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**|**Algorithms Mol Biol**|**v.5; 2010**|**PMC2930641

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Algorithms Mol Biol. 2010; 5: 30.

Published online 2010 August 16. doi: 10.1186/1748-7188-5-30

PMCID: PMC2930641

Zhenqiu Liu: ude.mmu@uilz; Laurence S Magder: ude.dnalyramu.ipe@redgaml; Terry Hyslop: ude.ujt.icj.liam@polsyht; Li Mao: ude.dnalyramu@oaml

Received 2010 January 7; Accepted 2010 August 16.

Copyright ©2010 Liu et al; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article has been cited by other articles in PMC.

It has been demonstrated that genes in a cell do not act independently. They interact with one another to complete certain biological processes or to implement certain molecular functions. How to incorporate biological pathways or functional groups into the model and identify survival associated gene pathways is still a challenging problem. In this paper, we propose a novel iterative gradient based method for survival analysis with group *L _{p }*penalized global AUC summary maximization. Unlike LASSO,

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., http://www.superarray.com; http://www.biocarta.com). 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 [3-8]. The *L*_{1 }and *L _{p }*(

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 }*=

Consider we have a set of *n *independent observations ${\{{t}_{i},{\delta}_{i},{\text{x}}_{i}\}}_{i=1}^{n}$, where *δ _{i }*is the censoring indicator and

$$\begin{array}{c}GAUCS\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\int AUC(t)g(t)S(t)dt\\ =\text{Pr}({M}_{j}>{M}_{k}|{t}_{j}<{t}_{k}),\end{array}$$

(1)

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

${\mathbb{R}}^{m}={\mathbb{R}}^{{m}_{1}}\times ...\times {\mathbb{R}}^{{m}_{r}}$, so that each data point **x*** _{i }*can be decomposed into

$${L}_{p}={\displaystyle \sum _{l=1}^{r}{d}_{l}}|{\text{w}}_{l}{|}^{p},$$

where within every group, an *L*_{2 }norm is used $(|{\text{w}}_{l}|={({\text{w}}_{l}^{T}{\text{w}}_{l})}^{1/2})$ and *d _{l }*can be set to be 1 if all clusters are equally important. Note that group

$$\begin{array}{l}\mathrm{max}\text{\hspace{0.17em}}GAUCS\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{max}Pr({M}_{j}>{M}_{k}|{t}_{j}<{t}_{k})\\ \text{s}.\text{t}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{p}<\beta ,\end{array}$$

(2)

where *M _{j }*=

*Pr*(*M _{j }*>

$$\begin{array}{c}GAUCS\text{\hspace{0.17em}}=\text{\hspace{0.17em}}Pr({M}_{j}>{M}_{k}|{t}_{j}<{t}_{k})\\ =\frac{{\displaystyle {\sum}_{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle {\sum}_{k=2}^{n}{1}_{{M}_{j}>{M}_{k}}}}}{{\displaystyle {\sum}_{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle {\sum}_{k=2}^{n}1}}},\end{array}$$

(3)

where **1**_{a>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 $\sigma (z)=\frac{1}{1+{e}^{-z}}$ and let $N={\displaystyle {\sum}_{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle {\sum}_{k=2}^{n}1}}$, then

$$GAUCS=\frac{{\displaystyle {\sum}_{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle {\sum}_{k=2}^{n}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))}}}{N}.$$

(4)

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

$$\frac{{\displaystyle {\sum}_{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle {\sum}_{k=2}^{n}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))}}}{N}\ge \frac{1}{N}{\displaystyle \prod _{\begin{array}{l}j<k\\ {\delta}_{j}=1\end{array}}{\displaystyle \prod _{k=2}^{n}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))}}.$$

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

$$\begin{array}{c}{E}_{p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{N}{\displaystyle \sum _{\begin{array}{l}j<k\\ \delta j=1\end{array}}{\displaystyle \sum _{k=2}^{n}\mathrm{log}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))-\lambda {L}_{p}}}\\ =\frac{1}{N}\text{\hspace{0.17em}}{\displaystyle \sum _{\begin{array}{l}j<k\\ \delta j=1\end{array}}{\displaystyle \sum _{k=2}^{n}\mathrm{log}\text{\hspace{0.17em}}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))-\lambda}}{\displaystyle \sum _{l=1}^{r}{d}_{l}|{\text{w}}_{l}{|}^{p}},\end{array}$$

(5)

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(

In order to find the **w **that maximizes *E _{p}*, we need to find the first order derivative. Since group

$$\begin{array}{c}f({\text{w}}_{l})\text{\hspace{0.17em}}=\text{\hspace{0.17em}}|{\text{w}}_{l}{|}^{p}=\underset{{\eta}_{l}}{\mathrm{min}}\{{\eta}_{l}|{\text{w}}_{l}{|}^{2}-g({\eta}_{l})\}\\ g({\eta}_{l})=\underset{\left|{\theta}_{l}\right|}{\mathrm{min}}\left\{{\eta}_{l}\right|{\theta}_{l}{|}^{2}-f({\theta}_{l})\},\end{array}$$

(6)

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

$$2{\eta}_{l}\left|{\theta}_{l}\right|-{f}^{\prime}({\theta}_{l})=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\eta}_{l}=\frac{{f}^{\prime}({\theta}_{l})}{2\left|{\theta}_{l}\right|},$$

and *f'*(*θ _{l}*) =

$$\begin{array}{l}|{\text{w}}_{l}{|}^{p}\le \frac{{f}^{\prime}({\theta}_{l})}{2{\theta}_{l}}(|{\text{w}}_{l}{|}^{2}-|{\theta}_{l}{|}^{2})+f({\theta}_{l})\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}\left\{p\right|{\theta}_{l}{|}^{p-2}|{\text{w}}_{l}{|}^{2}+(2-p)|{\theta}_{l}{|}^{p}\},\end{array}$$

(7)

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

$$\begin{array}{c}{E}_{p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{N}{\displaystyle \sum _{\begin{array}{l}j<k\\ \delta j=1\end{array}}{\displaystyle \sum _{k=2}^{n}\mathrm{log}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))-\lambda}}{\displaystyle \sum _{l=1}^{r}{d}_{l}|{\text{w}}_{l}{|}^{p}}\\ \ge \frac{1}{N}{\displaystyle \sum _{\begin{array}{l}j<k\\ \delta j=1\end{array}}{\displaystyle \sum _{k=2}^{n}\mathrm{log}\sigma ({\text{w}}^{T}({\text{x}}_{j}-{\text{x}}_{k}))}}\\ -\lambda {\displaystyle \sum _{l=1}^{r}\frac{{d}_{l}}{2}\left\{p\right|{\theta}_{l}{|}^{p-2}|{\text{w}}_{l}{|}^{2}+(2-p)|{\theta}_{l}{|}^{p}\}}\\ =E(\text{w},\theta ).\end{array}$$

(8)

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 *= (|*θ*_{1}|,|*θ*_{2}|,..., |*θ _{r}*|), with

Given *r *candidate pathways potentially associated with the survival time, *m _{l }*survival associated genes with the expression of

Given *p*, λ, and *ϵ *= 10^{-6}, initializing ${\text{w}}^{1}={({\text{w}}_{1}^{1},{\text{w}}_{2}^{1},...,{\text{w}}_{r}^{1})}^{T}$ randomly with nonzero ${\text{w}}_{l}^{1},\text{\hspace{0.17em}}l=1,...,\text{\hspace{0.17em}}r$, and set *θ*^{1 }= w^{1}.

Update w with *θ *fixed:

**w**^{t+1 }= **w**^{t }+ *α ^{t}d^{t}*, where

*d ^{t }*= g(

Update θ with w fixed:

*θ*^{t+1 }= **w**^{t+1}

Stop when |**w**^{t+1 }- **w**^{t}| <*ϵ *or maximal number of iterations exceeded.

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 (G*L*_{1}Cox), 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 G*L*_{1}Cox method with the step size of 0.1, as the *L _{p }*penalty goes to zero much quicker than

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 **x**_{1 }- **x**_{3}, **x**_{4 }- **x**_{6}, **x**_{7 }- **x**_{9},..., **x**_{298 }- **x**_{300 }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) (**x**_{1 }- **x**_{3}, **x**_{4 }- **x**_{6}, **x**_{7 }- **x**_{9}) 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(-**w**^{T }**x **+ *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 G*L*_{1}Cox, 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 G*L*_{1}Cox 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 Table1.1. Table Table11 shows that IGGAUCS with *p *= 0.1 outperforms the G*L*_{1}Cox in that IGGAUCS can identify the true group structures more frequently under different inner group correlation structures. Its performance is much better than G*L*_{1}Cox 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.

To compare more about the performance of IGGAUCS and G*L*_{1}Cox in parameter estimation, we show the results for each parameter with different inner correlation structures (0.1, 0.5, 0.9) in Figure Figure1.1. For each parameter in Figure Figure1,1, the left bar represents the parameter estimated from G*L*_{1}Cox, the middle bar is the true value of the parameter, and the right bar indicates parameter estimated from IGGAUCS. We observe that both the G*L*_{1}Cox and IGGAUCS methods estimated the sign of the parameters correctly for the first two groups. However both methods can only estimate the sign of *w*_{8 }correctly in group 3 with smaller coefficients. Moreover, **ŵ **estimated from IGGAUCS is much closer to the true **w **than that from G*L*_{1}Cox, especially when the covariates are larger. This indicates that the *L _{p }*(

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 **x**_{i }*R*(*t*, **x*** _{i}*) = |2

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

The eight KEGG pathways identified play an important role in patient survivals and they can be ranked with the average |*w _{j }*|:

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 Figure2,2, 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.

The authors declare that they have no competing interests.

ZL designed the method and drafted the manuscript. Both LSM and TH participated manuscript preparation and revised the manuscript critically. LM provided important help in its biological contents. All authors read and approved the final manuscript.

We thank the Associate Editor and the two anonymous referees for their constructive comments which helped improve the manuscript. ZL was partially supported by grant 1R03CA133899-01A210 from the National Cancer Institute.

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