Integrative Analysis of Gene Networks and Their Application to Lung Adenocarcinoma Studies

Abstract

The construction of gene regulatory networks (GRNs) is an essential component of biomedical research to determine disease mechanisms and identify treatment targets. Gaussian graphical models (GGMs) have been widely used for constructing GRNs by inferring conditional dependence among a set of gene expressions. In practice, GRNs obtained by the analysis of a single data set may not be reliable due to sample limitations. Therefore, it is important to integrate multiple data sets from comparable studies to improve the construction of a GRN. In this article, we introduce an equivalent measure of partial correlation coefficients in GGMs and then extend the method to construct a GRN by combining the equivalent measures from different sources. Furthermore, we develop a method for multiple data sets with a natural missing mechanism to accommodate the differences among different platforms in multiple sources of data. Simulation results show that this integrative analysis outperforms the standard methods and can detect hub genes in the true network. The proposed integrative method was applied to 12 lung adenocarcinoma data sets collected from different studies. The constructed network is consistent with the current biological knowledge and reveals new insights about lung adenocarcinoma.

Introduction

A gene regulatory network (GRN) describes the interactions among genes and how they work together to maintain life. The inference of GRNs leads to a systematic understanding of disease mechanisms at molecular levels and identification of potential therapeutic treatment targets for diseases. The development of high-throughput technologies has made it possible to simultaneously measure the activities of genes at the whole genome level, which greatly facilitates the study of GRNs. Gaussian graphical models (GGMs) have been widely used for inferring GRNs by estimating conditional dependence relationships among the expression levels of genes. The concept underlying GGMs is to use the partial correlation coefficient as a measure of dependency between the expressions of any 2 genes. Hence, inferring the GRN amounts to estimating its partial correlation coefficients or precision matrix (the inverse of covariance matrix) in GGMs, where partial correlation coefficients can be computed by entries in the precision matrix. However, applications of GGMs to high-dimensional data, where the number of genes $p$ is much larger than the number of patients $\mathrm{n}$, are not trivial because the sample covariance matrix is singular and thus the precision matrix cannot be directly estimated.

In recent literature, there has been much work on sparse estimation of the precision matrix in GGM using regularization methods with a Lasso penalty.^1–5 In addition to the regularization methods, other methods based on limited-order partial correlations have been proposed.^6-8 These methods are intended to use a low-order partial correlation in lieu of the full-order partial correlation. Recently, Liang et al⁹ proposed a new approach to construct a high-dimensional GGM based on an equivalent measure of partial correlation coefficients. However, in practical applications, GRNs constructed from the analysis of a single data set may lack reliability due to the limitation in sample size. An alternative approach is to use multiple data sets from comparable studies and conduct an integrative analysis.

Recent technologies have made it feasible to collect diverse and multiple genome-scale data sets in biomedical studies. For example, The Cancer Genome Atlas has collected genome, transcriptome, and epigenome data on more than 20 types of cancer from thousands of patients. The availability of such plentiful data enables researchers to construct more reliable GRNs to capture the heterogeneity of biological processes and phenotypes by borrowing strength across multiple sources of data. Guo et al¹⁰ proposed the joint estimation method of multiple graphical models that share the same genes and dependence relationships among genes. They used the likelihood-based method with a hierarchical lasso penalty¹¹ to encourage similar patterns of sparsity across multiple data sets. Similarly, Danaher et al¹² proposed the joint graphical lasso (JGL) which uses the sparse group Lasso¹³ or the fused lasso penalty.¹⁴ However, when integrating multiple data sets measured using different microarray platforms, these penalty-based methods may suffer from discrepancies among different platforms. For example, when the expressions of several genes are missing in all patients in a specific platform but not in the other platforms, the penalty-based methods are not applicable because they require a complete design matrix without any missing values from multiple data sets. Even though the complete design matrix can be obtained by the deletion or imputation of missing values, it may suffer from a loss of information and severe bias.

Liang et al⁹ proposed a new method, namely, the ψ-learning method, to construct a GGM based on an equivalent measure of partial correlation coefficients, and they briefly introduced ψ-learning with data integration for multiple data sets. However, they focused only on the standard ψ-learning method for a single data set.

In this article, we study the integrative ψ-learning method to construct a GRN by integrating multiple sources of data. This approach is based on the equivalent measures of partial correlation coefficients in GGMs, and hence, it could be applied to high-dimensional multiple data sets because equivalent measures are evaluated with a reduced conditional set. Moreover, we provide an extension to the integrative ψ-learning method for multiple data sets with a natural missing mechanism caused using different platforms in multiple sources of data. The proposed method outperformed other standard methods in simulation studies. Finally, the proposed method was applied to study the gene regulation of lung adenocarcinoma (ADC). In this study, we integrated gene expression profiles of 1246 patients with ADC collected from 12 different studies and measured across different platforms. It is thus far the largest study on GRNs of ADC. The resulting GRN reveals several important hub genes in ADC and leads to new biological insights on the disease and its potential therapeutic targets. Finally, our sensitivity analysis shows that the identified hub genes are robust against random noise and selection of genes.

Methods

The equivalent measure of partial correlation coefficients

Following Liang et al,⁹ we describe the GGM and some notation for defining the equivalent measure of partial correlation coefficients in GGMs. We then introduce the equivalent measure ψ of partial correlation coefficients and the ψ-learning algorithm to construct a network using a single data set.

Let $X={(X_1,\dots ,X_p)}^T$ denote a random vector drawn from the multivariate Gaussian distribution ${\mathrm{N}}_p$(µ,∑), where µ and $\Sigma$ are the mean vector and covariance matrix, respectively. The partial correlation coefficient between $X_i$ and $X_j$ is denoted by ${\rho }_{ij}|V\backslash \{i,j\}$, where $V=\{1,\dots ,p\}$ is the index set of all variables. It is well known that the partial correlation coefficient in GGM can be expressed as follows:

$${\rho }_{ij}|V\backslash \{i,j\}=-\frac{{\omega }_{ij}}{\sqrt{{\omega }_{ii}{\omega }_{jj}}}$$

where ${\omega }_{ij}$ is the $(i,j)$ entry of the precision matrix denoted by $\mathrm{\Omega }={\sum }^{-1}=({\omega }_{ij})$. The Gaussian random vector $X$ can be represented by the undirected graph $G=(V,E)$, where $V$ is the set of vertices corresponding to $p$ variables and $E=(e_{ij})$ is the adjacency matrix which specifies the edges included in the graph $G$. For a GGM, the following relation holds:

$$e_{ij}=e_{ji}=1\iff {\omega }_{ij}\ne 0\iff {\rho }_{ij}|V\backslash \{i,j\}\ne 0$$

In the context of GRNs, $X$ represents the expression levels of $p$ genes measured on each individual. Hence, constructing GRNs amounts to identifying their nonzero partial correlation coefficients.

Let $r_{ij}$ be the correlation coefficient between $X_i$ and $X_j$, and let $G_{ij}$ denote a reduced graph of $G$ with $e_{ij}$ being set to 0. We define $r_{G_{ij}}(i)$ as a set of vertices for which the corresponding variable is correlated with $X_i$ in $G_{ij}$, ie, $r_{G_{ij}}(i)=\{v:r_{iv}\ne 0\}\backslash \{j\}$. Similarly, we define $r_{G_{ij}}(j)=\{v:r_{jv}\ne 0\}\backslash \{i\}$. For any pair of vertices $i$ and $j$, Liang et al⁹ proposed the ψ_ij-partial correlation coefficient defined by the following equation:

$${\mathrm{\psi }}_{ij}={\mathrm{\rho }}_{ij}|S_{ij}$$ (1)

where $S_{ij}=r_{G_{ij}}(i)$ if $|r_{G_{ij}}(i)|<|r_{G_{ij}}(j)|$ and $S_{ij}=r_{G_{ij}}(j)$ otherwise, and $|D|$ is the cardinality of a set $D$. They showed that ${\mathrm{\psi }}_{ij}$ and ${\rho }_{ij}|V\backslash \{i,j\}$ are equivalent in the sense that

$${\mathrm{\psi }}_{ij}=0\iff {\rho }_{ij}|V\backslash \{i,j\}=0$$

under the faithfulness assumption for GGM. Based on the equivalent measure of partial correlation coefficient ${\mathrm{\psi }}_{ij}$, the Gaussian graphical network can be constructed by the ψ-learning algorithm summarized in Algorithm 1. Furthermore, Liang et al⁹ established the asymptotic consistency property of the ψ-learning method under mild conditions.

Algorithm 1

ψ-learning algorithm

• Step 1. (Correlation screening)—Conduct a multiple hypothesis test to identify the pairs of vertices for which the correlation coefficient is significantly different from zero.

• Step 2. (ψ-calculation)—For each pair of vertices $i$ and $j$, identify the conditional set $S_{ij}$ based on the results in Step 1 and calculate ${\mathrm{\psi }}_{ij}$ by inverting the sample covariance matrix of the variables indexed by $S_{ij}\cup \{i,j\}$.

• Step 3. (ψ-screening)—Conduct a multiple hypothesis test to identify the pairs of vertices for which ${\mathrm{\psi }}_{ij}$ is significantly different from zero.

Integrative ψ-learning method

The ψ-learning method described in Algorithm 1 focuses on constructing a network based on only a single data set. However, with the recent development of high-throughput technologies, it is common to gather multiple genome-scale data sets to study the mechanisms of a disease. In real-world applications, it is much more efficient to integrate such plentiful and compatible data, which could lead to more reliable GRNs. In this section, we describe how to construct a GRN by integrating multiple sources of data under the framework of the ψ-learning method.

Suppose that we have $K$ sources of data, all of which are normally distributed. Let ${\widehat{\mathrm{\psi }}}_{ij}^{(k)}$ be the estimated ψ-partial correlation coefficient in equation (1) from the $k\mathrm{th}$ source of data. We first apply the Fisher transformations to obtain the following equation:

$$z_{ij}^{(k)}=\frac{1}{2}\log \left(\frac{1+{\widehat{\mathrm{\psi }}}_{ij}^{(k)}}{1-{\widehat{\mathrm{\psi }}}_{ij}^{(k)}}\right)$$ (2)

whose distribution is approximately normally distributed with mean zero and variance $1/({\mathrm{n}}_k-|S_{ij}^{(k)}|-3)$ under the null hypothesis $H_0:{\mathrm{\psi }}_{ij}^{(k)}=0$, where ${\mathrm{n}}_k$ is the sample size of the $k\mathrm{th}$ source and $({\mathrm{n}}_k-|S_{ij}^{(k)}|-3)$ is called the effective sample size of the ψ-partial correlation coefficient.⁹ For convenience, we call the scaled z-score (equation (2)) a ψ_z-score defined by ${\tilde{z}}_{ij}^{(k)}=z_{ij}^{(k)}\sqrt{{\mathrm{n}}_k-|S_{ij}^{(k)}|-3}$, whose distribution approximately follows a standard normal distribution under $H_0$. We then combine the ψ_z-scores from different sources of data using the Stouffer meta-analysis method¹⁵ as follows:

$$Z_{ij}=\frac{{\displaystyle {\sum }_{k=1}^K{w}_k{\tilde{z}}_{ij}^{(k)}}}{\sqrt{{\displaystyle {\sum }_{k=1}^K{w}_k^2}}},\mathrm{\hspace{1em}}i,j=1,\dots ,p$$ (3)

where $w_k$ is a nonnegative weight assigned on the $k\mathrm{th}$ source of data. The assignment of $w_k$ may depend on the sample size or data quality for different sources known in advance. If a prior knowledge for each source of data is not available, we simply use the weight proportional to the sample size: for example, $w_k=n_k/{\displaystyle {\sum }_k{\mathrm{n}}_k}$. We notice that the weight $w_k$ for the $k\mathrm{th}$ source of data in equation (3) is set to be the same for all edges.

In real-world applications, it is common for different data sets to be collected using different microarray platforms. But this will create missing values for some genes in some of the data sets when combining all the data sets together due to the differences among platforms. In this case, a standard approach is to apply a method after deleting the patients or genes with missing values or to impute missing values. As a result, the network may suffer from a loss of information and severe bias. Moreover, both deletion and imputation of missing values are inappropriate in real applications because too many missing values exist for many genes in a source of data.

For the purpose of analyzing the data with missing values, we propose to use different weights for each edge in a source of data. Let ${\mathrm{n}}_i^{(k)}$ be the number of samples except for those with missing values for the $i\mathrm{th}$ gene in the $k\mathrm{th}$ source of data, and ${\mathrm{n}}_{ij}^{(k)}=\max \{{\mathrm{n}}_i^{(k)},{\mathrm{n}}_j^{(k)}\}$. We modify the method to combine ψ_z-scores from different sources in equation (3) in the following way:

$$Z_{ij}=\frac{{\displaystyle {\sum }_{k=1}^K{w}_{ij}^{(k)}{\tilde{z}}_{ij}^{(k)}}}{{\displaystyle {\sum }_{k=1}^K{(w_{ij}^{(k)})}^2}},\mathrm{\hspace{1em}}i,j=1,\dots ,p$$ (4)

where $w_{ij}^{(k)}$ denotes the nonnegative weight for the edge $e_{ij}$ assigned on the $k\mathrm{th}$ source of data. Similarly, if prior knowledge for each source of data is not available, we simply set the weight proportional to the sample size, $w_{ij}^{(k)}=n_{ij}^{(k)}/{\displaystyle {\sum }_k{\mathrm{n}}_{ij}^{(k)}}$. For each edge $e_{ij}$, $w_{ij}^{(k)}=0$ if the expressions of gene $i$ or gene $j$ are missing in source $k$, but $w_{ij}^{(k)}\ne 0$ otherwise. For a fixed $e_{ij}$, even if many missing values in genes $i$ or $j$ exist in a specific platform (a source of data), the integrative ψ-learning method can be applied to $Z_{ij}$ in equation (4) computed by other sources of data unless the expressions of the corresponding genes are missing in all sources of data. This enables us to partially use the information from the other sources of data, which is not achieved by the penalty-based joint estimation method because it requires a complete design matrix from all sources.

Note that $Z_{ij}$ approximately follows a standard normal distribution under the null hypothesis $H_0:e_{ij}=0$. Then, a multiple hypothesis test can be performed on $Z_{ij}\mathrm{s}$ to identify the pairs of vertices for which $Z_{ij}$ is differentially distributed from the standard normal $\mathrm{N}(0,1)$. The integrative ψ-learning algorithm is summarized in Algorithm 2.

Algorithm 2

Integrative ψ-learning algorithm.

• Step 1. (ψ_z-calculation)—Perform Steps 1 and 2 of the ψ-learning algorithm independently for each source of data.

• Step 2. (ψ_z-combination)—Calculate $Z_{ij}$ by combining ψ_z-scores in equations (2) and (4).

• Step 3. (ψ_z-screening)—Conduct a multiple hypothesis test to identify the pairs of vertices for which $Z_{ij}$ is differentially distributed from the standard normal $\mathrm{N}(0,1)$.

For the multiple hypothesis tests in Steps 1 and 3, we use the stochastic approximation–based method,¹⁶ which is also used in the ψ-learning method in Algorithm 1. Overall, this method works well under general dependence between test statistics. It is obvious that the correlation coefficients and ψ-partial correlation coefficients are generally dependent for GGMs. For the multiple hypothesis test procedure, one important issue is to choose significance levels used as cutoff values of correlation coefficients and ψ-partial correlation coefficients. We set the significance level in the Storey q value¹⁷ as in the ψ-learning of Liang et al.⁹ For correlation screening, we generally use a high q value (eg, 0.05 or even larger). When $p$ is extremely large with sample size $\mathrm{n}$ and the q value is large, the case $|S_{ij}^{(k)}|>n_{ij}^{(k)}-3$ might occur where ${\mathrm{\psi }}_{ij}^{(k)}$ is incalculable. A small q value would reduce the computational complexity of ψ-partial correlation coefficients by the small size of the conditional set $S_{ij}$, but the calculated ψ-partial correlation coefficients may be less reliable. In this article, we set the q value to 0.05 for correlation screening. For ψ-screening, a large q value produces a dense network, whereas a small q value leads to a sparse network. Similar to regularization methods such as the glasso,³ a network path can be obtained with a monotone sequence of q values for ψ-screening.

Finally, it is worth noting that the Stouffer meta-analysis method in equation (3) could also be replaced by other probability test methods, such as the Fisher method and the Pearson method.¹⁸ This can be done by combining the corresponding P values of each hypothesis instead of the z-score. The multiple hypothesis tests in Steps 1 and 3 can be also performed in various methods, such as the positive false discovery rate method,¹⁹ empirical Bayesian method,²⁰ 2-stage method,²¹ and principal factor approximation method.²²

Results and Discussion

Simulation studies

In this section, we investigate the performance of the integrative ψ-learning method on simulated data based on 2 real protein-protein interaction networks.²³ The underlying network structures were partially selected from the human protein reference database and form an approximate scale-free topology, as shown in Figure 1.²⁴ The strengths of dependencies between 2 proteins with interaction are generated from a normal distribution with mean 0.5 and variance 0.2, and then the sign of dependencies is randomly decided by Bernoulli random variables with probability .5 so that we allow both positive and negative regulations. The expression data are generated from the conditional normal distribution. Specifically, the expression of gene $j$ is simulated from the following equation:

Figure 1.

The topology of 2 networks with different sizes used in the simulation. The large nodes with red color represent the hub gene node whose node degrees are greater than upper 95% quantile for each network: (A) small size network and (B) large size network.

$$X_i|X_{\backslash i}~\mathrm{N}\left(\sum _{j\in {\mathcal{A}}_i}{\beta }_{ij}{X}_j,{\sigma }^2\right)$$

where $X_i$ is the expression level of gene $i$, ${\mathcal{A}}_i$ is the set of genes with connection to gene $i$ in the true underlying network (ie, the set of genes that regulate gene $i$), and ${\beta }_{ij}$ is the strength of dependency between genes $i$ and $j$.

We consider 2 networks with small $(p=83)$ and large $(p=612)$ numbers of genes, in which there are 114 and 911 connections, respectively. For each network, we generate 3 data sets with size of $\mathrm{n}\in \{100,300\}$ in which the noise levels ${\sigma }^2$ are set to be 1, 2, and 4 for the first, second, and third data set, respectively. Note that the underlying network structures for the 3 data sets are the same, but the noise levels are different. In summary, we have 4 simulation scenarios with different $\mathrm{n}$ and $p$, where each scenario is replicated 20 times.

For comparison, we consider the ψ-learning to be applied separately to each data set (ψ_k-learning, $k=1,2,3$) and ψ-learning to be simply applied to the pooled data set with a size of $3\mathrm{n}$ (ψ_p-learning) along with the integrative ψ-learning method (ψ_i learning). We further consider the JGL method with the group lasso penalty,¹² for which the final network is constructed by the nonzero ${\ell }_2-\mathrm{norms}$ of coefficients across 3 sources of data for each edge.

Figure 2 displays the receiver operating characteristic (ROC) and partial ROC curves of all methods for a sample data set in the simulation with a small size network. The partial ROC curve represents the ROC curve on the region where the false-positive rate is less than 0.05. Figure 2 shows that the ψ_i-learning method performs much better than other methods. For the separate ψ-learning methods based on each data set, as expected, the ψ₁-learning method performs better than other ones from data sets with larger noise levels. It is interesting to note that the ψ_p-learning method performs worse than the ψ₁-learning and ψ₂-learning methods, although the ψ_p-learning method uses more samples. This result might be caused by the third data set’s poor quality, which means that simply pooling data sets from multiple sources may not be a good choice, whereas the proposed integrative analysis provides an effective way to improve the performances of the ψ-learning method using more information.

Figure 2.

ROC curve and partial ROC curve under $\mathrm{FPR}<0.05$ for all methods where the sample and network sizes are $\mathrm{n}=100$ and $p=83$.FPR indicates false-positive rate; JGL, joint graphical lasso; ROC, receiver operating characteristic curve.

For the performance measures, we consider the numbers of true-positive edges and false-positive edges (TPE and FPE), sensitivity and specificity (SEN and SPE), the rate of misspecified edges (MIS), and the area under ROC and partial ROC curves under $\mathrm{FPR}<0.05$ (AUC and PAUC). Table 1 displays the average values of performance measures over 20 replications in which the levels of sparsity for the ψ-learning methods are selected by the Storey q value in multiple hypothesis tests. For the JGL method, we selected a level of sparsity similar to the ψ_i-learning method because the JGL method using cross-validation or information criteria tends to produce too dense networks. The results indicate that the ψ_i-learning method performs much better than all the other methods. The ψ₃-learning from data with the highest noise shows the worst performance, and the ψ_p-learning method also shows worse performance than the ψ₁-learning method.

Table 1.

Comparison of the proposed method with the JGL, separate, and pooled ψ-learning methods in the simulation on small size network.

Q value	$\mathrm{N}$	Method	TPE	FPE	SEN	SPE	MIS	AUC	PAUC
0.1	100	JGL	30.95	74.90	0.3005	0.9773	0.0432	0.8185	0.0087
		ψi-learning	61.40	10.05	0.5961	0.9970	0.0152	0.9464	0.0290
		ψp-learning	30.95	8.95	0.3005	0.9973	0.0238	0.8458	0.0211
		ψ1-learning	38.60	6.05	0.3748	0.9982	0.0207	0.8928	0.0233
		ψ2-learning	26.95	3.65	0.2617	0.9989	0.0234	0.8545	0.0223
		ψ3-learning	0.75	0.70	0.0073	0.9998	0.0303	0.6645	0.0062
	300	JGL	63.30	68.10	0.6146	0.9794	0.0317	0.9281	0.0214
		ψi-learning	90.05	16.30	0.8743	0.9951	0.0086	0.9868	0.0360
		ψp-learning	64.50	11.00	0.6262	0.9967	0.0145	0.9395	0.0284
		ψ1-learning	74.05	16.85	0.7189	0.9949	0.0135	0.9608	0.0326
		ψ2-learning	62.75	5.75	0.6092	0.9983	0.0135	0.9342	0.0326
		ψ3-learning	16.75	2.40	0.1626	0.9993	0.0261	0.8239	0.0194
0.3	100	JGL	36.40	112.25	0.3534	0.9660	0.0526	0.8185	0.0087
		ψi-learning	70.90	32.95	0.6883	0.9900	0.0191	0.9464	0.0290
		ψp-learning	43.50	36.65	0.4223	0.9889	0.0283	0.8458	0.0211
		ψ1-learning	50.50	22.10	0.4903	0.9933	0.0219	0.8928	0.0233
		ψ2-learning	40.25	18.75	0.3908	0.9943	0.0239	0.8545	0.0223
		ψ3-learning	1.15	1.35	0.0112	0.9996	0.0303	0.6645	0.0062
	300	JGL	69.95	122.10	0.6791	0.9630	0.0456	0.9281	0.0214
		ψi-learning	94.60	48.55	0.9184	0.9853	0.0167	0.9868	0.0360
		ψp-learning	72.45	39.75	0.7034	0.9880	0.0207	0.9395	0.0284
		ψ1-learning	82.65	47.20	0.8024	0.9857	0.0199	0.9608	0.0326
		ψ2-learning	72.15	28.65	0.7005	0.9913	0.0175	0.9342	0.0326
		ψ3-learning	31.80	14.20	0.3087	0.9957	0.0251	0.8239	0.0194

Abbreviations: AUC, area under ROC; FPE, false-positive edges; JGL, joint graphical lasso; MIS, misspecified edges; PAUC, area under the partial ROC; ROC, receiver operating characteristic curve; SEN, sensitivity; SPE, specificity; TPE, true-positive edges.

Table reports the average values over 20 replications for each measure.

Figure 3 and Table 2 show the results for the simulations based on a large network. We again see that the ψ_i-learning method overall outperforms all other methods and works well for the high-dimensional data. Figure 4 displays the network paths constructed by various q values for the ψ_i-learning method. The ψ_i-learning method with a large q value produces a sparse network, whereas one with a small q value leads to a dense network, as shown in Figure 4. For a small q value, the resulting network not only shows a lower number of FPEs but also selects a lower number of TPEs. In contrast, the network constructed by a large q value not only identifies the TPEs well but also selects more irrelevant edges, as seen in Tables 1 and 2. Figure 4 indicates that the ψ_i-learning method with an appropriate q value detects the hub genes well in the true network. These results suggest that the integrative analysis for multiple data sets would improve the performance of the ψ-learning method and would enable us to construct more reliable GRNs.

Figure 3.

ROC curve and partial ROC curve under $\mathrm{FPR}<0.05$ for all methods where the sample and network sizes are $\mathrm{n}=100$ and $p=612$, respectively. FPR indicates false-positive rate; JGL, joint graphical lasso; ROC, receiver operating characteristic curve.

Table 2.

Comparison of the proposed method with the JGL, separate, and pooled ψ-learning methods in the simulation on large size network.

Q value	$\mathrm{N}$	Method	TPE	FPE	SEN	SPE	MIS	AUC	PAUC
0.1	100	JGL	156.05	755.00	0.1864	0.9959	0.0077	0.6055	0.0095
		ψi-learning	384.55	67.70	0.4594	0.9996	0.0028	0.9264	0.0273
		ψi-learning	191.70	144.35	0.2290	0.9992	0.0042	0.8381	0.0185
		ψ1-learning	267.45	52.60	0.3195	0.9997	0.0033	0.8862	0.0207
		ψ2-learning	139.00	26.90	0.1661	0.9999	0.0039	0.8473	0.0198
		ψ3-learning	0.15	1.00	0.0002	1.0000	0.0045	0.6002	0.0036
	300	JGL	458.35	1183.30	0.5476	0.9936	0.0084	0.9173	0.0269
		ψi-learning	675.65	162.65	0.8072	0.9991	0.0017	0.9811	0.0384
		ψp-learning	477.80	204.60	0.5708	0.9989	0.0030	0.9326	0.0295
		ψ1-learning	532.50	156.05	0.6362	0.9992	0.0025	0.9488	0.0281
		ψ2-learning	449.90	61.70	0.5375	0.9997	0.0024	0.9267	0.0295
		ψ3-learning	85.05	13.55	0.1016	0.9999	0.0041	0.8310	0.0183
0.3	100	JGL	156.05	755.00	0.1864	0.9959	0.0077	0.6055	0.0095
		ψi-learning	454.25	276.40	0.5427	0.9985	0.0035	0.9264	0.0273
		ψp-learning	280.80	766.60	0.3355	0.9959	0.0071	0.8381	0.0185
		ψ1-learning	335.80	220.35	0.4012	0.9988	0.0039	0.8862	0.0207
		ψ2-learning	220.85	150.40	0.2639	0.9992	0.0041	0.8473	0.0198
		ψ3-learning	0.45	3.40	0.0005	1.0000	0.0045	0.6002	0.0036
	300	JGL	458.35	1183.30	0.5476	0.9936	0.0084	0.9173	0.0269
		ψi-learning	711.75	490.85	0.8504	0.9974	0.0033	0.9811	0.0384
		ψp-learning	539.75	825.30	0.6449	0.9956	0.0060	0.9326	0.0295
		ψ1-learning	587.75	470.20	0.7022	0.9975	0.0038	0.9488	0.0281
		ψ2-learning	502.35	255.45	0.6002	0.9986	0.0032	0.9267	0.0295
		ψ3-learning	151.60	83.45	0.1811	0.9996	0.0041	0.8310	0.0183

Abbreviations: AUC, area under ROC; FPE, false-positive edges; JGL, joint graphical lasso; MIS, misspecified edges; PAUC, area under the partial ROC; ROC, receiver operating characteristic curve; SEN, sensitivity; SPE, specificity; TPE, true-positive edges.

Table reports the average values over 20 replications for each measure.

Figure 4.

Path of networks constructed by various q values in the simulation on large size network: (A) true, (B) q value = 0.000001, (C) q value = 0.001, (D) q value = 0.01, (E) q value = 0.1, and (F) q value = 0.3, where the large nodes with red color represent the hub gene node whose node degrees are greater than 9.

Finally, we investigate the robustness of the integrative ψ-learning method for data sets with missing values. To do this, we consider the large size network with $\mathrm{n}=100$ and $p=612$. For each simulation, we randomly pick 1 data set and select 10 genes in this data set and set their expression values to be missing in all samples. Note that it mimics the multiple data sets to be collected using different microarray platforms, and other methods except the proposed method are inappropriate for data sets with missing values as mentioned previously. Figure 5 displays the proportion of each true hub gene being detected by the integrative ψ-learning method among 100 simulations, and Table 3 compares the integrative ψ-learning methods under the situations with and without missing values. These results indicate that the integrative ψ-learning method is insensitive for multiple data sets with missing values.

Figure 5.

Proportion of each true hub gene being detected by the proposed method in the simulation on large size network: (upper panel) q value = 0.1; (bottom panel) q value = 0.3.

Table 3.

Comparison of the integrative ψ-learning methods under the situations with and without missing values in the simulation on large size network.

Q value	Missing	TPE	FPE	SEN	SPE	MIS	AUC	PAUC
0.1	Without	384.55	67.70	0.4594	0.9996	0.0028	0.9264	0.0273
	With	368.00	64.65	0.4397	0.9996	0.0028	0.9236	0.0275
0.3	Without	454.25	276.40	0.5427	0.9985	0.0035	0.9264	0.0273
	With	444.65	259.65	0.5312	0.9986	0.0035	0.9236	0.0275

Abbreviations: AUC, area under ROC; FPE, false-positive edges; JGL, joint graphical lasso; MIS, misspecified edges; PAUC, area under the partial ROC; ROC, receiver operating characteristic curve; SEN, sensitivity; SPE, specificity; TPE, true-positive edges.

Application to multiple lung cancer adenocarcinoma data sets

Lung cancer is one of the most prominent types of cancer. It is the leading cause of cancer death in the United States in both men and men.²⁵ Non–small-cell lung cancer is the most common cause of lung cancer death, accounting for up to 85% of deaths from lung cancer. About 40% of all lung cancers are ADCs. In this study, we constructed a lung cancer adenocarcinoma–specific gene network by integrating several messenger RNA (mRNA) expression data sets collected from the public domain to better understand the molecular mechanisms associated with patient survival.

Data sets with patients annotated as ADC were selected from Lung Cancer Explorer database (http://qbrc.swmed.edu/lce/). Twelve data sets were collected; their inclusion criteria and details are summarized in Table 4. This study integrates mRNA expression data on patients with lung ADC from 12 data sets. The total data set consists of the gene expression levels of 21 353 genes in 1281 patients with lung ADC. Thus far, this is the largest study to use mRNA expression data to construct a gene network in lung ADC. Genes from different genome-wide platforms are mapped using Probemapper.²⁶ When multiple probes were mapped to a gene, the arithmetic mean of the probe-level expression was computed as the expression level for the gene. The gene expression profiles of each sample were log₂-transformed and standardized to have zero median and unit variance. Following Tang et al,²⁷ we first selected 711 genes that are significantly associated with overall patient survival times after adjusting for clinical factors such as age, gender, and tumor stage in the Lung Cancer Consortium study.²⁸ In this study, we focused on the construction of the global GRN based on the selected 711 genes, and thus, the data set for the analysis had 1281 patients from 12 different sources and 711 genes. Note that for each data source, the number of patients is still less than the number of genes.

Table 4.

Summary of 12 lung ADC data sets from different sources.

Source	Samples	$p_{mis}/{\mathrm{n}}_{mis}$	Source	Samples	$p_{mis}/{\mathrm{n}}_{mis}$ $p_{mis}/{\mathrm{n}}_{mis}$
Shedden et al28	442	2/256	Matsuyama et al29	94	160/94
Kuner et al30	40	—	Bhattacharjee et al31	138	124/138
Zhu et al32	28	—	Tang et al27	133	87/133
Hou et al33	50	—	Wright et al34	36	221/+5 (176/36)
Lee et al35	62	—	Larsen et al36	48	226/+5 (176/48)
Bild et al37	58	—	Tomida et al38	117	112/+5

$p_{mis}/{\mathrm{n}}_{mis}$ means that $p_{mis}$ genes are missing in ${\mathrm{n}}_{mis}$ patients, and +5 stands for 5 more patients.

As mentioned previously, the total data set had too many missing values caused by the differences in genome-wide platforms from different studies. Table 4 displays the missing information for each source of data. For some data sets, the expression levels of many genes were missing in all patients for each source. In this case, it is not reasonable to delete missing samples or genes because there were too many. Moreover, the imputation of such expression levels based on information from other sources without missing values may cause severe biases due to discrepancies in the selection of different platforms. Thus, penalty-based joint estimation methods such as the JGL are not applicable for this type of data because it requires a complete design matrix from all sources.

However, the integrative ψ-learning method is appropriate for multiple data sets with a natural missing mechanism because it is based on meta-analysis using a combination of separate results from each data set. This approach enables us to partially use the information from each source for each edge, which could not be achieved by the penalty-based joint estimation methods because it requires a complete design matrix from all sources. For example, for GAPDH and ANKRD46 genes, there are many missing values in some data sets,^28,29,34,36 but there are no missing values in other data sets. Although the ψ_z-scores in equation (2) cannot be computed in the data sets with missing values, the combined ψ_z-score for the relationship between GAPDH and ANKRD46 can be obtained by combining the ψ_z-scores from other sources, where the corresponding weights $w_{ij}^{(k)}$ for data sets with missing values are set to zero in equation (4).

Figure 6 displays the structure of networks constructed by the integrative ψ-learning method at the q value of 0.0001, in which 602 edges were detected for 456 genes. The constructed networks consisted of many genes with few connections and a few hub genes with many connections. The activity of hub genes may affect many genes in the biological network, and hence, it is expected to play an important role in biology. In the constructed network, we identified 24 potential hub genes whose degrees are greater than 95% quantile of node degree distribution.

Figure 6.

Networks constructed by the integrative ψ-learning method: (A) q value = 0.0001, $\tau =7$ and (B) q value = 0.001, $\tau =9$. The large nodes with red color represent the hub genes whose node degrees are greater than 95% quantile $\tau$ of node degrees for each network.

Table 5 summarizes the hub genes identified by the method at the q value of 0.0001. Among the identified genes, most are cancer-related genes previously identified in the literature, and several are known lung cancer genes. For example, CDH1 is a classical tumor suppressor gene and it encodes for epithelial cadherin, which is important for cell-cell adhesion. Disruption of the cadherin-catenin complex at the cell-cell junction from CDH1 loss or mutation often leads to decreased cell-cell adhesion, increased mobility, and induction of epithelial-mesenchymal transition, which favors the formation of metastasis.³⁹ SMAD5 encodes for one of the Smad proteins that are transcription factors downstream of the transforming growth factor β (TGF-β) pathway. The TGF-β pathway is often dysregulated in cancer and modulates multiple processes in cancer development.⁴⁰ MST1R encodes for a receptor tyrosine kinase (RON) that uses macrophage-stimulating proteins (MSPs) as ligands. The MSP-RON signaling has been implicated in the invasive growth of several types of cancer.⁴¹ DUSP6 encodes for dual specific phosphatase 6 that dephosphorylates and inactivates ERK2 to inhibit mitogen-activated protein kinase signaling. The expression of DUSP6 is regulated by epidermal growth factor receptor (EGFR) signaling, and depletion of DUSP6 from EGFR inhibition has been implicated in resistance to EGFR inhibitors.⁴² In summary, the proposed method seems appropriate for multiple data sets and should perform well in real applications.

Table 5.

List of hub genes identified by the integrative ψ-learning method at the level of q value = 0.0001.

Index	Gene symbol	Degree	Index	Gene symbol	Degree
1	CDH1	45	13	GGCX	8
2	AHNAK	38	14	LDHA	8
3	TMEM9B	33	15	PI4KA	8
4	UBE2D2	30	16	ZNF3	8
5	SMAD5	29	17	TPX2	8
6	MST1R	23	18	EPHX1	7
7	COL10A1	15	19	FOXM1	7
8	MUC5AC	14	20	MCM2	7
9	DUSP6	11	21	PTP4A2	7
10	EIF4A1	10	22	MELK	7
11	BIRC5	9	23	SIVA1	7
12	TMEM183A	9	24	WSB1	7

Finally, we performed sensitivity analysis to test the stability of the identified hub genes. We randomly selected 100 genes and generated their expression levels by randomly shuffling their original expression across different patients. Then, we added these 100 random genes to the 711 survival-related genes and constructed the network of the total 811 (711 + 100) genes. At the similar sparsity level (605 edges detected for 444 genes among the 811 genes, compared with 602 edges detected for 456 genes among the 711 genes in the original analysis), we found that 20 of the 24 hub genes (except FOXM1, LDHA, PI4KA, ZNF3) in the original analysis remain to be hub genes when adding the 100 random genes, which indicates that the identified hub genes are reasonably robust against random noise and selection of genes.

Conclusions

In this article, we developed an approach for constructing a GRN by integrating multiple sources of data. We have introduced an integrative ψ-learning method for construction of GRNs using multiple data sets. Numerical results show that the integrative ψ-learning method improves the performances of the standard ψ-learning method and identifies hub genes well in the true network. Furthermore, we have proposed an extension of the integrative ψ-learning method for the data with a natural missing mechanism caused using different platforms for each source of data. The proposed integrative ψ-learning method was applied to multiple lung cancer adenocarcinoma data sets with many missing values and detected hub genes well that are more likely to be meaningful in biological networks.