Biologically complex diseases such as cancer are caused by mutations in biological pathways or functional groups instead of individual genes. Statistically, genes sharing the same pathway have high correlations and form functional groups or biological pathways. Many databases about biological knowledge or pathway information are available in the public domain after many years of intensive biomedical research. Such databases are often named metadata, which means data about data. Examples of such databases include the gene ontology (GO) databases (Gene Ontology Consortium, 2001), the Kyoto Encyclopedia of Genes and Genomes (KEGG) database [2], and several other pathways on the internet (e.g.,; Most current methods, however, are developed purely from computational points without utilizing any prior biological knowledge or information. Gene selections with survival outcome data in the statistical literature are mainly within the penalized Cox or additive risk regression framework [38]. The L1 and L p (p < 1) penalized Cox regressions can work for simultaneous individual gene selection and survival prediction and have been extensively studied in statistics and bioinformatics literature [811]. The performance of the survival model is evaluated by the global area under the ROC curve summary (GAUCS) [12]. Unfortunately, those methods are mainly for individual gene selections and cannot be used to identify pathways directly. In microarray analysis, several popular tools for pathway analysis, including GENMAP, CHIPINFO, and GOMINER, are used to identify pathways that are over-expressed by differentially expressed genes. These gene set enrichment analysis (GSEA)methods are very informative and are potentially useful for identifying pathways that related to disease status [13]. One drawback with GSEA is that it considers each pathway separately and the pathway information is not utilized in the modeling stage. Wei and Li [14] proposed a boosting algorithm incorporating related pathway information for classification and regression. However, their method can only be applied for binary phenotypes. Since most complex diseases such as cancer are believed to be associated with the activities of multiple pathways, new statistical methods are required to select multiple pathways simultaneously with the time-to-event phenotypes.

A ROC curve provides complete information on the set of all possible combinations of true-positive and false-positive rates, but is also more generally useful as a graphic characterization of the magnitude of separation between the case and control distributions. AUC is known to measure the probability that the marker value (score) for a randomly selected case exceeds the marker value for a randomly selected control and is directly related to the Mann-Whitney U statistic [15, 16]. In survival analysis, a survival time can be viewed as a time-varying binary outcome. Given a fixed time t, the instances for which t i = t are regarded as cases and samples with t i > t are controls. The global AUC summary (GAUCS) is then defined as GAUCS = P (M j > M k |t j < t k ), which indicates that subject who died at an earlier time has a larger score value, where M is a score function. Heagerty and Zheng [12] have shown that GAUCS is a weighted average of the area under time-specific ROC curves. Liu et al. [17] proposed a L1 penalized quadratic support vector machine (SVM) method for GAUCS maximization and individual gene selection with survival outcomes and showed the method outperformed the Cox regression. However, that method is only for gene selections and can not be directly used for identifying pathways without additional criteria. Group LASSO (L1) related penalized methods have been extensively studied recently in logistic regression [18], multiple kernel learning [19], and microarray analysis [20] with binary phenotypes. The methods are designed for selecting groups of variables and identifying important covariate groups (pathways). However, LASSO is biased. L p (p ≤ 1) (with one specific implementation entitled adaptive LASSO [1, 21]) is asymptotic unbiased and has oracle properties. Therefore, it is reasonable to extend the L p to group L p for pathway identifications. In this paper, we therefore extend L p to group L p penalty and develop a novel iterative gradient based algorithm for GAUCS maximization (IGGAUCS), which can effectively integrate genomic data and biological pathway information and identify disease associated pathways with right censored survival data. In Section 2, we formulate the GAUCS maximization and group L p penalty model and propose an efficient EM algorithm for survival prediction. Its performance is compared with its group lasso penalized Cox regression (implemented by ourselves). We also propose an integrated algorithm with GAUCS for microarray data analysis. The proposed approach is demonstrated with simulation and gene expression examples in Section 3. Concluding remarks are discussed in Section 4.

Group L p Penalized GAUCS Maximization

Consider we have a set of n independent observations { t i , δ i , x i } i = 1 n , where δ i is the censoring indicator and t i is the survival time (event time) if δ i = 1 or censoring time if δ i = 0, and x i is the m-dimensional input vector of the i th sample. We denote N i * ( t ) = 1 ( t i t ) and the corresponding increment d N i * ( t ) = N i * ( t ) N i * ( t ) . The time-dependent sensitivity and specificity are defined by sensitivity (c, t): Pr ( M i > c | t i = t ) = Pr ( M i > c | d N i * ( t ) = 1 ) and specificity (c, t): ( Pr ( M i c | t i > t ) = Pr ( M i c | N i * ( t ) = 0 ) . Here sensitivity measures the expected fraction of subjects with a marker greater than c among the subpopulation of individuals who die (cases) at time t, while specificity measures the fraction of subjects with a marker less than or equal to c among those who survive (controls) beyond time t. With this definition, a subject can play the role of a control for an early time, t < t i , but then play the role of case when t = t i . Then, ROC curves are defined as ROC t (q) = TP t {[FP t ]-1(q)} for q ϵ [0, 1], and the area under the ROC curve for time t is A U C ( t ) = 0 1 R O C t ( q ) d q , where TP t and FP t are the true and false positive rate at time t respectively, and [FP t ]-1(q) = inf c {c : FP t (c) ≤ q}. ROC methods can be used to characterize the ability of a marker to distinguish cases at time t from controls at time t. However, in many applications there is no prior time t identified and thus a global accuracy summary is defined by averaging over t:

G A U C S = 2 A U C ( t ) g ( t ) S ( t ) d t = Pr ( M j > M k | t j < t k ) ,

which indicates the probability that the subject who died (cases) at the early time has a larger value of the marker, where S(t) and g(t) are the survival and corresponding density functions, respectively.

Assuming there are r clusters in the input covariates, our primary aim is to identify a small number of clusters associated with survival time t i . Mathematically, for each input x i ϵ ℝm, we are given a decomposition of ℝm as a product of r clusters:

m = m 1 × ... × m r , so that each data point x i can be decomposed into r cluster components, i.e. x i = (xi 1,...,x ir ), where each x il is in general a vector. We define M(x) = wT x to be the risk score function, where w = ( w 1 , w 2 , ... , w r ) T m 1 + , ... , + m r is the vector of coefficients that has the same cluster decomposition as x i . We denote M i = M(x i ) for simplicity. Our goal is to encourage the sparsity of vector w at the level of clusters; in particular, we want most of its multivariate components w l to be zero. The natural way to achieve this is to explore the combination of L p (0 ≤ p ≤ 1) norm and L2 norm. Since w is defined by clusters, we define a weighted group L p norm

L p = l = 1 r d l | w l | p ,

where within every group, an L2 norm is used ( | w l | = ( w l T w l ) 1 / 2 ) and d l can be set to be 1 if all clusters are equally important. Note that group L p = L2 if r = 1 and d l = 1, and group L p = L p when r = m and d l = 1. We can define the optimization problem

max G A U C S = max P r ( M j > M k | t j < t k ) s . t . L p < β ,

where M j = wT x j . The ideal situation is that M(x j ) > M(x k ) or wT (x j - x k ) > 0, ∀ couple (x j , x k ) with corresponding times t j < t k (or j < k) and δ j = 1.

Pr(M j > M k |t j < t k ) can be estimated as

G A U C S = P r ( M j > M k | t j < t k ) = j < k δ j = 1 k = 2 n 1 M j > M k j < k δ j = 1 k = 2 n 1 ,

where 1a>b = 1 if a > b, and 0 otherwise. Obviously, GAUCS is a measure to rank the patients' survival time. The perfect GAUCS = 1 indicates that the order of all patients' survival time are predicted correctly and GAUCS = 0.5 indicates for a completely random choice.

One way to approximate step function 1 M j > M k is to use a sigmoid function σ ( z ) = 1 1 + e z and let N = j < k δ j = 1 k = 2 n 1 , then

G A U C S = j < k δ j = 1 k = 2 n σ ( w T ( x j x k ) ) N .

Equation (4) is nonconvex function and can only be solved with the conjugate gradient method to find a local minimum. Based on the property that the arithmetic average is greater than the geometric average, we have

j < k δ j = 1 k = 2 n σ ( w T ( x j x k ) ) N 1 N j < k δ j = 1 k = 2 n σ ( w T ( x j x k ) ) .

We can, therefore, maximize the following log likelihood lower bound of equation (4).

E p = 1 N j < k δ j = 1 k = 2 n log σ ( w T ( x j x k ) ) λ L p = 1 N j < k δ j = 1 k = 2 n log σ ( w T ( x j x k ) ) λ l = 1 r d l | w l | p ,

where λ is a penalized parameter controlling model complexity. Equation (5) is the maximum a posterior (MAP) estimator of w with Laplace prior provided we treat the sigmoid function as the pair-wise probability, i.e. Pr(M j > M k ) = σ(wT (x j - x k )). When p = 1, E p is a convex function. A global optimal solution is guaranteed.

The IGGAUCS Algorithm

In order to find the w that maximizes E p , we need to find the first order derivative. Since group L p with p ≤ 1 is not differentiable at |w l | = 0, differentiable approximations of group L p is required. We propose a local quadratic approximation for group |w l |p based on convex duality and local variational methods [23, 24]. Fan and Li [25] proposed a similar approximation for single variable. The adaptive LASSO approach proposed by Zou and Li [21] is a LASSO (linear bound) approximation for L p penalty. The drawback with that approach is that LASSO itself is not differentiable at |w l | = 0. Since |w l |p is concave when p < 1, we can have

f ( w l ) = | w l | p = min η l { η l | w l | 2 g ( η l ) } g ( η l ) = min | θ l | { η l | θ l | 2 f ( θ l ) } ,

where the function g(.) is the dual function of f(.) in variational analysis. Geometrically, g(η l ) represents the amounts of vertical shift applied to η l |w l |2 to obtain a quadratic upper bound with precision parameter η l , that touches f(w l ). Taking the first order derivative for η l θ l 2 f ( θ l ) , the minimum occurs at a solution of stationary equation when θ l ≠ 0,

2 η l | θ l | f ( θ l ) = 0 η l = f ( θ l ) 2 | θ l | ,

and f'(θ l ) = p|θ l |p-1. Substituting into f(w l ), we have the variational bound:

| w l | p f ( θ l ) 2 θ l ( | w l | 2 | θ l | 2 ) + f ( θ l ) = 1 2 { p | θ l | p 2 | w l | 2 + ( 2 p ) | θ l | p } ,

where θ l denote variational parameters. With the local quadratic bound, we have the following smooth lower bound.

E p = 1 N j < k δ j = 1 k = 2 n log σ ( w T ( x j x k ) ) λ l = 1 r d l | w l | p 1 N j < k δ j = 1 k = 2 n log σ ( w T ( x j x k ) ) λ l = 1 r d l 2 { p | θ l | p 2 | w l | 2 + ( 2 p ) | θ l | p } = E ( w , θ ) .

In equation (8), the lower bound E(w, θ) is differentiable w.r.t both w and θ. We therefore propose a EM algorithm to maximize E(w, θ) w.r.t w while keeping θ fixed and maximize E(w, θ) w.r.t the variational parameter θ to tighten the variational bound while keeping w fixed. Convergence to the local optimum is guaranteed. Since maximization w.r.t the variational parameters 0 = (|θ1|,|θ2|,..., |θ r |), with w being fixed, can be solved with the stationary equation E ( w , θ ) | θ l | = 0 we have θ l = w l , for l = 1, 2,..., r.

Given r candidate pathways potentially associated with the survival time, m l survival associated genes with the expression of x l on each pathway l (l = 1, 2,..., r), and letting w = (w1, w2,..., w r )Tbe a vector of the corresponding coefficients and g ( w ) = w E ( w , θ ) w , we have the following iterative gradient algorithm for E(w, θ) maximization:

The IGGAUCS Algorithm

Given p, λ, and ϵ = 10-6, initializing w 1 = ( w 1 1 , w 2 1 , ... , w r 1 ) T randomly with nonzero w l 1 , l = 1 , ... , r , and set θ1 = w1.

Update w with θ fixed:

wt+1= wt+ αtdt, where t: the number of iterations and αt: the step size, dt is updated with the conjugate gradient method:

dt = g(wt) + utdt and u t = [ g ( w t ) g ( w t 1 ) ] T g ( w t ) g ( w t 1 ) T g ( w t 1 ) .

Update θ with w fixed:

θt+1= wt+1

Stop when |wt+1- wt| < ϵ or maximal number of iterations exceeded.

Choice of Parameters

There are two parameters p and λ in this method, which can be determined through 10-fold cross validation. One efficient way is to set p = 0.1, 0.2,..., and 1 respectively, and search for an optimal λ for each p using cross validation. The best (p, λ) pair will be found with the maximal test GAUCS value. Theoretically when p = 1, E(w, θ) is convex and we can find the global maximum easily, but the solution is biased and small values of p would lead to better asymptotic unbiased solutions. Our results with limited experiments show that optimal p usually happens at a small p such as p = 0.1. For comparison purposes, we implement the popular Cox regression with group LASSO (GL1Cox), since there is no software available in the literature. Our implementation is based on group LASSO penalized partial log-likelihood maximization. The best λ is searched from λ ϵ [0.1, 25] for IGGAUCS and from λ ϵ [0.1, 40] for GL1Cox method with the step size of 0.1, as the L p penalty goes to zero much quicker than L1. We suggest that the larger step size such as 0.5 can be used for most applications, since the test GAUCS does not change dramatically with a small change of λ.

Computational Results

Simulation Data

We first perform simulation studies to evaluate how well the IGGAUCS procedure performs when input data has a block structure. We focus on whether the important variable groups that are associated with survival outcomes can be selected using the IGGAUCS procedure and how well the model can be used for predicting the survival time for future patients. In our simulation studies, we simulate a data set with a sample size of 100 and 300 input variables with 100 groups (clusters). The triple variables x1 - x3, x4 - x6, x7 - x9,..., x298 - x300 within each group are highly correlated with a common correlation γ and there are no correlations between groups. We set γ = 0.1 for weak correlation, γ = 0.5 for moderate, and γ = 0.9 for strong correlation in each triple group and generate training and test data sets of sample size 100 with each γ respectively from a normal distribution with the band correlation structure. We assume that the first three groups(9 covariates) (x1 - x3, x4 - x6, x7 - x9) are associated with survival and the 9 covariates are set to be w = [-2.9 2.1 2.4 1.6 -1.8 1.4 0.4 0.8 -0.5]t. With this setting, 3 covariates in the first group have the strongest association (largest covariate values) with survival time and 3 covariates in group 3 have less association with survival time. The survival time is generated with H = 100 exp(-wTx + ε) and the Weibull distribution, and the census time is generated from 0.8*median(time) plus a random noise. Based on this setting, we would expect about 25% - 35% censoring. To compare the performance of IGGAUCS and GL1Cox, we build the model based on training data set and evaluate the model with the test data set. We repeat this procedure 100 times and use the time-independent GAUCS to assess the predictive performance.

We first compare the performance of IGGAUCS and GL1Cox methods with the frequency of each of these three groups being selected under two different correlation structures based on 100 replications. The results are in Table 1. Table 1 shows that IGGAUCS with p = 0.1 outperforms the GL1Cox in that IGGAUCS can identify the true group structures more frequently under different inner group correlation structures. Its performance is much better than GL1Cox regression, when the inner correlation in a group is high (γ = 0.9) and the variables within a group have weak association with survival time.

Table 1 Frequency of Three Survival Associated Groups Selected in 100 Replications

To compare more about the performance of IGGAUCS and GL1Cox in parameter estimation, we show the results for each parameter with different inner correlation structures (0.1, 0.5, 0.9) in Figure 1. For each parameter in Figure 1, the left bar represents the parameter estimated from GL1Cox, the middle bar is the true value of the parameter, and the right bar indicates parameter estimated from IGGAUCS. We observe that both the GL1Cox and IGGAUCS methods estimated the sign of the parameters correctly for the first two groups. However both methods can only estimate the sign of w8 correctly in group 3 with smaller coefficients. Moreover, ŵ estimated from IGGAUCS is much closer to the true w than that from GL1Cox, especially when the covariates are larger. This indicates that the L p (p = 0.1) penalty is less biased than the L1 penalty. The estimators of IGGAUCS are larger than that of GL1Cox with weak, moderate, and strong correlations. Finally, the test global AUC summaries (GAUCSs) of IGGAUCS and GL 1Cox with 100 replications are shown in Table 2. Table 2 shows that IGGAUCS performs better than GL1Cox regression. This is reasonable, since our method, unlike Cox regression which maximizes a partial log likelihood, directly maximizes the penalized GAUCS. One interesting result is that the test GAUCSs become smaller as the inner group correlation coefficient γ increases from 0.1 to 0.9. We also apply the gene harvesting method proposed by Hastie et al. (2001) and discussed by Segal (2006) [7, 26] to the simulation data, but don't show the results in Table 2. The gene harvest method uses the average gene expression in each group (cluster) and ignore the variance among genes within the group. The prediction performances are poor with test GAUCS of 0.57 ± 0.02 and 0.65 ± 0.016 respectively, when the correlations among genes are weak (γ = 0.1) and moderate (γ = 0.5). Its performance is slightly better with the test GAUCS of 0.75 ± 0.024, when γ = 0.9, but this4 performance is still not as good as either IGGAUCS or GL1Cox. One explantation is that the group is more heterogeneous with weaker correlations among variables, and the average does not provide a meaningful summary. Moreover, we cannot identify the survival association of individual variables using gene harvesting.

Table 2 Test GAUCS of Simulated Data w ith Different Correlation Structures
Figure 1
figure 1

True and Estimated Parameters. The true and estimated parameters with the simulation data are shown in Figure 2. The left bars represent each parameter estimated from GL1Cox, the middle bars are the true value of the parameter, and the right bars indicate parameters estimated form IGGAUCS.

Follicular Lymphoma (FL) Data

Follicular lymphoma is a common type of Non-Hodgkin Lymphoma (NHL). It is a slow growing lymphoma that arises from B-cells, a type of white blood cell. It is also called an "indolent" or "low-grad" lymphoma for its slow nature, both in terms of its behavior and how it looks under the microscope. A study was conducted to predict the survival probability of patients with gene expression profiles of tumors at diagnosis [27].

Fresh-frozen tumor biopsy specimens and clinical data were obtained from 191 untreated patients who had received a diagnosis of follicular lymphoma between 1974 and 2001. The median age of patients at diagnosis was 51 years (range 23 - 81) and the median follow up time was 6.6 years (range less than 1.0 - 28.2). The median follow up time among patients alive was 8.1 years. Four records with missing survival information were excluded from the analysis. Affymetrix U133A and U133B microarray gene chips were used to measure gene expression levels from RNA samples. A log 2 transformation was applied to the Affymetrix measurement. Detailed experimental protocol can be found in Dave et al. 2004. The data set was normalized for each gene to have mean 0 and variance 1. Because the data is very large and there are many genes with their expressions that either do not change cross samples or change randomly, we filter out the genes by defining a correlation measure with GAUCS for each gene xi R(t, x i ) = |2GAUCS(t, x i ) - 1|, where R = 1 when GAUCS = 0, or 1 and R = 0 when GAUCS = 0.5 (gene x i is not associated with the survival time). We perform the permutation test 1000 times for R to identify 2150 probes associated with survival time. We then identify 49 candidate pathways with 5 and more genes using DAVID. There are total 523 genes on the candidate pathways. Since a gene can be involved in more than one pathway, the number of distinguished genes should be a little less than 500. The 49 biological pathways are given in Table 3. We finally apply IGGAUCS to identify the small number of biological pathways associated with survival phenotypes. We first randomly divide the data into two subsets, one for training with 137 samples, and the other for testing with 50 samples. To avoid overfitting and bias from a particular partition, we randomly partition the data 50 times to estimate the performance of the model with the average of the test GAUCS. The regularization parameter λ is tuned using 10-fold cross-validation with training data only.

Table 3 Candidate Survival Associated Pathways

Since it is possible different pathways may be selected in the cross validation procedure, the relevance count concept [28] was utilized to count how many times a pathway is selected in the cross validation. Clearly, the maximum relevance count for a pathway is 200 with the 10-fold cross validation and 20 repeating. We have selected 8 survival associated pathways with IGGAUCS. The average test GAUCS is 0.892 ± 0.013. Moreover, the parameters (weights) w i and corresponding genes on each pathway indicate the association strength and direction between genes and the survival time. Positive w i s indicate that patients with high expression level die earlier and negative w i s represent that patients live longer with relatively high expression levels. The absolute values of |w i | indicate the strength of association between survival time and the specific gene. Genes on the pathway, estimated parameters, and relevance accounts are given in Table 4.

Table 4 Genes on pathway, relevance accounts, and estimated parameters

The eight KEGG pathways identified play an important role in patient survivals and they can be ranked with the average |w j |: j |w j |/L, where L is the number of genes on a pathway as shown in Table 5. The identified 8 KEGG pathways fall into three categories (i) signaling molecules and interaction including the MAPK signaling pathway, the Calcium signaling pathway, Focal adhesion, ECM-receptor interactions and neuroactive ligand-receptor interactions, (ii) metabolic pathways including Fatty acid metabolism and Porphyrin and chlorophyll metabolism, and (iii) nitric oxide and cell stress including Ubiquitin mediated proteolysis. These pathways are involved in different aspects of genetic functions and are vital for cancer patient survivals. We only discuss the MAPK signaling pathway but others can be analyzed in a similar fashion. The top-rank MAPK signaling pathway transduces a large variety of external signals, leading to a wide range of cellular responses, including growth, differentiation, inflammation, and apoptosis invasiveness and ability to induce neovascularization. MAPK signaling pathway has been linked to different cancers including follicular lymphoma [29]. The pathway and genes on pathways are given in Figure 2.

Table 5 Pathway Ranks
Figure 2
figure 2

MAPK Signaling Pathway. MAPK signaling pathway and the associated genes. Genes in red are highly expressed in patients who died earlier and genes in yellow are highly expressed in patients who lived longer.

Genes in red color are highly expressed in patients with aggressive FL and genes in yellow are highly expressed in the earlier stage of FL cancers. Many important cancer related genes are identified with our methods. For example, SOS1, one of the RAS genes (e.g., MIM 190020), encodes membrane-bound guanine nucleotide-binding proteins that function in the transduction of signals that control cell growth and differentiation. Binding of GTP activates RAS proteins, and subsequent hydrolysis of the bound GTP to GDP and phosphate inactivates signaling by these proteins. GTP binding can be catalyzed by guanine nucleotide exchange factors for RAS, and GTP hydrolysis can be accelerated by GTPase-activating proteins (GAPs). SOS1 plays a crucial role in the coupling of RTKs and also intracellular tyrosine kinases to RAS activation. The deregulation of receptor tyrosine kinases (RTKs) or intracellular tyrosine kinases coupled to RAS activation has been involved in the development of a number of tumors, such as those in breast cancer, ovarian cancer and leukemia. Another gene, IL1B, is one of a group of related proteins made by leukocytes (white blood cells) and other cells in the body. IL1B, one form of IL1, is made mainly by one type of white blood cell, the macrophage, and helps another type of white blood cell, the lymphocyte, fight infections. It also helps leukocytes pass through blood vessel walls to sites of infection and causes fever by affecting areas of the brain that control body temperature. IL1B made in the laboratory is used as a biological response modifier to boost the immune system in cancer therapy.

As shown in Figure 2, the genes SOS1, IL1B, RAS, CACNB1, MEF2C, JUND, and MAPKAPK5 are highly expressed in patients who were diagnosed earlier and lived longer and the genes FGF14, PTPN5, MOS, RAF1, CD14 are highly expressed in patients who were diagnosed at more aggressive stages and died earlier, which may indicate that oncogenes such such SOS1, JUND, and RAS may initialize FL cancer and genes such as MOS, IKK, and CD14 may cause FL cancer to be more aggressive. There are several causal relations among the identified genes on MAPK. For instance, the down-expressed SOS and RAS cause the up-expressed RAF1 and MOS and the up-stream gene IL1 is coordinately expressed with CASP and the gene MST1/2.


Since a large amount of biological information on various aspects of systems and pathways is available in public databases, we are able to utilize this information in modeling genomic data and identifying pathways and genes and their interactions that might be related to patient survival. In this study, we have developed a novel iterative gradient algorithm for group L p penalized global AUC summary (IGGAUCS) maximization methods for gene and pathway identification, and for survival prediction with right censored survival data and high dimensional gene expression profile. We have demonstrated the applications of the proposed method with both simulation and the FL cancer data set. Empirical studies have shown the proposed approach is able to identify a small number of pathways with nice prediction performance. Unlike traditional statistical models, the proposed method naturally incorporates biological pathways information and it is also different from the commonly used Gene Set Enrichment Analysis (GSEA) in that it simultaneously considers multiple pathways associated with survival phenotypes.

With comprehensive knowledge of pathways and mammalian biology, we can greatly reduce the hypothesis space. By knowing the pathway and the genes that belong to particular pathways, we can limit the number of genes and gene-gene interactions that need to be considered in modeling high dimensional microarray data. The proposed method can efficiently handle thousands of genes and hundreds of pathways as shown in our analysis of the FL cancer data set.

There are several directions for our future investigations. For instance, we may want to further investigate the sensitivity of the proposed methods to the misspecification of the pathway information and misspecification of the model. We may also extend our method to incorporate gene-gene interactions and gene (pathway)- environmental interactions.

Even though we have only applied our methods to gene expression data, it is straightforward to extend our methods to SNP, miRNA CGH, and other genomic data without much modification.