A method for analyzing censored survival phenotype with gene expression data
- Tongtong Wu†^{1},
- Wei Sun†^{2},
- Shinsheng Yuan^{3},
- Chun-Houh Chen^{3} and
- Ker-Chau Li^{3, 4}Email author
https://doi.org/10.1186/1471-2105-9-417
© Wu et al; licensee BioMed Central Ltd. 2008
Received: 13 March 2008
Accepted: 06 October 2008
Published: 06 October 2008
Abstract
Background
Survival time is an important clinical trait for many disease studies. Previous works have shown certain relationship between patients' gene expression profiles and survival time. However, due to the censoring effects of survival time and the high dimensionality of gene expression data, effective and unbiased selection of a gene expression signature to predict survival probabilities requires further study.
Method
We propose a method for an integrated study of survival time and gene expression. This method can be summarized as a two-step procedure: in the first step, a moderate number of genes are pre-selected using correlation or liquid association (LA). Imputation and transformation methods are employed for the correlation/LA calculation. In the second step, the dimension of the predictors is further reduced using the modified sliced inverse regression for censored data (censorSIR).
Results
The new method is tested via both simulated and real data. For the real data application, we employed a set of 295 breast cancer patients and found a linear combination of 22 gene expression profiles that are significantly correlated with patients' survival rate.
Conclusion
By an appropriate combination of feature selection and dimension reduction, we find a method of identifying gene expression signatures which is effective for survival prediction.
Background
The DNA microarray technique allows researchers to simultaneously interrogate the expression levels of all genes in an organism. It has been widely applied in disease studies, such as cancer subtype discovery, cancer/normal sample discrimination, disease gene identification [1–3]. Recently several studies have focused on dissecting the relation between survival time and gene expression [4–7]. One difficulty of these studies is that the survival times are often right-censored. For example, at the ends of the studies, some patients may still be alive. We only know that their survival times are greater than the last follow-up time, but not the exact survival times. Thus, treating these censored survival times as the true life times without adjustment will lead to systematic bias. Another difficulty is that microarray gene expression data are often measured at the full genome scale with tens of thousands of gene expression profiles, while the number of patients under study is relatively small, thereby presenting a difficult variable selection problem.
Many methods of correlating patient survival with gene expression are one-step procedures. Individual genes, gene clusters, or linear combinations of genes are selected by unsupervised or supervised methods. The selected gene expression signatures are then directly used to predict survival probabilities. Unsupervised approaches, such as selecting a small number gene clusters by unsupervised clustering [8], has the disadvantage that survival phenotype information is completely ignored in the feature selection step. Most currently available methods are supervised methods. Nguyen et al. [9] proposed to use the standard partial least square (PLS) method in selecting linear combinations of genes. The survival phenotype is utilized because the PLS method selects linear combinations of genes by maximizing their covariances with the survival time. However, the censoring information has been ignored in this procedure. Li et al. [4] proposed to select linear combinations of genes by a partial Cox regression (PCR) method, which is an extension of PLS method for censoring data. The results of PLS and PCR are linear combinations of thousands of genes, which may be difficult to interpret. Furthermore, the appropriateness of the proportional hazard assumption underlying the Cox regression methodology is not free of challenge. A Bayesian variable selection approach based on the accelerated failure time (AFT) model was introduced by Sha et al. [6], but the performance of this method when the AFT assumption itself is violated has not yet been extensively studied.
Recently, several two-step procedure has been introduced. The first step is the preliminary gene filtering, and the second step is to model the survival time with the pre-selected genes. Li et al. [5] used principal component analysis (PCA) for pre-selection. In the PCA step, linear combinations of genes (called the principal components, PC) are sequentially identified by maximizing the variances explained by the PCs, and a small number of PCs that explain most variances are selected. The second step is to apply SIR to identify some linear combinations of PCs to further reduce the dimensionality. This two-step procedure overcomes the difficulty of handling thousands of genes simultaneously and has a good prediction performance for patients' survival probabilities. However, its performance and interpretability could be improved. First, although PCA can effectively reduce the dimension, it ignores the survival phenotype information. In addition, with principal components selected in the first step, the final results are linear combinations of linear combinations of gene expression profiles, of which the meanings are difficult to interpret. Secondly, although Li et al. [5] used SIR to identify the joint space of life time and censor time, they did not conduct the recovery of the life time space, which should be of the primary interest (see [10] for more details about the distinction between life time space and censor time space). Ma et al. [7] proposed a Lasso (least absolute shrinkage and selection operator, see [11]) type of approach for simultaneous gene selection and parameter estimation based on an additive risk model. In their method, a pre-selection step is used to select gene expression profiles correlated with survival time among those patients without censoring. However, in many real data, censoring rate is often very high. For example, in the real data we study in this paper, more than 70% of the survival times are censored. Therefore ignoring the censored patients in pre-selection step may limit the power of this method.
In this paper, we propose a different two-step procedure to identify a gene signature to predict patients' survival probabilities. In the first step, we use a nonparametric approach to impute the survival probabilities for the censored patients based on the well-known Kaplan Meier estimate. We then use the imputed survival probabilities together with the uncensored survival probabilities to pre-select genes via either the correlation or the liquid association (LA) method [12]. In the second step, we apply the modified SIR for censored data (censorSIR, [10]) to further reduce the dimensionality of the selected genes. CensorSIR found projection directions in life time space without imposing any assumption of the structure model (such as the proportion hazard or AFT) between survival time and gene expression profiles. We can use these projection directions to predict survival probabilities or classify newly diagnosed cancer patients.
The rest of this paper is organized as follows. The method section will be arranged into two parts: preliminary gene screening and dimension reduction – censorSIR. Both simulated and real data will be analyzed to illustrate and evaluate our method in the result section. A discussion section is provided at the end.
Method
Preliminary Gene Filtering
For effective use of the current SIR methodology, the number of genes (G) need to be much smaller than the number of samples (N). It is not even appropriate to allow G to be in the same magnitude as N. However, for most microarray data, G is much larger than N. We therefore need to first conduct gene screening to reduce the number of genes. In order to achieve this end, we employ both liquid association (LA) [12, 13] and correlation to select candidate genes. From the parsimonious predictive model building point of view, we prefer to start from a smaller gene set as long as a reasonable power of prediction can be obtained. Nevertheless, as pointed out by an anonymous referee, in radiation or carcinogen experiments where tissues are exposed, broad and more global changes of gene expressions are expected across the genome. Therefore, if the sample size is small, our preliminary gene filtering may miss some important genes.
Imputation of Survival Probabilities
- 1.
Calculate ${\widehat{S}}_{i}$ the Kaplan-Meier estimate of the survival probability [14]. Specifically, ${\widehat{S}}_{i}={\displaystyle {\prod}_{{t}_{j}\le {t}_{i}}\frac{{n}_{j}-{d}_{j}}{{n}_{j}}}$ where n_{ j }is the number of individuals at-risk just prior to time t_{ j }, and d_{ j }is the number of deaths at time t_{ j };
- 2.Impute the survival probability by the predicted conditional median${\tilde{S}}_{i}=\{\begin{array}{ll}{\widehat{S}}_{i}\hfill & \text{if}{\delta}_{i}=1\hfill \\ {\widehat{S}}_{i}/2\hfill & \text{if}{\delta}_{i}=0\hfill \end{array};$
- 3.
Calculate the percentile ${p}_{i}=1-{\tilde{S}}_{i}$;
- 4.
Perform the normal quantile transformation on p_{ i }.
The normal quantile transformation is necessary for the LA calculation (see next section) and make our procedure robust against outliers. Specifically, it is carried out as follows. For any variable Z observed in the N patients, we rank all Z_{ i }, i = 1,...,N and denote the rank as R_{ i }. The normally transformed profile is then defined as Φ^{-1} (R_{ i }/(N + 1)), where Φ(.) is the cumulative standard normal distribution. Notice that, instead of imputing the survival time, we actually impute the survival probability.
A natural estimate of S(z) is to use the Kaplan-Meier estimate of the survival function. However, the resulting estimator is inappropriate if the last observation is censored, because the Kaplan-Meier estimator is undefined beyond the largest uncensored survival time and the integral will be infinite [15]. In practice, for many real data, including the ones we will analyze in this paper, the last observation is censored. Therefore, we will not adopt this conditional mean estimate.
Now we can estimate the survival probability of ${\tilde{T}}_{i}$ by $\widehat{S}({T}_{i})/2$, where $\widehat{S}(t)$ is the Kaplan-Meier estimate of survival function.
It is worth to emphasize that the imputed survival probabilities are only used in the preliminary gene filtering step. The observed survival time with censoring information are used in the dimension reduction step by censorSIR.
Liquid Association
LA was originally introduced for studying coexpression patterns between three genes. Specifically, we assume that the correlation of two genes (X and Y) may vary, depending on the underlying cellular states. For example, X and Y may be positively correlated at state 1, and negatively correlated at state 2. The overall correlation coefficient could be around zero because the positive and negative correlations might cancel each other out. If the expression of another gene, denoted as Z, can reflect the change of cellular state, the correlation between X and Y can be detected by conditioning on Z. Suppose that when Z is lowly expressed, X and Y are positively correlated and when Z is highly expressed, X and Y are negatively correlated. In other words, the increase in the expression of Z is associated with the decrease of the correlation between X and Y . Then the pair (X, Y) is called a negative LA pair (LAP) of Z and a negative score is assigned. Similarly, if the increase in the expression of Z is associated with the increase of the correlation between X and Y, a positive LA score is assigned. Extreme LA scores, either positive or negative, are of interest. In this context of survival studies, we take the survival probability (after imputation for censored cases) as the third variable to find gene pairs whose coexpression pattern may vary as the survival probability changes. Biologically we expect these genes detected may be associated with molecular pathways related to survival. Therefore we wish to select genes with highest LA scores as candidates for constructing gene signatures to predict survival phenotype.
Based on [12], the LA of X and Y with respect to Z, which measures how the conditional expectation of XY given Z = z varies as z varies, is given by
LA(X, Y|Z) = Eg'(Z),
whereg(z) = E(XY|Z = z).
where N is the number of samples. Note that due to the normality assumption of Z, a normal quantile transformation should be performed before the LA computation [12]. In this study, we normalize both the survival probability and gene expression profiles by normal quantile transformation to ensure the robustness of our method.
Feature selection
After imputation of survival probability and normal quantile transformation of both survival probability and gene expression profiles, we can now calculate the correlation between survival probability Z and gene expression profile X, and the LA score LA(X, Y|Z), where Y is another gene expression profile. The gene pair (X, Y) is chosen from the whole genome, therefore we calculate LA scores for all the G^{2} gene pairs. Both LA and correlation calculation can be conducted in the LA website (http://kiefer.stat2.sinica.edu.tw/LAP3/index.php). The candidate genes can be selected from both the correlation and LA results. Because of the large number of comparisons in the LA results, the signals may be difficult to detect by examining each individual LA pair (LAP). One alternative strategy is to examine a subset of LAPs with the most extreme LA scores and extract the recurrent patterns of some genes. We refer those recurrent genes as LA hub genes. The effectiveness of this strategy is demonstrated in the Result section.
Dimension Reduction: SIR and Modified SIR for Censored Data (censorSIR)
We will briefly introduce SIR and censorSIR in this section and discuss the related issues of applying censorSIR to gene expression data. The theoretical derivation and implementation of censorSIR, as well as a simulation example are presented in our supplementary materials. Interested readers are referred to [17] and [10] for more details. An R package of censorSIR is available at http://www.bios.unc.edu/~wsun/software.htm.
where Y is the response variable, Σ_{ X }is the covariance matrix of X, and the random error ϵ follows an unknown distribution (we do not need any prior assumption about the distribution of ϵ in order to estimate β_{1},...,β_{ k }) and is independent of X. If f is known, then equation (1) is not much different from a simple neural network model or a nonlinear regression model. But what makes SIR special is that β_{1},...,β_{ k }can be estimated while even if f is unknown. The space spanned by ${{\beta}^{\prime}}_{1}X,\mathrm{...},{{\beta}^{\prime}}_{k}X$, which is a subspace spanned by all the columns in X, is called the effective dimension reduction (e.d.r.) space.
SIR can be implemented as follows. First the response variable Y is sliced into h intervals (SIR is insensitive to the choice of h as long as h > k). Then the inverse mean E(X|Y) is estimated by taking the average of all the values of X in each slice, where X is one predictor, i.e., one column of X. Finally the projection directions β_{1},...,β_{ k }can be identified as the eigenvectors in the eigenvalue decomposition of the between-slice covariance matrix Σ_{X|Y}= Cov[E(X|Y)] with respect to Σ_{ X }= Cov(X) (see [17] for the proof). The number of significant projections, denoted by k, can be determined by an asymptotic Chi-square test, which tests the hypotheses k = m versus k > m for m = 0,...,p - 1, where p is the number of covariates.
- 1.
C is independent of X and Y^{0};
- 2.
Conditional on X, C is independent of Y^{0}.
respectively. First, the uncensored observations and the censored observations will be sliced separately, namely the double slicing procedure. Then the joint e.d.r. space of the underlying survival time and censoring time will be obtained by taking the eigenvalue decomposition on the between-slice covariance matrix with respect to the covariance matrix of X. The leading eigenvectors will serve as the projection directions
After identifying the gene expression signatures (the projection directions), one can use scatter plots (2d or 3d) or non-parametric fittings (e.g., splines) to explore the possible forms of the function g in equation (2). Predictive model can also be build based on the reduced expression data. For example, survival models, e.g., Cox proportional hazard model, can be fitted with the projected directions as the explanatory variables first and then the fitted model can be used to predict the survival probability of a newly diagnosed patient. These gene expression signatures can also be used to classify the cancer patients into different treatment groups for better clinical outcomes.
Results and Discussion
Simulation
Clustering is an important feature in gene expression data. Genes involved in the same or related biological process are likely to coexpress, so that the expression profiles of these genes form a cluster. Thus besides simulating independent expression profiles, we also simulate gene expression clusters. Specifically, we simulate four clusters with the corresponding gene indexes as: (11–15), (16–20), (3, 31–34), (4, 41–44). In each cluster, each vector X_{ i }∈ R^{ p }of gene expression is simulated from a multivariate normal distribution whose marginal distributions are standard normal, and the five genes are correlated with each other with a common correlation coefficient ρ. The rest of the genes are uncorrelated and are simulated by standard normal distribution. The true survival time is generated by
Y^{0} = exp (W_{1}β_{1} + W_{2}β_{2} + X_{3}β_{3} + X_{4}β_{4}),
where β_{ k }= 0.5 for k = 1,...,4 andW_{1} = (0.15X_{11} + 0.3X_{12} + 0.45X_{13} + 0.6X_{14} + 0.75X_{15} + 0.5) · X_{1}W_{2} = (0.15X_{16} + 0.3X_{17} + 0.45X_{18} + 0.6X_{19} + 0.75X_{20} + 0.5) · X_{2},
The censoring time is generated by
C = exp (X_{6}γ_{1} + X_{7}γ_{2} + X_{8}γ_{3} + X_{10}γ_{5}),
where γ_{ l } = 0.1 for l = 1,...,5. The observed survival time is the minimum of Y^{0} and C. With n = 500, p = 10, 000, we generate 50 random samples for ρ = 0.4, and 0.8, representing the modest and high correlation cases respectively. The censoring rate is about 50% in both cases.
Average numbers of selected and true predictors found in simulation
ρ | n _{1} | K _{true,1} | n _{2} | K _{true,2} | K _{cluster,2} |
---|---|---|---|---|---|
0.8 | 3.56 | 2 | 10 | 2 | 7.22 |
0.4 | 2.54 | 2 | 10 | 2 | 2.6 |
Quantiles of the p-values (in log_{10} scale) of the log rank test for testing data
ρ | 0% | 25% | 50% | 75% | 100% |
---|---|---|---|---|---|
0.8 | -48.6007 | -41.2775 | -36.3042 | -28.5859 | -22.0066 |
0.4 | -52.4528 | -45.4521 | -38.5598 | -33.7795 | -22.1615 |
Coefficients in the censorSIR projection direction in simulation
ρ | X _{1} | X _{2} | X _{3} | X _{4} | Cluster | Other |
---|---|---|---|---|---|---|
0.8 | 0.2441 | 0.2299 | 0.6222 | 0.5825 | 0.1321 | 0.1133 |
0.4 | 0.2200 | 0.2331 | 0.5347 | 0.5240 | 0.0620 | 0.1591 |
Application in NKI breast cancer data
In this section, we present the results of analyzing the NKI breast cancer dataset [18] using our method. The data can be downloaded from http://microarray-pubs.stanford.edu/wound_NKI. Because this data set was initially generated by the Netherlands Cancer Institute (Nederlands Kanker Instituut in Dutch, NKI), it is referred to as NKI breast cancer data or NKI-295. Seventy-nine out of the 295 patients died before the study ended, which yields 79 real survival times and a heavy censoring rate of 73.2%. The expression levels of 24,481 genes were measured for each of the 295 patients.
Gene Signature Identification
Twenty-two genes selected by LA and correlations
Official Symbol | Official Name | Annotation |
---|---|---|
ABCG1 (3) | ATP-binding cassette, sub-family G (WHITE), member 1 | ATP binding; cholesterol homeostasis |
BIRC5 (-0.31) | baculoviral IAP repeat-containing 5 (survivin) | Colorectal cancer; apoptosis |
C5orf30 (3) | chromosome 5 open reading frame 30 | |
CENPA (-0.32) | centromere protein A | chromosome organization and biogenesis |
CTSL2 (-0.33) | cathepsin L2 | cathepsin L activity; proteolysis |
E2F7 (-0.31) | E2F transcription factor 7 | breast cancer cell growth [30]. |
ERBB2 (3) | v-erb-b2 erythroblastic leukemia viral oncogene homolog 2 | member of the epidermal growth factor (EGF) receptor family of receptor tyrosine kinases; Amplification and/or overexpression in numerous cancers, including breast and ovarian tumors |
FAM150B (3) | family with sequence similarity 150, member B | |
H06509 (3) | mRNA sequence | |
HJURP (-0.31) | Holliday junction recognition protein | up-regulated in lung cancer |
KIF20A (-0.30) | kinesin family member 20A | Collaboration of KIF20A and disc large homologue 5 is likely to be involved in pancre atic cancer [31] |
KIFC1 (-0.30) | kinesin family member C1 | mitotic sister chromatid segregation |
KRT6B (4) | keratin 6B | Cell Communication; ectoderm development |
LOC284072 (3) | hypothetical protein | |
ORMDL2 (4) | ORM1 (S. cerevisiae)-like 2 | expressed in normal aorta |
PDGFRA (4) | platelet-derived growth factor receptor, alpha polypeptide | Prostate cancer; cell proliferation |
PELI1 (3) | pellino homolog 1 (Drosophila) | role in interleukin-1-mediated signaling through interaction with interleukin-1 receptor-associated kinase 4-IRAK-tumor necrosis factor receptor-associated factor 6 complex [32] |
PERLD1 (5) | per1-like domain containing 1 | gastric cancer [33] |
PRR11 (-0.32) | proline rich 11 | interact with E2F1, E2F4 |
PTTG2 (-0.33) | pituitary tumor-transforming 2 | chromosome organization and biogenesis |
QSOX2 (-0.33) | quiescin Q6 sulfhydryl oxidase 2 | oxidoreductase activity; cell redox homeostasis |
TROAP (-0.32) | trophinin associated protein (tastin) | cell adhesion |
The estimated coefficient $\widehat{a}$ is -0.93 (p-value = 2.2e^{-16}), with 95% confidence interval of hazard rate [0.313, 0.496]. From the coefficients of the 22 genes in the e.d.r. direction, i.e., β_{1}, we can see that both genes recruited via correlation and liquid association can have large impacts on the projection direction (Figure 1). Both positive and negative coefficients are observed. Since the projection is positively correlated with survival time, genes with negative coefficients are possible "oncogenes", meaning higher expression is associated with higher risk. In contrast, genes with positive coefficients are possible "tumor repressor genes", meaning higher expression is associated with lower risk. We also use Cox's model to test the efficacy of using each single gene of the 22 genes to predict the survival probability. The most significant p-value that can be achieved by a single gene is 4e^{-11}, which is significant, but much less significant than the result using their linear combination.
Prediction and Cross-validation
The baseline hazard function h_{0}(t) can be estimated by the Breslow estimators after the coefficients of the Cox's model have been estimated [15, 19, 20]. Other methods, including the exact method [21], the discrete method [22] and the Efron's method [23], can handle observations that have tied survival times. Most statistical software provides the baseline cumulative hazard function, for example, the function "basehaz" in R.
To assess the prediction ability of our method, we carry out 100 cross-validations. In each cross-validation, we randomly divide the 295 patients into training and testing data of sizes 148 and 147, respectively. Only the training data are used to identify the gene signatures (from survival time imputation to the censorSIR), and then the identified gene signature is tested in the testing data. Based on the training data, genes appear at least three times in the top/bottom 50 LAPs are selected. The median number of genes selected via LA is six with 1st/3rd quantiles as four and eight, respectively. In addition, 10 genes that have the highest absolute correlation with the processed survival time are also selected. Five out of 100 cross-validations have no significant e.d.r. directions (p > 0.05). For the rest 95 cases, we conduct the log-rank test to test how well the most significant SIR directions differentiate the survival rates in both training and testing data. For the training data, the median, 1st, and 3rd quantile of log-rank p-values are 4.0e^{-10}, 3.2e^{-11}, and 1.0e^{-8}, respectively. For the testing data, the median, 1st, and 3rd quantile of log-rank p-values are 4.4e^{-3}, 7.2e^{-4}, and 1.4e^{-2}, respectively. Among the 95 cross-validations, 84 of them yield a testing p-value smaller than 0.05. We conclude that our method has a reasonably good prediction power even for such a small training sample with a high censoring rate.
Discussion
In this paper, we have introduced a two-step method for the joint analysis of survival time and gene expression data. The first step is the pre-selection of gene expression profiles. In order to offset the bias of the censored survival time, we employ a nonparametric method to impute the censored survival time. This method is simple to implement but we agree with an anonymous referee that future improvement would be desirable. Both correlation and LA are then used as the criteria to pre-select genes related with the imputed survival time. In the second step, with these selected genes, the modified SIR for censored data is conducted to further reduce the dimension of the gene expression data by identifying a few projection directions. Two major advantages of censorSIR over other methods are: 1) it employs the information of both survival time and gene expressions; 2) it does not require any function form for the relation between survival time and the projection directions. No explicit parametric assumptions are needed in the whole dimension reduction procedure.
A data set of 295 breast cancer patients was analyzed using the proposed method. A single projection direction (linear combination of 22 gene expression profiles) was identified that is significantly related to the survival time. Several studies have been done on this data set ([18, 24–26]). One gene signature of 70 genes is identified by maximizing the accuracy of classifying the patient with distant metastases within 5 years or not [24]. van de Vijver et al. [18] showed that this signature of 70 genes can also be used to predict the survival probabilities of the 295 patients. Chang et al. [25] employed 442 "core serum response" (CSR) genes for the survival probability prediction. Perou et al. [2, 27] have used clustering strategy to identify a set of 1410 "intrinsic genes", which can be used to classify patients into five subtypes. Each subtype has different levels of risk, therefore can be related with survival time [26]. Although these three gene sets (70-gene, CSR genes, intrinsic genes) only overlap slightly, they yield similar prediction strength, which leads to the conclusion that overlaps between gene expression signatures might not be a necessary measure of reproducibility [26]. Majority of the 22 genes we identified have not been included in these three gene sets. Nevertheless, the high accuracy of classifying patients in terms of their survival probabilities by only 22 genes demonstrates the efficacy of our method.
We have conducted simulation studies to demonstrate the effectiveness of our method. Our simulation setting intends to address the dependence issue between gene profiles. As the results have shown, it is possible that correlated genes would be selected as the surrogated genes for obtaining gene signatures. This is biologically meaningful because biologically correlated genes are likely to participate in the same pathways. The LA hub genes represent higher order interaction structures which would not be easy to detect by correlation method; see [13] for an illustration in multiple sclerosis candidate gene finding.
One subtle question is how to choose the cutoffs of correlation and liquid association in the pre-selection step. This would surely depend on the size of dataset. The permutation test of correlations/LA scores as developed before can be applied. On the other hand, because SIR requires the inverting of the covariance matrix. Although there are many on-going studies to sidestep this difficulties (for example [28, 29]), like most model or variable section procedures, their validity should be taken with caution because some user-specified tuning parameters are always hidden. Our preference is to preset a relatively smaller number of genes (so as to assure the stability of covariance estimation) to begin with. This can be coupled with the procedure of cross-validation to circumvent the overfitting problem to some extent. On the other hand, biological network affecting the survival phenotype is complex and is likely to involve multiple pathways. Thus practically it is unlikely for any method to claim the ability of identifying all genes involved in the survival network. An effective gene signature can still be obtained without having to find every single gene in generating the survival time.
Conclusion
In summary, we propose an effective dimension reduction and variable selection method to dissect the relationship between gene expression profiles and survival times. Compared with most available methods, our method has two major advantages. Firstly, not only the correlation but also the dynamic correlation between gene expression and survival time are explored by employing liquid association (LA) as part of the feature selection criteria; Secondly, no model assumption is required on the relationship between gene expression and survival time.
Acknowledgements
Authors would like to thank Dr. Charles Perou and Mr. Cheng Fan for helpful discussions. Research conducted by Li is supported in part by NSF grants DMS0201005, DMS0406091, and DMS-0707160. Li and Yuan were also supported in part by MIB, Institute of Statistical Science, Academia Sinica and grant NSC95-3114-P-002-005-Y and NSC97-2627-P-001-003. Research conducted by Sun is supported in part by NIEHS grant 5 P42 ES05948-15 and 5 P30 ES10126-07.
Notes
Declarations
Authors’ Affiliations
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