Vert w_{\mathit{\text{temp}}}\Vert F}$ and w₁ = w_temp/s^G.
  3. Set

     $$u_{\text{temp}}=\frac{1}{2}\mathit{\text{sign}}\left[2R_{\mathit{\text{diff}}}{\tilde{z}}_1+{\lambda }_3{C}_4{\left({\tilde{u}}_1\right)}^{\top }{T}^{\top }I\left(v_1\ne 0\right)\right]\cdot {\left[\left|2R_{\mathit{\text{diff}}}{\tilde{z}}_1+{\lambda }_3{C}_4{\left({\tilde{u}}_1\right)}^{\top }{T}^{\top }I\left(v_1\ne 0\right)\right|-\dot{\rho }\left(\left|{\tilde{u}}_1\right|;{\lambda }_2,a\right)\right]}_{+}.$$
     Let $s^R=\sqrt{\Vert u_{\mathit{\text{temp}}}\Vert F\cdot \Vert {\tilde{z}}_1\Vert F}$ and u₁ = u_temp/s^R.
  4. Set $z_{\text{temp}}=\frac{1}{2}\mathit{\text{sign}}\left[R_{\mathit{\text{diff}}}{u}_1\right]\cdot {\left[2R_{\mathit{\text{diff}}}{u}_1-\dot{\rho }\left(\left|{\tilde{z}}_1\right|;{\lambda }_2,a\right)\right]}_{+}$. Let $s^R=\sqrt{\Vert u_1\Vert F\cdot \Vert z_{\mathit{\text{temp}}}\Vert F}$ and z₁ = z_temp/s^R.

• Step 3. Set ${\tilde{v}}_1=v_1,{\tilde{w}}_1=w_1,{\tilde{u}}_1=u_1$, and ${\tilde{z}}_1=z_1$. Repeat Step 2 until convergence.

In numerical analysis, we conclude convergence when the difference between the estimates from two consecutive iterations is smaller than a prespecified cutoff. The objective function is bounded below, and each iteration reduces its value. As such, convergence can be established. It is noted that several published studies have examined the convergence of SSVD, and their arguments are also applicable here. In our numerical studies, convergence is usually achieved within 20 iterations. The proposed approach involves three tuning parameters λ₁, λ₂, and λ₃. When p and q are very large, one can set λ₁ = λ₂ to reduce computer time. As the proposed algorithm only involves simple updates, it is very affordable to conduct a grid search of three tunings. For one simulated dataset with p = 50, q = 100, and n = 200 (details described below), the analysis with fixed tunings takes a few seconds on a standard desktop. To facilitate data analysis within and beyond this article, we have developed R code and made it publicly available at https://github.com/DeniseYi/Assisted_Differential_Network.

3. Simulation

Simulation is conducted to gain more insight into the proposed approach and compare against relevant alternatives. We consider two sample groups with sizes n₁ = n₂ and denote n = n₁ + n₂. We generate all components of X₂ and Y₂ from N(0, 1). This leads to cor(Y₂, X₂) ~ 0 and the “average regulations” driven by cor(Y₁, X₁). Although biologically this setup may not be sensible, statistically, it is perhaps the most lucid – all differences between the two groups are caused by properties of the first group. We have also experimented with setups under which both groups have “more meaningful” distributions. Their differences are not as lucid, but observations on performance of the proposed and alternative approaches are similar to below.

For modeling the relationship between X₁ and Y₁, we consider:

$$Y_1=B^{\top }{X}_1+W_1.$$ (2)

Here we adopt linear regression to model the regulations between gene expressions and regulators. Although in reality there can be nonlinear regulations, this linear approach has been popular (Chai et al. 2017; Shi et al. 2015), with considerations on dimensionality, computational cost, and stability. B is the q × p matrix of regression coefficients. The length-p vector W₁ accommodates both “random errors” and regulation mechanisms not measured and not known.

For ${\Sigma }_{X_1}$, the covariance matrix of X₁, we consider two scenarios: A1) a block diagonal structure with block sizes ten. One block has an Erdos-Renyi structure, under which the probability of drawing an edge between any two nodes is 0.05. The remaining blocks are equal to the identity matrix. Under this scenario, the difference between the two groups is attributable to the first special block. And, A2) this scenario is built on A1). Specifically, we “contaminate” the identity matrix blocks by randomly adding a small number of nonzero values to the off-diagonal elements. This scenario reflects that differences can be additionally caused by “scattered” signals. For the regression coefficient matrix B, we consider four different structures. B1) Strong-effect and block-diagonal. B has a block-diagonal structure, and the dimensions of the blocks match those of the regulator covariance matrix. All elements in the blocks are generated from a uniform distribution U(0.9, 1). B2) Strong-effect and “milky way”. Built on B1), a small portion (2%) of the off-block-diagonal elements are randomly generated from the same uniform distribution U(0.9, 1). The positions of the additional signals are random. B3) Weak-effect and block-diagonal. This is similar to B1), with the difference that all nonzero elements are generated from U(0.27, 0.3). B4) Weak-effect and “milky way”. Built on B3), a small portion (2%) of the off-block-diagonal elements are randomly generated from U(0.27, 0.3). W₁ has a multivariate normal distribution with marginal means 0 and marginal variances 1. With respect to its variance-covariance, we consider the following two structures. C1) Independence. This corresponds to “standard” regression, where W₁ only accommodates “random noises”. C2) W₁ has a block structure that matches the above. Different blocks are independent, and components i and j within the same block have correlation ρ^|i−j| with ρ = 0.3.

Once X₁ and W₁ are simulated, Y₁ is computed from the regulation regression model. We set n = 200 and consider (p, q) = (50,100) and (50,50). Here is noted that, although the values of p and q are not “impressively large”, the numbers of parameters to be estimated are much larger than the sample size. In published studies with a single network, similar p and q values have been considered.

With the proposed approach, as in many published SVD and SSVD studies, we focus on identifying gene expressions involved in the first singular vector. To better gauge its performance, we consider the following alternatives:

• Alt.1 The standard SVD is conducted with G_diff and R_diff separately. This approach is dense, with all gene expressions and regulators involved in the first singular vectors. To be compatible, we apply thresholding and identify those with large magnitudes as important.

• Alt.2 The IRLBA approach is applied to G_diff and R_diff separately. It is realized using the augmented implicitly restarted Lanczos bidiagonalization technique (Baglama and Reichel, 2005) and R package ‘irlba’. The resulted singular vectors are sparse, and the important gene expressions and regulators can be directly identified.

• Alt.3 BSSVD, the biclustering SVD approach, is applied to G_diff and R_diff separately. It uses Lasso to achieve sparsity. Some of its properties are similar to Alt.2, but the adopted techniques are significantly different.

Here we note that the alternatives all analyze gene expressions and regulators separately – to the best of our knowledge, there is no existing assisted or joint analysis approach that can achieve the same goal as the proposed approach. It is also recognized that there are other SVD-based approaches. However, there is a lack of evidence that they are superior to the above.

The proposed and alternative approaches involve different numbers of tuning parameters. It is noted that with Alt.1, the thresholds (imposed for identifying important gene expressions and regulators) can be viewed as tunings. It may not be fair to compare performance with selected tuning values – this may favor the proposed approach which has more tunings. To resolve this problem, we follow published literature and adopt a ROC (Receiver Operating Characteristic)-based approach to evaluate identification accuracy. Specifically, we consider a sequence of tunings, compute TP/FP (true and false positive rates) at each set of tuning values, construct an ROC curve, and use AUC (area under the ROC curve) to evaluate identification accuracy. In the proposed analysis, the key interest lies in identifying important gene expressions. We also evaluate the byproduct of identifying important regulators.

A total of 32 simulation settings are considered. Under each setting, we simulate 200 replicates. With each simulated dataset, we conduct both unconditional differential network analysis with covariance matrices and conditional analysis with precision matrices. Results for p = 50 and q = 100 are shown in Tables 1 and 2, and those for p = 50 and q = 50 are shown in Tables 5 and 6 (Appendix). Consider for example Table 1, the scenario with A1), B1), and C2). For the unconditional differential network analysis on gene expressions, the average AUC values (×100) are 52.8 (proposed), 48.7 (Alt.1), 53.3 (Alt.2), and 52.3 (Alt.3); for the conditional differential network analysis on gene expressions, the average AUC values (×100) are 56.1 (proposed), 40.8 (Alt.1), 47.2 (Alt.2), and 47.4 (Alt.3). For regulators, the unconditional analysis has the average AUC values (×100) 60.4 (proposed), 55.5 (Alt.1), 59.0 (Alt.2), and 60.0 (Alt.3); and the conditional analysis has the average AUC values (×100) 70.0 (proposed), 61.6 (Alt.1), 65.4 (Alt.2), and 66.2 (Alt.3). This set of results is representative: across all settings and for all measures, the proposed approach either has the best or close to the best performance. More improvement is observed in conditional analysis, which is sensible as it is more challenging than unconditional analysis and hence more strongly demands additional information. It is interesting to note that the proposed analysis can be effective even when the gene-expression-regulator relationship is weak.

Table 1:

Simulation results of AUC: mean×100 (sd×100) for p = 50, q = 100.

		Unconditional		Conditional
Scenario	Approach	X	Y	X	Y
A1)B1)C1)	Proposed	64.3 (13.0)	48.5 (18.5)	67.0 (15.0)	60.5 (15.1)
	Alt.1	58.8 (14.0)	40.1 (25.7)	62.3 (15.0)	42.9 (13.5)
	Alt.2	62.8 (14.7)	43.7 (28.2)	65.8 (15.4)	49.2 (15.2)
	Alt.3	62.8 (14.6)	46.6 (29.0)	65.8 (15.0)	49.6 (13.4)
A2)B1)C1)	Proposed	76.4 (7.8)	81.2 (17.1)	45.5 (11.0)	57.9 (15.0)
	Alt.1	67.9 (8.2)	73.7 (21.0)	41.5 (10.1)	39.3 (12.4)
	Alt.2	72.3 (8.5)	80.1 (26.6)	45.2 (10.5)	49.3 (15.3)
	Alt.3	74.8 (8.1)	80.5 (26.7)	46.3 (9.9)	47.9 (15.4)
A1)B2)C1)	Proposed	61.8 (17.7)	52.1 (12.5)	69.1 (13.5)	50.4 (9.8)
	Alt.1	57.3 (17.1)	48.9 (17.7)	61.7 (14.2)	35.1 (9.7)
	Alt.2	61.2 (18.0)	54.3 (18.7)	66.8 (14.5)	40.9 (10.7)
	Alt.3	63.4 (17.8)	53.1 (17.1)	68.5 (13.7)	39.7 (10.7)
A2)B2)C1)	Proposed	75.0 (7.6)	80.6 (17.0)	46.2 (9.9)	54.5 (13.9)
	Alt.1	69.3 (8.7)	67.6 (25.3)	41.7 (9.4)	35.4 (12.9)
	Alt.2	73.4 (9.4)	76.7 (28.7)	45.4 (9.6)	44.1 (15.3)
	Alt.3	76.0 (8.6)	80.5 (25.3)	45.9 (9.5)	45.9 (12.5)
A1)B1)C2)	Proposed	60.4 (14.3)	52.8 (20.7)	70.0 (12.1)	56.1 (14.5)
	Alt.1	55.5 (15.5)	48.7 (23.7)	61.6 (13.8)	40.8 (14.8)
	Alt.2	59.0 (16.3)	53.3 (26.9)	65.4 (14.4)	47.2 (17.0)
	Alt.3	60.0 (15.3)	52.3 (27.5)	66.2 (12.4)	47.4 (16.7)
A2)B1)C2)	Proposed	78.3 (8.3)	86.5 (14.4)	44.8 (9.6)	65.2 (18.4)
	Alt.1	71.6 (8.3)	74.6 (21.4)	40.3 (9.6)	45.4 (16.1)
	Alt.2	76.4 (8.7)	83.6 (26.9)	44.5 (10.1)	54.6 (18.7)
	Alt.3	77.6 (7.7)	86.3 (24.0)	45.4 (10.1)	55.1 (18.3)
A1)B2)C2)	Proposed	60.2 (16.4)	52.3 (14.4)	65.8 (13.2)	47.1 (12.8)
	Alt.1	55.6 (17.1)	48.0 (18.4)	62.1 (12.8)	31.1 (8.7)
	Alt.2	58.9 (17.5)	54.3 (17.9)	65.9 (13.3)	35.7 (10.0)
	Alt.3	61.2 (17.6)	51.6 (17.2)	67.4 (13.0)	37.6 (10.3)
A2)B2)C2)	Proposed	76.4 (7.4)	78.0 (13.5)	44.3 (9.3)	59.1 (13.2)
	Alt.1	70.2 (7.7)	69.0 (19.0)	40.1 (8.9)	43.6 (14.1)
	Alt.2	74.7 (7.9)	74.2 (23.1)	43.4 (9.3)	51.8 (15.8)
	Alt.3	76.2 (7.2)	76.0 (23.4)	45.4 (9.7)	52.0 (15.7)

Table 2:

Simulation results of AUC: mean×100 (sd×100) for p = 50, q = 100 (cont.).

		Unconditional		Conditional
Scenario	Approach	X	Y	X	Y
A1)B3)C1)	Proposed	70.3 (15.2)	46.7 (18.1)	68.2 (13.3)	65.7 (13.8)
	Alt.1	55.4 (17.7)	37.9 (20.2)	62.0 (14.6)	46.1 (12.3)
	Alt. 2	59.3 (18.1)	43.6 (19.1)	66.0 (14.5)	54.3 (14.4)
	Alt.3t	62.1 (17.5)	43.9 (20.0)	67.8 (12.4)	53.3 (15.4)
A2)B3)C1)	Proposed	83.6 (6.1)	82.3 (19.3)	45.7 (10.9)	59.3 (11.9)
	Alt.1	71.3 (7.7)	62.0 (23.2)	41.5 (10.5)	39.9 (10.0)
	Alt. 2	75.9 (8.2)	70.5 (27.6)	45.5 (10.8)	48.3 (11.7)
	Alt.3	77.7 (6.8)	70.0 (26.7)	45.9 (10.6)	51.3 (11.0)
A1)B4)C1)	Proposed	72.0 (12.7)	57.9 (16.4)	65.2 (14.2)	59.2 (10.1)
	Alt.1	55.6 (14.2)	50.8 (17.1)	60.8 (14.6)	41.4 (10.7)
	Alt.2	58.4 (15.0)	56.0 (17.6)	64.6 (15.1)	47.0 (10.4)
	Alt.3	59.7 (16.0)	52.4 (15.6)	66.1 (15.6)	48.1 (10.6)
A2)B4)C1)	Proposed	82.0 (7.7)	83.0 (21.1)	45.5 (10.7)	56.2 (14.1)
	Alt.1	71.5 (9.1)	65.4 (26.0)	41.5 (11.6)	36.3 (14.4)
	Alt. 2	76.6 (9.0)	75.4 (28.9)	44.9 (12.0)	44.1 (16.0)
	Alt.3	78.0 (8.3)	76.0 (28.4)	45.0 (10.9)	46.3 (15.4)
A1)B3)C2)	Proposed	69.4 (14.7)	49.8 (18.9)	69.6 (13.3)	57.5 (15.0)
	Alt.1	54.9 (15.5)	43.0 (21.6)	64.7 (13.0)	44.2 (14.0)
	Alt. 2	58.9 (16.9)	48.6 (21.8)	68.9 (14.2)	51.3 (16.4)
	Alt.3	60.4 (15.3)	48.1 (22.9)	68.9 (14.1)	50.4 (14.9)
A2)B3)C2)	Proposed	82.8 (6.4)	84.5 (19.4)	44.8 (10.6)	55.4 (18.4)
	Alt.1	69.8 (7.5)	65.2 (24.1)	40.3 (10.8)	37.9 (16.5)
	Alt.2	73.9 (7.9)	73.1 (28.0)	44.0 (10.8)	46.7 (20.0)
	Alt.3	74.8 (7.4)	75.6 (25.4)	45.6 (11.1)	46.8 (19.1)
A1)B4)C2)	Proposed	73.3 (13.8)	55.9 (15.0)	68.2 (12.1)	55.6 (11.5)
	Alt.1	58.0 (14.8)	48.2 (15.0)	61.9 (13.3)	40.3 (10.9)
	Alt.2	62.4 (15.4)	52.3 (14.9)	66.6 (13.3)	45.7 (11.8)
	Alt.3	63.7 (16.6)	54.5 (15.9)	68.4 (12.1)	46.4 (10.8)
A2)B4)C2)	Proposed	80.0 (6.8)	74.1 (17.6)	46.5 (9.8)	60.5 (14.2)
	Alt.1	70.3 (7.8)	57.6 (20.4)	42.0 (10.1)	44.2 (14.2)
	Alt.2	74.4 (8.2)	64.4 (22.8)	45.6 (10.3)	52.2 (14.6)
	Alt.3	75.9 (8.0)	64.8 (22.8)	46.7 (10.2)	51.7 (14.4)

It is noted that the AUC values are in general low for both the proposed and alternative approaches, which is partly caused by the challenge of differential network analysis and low signals of the simulated settings. In addition, the ROC and AUC adopted here have different operating characteristics from their “standard” counterparts. For example, with the “standard” ROC, if an AUC below 0.5 is obtained, simply flipping the “direction” can lead to an AUC above 0.5 (Pepe 2004). However, this is not true when applying ROC to variable selection evaluation. The proposed approach provides small to moderate improvement over the alternatives. This observation is in line with those in the literature (Li et al. 2018; Chai et al. 2017). This is sensible as the main data properties and analysis results still largely hinge on gene expression data itself. With the increasing popularity of gene expression studies that also measure regulators, the proposed approach can provide a cost-effective way of improving performance.

4. Data Analysis

TCGA (The Cancer Genome Atlas) is one of the largest and most comprehensive cancer projects jointly organized by the NCI and NHGRI. For over thirty cancer types, it has published comprehensive molecular and other types of data. We analyze TCGA data because of its high quality, easy accessibility, and broad scientific impact. In particular, we analyze the TCGA data on LUAD (lung adenocarcinoma), a major histological subtype of lung cancer. Data on gene expressions and copy number variations are downloaded from the TCGA data portal. The total sample size is 512. We consider genes in the KEGG-CELL-CYCLE-PATHWAY, which has been established as playing critical roles in cell cycle and lung cancer prognosis. Limiting to one pathway and hence a smaller number of genes can improve reliability and reduce computational cost. In addition, network analysis can be more sensible for interconnected genes (in this case, in the same pathway and sharing the same biological functions). By matching the data with KEGG, a total of 224 gene expressions and 228 copy number variations are identified as belonging to this pathway. In this analysis, we are interested in the differences between stages, which describe cancer severity and are associated with prognosis and many other outcomes. Specifically, Stage I, Stage IA, and Stage IB patients are grouped together, and the higher-stage patients form the second group. To further improve reliability and reduce cost, we conduct a marginal screening and identify the 57 most relevant genes based on p-values. Realizing that in many applications not all genes are “interesting”, and to further test performance of the proposed approach, we additionally include 23 least relevant genes (with the largest marginal p-values) to mimic “noises”. The final dataset thus contains 80 gene expressions and their cis-acting copy number variations.

Using the proposed and three alternative methods, we conduct both unconditional and conditional differential network analysis. Different from in simulation, to report single sets of findings, we determine the optimal tuning parameters using BIC. The estimation results for gene expressions are summarized in Tables 3 (unconditional) and 4 (conditional), respectively. Those for copy number variations are available from the authors. The identification results are also summarized in Table 7 (Appendix). Overall, it is observed that different approaches lead to quite similar identification results. In the unconditional analysis, the proposed approach identifies 26 genes, all of which are also identified by Alt.1 and Alt.3. Alt.2 misses one gene. The alternatives may also have additional findings. In the conditional analysis, the proposed approach identifies 16 genes, with 15, 15, and 14 overlapping with the alternatives. In addition, the alternatives are also observed to have similar identification results. The estimation results, which indicate the relative contributions of different genes, are much less consistent across approaches. It is worth noting that all the identified genes belong to the set of 57 with the smallest p-values, which may provide some support to the validity of analysis. In Figure 2 (Appendix), for the genes identified by the proposed approach in the unconditional analysis, we present the correlation heatmaps, which clearly show differences. It is also noted that the unconditional and conditional analyses lead to significantly different findings, which is not surprising considering the fundamental differences between unconditional and conditional dependence.

Table 3:

Unconditional differential network analysis: identified genes and estimates.

	Proposed	Alt.1	Alt.2	Alt.3
CHUK	−0.011	−0.014	−0.002	−0.003
ELK1	0.012	0.014	0.003	0.003
FGFR3	0.165	0.049	0.038	0.035
FGFR2	0.127	0.039	0.028	0.027
IKBKB	0.025	0.018	0.006	0.006
MET	0.038	0.020	0.008	0.008
MAP2K1	−0.006	−0.014	−0.002	−0.002
RGL2	0.032	0.020	0.008	0.007
RALA	−0.091	−0.031	−0.020	−0.020
ZAP70	0	−0.012	0	0
SYNGAP1	0.011	0.015	0.003	0.003
RAPGEF5	−0.002	−0.012	0	−0.001
RASSF1	−0.005	−0.013	−0.002	−0.002
RRAS2	4.780	0.993	0.998	0.998
MRAS	0.059	0.024	0.013	0.013
PLA2G2D	0	−0.013	−0.001	0
PLCE1	0.109	0.036	0.024	0.023
GNG2	−0.002	−0.013	−0.002	−0.001
CALM2	−0.013	−0.016	−0.005	−0.003
RASA3	−0.010	−0.015	−0.003	−0.002
PLA2G4E	0.036	0.019	0.008	0.008
MAPK9	−0.028	−0.018	−0.007	−0.006
PDGFA	0.122	0.038	0.026	0.026
SHC4	0.028	0.018	0.006	0.006
GNG10	−0.027	−0.019	−0.007	−0.006
ETS1	−0.005	−0.014	−0.002	−0.002
NGF	0.005	0.013	0.001	0.002
PIK3CA	0.009	0.015	0.003	0.002

The identified contributing genes have sensible biological implications. For example, genes CHUK, MET, PIK3CA, and ELK1 have been suggested as associated with lung cancer severity by multiple studies. In particular, the loss of CHUK mRNA expression in lung adenomas was confirmed by eRT-PCR analysis of CHUK exons 6 and 7, and two models were established showing that CHUK is a major NSCLC tumor suppressor (Chavdoula et al., 2019). Paik and others found that responses to MET inhibitors in patients with stage IV lung adenocarcinomas harboring MET mutations cause exon 14 skipping (Paik et al., 2015). Yamamoto and others analyzed PIK3CA mutations in exons 9 and 20 in lung cancer cell lines and tumors, and identified PIK3CA mutations among all the major histologic subtypes (Yamamoto et al., 2008). Sheng and others found that ELK1-induced upregulation of HOXA10-AS improves LUAD progression through increasing Wnt/β-catenin signaling (Sheng et al., 2018). Genes VEGFC and RASSF1 are identified in the conditional analysis. In published studies, VEGFC/Flt-4-mediated invasion and metastasis of lung cancer cells were found to require upregulation of the neural cell adhesion molecule contactin-1 through activation of the Src-p38 MAPK-C/EBP-dependent pathway (Su et al., 2006). Gene RASSF1 locates in the chromosomal segment of 3p21.3. The high allelic loss in a variety of cancers suggests a crucial role of this region in tumorigenesis. Re-expression of RASSF1A reduces the growth of human cancer cells, supporting a role for RASSF1 as a tumor suppressor gene. RASSF1A inactivation and K-ras activation are mutually exclusive in the development of certain carcinomas (Dammann et al., 2003).

5. Discussion

Differential network analysis with gene expressions has important applications. In such analysis, a major goal is to identify key contributing genes. In this study, we have further advanced this field by developing an assisted analysis strategy, which has been shown as effective for simpler settings such as clustering and regression. The proposed approach has an intuitive strategy: information from regulators is borrowed by promoting interconnected important gene expressions and regulators. In simulation, the proposed approach has improved identification accuracy over three highly relevant competitors. Although the improvement is not dramatic, the proposed approach can still be highly valuable with its efficiency and cost-effectiveness, especially considering the growing popularity of multidimensional profiling data. In our data analysis, the identification results using the proposed approach are similar to those using the alternative methods – this is sensible given the observations made in simulation and also provides support to the validity of the findings. In the analysis, the added “noises” are successfully screened out, and the identified genes have sound interpretations. Significant differences have been observed in estimation – we postpone investigating this implication to future research.

This study can be potentially extended in multiple directions. Theoretical investigation can be pursued. It is expected that certain consistency properties can follow those of SSVD using penalization. More scenarios can be considered in simulation. With the consistency observed across simulation settings, it is expected that the proposed approach will have sensible performance under other settings. In TCGA, data on other types of regulators, for example methylation, is also available. Copy number variation is one of the most essential gene expression regulators, and the current data analysis may be sufficient to demonstrate the practical application of the proposed approach. It may still be of interest to include more regulator data. Last but not least, all the existing assisted analyses are based on raw measurements. It may be of interest to extend assisted analysis to the scenario where only summary statistics are available. One possibility is to re-estimate the summary statistics for gene expressions (and possibly those for regulators too) to make the important summary statistics connected by the regulations. With significant methodological developments needed, this will be postponed to future research.

Table 4:

Conditional differential network analysis: identified genes and estimates.

	Proposed	Alt.1	Alt.2	Alt.3
GNB3	−0.223	−0.105	−0.048	−0.051
GNG7	0	−0.080	−0.003	0
PIK3CD	−0.083	−0.109	−0.043	−0.023
PLCG2	0	0.078	0.008	0
PRKCB	0.136	0.114	0.057	0.044
MAPK10	0.090	0.115	0.050	0.016
VEGFC	0.339	0.116	0.055	0.060
ZAP70	2.851	0.612	0.699	0.700
SYNGAP1	−0.031	0	0	−0.002
RASGRP2	0.068	0.122	0.059	0.015
RRAS2	−0.024	−0.079	−0.011	0
PLA2G2D	−0.306	−0.134	−0.084	−0.088
PAK7	−1.681	−0.323	−0.317	−0.387
RASAL3	−2.377	−0.532	−0.617	−0.581
RASSF5	0.016	0.073	0.0073	0
RASA3	−0.254	−0.137	−0.074	−0.069
FGF22	−0.063	−0.081	−0.014	−0.010
REL	0	−0.072	0	0
NGF	−0.139	−0.087	−0.020	−0.013

Appendix

Table 5:

Simulation results of AUC: mean×100 (sd×100) for p = 50, q = 50.

		Unconditional		Conditional
Scenario	Approach	X	Y	X	Y
A1)B1)C1)	Proposed	75.7 (9.9)	83.5 (27.2)	80.6 (12.6)	62.1 (11.5)
	Alt.1	67.4 (10.5)	81.8 (26.8)	67.7 (13.9)	44.7 (11.1)
	Alt.2	70.3 (11.1)	84.9 (30.8)	73.2 (13.8)	47.5 (11.8)
	Alt.3	70.8 (9.2)	84.3 (29.7)	76.5 (13.3)	48.1 (11.4)
A2)B1)C1)	Proposed	76.6 (8.9)	66.9 (26.9)	49.4 (8.7)	54.6 (11.0)
	Alt.1	68.4 (12.3)	55.9 (28.4)	41.9 (8.4)	48.0 (11.8)
	Alt.2	73.3 (13.3)	58.3 (31.1)	45.4 (9.2)	51.3 (12.4)
	Alt.3	73.9 (11.1)	57.4 (29.5)	45.2 (9.9)	51.5 (11.3)
A1)B2)C1)	Proposed	86.9 (8.1)	83.3 (23.0)	81.7 (13.7)	67.2 (12.4)
	Alt.1	68.6 (11.2)	78.7 (23.3)	69.1 (15.2)	48.3 (13.3)
	Alt.2	70.5 (12.8)	82.8 (27.4)	73.1 (16.0)	52.1 (13.7)
	Alt.3	71.2 (11.4)	83.1 (25.3)	75.3 (16.3)	52.7 (12.4)
A2)B2)C1)	Proposed	82.0 (11.2)	57.1 (24.7)	45.1 (11.1)	53.2 (9.7)
	Alt.1	70.3 (12.7)	52.0 (24.3)	39.4 (10.7)	44.9 (9.6)
	Alt.2	74.3 (13.2)	55.5 (26.6)	42.4 (11.1)	48.9 (10.7)
	Alt.3	73.5 (12.7)	54.4 (24.9)	43.6 (11.2)	46.9 (8.8)
A1)B1)C2)	Proposed	74.8 (10.3)	88.1 (21.7)	84.0 (11.4)	61.1 (13.4)
	Alt.1	67.0 (11.3)	85.2 (25.1)	72.6 (12.6)	44.9 (11.3)
	Alt.2	69.1 (11.6)	88.7 (29.5)	77.1 (13.3)	49.3 (12.3)
	Alt.3	70.2 (10.4)	90.5 (22.4)	80.1 (13.4)	49.1 (10.7)
A2)B1)C2)	Proposed	74.7 (9.7)	67.0 (23.7)	50.1 (8.1)	58.2 (9.5)
	Alt.1	68.6 (10.9)	55.1 (24.3)	40.6 (7.9)	50.4 (10.2)
	Alt.2	72.1 (12.6)	54.9 (25.6)	43.7 (8.0)	54.2 (11.1)
	Alt.3	73.1 (11.1)	58.4 (27.2)	44.8 (7.7)	53.7 (9.7)
A1)B2)C2)	Proposed	86.0 (9.0)	78.4 (31.1)	81.1 (10.1)	65.6 (11.7)
	Alt.1	67.7 (12.4)	70.9 (31.6)	66.9 (12.2)	48.0 (12.4)
	Alt.2	68.9 (13.8)	75.4 (33.9)	71.4 (12.3)	51.5 (11.7)
	Alt.3	70.8 (12.9)	73.2 (30.6)	75.8 (11.8)	52.0 (10.2)
A2)B2)C2)	Proposed	83.1 (9.4)	50.7 (23.8)	48.7 (12.2)	56.7 (13.7)
	Alt.1	69.9 (12.6)	45.3 (21.8)	43.4 (12.6)	49.7 (13.9)
	Alt.2	74.8 (13.0)	47.4 (24.6)	46.8 (12.7)	53.1 (14.7)
	Alt.3	74.7 (11.5)	46.8 (24.0)	46.9 (11.3)	52.3 (13.8)

Table 6:

Simulation results of AUC: mean×100 (sd×100) for p = 50, q = 50 (cont.).

		Unconditional		Conditional
Scenario	Approach	X	Y	X	Y
A1)B3)C1)	Proposed	80.3 (7.7)	94.4 (15.5)	77.1 (15.5)	63.7 (10.4)
	Alt.1	65.9 (9.9)	82.8 (20.0)	68.4 (15.7)	47.8 (9.0)
	Alt.2	69.1 (10.4)	86.2 (22.6)	72.6 (16.6)	51.0 (9.2)
	Alt. 3	68.1 (10.4)	84.1 (21.8)	75.0 (16.3)	49.6 (8.7)
A2)B3)C1)	Proposed	76.1 (9.6)	55.9 (12.0)	47.6 (11.8)	50.8 (8.3)
	Alt.1	69.6 (11.4)	55.0 (21.1)	42.6 (10.6)	43.6 (7.2)
	Alt.2	73.1 (12.4)	57.6 (21.8)	45.6 (11.2)	46.7 (7.6)
	Alt.3	74.5 (10.7)	57.7 (22.1)	45.8 (11.7)	46.5 (8.1)
A1)B4)C1)	Proposed	84.5 (10.2)	83.0 (25.0)	79.1 (11.6)	66.7 (11.2)
	Alt.1	64.7 (10.8)	73.4 (26.8)	69.5 (12.6)	48.3 (11.8)
	Alt.2	67.2 (11.5)	77.8 (28.3)	73.7 (13.0)	52.3 (12.6)
	Alt.3	69.0 (13.5)	77.7 (27.3)	76.3 (13.0)	51.6 (11.0)
A2)B4)C1)	Proposed	81.8 (7.4)	52.4 (15.4)	49.0 (10.4)	55.2 (7.6)
	Alt.1	68.2 (13.4)	50.6 (17.7)	43.1 (9.4)	49.0 (7.9)
	Alt.2	72.7 (14.3)	52.4 (17.9)	46.4 (9.6)	52.0 (7.6)
	Alt.3	73.7 (9.9)	49.0 (15.0)	47.2 (9.3)	50.3 (7.5)
A1)B3)C2)	Proposed	84.2 (8.4)	95.4 (13.0)	77.8 (13.2)	61.4 (10.4)
	Alt.1	66.5 (10.7)	85.6 (20.0)	66.9 (14.0)	45.3 (10.0)
	Alt.2	68.4 (11.4)	90.0 (21.4)	71.7 (15.3)	48.8 (10.4)
	Alt.3	67.8 (12.7)	87.8 (21.6)	74.2 (16.0)	49.2 (10.2)
A2)B3)C2)	Proposed	73.2 (11.9)	63.8 (16.4)	48.9 (10.7)	57.6 (9.6)
	Alt.1	66.1 (12.6)	58.7 (18.4)	44.5 (9.9)	48.4 (10.4)
	Alt.2	69.7 (14.0)	59.9 (19.1)	47.7 (10.6)	50.8 (11.1)
	Alt.3	70.6 (13.1)	62.3 (20.3)	47.4 (11.1)	52.2 (9.4)
A1)B4)C2)	Proposed	85.9 (7.5)	90.1 (20.4)	77.7 (15.6)	64.9 (10.4)
	Alt.1	68.1 (8.9)	78.8 (25.2)	67.5 (15.4)	48.5 (10.9)
	Alt.2	70.2 (9.7)	85.6 (27.1)	71.8 (16.3)	51.7 (11.8)
	Alt.3	70.2 (8.8)	82.3 (25.3)	74.3 (17.2)	52.7 (11.1)
A2)B4)C2)	Proposed	79.3 (11.4)	55.3 (15.2)	45.2 (9.3)	54.3 (9.8)
	Alt.1	66.9 (11.7)	51.7 (17.1)	40.6 (9.0)	48.4 (9.7)
	Alt.2	70.9 (12.8)	53.5 (17.5)	44.0 (9.6)	51.2 (10.1)
	Alt.3	71.0 (12.7)	52.5 (17.0)	44.6 (8.9)	50.6 (9.8)

Table 7:

Numbers of overlapping findings between different approaches.

	Proposed	Alt.1	Alt.2	Alt.3
	Unconditional analysis
Proposed	26	26	25	26
Alt.1		28	26	26
Alt.2			26	24
Alt.3				26
	Conditional analysis
Proposed	16	15	15	14
Alt.1		18	17	13
Alt.2			17	13
Alt.3				14

Figure 1:

Toy example. Upper-left: true G_diff; Upper-right: estimated G_diff; Lower-left: regulation coefficient matrix; Lower-right: true and identified important gene expressions (from left to right: true, Alt.1, Alt.2, Alt.3, and proposed).

Figure 2:

Correlation heatmaps for genes identified by the proposed approach in the unconditional analysis: sample groups with lower (left) and higher (right) stages.