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  <cit> .
recently several studies have focused on dissecting the relation between survival time and gene expression  <cit> .
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  <cit> , 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.
<cit>  proposed to use the standard partial least square  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.
<cit>  proposed to select linear combinations of genes by a partial cox regression  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  model was introduced by sha et al.
<cit> , 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.
<cit>  used principal component analysis  for pre-selection.
in the pca step, linear combinations of genes  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.
<cit>  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 .
ma et al.
<cit>  proposed a lasso  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  method  <cit> .
in the second step, we apply the modified sir for censored data  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  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  need to be much smaller than the number of samples .
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   <cit>  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
both la and correlation computation cannot be applied directly to survival data due to the presence of censoring.
to temper the influence of censoring bias, we propose to apply the following nonparametric imputation method for correcting the right-censored survival time before la/correlation computation.
let ti, i ∈ { <dig> ...,n} be the survival time for patient i and δi be the censoring indicator such that δi =  <dig> indicates censoring and  <dig> indicates actual death.
the imputation procedure can be summarized as:
<dig>  calculate s^i the kaplan-meier estimate of the survival probability  <cit> .
specifically, s^i=∏tj≤tinj−djnj where nj is the number of individuals at-risk just prior to time tj, and dj is the number of deaths at time tj;
<dig>  impute the survival probability by the predicted conditional median
s˜i={s^iif δi=1s^i/2if δi=0;
<dig>  calculate the percentile pi=1−s˜i;
<dig>  perform the normal quantile transformation on pi.
the normal quantile transformation is necessary for the la calculation  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 zi, i =  <dig> ...,n and denote the rank as ri.
the normally transformed profile is then defined as Φ- <dig> ), where Φ is the cumulative standard normal distribution.
notice that, instead of imputing the survival time, we actually impute the survival probability.
a justification for the above imputation procedure is given next.
suppose that z is the true survival time and its density and survival functions are f and s ≡ p, respectively.
if the squared error loss is used, to predict/impute the true survival time z for a censored patient with censoring time ti, we can compute the conditional mean given z > ti
e=∫z>tizfdzs=ti+∫z>tisdzs.
a natural estimate of s 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  <cit> .
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.
if the absolute value of error is used, we can predict the true survival time z for a censored observation by the conditional median t˜i=median .
this means
p/p=1/ <dig>
which leads to
s=s/ <dig>
now we can estimate the survival probability of t˜i by s^/ <dig>  where s^ 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  may vary, depending on the underlying cellular states.
for example, x and y may be positively correlated at state  <dig>  and negatively correlated at state  <dig>  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  is called a negative la pair  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  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  <cit> , 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 = eg',
where
g = e.
if z follows standard normal distribution, the liquid association can be easily computed by stein's lemma  <cit> .
la=eg′=egz=eez=e≈∑i=1mxiyizin,
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  <cit> .
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, where y is another gene expression profile.
the gene pair  is chosen from the whole genome, therefore we calculate la scores for all the g <dig> gene pairs.
both la and correlation calculation can be conducted in the la website ().
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 .
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
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  <cit>  and  <cit>  for more details.
an r package of censorsir is available at .
the original sliced inverse regression   <cit>  is a dimension reduction method for regression problems.
it reduces the dimension of covariates  by identifying the projection directions β <dig> ...,βk in the following model
y=f,withβ′iΣxβi= <dig>
where y is the response variable, Σx is the covariance matrix of x, and the random error ϵ follows an unknown distribution  and is independent of x. if f is known, then equation  is not much different from a simple neural network model or a nonlinear regression model.
but what makes sir special is that β <dig> ...,βk can be estimated while even if f is unknown.
the space spanned by β′1x,...,β′kx, which is a subspace spanned by all the columns in x, is called the effective dimension reduction  space.
sir can be implemented as follows.
first the response variable y is sliced into h intervals .
then the inverse mean e 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 <dig> ...,βk can be identified as the eigenvectors in the eigenvalue decomposition of the between-slice covariance matrix Σx|y = cov with respect to Σx = cov .
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 =  <dig> ...,p -  <dig>  where p is the number of covariates.
the paper by li et al.
<cit>  extended the original sir to censored data.
denote y <dig> as the underlying true survival time, c as the censoring time, and t = min{y <dig>  c} as the observed survival time.
two censoring mechanisms are discussed:
<dig>  c is independent of x and y0;
<dig>  conditional on x, c is independent of y <dig>
in the first situation, the general theory of sir is applicable without modification  <cit> .
however, the second censoring mechanism, which is more common, will introduce bias to the sir estimations.
to address this question, li et al.
<cit>  introduced a two-step procedure.
the basic idea is to estimate a weight function ω for the patient with censoring time y and covariate x at time t with t > y. specifically, ω = s0/s <dig>  and s <dig> = p. the weight function can be estimated by any kernel method.
effective dimension reduction  space of life time can be identified given this weight function.
because kernel estimation is more efficient in low-dimension spaces, one initial dimension reduction step is required.
assume that the true underlying survival time y <dig> and the censoring time c have dimension reduction structures given by
y0=g
and
c=h,
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 , one can use scatter plots  or non-parametric fittings  to explore the possible forms of the function g in equation .
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
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: , , , .
in each cluster, each vector xi ∈ rp 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 <dig> = exp ,
where βk =  <dig>  for k =  <dig> ..., <dig> and
w <dig> =  · x <dig>
w <dig> =  · x <dig>
the censoring time is generated by
c = exp ,
where γl =  <dig>  for l =  <dig> ..., <dig>  the observed survival time is the minimum of y <dig> and c. with n =  <dig>  p =  <dig>   <dig>  we generate  <dig> random samples for ρ =  <dig> , and  <dig> , representing the modest and high correlation cases respectively.
the censoring rate is about 50% in both cases.
in this setup, the two genes x <dig> and x <dig> act as the la hub genes.
we analyze the simulated data in two steps.
in step one, survival probabilities are imputed and normalized by normal quantile transformation and candidate genes are selected based on the la and the correlation scores.
genes that appear at least three times in top  <dig> positive and negative la pairs are selected.
the first  <dig> genes with the highest correlations with survival probability in absolute values are also selected.
table  <dig> reports the average number of genes selected from la  and correlation , the average number of correct genes ktrue, <dig> and ktrue, <dig> selected, and the average number of predictors kcluster, <dig> found in the clusters  <dig> –  <dig> and  <dig> –  <dig>  we took ρ =  <dig>  and  <dig>  to represent modest and high co-regulation within a cluster of genes.
the la hub genes x <dig> and x <dig> can be found by la every time.
correlation also works well since the genes in the same clusters of x <dig> and x <dig> were also picked up.
this table reports the average numbers of genes selected from la  and correlation , the average numbers of correct genes selected from la  and correlation , and the average numbers of predictors kcluster, <dig> found in the clusters  <dig> –  <dig> and  <dig> –  <dig> for ρ =  <dig>  and  <dig> .
the sample size is  = .
in step two, the dimension reduction is performed on the selected genes, followed by using the first sir component as the gene signature score for survival prediction.
to test the effectiveness of gene signature, we want to test if the gene signature can separate the patients into high and low risk groups.
we define the cutoff as the median of the gene signature, i.e.
x˜=x′b,
where x is the matrix of genes selected from step one, and b is the first sir direction.
a testing data set of  <dig>  subjects are generated similarly as the training data, we divide the  <dig>  subjects into higher and lower groups based on whether the score x˜ is higher or lower than the cutoff and apply the log rank test.
the simulation result shows that the p-value of the log rank test is smaller than 10- <dig> in every simulation run.
table  <dig> reports the quantiles of the p-values .
this table reports the average absolute values of the coefficients for genes 1– <dig>  genes in the clusters  and , and noise genes in censorsir directions.
application in nki breast cancer data
in this section, we present the results of analyzing the nki breast cancer dataset  <cit>  using our method.
the data can be downloaded from .
because this data set was initially generated by the netherlands cancer institute , it is referred to as nki breast cancer data or nki- <dig>  seventy-nine out of the  <dig> patients died before the study ended, which yields  <dig> real survival times and a heavy censoring rate of  <dig> %.
the expression levels of  <dig>  genes were measured for each of the  <dig> patients.
gene signature identification
following the steps described in the method section, we first impute survival probabilities and normalize the imputed survival probabilities and gene expression profiles by normal quantile transformation.
assigning the processed survival time as z, we apply la with all the  <dig>  genes serving as x and y to calculate la score la.
we select  <dig> genes that appear at least  <dig> times in the top  <dig>  or bottom  <dig>  la pairs, as well as  <dig> genes of which the absolute value of the correlation coefficient with the processed survival time is bigger than  <dig>  .
these cutoffs are determined based on the results of  <dig> permutations.
in each permutation, we permute the survival time and censoring time of the  <dig> patients simultaneously, and then calculate the la scores and correlation scores as for un-permuted data.
we use the number of reoccurrences of one gene in top  <dig> or bottom  <dig> la pairs to evaluate the importance of that gene.
a threshold of three reoccurrences is used to select la hub genes.
our simulation tells how many la hub genes can occur by chance alone.
it turns out that among the  <dig> permutations, on the average there are only four la hub genes present, as compared to the  <dig> la hub genes detected in the real data.
to evaluate the significance of correlation score, we record the highest correlation  between the processed survival time and all the  <dig>  gene expression profiles.
in only  <dig> out of the  <dig> permutations, we observe a correlation score greater than  <dig>  .
the permutation results for both la and correlation scores indicate that at least the majority of the genes selected are significantly related with survival time after the genome-wise multiple testing correction.
it is possible to select more genes based on the correlation criterion, but we prefer a parsimonious model to serve the purpose of prediction and avoid overfitting.
the numbers in the parenthesis are either the number of times the gene appear in top/bottom  <dig> la pairs or the correlation coefficient.
only genes with negative correlations are selected because positive correlations have smaller absolute value.
the highest positive correlation is  <dig> , which is ranked as 31st by absolute values.
the next step is to perform censorsir on the  <dig> selected genes.
first, the double-slicing procedure is used to find the joint e.d.r.
space of the true survival time y <dig> and the censoring time c .
the χ <dig> test shows that the first projection direction is significant , and the second direction is mildly significant , while the other directions are not significant .
therefore, we take the first two eigenvectors as the projection directions in the joint e.d.r.
space and continue to recover the e.d.r.
space of true survival time y <dig>  now with only the first eigenvalue being significant , we have reduced the survival time space to one dimension  .
in the following discussion, we only consider the projection of gene expression data in the first projection direction: x˜=β′1x
we fit a cox's model with the covariate x˜ such that
h=h0exp⁡
the estimated coefficient a^ is - <dig>  , with 95% confidence interval of hazard rate .
from the coefficients of the  <dig> genes in the e.d.r.
direction, i.e., β <dig>  we can see that both genes recruited via correlation and liquid association can have large impacts on the projection direction .
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  <dig> genes to predict the survival probability.
the most significant p-value that can be achieved by a single gene is 4e- <dig>  which is significant, but much less significant than the result using their linear combination.
prediction and cross-validation
to predict the survival probability of a newly diagnosed breast cancer patient, we will only need to check the expression levels of the  <dig> genes and then compute x˜.
the survival probability of this patient can be easily predicted using the formula
s=exp⁡e− <dig> x˜du).
the baseline hazard function h <dig> can be estimated by the breslow estimators after the coefficients of the cox's model have been estimated  <cit> .
other methods, including the exact method  <cit> , the discrete method  <cit>  and the efron's method  <cit> , can handle observations that have tied survival times.
most statistical software provides the baseline cumulative hazard function, for example, the function "basehaz" in r.
we divide the patients into two groups of approximately equal sizes  based on the gene expression level .
the log-rank test shows that the survival rates of these two groups are significantly different .
here a note of caution in interpreting the face value of this significance level is important.
as suggested by one referee, it is instructive to construct a null situation where the survival data and the gene expression data are completely unlinked.
we took this suggestion by first randomly permuting the gene expression data and then carrying out the same steps of our gene signature procedure for correlating the permuted gene expression data with the un-permuted survival data.
it turns out that after the steps of candidate gene selection, the sir analysis does not detect any significant direction  .
this is the correct conclusion and if this was the real life situation, we should stop here.
we would not recommend using the first sir direction for constructing gene signature and splitting the patients into high and low risk groups.
however, if one ignored the sir significance test and went on to split the patients, then a statistically significant separation was observed with a significant p-value of p = 6e- <dig> .
this artifact is largely due to the smallness of the sample size which leads to the chance of overfitting in the permuted data, a phenomenon similar to the one commonly faced in multiple testing without adjustment.
this result speaks for the needs of conducting the sir significance test.
in the real data we did find that the first sir direction has a significant p-value  and the p-value for the log rank test in comparing high and low risk groups is also much lower .
to assess the prediction ability of our method, we carry out  <dig> cross-validations.
in each cross-validation, we randomly divide the  <dig> patients into training and testing data of sizes  <dig> and  <dig>  respectively.
only the training data are used to identify the gene signatures , 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  <dig> laps are selected.
the median number of genes selected via la is six with 1st/3rd quantiles as four and eight, respectively.
in addition,  <dig> genes that have the highest absolute correlation with the processed survival time are also selected.
five out of  <dig> cross-validations have no significant e.d.r.
directions .
for the rest  <dig> 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  <dig> e- <dig>   <dig> e- <dig>  and  <dig> e- <dig>  respectively.
for the testing data, the median, 1st, and 3rd quantile of log-rank p-values are  <dig> e- <dig>   <dig> e- <dig>  and  <dig> e- <dig>  respectively.
among the  <dig> cross-validations,  <dig> of them yield a testing p-value smaller than  <dig> .
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  <dig> breast cancer patients was analyzed using the proposed method.
a single projection direction  was identified that is significantly related to the survival time.
several studies have been done on this data set .
one gene signature of  <dig> genes is identified by maximizing the accuracy of classifying the patient with distant metastases within  <dig> years or not  <cit> .
van de vijver et al.
<cit>  showed that this signature of  <dig> genes can also be used to predict the survival probabilities of the  <dig> patients.
chang et al.
<cit>  employed  <dig> "core serum response"  genes for the survival probability prediction.
perou et al.
<cit>  have used clustering strategy to identify a set of  <dig> "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  <cit> .
although these three gene sets  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  <cit> .
majority of the  <dig> 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  <dig> 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  <cit>  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 , 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  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.
CONCLUSIONS
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  as part of the feature selection criteria; secondly, no model assumption is required on the relationship between gene expression and survival time.
authors' contributions
wu and sun shared equally as the co-first author, with ordering by randomization.
design of the study, problem conceptualization, and envision of statistical methods were contributed by li, wu and sun.
implementation of methods were contributed by sun and wu.
liquid association computing is facilitated by an on-line computing system designed by yuan and li.
development of censor-sir methodology and programming were contributed by chen and li.
writing was contributed by wu, sun and li.

 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. recently several studies have focused on dissecting the relation between survival time and gene expression. 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, 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. proposed to use the standard partial least square 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. proposed to select linear combinations of genes by a partial cox regression 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 model was introduced by sha et al., 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. used principal component analysis for pre-selection. in the pca step, linear combinations of genes 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. 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. ma et al. proposed a lasso 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 method. in the second step, we apply the modified sir for censored data 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 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 need to be much smaller than the number of samples. 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 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 both la and correlation computation cannot be applied directly to survival data due to the presence of censoring. to temper the influence of censoring bias, we propose to apply the following nonparametric imputation method for correcting the right-censored survival time before la/correlation computation. let ti, i ∈ {...,n} be the survival time for patient i and δi be the censoring indicator such that δi = indicates censoring and indicates actual death. the imputation procedure can be summarized as: calculate s^i the kaplan-meier estimate of the survival probability. specifically, s^i=∏tj≤tinj−djnj where nj is the number of individuals at-risk just prior to time tj, and dj is the number of deaths at time tj; impute the survival probability by the predicted conditional median s˜i={s^iif δi=1s^i/2if δi=0; calculate the percentile pi=1−s˜i; perform the normal quantile transformation on pi. the normal quantile transformation is necessary for the la calculation 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 zi, i =...,n and denote the rank as ri. the normally transformed profile is then defined as Φ- ), where Φ is the cumulative standard normal distribution. notice that, instead of imputing the survival time, we actually impute the survival probability. a justification for the above imputation procedure is given next. suppose that z is the true survival time and its density and survival functions are f and s ≡ p, respectively. if the squared error loss is used, to predict/impute the true survival time z for a censored patient with censoring time ti, we can compute the conditional mean given z > ti e=∫z>tizfdzs=ti+∫z>tisdzs. a natural estimate of s 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. 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. if the absolute value of error is used, we can predict the true survival time z for a censored observation by the conditional median t˜i=median. this means p/p=1/ which leads to s=s/ now we can estimate the survival probability of t˜i by s^/ where s^ 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 may vary, depending on the underlying cellular states. for example, x and y may be positively correlated at state and negatively correlated at state 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 is called a negative la pair 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 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, 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 = eg', where g = e. if z follows standard normal distribution, the liquid association can be easily computed by stein's lemma. la=eg′=egz=eez=e≈∑i=1mxiyizin, 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. 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, where y is another gene expression profile. the gene pair is chosen from the whole genome, therefore we calculate la scores for all the g gene pairs. both la and correlation calculation can be conducted in the la website (). 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. 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 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 and for more details. an r package of censorsir is available at. the original sliced inverse regression is a dimension reduction method for regression problems. it reduces the dimension of covariates by identifying the projection directions β...,βk in the following model y=f,withβ′iΣxβi= where y is the response variable, Σx is the covariance matrix of x, and the random error ϵ follows an unknown distribution and is independent of x. if f is known, then equation is not much different from a simple neural network model or a nonlinear regression model. but what makes sir special is that β...,βk can be estimated while even if f is unknown. the space spanned by β′1x,...,β′kx, which is a subspace spanned by all the columns in x, is called the effective dimension reduction space. sir can be implemented as follows. first the response variable y is sliced into h intervals. then the inverse mean e 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...,βk can be identified as the eigenvectors in the eigenvalue decomposition of the between-slice covariance matrix Σx|y = cov with respect to Σx = cov. 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 =...,p - where p is the number of covariates. the paper by li et al. extended the original sir to censored data. denote y as the underlying true survival time, c as the censoring time, and t = min{y c} as the observed survival time. two censoring mechanisms are discussed: c is independent of x and y0; conditional on x, c is independent of y in the first situation, the general theory of sir is applicable without modification. however, the second censoring mechanism, which is more common, will introduce bias to the sir estimations. to address this question, li et al. introduced a two-step procedure. the basic idea is to estimate a weight function ω for the patient with censoring time y and covariate x at time t with t > y. specifically, ω = s0/s and s = p. the weight function can be estimated by any kernel method. effective dimension reduction space of life time can be identified given this weight function. because kernel estimation is more efficient in low-dimension spaces, one initial dimension reduction step is required. assume that the true underlying survival time y and the censoring time c have dimension reduction structures given by y0=g and c=h, 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, one can use scatter plots or non-parametric fittings to explore the possible forms of the function g in equation. 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.  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:,,,. in each cluster, each vector xi ∈ rp 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 = exp, where βk = for k =..., and w = · x w = · x the censoring time is generated by c = exp, where γl = for l =..., the observed survival time is the minimum of y and c. with n = p = we generate random samples for ρ =, and, representing the modest and high correlation cases respectively. the censoring rate is about 50% in both cases. in this setup, the two genes x and x act as the la hub genes. we analyze the simulated data in two steps. in step one, survival probabilities are imputed and normalized by normal quantile transformation and candidate genes are selected based on the la and the correlation scores. genes that appear at least three times in top positive and negative la pairs are selected. the first genes with the highest correlations with survival probability in absolute values are also selected. table reports the average number of genes selected from la and correlation, the average number of correct genes ktrue, and ktrue, selected, and the average number of predictors kcluster, found in the clusters – and – we took ρ = and to represent modest and high co-regulation within a cluster of genes. the la hub genes x and x can be found by la every time. correlation also works well since the genes in the same clusters of x and x were also picked up. this table reports the average numbers of genes selected from la and correlation, the average numbers of correct genes selected from la and correlation, and the average numbers of predictors kcluster, found in the clusters – and – for ρ = and. the sample size is =. in step two, the dimension reduction is performed on the selected genes, followed by using the first sir component as the gene signature score for survival prediction. to test the effectiveness of gene signature, we want to test if the gene signature can separate the patients into high and low risk groups. we define the cutoff as the median of the gene signature, i.e. x˜=x′b, where x is the matrix of genes selected from step one, and b is the first sir direction. a testing data set of subjects are generated similarly as the training data, we divide the subjects into higher and lower groups based on whether the score x˜ is higher or lower than the cutoff and apply the log rank test. the simulation result shows that the p-value of the log rank test is smaller than 10- in every simulation run. table reports the quantiles of the p-values. this table reports the average absolute values of the coefficients for genes 1– genes in the clusters and, and noise genes in censorsir directions. application in nki breast cancer data in this section, we present the results of analyzing the nki breast cancer dataset using our method. the data can be downloaded from. because this data set was initially generated by the netherlands cancer institute, it is referred to as nki breast cancer data or nki- seventy-nine out of the patients died before the study ended, which yields real survival times and a heavy censoring rate of %. the expression levels of genes were measured for each of the patients. gene signature identification following the steps described in the method section, we first impute survival probabilities and normalize the imputed survival probabilities and gene expression profiles by normal quantile transformation. assigning the processed survival time as z, we apply la with all the genes serving as x and y to calculate la score la. we select genes that appear at least times in the top or bottom la pairs, as well as genes of which the absolute value of the correlation coefficient with the processed survival time is bigger than. these cutoffs are determined based on the results of permutations. in each permutation, we permute the survival time and censoring time of the patients simultaneously, and then calculate the la scores and correlation scores as for un-permuted data. we use the number of reoccurrences of one gene in top or bottom la pairs to evaluate the importance of that gene. a threshold of three reoccurrences is used to select la hub genes. our simulation tells how many la hub genes can occur by chance alone. it turns out that among the permutations, on the average there are only four la hub genes present, as compared to the la hub genes detected in the real data. to evaluate the significance of correlation score, we record the highest correlation between the processed survival time and all the gene expression profiles. in only out of the permutations, we observe a correlation score greater than. the permutation results for both la and correlation scores indicate that at least the majority of the genes selected are significantly related with survival time after the genome-wise multiple testing correction. it is possible to select more genes based on the correlation criterion, but we prefer a parsimonious model to serve the purpose of prediction and avoid overfitting. the numbers in the parenthesis are either the number of times the gene appear in top/bottom la pairs or the correlation coefficient. only genes with negative correlations are selected because positive correlations have smaller absolute value. the highest positive correlation is, which is ranked as 31st by absolute values. the next step is to perform censorsir on the selected genes. first, the double-slicing procedure is used to find the joint e.d.r. space of the true survival time y and the censoring time c. the χ test shows that the first projection direction is significant, and the second direction is mildly significant, while the other directions are not significant. therefore, we take the first two eigenvectors as the projection directions in the joint e.d.r. space and continue to recover the e.d.r. space of true survival time y now with only the first eigenvalue being significant, we have reduced the survival time space to one dimension. in the following discussion, we only consider the projection of gene expression data in the first projection direction: x˜=β′1x we fit a cox's model with the covariate x˜ such that h=h0exp⁡ the estimated coefficient a^ is -, with 95% confidence interval of hazard rate. from the coefficients of the genes in the e.d.r. direction, i.e., β we can see that both genes recruited via correlation and liquid association can have large impacts on the projection direction. 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 genes to predict the survival probability. the most significant p-value that can be achieved by a single gene is 4e- which is significant, but much less significant than the result using their linear combination. prediction and cross-validation to predict the survival probability of a newly diagnosed breast cancer patient, we will only need to check the expression levels of the genes and then compute x˜. the survival probability of this patient can be easily predicted using the formula s=exp⁡e− x˜du). the baseline hazard function h can be estimated by the breslow estimators after the coefficients of the cox's model have been estimated. other methods, including the exact method, the discrete method and the efron's method, can handle observations that have tied survival times. most statistical software provides the baseline cumulative hazard function, for example, the function "basehaz" in r. we divide the patients into two groups of approximately equal sizes based on the gene expression level. the log-rank test shows that the survival rates of these two groups are significantly different. here a note of caution in interpreting the face value of this significance level is important. as suggested by one referee, it is instructive to construct a null situation where the survival data and the gene expression data are completely unlinked. we took this suggestion by first randomly permuting the gene expression data and then carrying out the same steps of our gene signature procedure for correlating the permuted gene expression data with the un-permuted survival data. it turns out that after the steps of candidate gene selection, the sir analysis does not detect any significant direction. this is the correct conclusion and if this was the real life situation, we should stop here. we would not recommend using the first sir direction for constructing gene signature and splitting the patients into high and low risk groups. however, if one ignored the sir significance test and went on to split the patients, then a statistically significant separation was observed with a significant p-value of p = 6e-. this artifact is largely due to the smallness of the sample size which leads to the chance of overfitting in the permuted data, a phenomenon similar to the one commonly faced in multiple testing without adjustment. this result speaks for the needs of conducting the sir significance test. in the real data we did find that the first sir direction has a significant p-value and the p-value for the log rank test in comparing high and low risk groups is also much lower. to assess the prediction ability of our method, we carry out cross-validations. in each cross-validation, we randomly divide the patients into training and testing data of sizes and respectively. only the training data are used to identify the gene signatures, 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 laps are selected. the median number of genes selected via la is six with 1st/3rd quantiles as four and eight, respectively. in addition, genes that have the highest absolute correlation with the processed survival time are also selected. five out of cross-validations have no significant e.d.r. directions. for the rest 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 e- e- and e- respectively. for the testing data, the median, 1st, and 3rd quantile of log-rank p-values are e- e- and e- respectively. among the cross-validations, of them yield a testing p-value smaller than. 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 breast cancer patients was analyzed using the proposed method. a single projection direction was identified that is significantly related to the survival time. several studies have been done on this data set. one gene signature of genes is identified by maximizing the accuracy of classifying the patient with distant metastases within years or not. van de vijver et al. showed that this signature of genes can also be used to predict the survival probabilities of the patients. chang et al. employed "core serum response" genes for the survival probability prediction. perou et al. have used clustering strategy to identify a set of "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. although these three gene sets 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. majority of the 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 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 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, 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 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.  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 as part of the feature selection criteria; secondly, no model assumption is required on the relationship between gene expression and survival time. authors' contributions wu and sun shared equally as the co-first author, with ordering by randomization. design of the study, problem conceptualization, and envision of statistical methods were contributed by li, wu and sun. implementation of methods were contributed by sun and wu. liquid association computing is facilitated by an on-line computing system designed by yuan and li. development of censor-sir methodology and programming were contributed by chen and li. writing was contributed by wu, sun and li. 
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. 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. we propose a method for an integrated study of survival time and gene expression. 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. for the real data application, we employed a set of breast cancer patients and found a linear combination of gene expression profiles that are significantly correlated with patients' survival rate. the new method is tested via both simulated and real data.
 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. 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.  the new method is tested via both simulated and real data. for the real data application, we employed a set of breast cancer patients and found a linear combination of gene expression profiles that are significantly correlated with patients' survival rate.  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. 