BACKGROUND
a genome-wide association study  involves an examination of the entire genome, typically single-nucleotide polymorphisms , of different individuals to determine whether any variant is associated with a particular clinical outcome.
many researchers have considered the design and analysis of gwass with respect to binary clinical outcomes such as case/control or response/non-response ones  <cit> .
in clinical cancer research, the primary endpoint of interest is usually a time-to-event trait subject to censoring.
in calgb  <dig>  for example, germline dnas are collected, together with time to progression and overall survival data, from  <dig> advanced pancreatic cancer patients.
one objective of an snp correlative study is to discover snp markers that are correlated with such time-to-event endpoints.
one of the first objectives of a statistical analysis in a gwas is the discovery of snp markers that are associated with a particular trait.
the major statistical issue in marker discovery is multiple testing to avoid enlarged type i error probability due to the large number of univariate tests  <cit> .
each prognostic snp has two or three possible outcomes depending on its genetic model, and the efficiency of a statistical method in associating it with a trait is maximized when the true genetic model is known.
for most snps, however, the true genetic model is unknown.
to identify the true genetic model of each snp and optimize the association analysis, many researchers have considered some candidate genetic models for a given trait and derived a null distribution of the maximum of test statistics specific to individual genetic models  <cit> .
this test is referred to as the maxtest.
these methods have been developed for binary traits such as case/control or response/non-response ones.
we develop a maxtest to identify the genetic model of each snp when the trait is a survival endpoint, e.g., the time to tumor progression or death.
another major objective of a gwas is to predict a trait of interest by using snps.
prediction methods using microarray data have been widely investigated  <cit> , but cannot be directly applied to snp-based predictions.
the number of snp markers in genome-wide snp data far exceeds that of gene markers  in microarray data, e.g., 1m vs. 20k.
in addition, although gene expression data in microarray studies are continuous, genome-wide snp data are discrete, taking only three different values at most and showing different values depending on the genetic model.
this paper presents a method for predicting a survival outcome that uses genome-wide snp data but can be easily modified for any type of trait, including binary or continuous outcomes.
the proposed method uses the gradient lasso method  <cit> , which has been developed for microarray data.
some investigators fit a prediction model while ignoring the genetic model of each snp  <cit> .
we also propose a maxtest associating between a time-to-event trait and a snp accounting for its possible genetic model and identifies the genetic model of each candidate prognostic snp by using the proposed maxtest before fitting a prediction model.
the simulation results show that this procedure improves prediction efficiency and prognostic power.
for computational efficiency, nonsignificant snps are excluded using the maxtest before starting the gradient lasso.
for the facilitation of the proposed maxtest and prediction method, glcoxphsnp r packages  are provided.
methods
genetic models of snps
suppose that the genotype for an snp is encoded as aa, ab, or bb.
let g denote the number of copies of the b allele.
that is, g= <dig>   <dig> or  <dig> if the genotype is aa, ab, or bb, respectively.
let λg denote the hazard function of genotype g. without loss of generality, assume that b is the risk allele in the sense that having b increases the risk of an event.
more specifically, assume that λ0≤λ1≤λ <dig> for all t≥ <dig>   we now consider the following three popular genetic models:
•recessive model: λ0=λ1<λ <dig>
•dominant model: λ0<λ1=λ <dig>
•multiplicative model: λ2/λ1=λ1/λ <dig>  or equivalently λ1=γλ <dig> and λ2=γ2λ <dig> for γ> <dig>
for a chosen score cg, we consider a proportional hazard model , λg=λ <dig> exp.
then cox’s partial maximum likelihood test has optimal power with = for a recessive model,  for a dominant model, and  for a multiplicative model  <cit> .
note that the phm is invariant to the linear transformation of the covariate .
maxtest
suppose that we want to test whether an snp is associated with a given clinical outcome.
the test statistic is dependent on the true genetic model of the snp.
at the time of testing, however, we usually have no knowledge of the true genetic model.
in this case, a naive approach is to conduct all statistical tests by assuming different genetic models and choose the lowest p-value as the measurement of the association.
this approach can lead to an enlarged type i error because of multiple tests.
to adjust for multiple tests, investigators have proposed a method considering the maximum of test statistics with respect to all candidate genetic models under consideration, namely the maxtest.
many studies have considered the maxtest for binary clinical outcomes.
zheng et al.
<cit>  propose a robust ranking method when the underlying genetic model is unknown, namely the max-rank test.
conneely and boehnke  <cit>  propose a method for computing p-values that adjusts for correlated tests and show that the method can improve the accuracy of permutation tests with greater computational efficiency.
li et al.
<cit>  propose a method for approximating the p-value for the maxtest with or without covariates adjusted for, namely the p-rank test.
li et al.
<cit>  compare the results of the max-rank and p-rank tests.
hoggart et al.
<cit>  formulate the problem as variable selection in a logistic regression analysis including a covariate for each snp and find the posterior mode for shrinkage priors based on a stochastic search on a penalized likelihood function.
we propose a maxtest for survival endpoints.
here we assign numeric scores to three genotypes based on their genetic model: = for a recessive model, = for a dominant model, and = for a multiplicative model.
for a given score cg assigned to the genotype g  of an snp, we consider the cox propotional hazard model,
λg=λ0exp.
for patient i , let ti and ci denote the survival and censoring times, respectively.
we observe that xi= min and δi=i, where i indicates the indicator function.
in addition, for t≥ <dig>  let yi=i and ni=δii denote the at-risk and event processes, respectively.
for a given score, we set zi=cg if patient i has genotype g. let sk=∑i=1nzikyi, k= <dig> , <dig>  then, the partial score test statistic by cox  <cit> ,
w=n−1/2∑i=1n∫0∞{zi−s1/s0}dni
is asymptotically normal with mean  <dig> and variance that can be consistently estimated by
σ^2=n−1∑i=1n∫0∞s2s0−s12s02dni
under h0:λ0=λ1=λ <dig> .
by combining the statistics with respect to the three candidate genetic models, we can derive a maxtest statistic.
let wl and σ^l <dig> denote the test statistic and the variance estimator with respect to genetic model l .
then we can define the proposed maxtest as q= max, where tl=wl/σ^l.
under h <dig>  σll′=cov is consistently estimated by
σ^ll′=n−1∑i=1n∫0∞zli−sl1sl0zl′i−sl′1sl′0yiydn
where zli and slk denote zi and sk, respectively, for genetic model l; y=∑i=1nyi, n=∑i=1nni.
let Σ^=1≤l,l′≤ <dig>  where ρ^ll′=σ^ll′/σ^lσ^l′.
then we can obtain the critical value of q by a numerical method or a simulation method from the n distribution.
this is a survival trait counterpart for the maxtest with a binary trait, as discussed in several studies  <cit> .
we can construct an alternative test based on the quadratic form w2=stΣ^−1s, where s=t.
in addition to recessive, dominant, and multiplicative genetic models, we can consider other models to develop a test statistic to measure the relationship between an snp and a survival trait.
for example, we may consider the long-rank test based on the one-way anova in  <cit>  or the test based on the wilcoxon rank-sum test in  <cit> , which require no specific genetic model assumptions.
in particular, the anova-type test is a reasonable option if the monotone trend in genotypes g =  <dig>   <dig>  and  <dig> is doubtful.
cox model with a lasso penalty
in an analysis using snp data, we may face a problem in which the number of snps exceeds the size of data, that is, m≫p, which frequently occurs even when a smaller number of snps are selected through a prescreening step.
this may lead to a serious collinearity problem when directly applying the partial likelihood estimation to the cox model.
to address this problem, tibshirani <cit>  estimates the parameters of the cox model under the l <dig> constraint as follows:
β^=argmaxβ,sl,subject to∑j=1m|βj|≤s,
where l is the partial likelihood function  <cit> .
the above optimization problem is suitable for reducing the dimension of covariates but is computationally difficult because the l <dig> objective function is not differentiable.
to address this computational problem, previous studies have proposed many algorithms  <cit> .
tibshirani <cit>  proposes an algorithm using quadratic programming within an iterative procedure.
gui and li <cit>  propose an las-cox procedure applying the choleski decomposition and the lars procedure.
however, these algorithms can be computationally burdensome and sometimes fail to converge to an optimum because they involve quadratic programming and/or matrix inversions.
sohn et al.
<cit>  propose glcoxph for the cox model by using the gradient lasso algorithm in  <cit> .
this glcoxph employs a coordinate-wise gradient decent with a deletion step and requires only univariate optimization in each iteration.
its convergence speed is almost independent of the number of input variables, and it does not require a matrix inversion, which makes it scalable to high-dimensional data and allows it to converge to a global optimum faster.
glmpath  <cit>  provides the entire penalization path for the cox model in a forward stagewise manner.
because it requires matrix inversions only for active sets, it is faster and more stable than other methods.
sohn et al.
<cit>  provided a comparative analysis using real and simulated data and show that the gradient lasso algorithm outperforms glmpath in analyzing high-dimensional survival data in terms of the sparsity, predictability, and computational efficiency of the final prediction model.
therefore, the following gradient lasso algorithm can be a useful alternative for fitting the cox model to predict the survival time of patients based on high-dimensional snp data:
<dig>  initialize: β= <dig> and k= <dig>
<dig>  do until convergence
addition:
compute the gradient ∇l.
find the γj maximizing |∂l/∂βj| for j= <dig> …,p and γ^=−s×sign/∂βĵ).
let v be a p-dimensional vector such that its γj-th element γ^ and other elements are zeros.
find α^=argminα∈ <cit> lβ+αv.
update β=β+α^v.
deletion:
calculate hσ=−∇l+θσ∇ltθσ/|σ|, where σ={j:βj≠0}.
find δ^=argminδ∈l, where h=hσ <dig> and u= mink∈σ{−βk/hk:βkhk<0}.
update β=β+δ^h.
set m=m+ <dig>
<dig>  returnβ.
proposed algorithm for predicting a survival trait
we propose a new algorithm for fitting a cox regression model using snp data.
the proposed algorithm consists of the following four steps: we  select significant snps by the maxtest, as in section “example using real data”,  convert these snps into corresponding scores by genetic models identified by the maxtest,  standardize these scores, and  apply the gradient lasso algorithm  <cit>  to selected snps.
we summarize the algorithm in greater detail as follows:
<dig>  read in the clinical data {,i= <dig> ...,n} and snp data {,i= <dig> ...,n}, where sij denotes the number of b alleles for snp j .
<dig>  for snp j , calculate the variance and covarince matrix Σ^j, and generate the null distribution of the maxtest as follows.
for b= <dig> ...,b , generate ,t2j,t3j) from n.
let qj=max|,|t2j|,|t3j|) for b= <dig> ...,b.
<dig>  for snp j ,
using original data, calculate the test statistics , the maxtest statistic qj= max, and two-sided p-values p1j,p2j,p3j from the marginal test for respective genetic models.
approximate the p-value of the maxtest by
pj=b−1∑b=1bi≥qj)
<dig>  snp screening: select j  snps with pj<α for specified a α .
<dig>  for selected snps j , identify the genetic model  by the lowest marginal p-value from p1j,p2j,p3j or the largest test statistic from t1j,t2j,t3j.
<dig>  for patient i , define covariates  by the identified genetic model and the corresponding score.
<dig>  standardize the covariates:
zij′=zij−z¯jsj,
where z¯j=n−1∑i=1nzij and sj2=n−1∑i=1n <dig>
<dig>  apply the gradient lasso to the cox regression model with response data {,i= <dig> ...,n) and standardized covariates {,i= <dig> ...,n}.
RESULTS
simulation study
we provide a simulation study.
the data generation scheme is as follows: we generate snp data z <dig> ...,zm from n random variables with an ar correlation structure with the autocorrelation coefficient ρ, x <dig> ...,xm.
due to linkage disequilibrium, snps which lay in close vicinity within chromosomes tend to have a stronger association.
in this sense, an ar correlation structure is a reasonable correlation structure for the continuous random variables generating snp data.
let x1=ε <dig> and xj=ρxj−1+1−ρ2εj for j= <dig> ...,m, where ε <dig> ...,εm∼iidn random numbers.
the cutoff values for xj for generating zj determine allele frequency.
for each snp, let f <dig> f <dig> f <dig>  denote the frequency of aa, ab, and bb genotypes, respectively, where b denotes the risk allele.
note that marginally xj∼n.
the true model for the survival times is given as
Λ=Λ0exp∑j=1dβjzij,
where d denotes the number of prognostic snps.
for the experiment, we set m= <dig>  n= <dig>  d= <dig>  ρ= <dig> or  <dig> , βj= <dig>  , and a uniform censoring distribution for 15% or 30% of censoring.
all six prognostic snps have =.
snp  <dig> and snp  <dig> have a dominant model; snp  <dig> and snp  <dig>  a recessive model; and snp  <dig> and snp  <dig>  a multiplicative model.
each of the remaining  <dig> snps has  with =.
to evaluate the performance of the proposed method, we generate  <dig> random samples and divide them into a training set  and a test set .
we calculate the maxtest p-value of each snp by using b= <dig>  permutations from the training set and identify the genetic model for each snp.
we select snps with p-values less than α= <dig>  and convert selected snps into corresponding scores by their genetic models.
we apply the gradient lasso to the selected snps to fit the prediction model.
let snps j  be included in the fitted prediction model with corresponding regression estimates β^ <dig> ...,β^k.
then we can define the risk score for sample i as ri=β^1zi1+⋯+β^kzik.
using the median risk score from the test set as a cutoff value, we divide the patients in the test set into high- and low-risk groups.
we apply a two-sample log-rank test to compare the survival distribution between these two risk groups.
we repeat this procedure  <dig> times and count the number of snps and that of prognostic snps included in each fitted prediction model by the gradient lasso.
we summarize the distribution of log-rank p-values from the test set, and for comparison purposes, we consider the prediction methods by assuming that all m snps have the same genetic model.
example using real data
we apply the proposed method to the gwas data in choi et al.
<cit> , who provide a gwas of  <dig> patients with normal karyotype acute myeloid leukemia  by using affymetric genome-wide human snp arrays  <dig>  .
we exclude those snps with missing genotype data for any patient.
we also exclude those snps with only one genotype for the  <dig> patients.
the final data set for the analysis includes m =  <dig>   <dig> autosomal snps from n =  <dig> patients.
the primary endpoint in this analysis is event-free survival , which is defined as the interval between the registration and the end of induction chemotherapy for patients showing no complete response , a relapse after achieving a cr to induction chemotherapy, or death.
we employ the leave-one-out cross-validation  procedure to evaluate the predictive performance of the proposed method for the data set.
from a training set of size n− <dig> =  <dig>  we calculate the maxtest p-value of each snp based on b= <dig>  permutations, select j candidate snps with p-values less than α= <dig>  by maxtest, and apply the gradient lasso to candidate snps to obtain a prediction model.
using the median risk score for the patients in the training set, we allocate those patients who are left out for the validation to the high- or low-risk group.
we repeat this procedure n times and calculate the log-rank p-value to compare the efs between the two risk groups.
figure  <dig> shows the kaplan-meier curves for the high- and low-risk groups classified by the loocv procedure.
the five-year efs rate for the low-risk group  is much higher than that for the high-risk group  with the estimated hazard ratio of  <dig>  , and the log-rank p-value is  <dig> .
a standard approach may be to fit a prediction model assuming a multiplicative genetic models for all snps, e.g.
tan et al.
<cit> .
we analyzed this data set using the same method as above except that all snps were assumed to have a multiplicative model.
figure  <dig> displays the loocv results.
note that the fitted prediction models do not significantly partition the test set into high- and low-risk groups by ignoring the possible genetic models.
we also apply our prediction procedure to the whole data set with n= <dig>  using α= <dig> , we select j= <dig> candidate snps, among which  <dig>  are shown to have a recessive model,  <dig>  a dominant model, and  <dig>  a multiplicative model.
by applying the gradient lasso to the selected  <dig> snps, we obtain the final prediction model including k= <dig> snps.
table  <dig> lists the rs ids, the chromosome numbers, the base-pair position and the gene name of the  <dig> snps included in the final prediction model.
for each of the  <dig> snps, we report the genetic model  identified by the maxtest, the marginal maxtest p-value and number of times  that each snp is included in the prediction models during the loocv.
note that the first four snps in table  <dig> are included in all  <dig> prediction models during loocv.
the rgs <dig> gene  is associated with treatment outcomes in aml-nk patients.
rgs <dig>  a regulator of g-protein signaling  <dig>  modulates the g-protein function in the signaling pathway by activating the intrinsic gtpase activity of alpha subunits  <cit> .
an snp on rgs <dig> has been found to modulate the risk of bladder cancer  <cit> .
in addition, it is known that rgs <dig> induces apoptosis through a mitochondrial-dependent pathway, which implies that rgs <dig> may be involved in cancer progression  <cit> .
further, membrane drug transporters, including slc25a <dig>  and slc24a <dig> , are known to be associated with event-free survival.
slc25a <dig>  solute carrier family  <dig> , member  <dig>  is known to interact with the bcl-2-associated x protein, which is involved in the apoptosis pathway  <cit> .
the rs <dig> snp on family with sequence similarity  <dig>  member c, fam5c, is selected by the proposed model.
a loss of hypermethylated fam5c is known to be associated with the development of tongue squamous cell carcinoma or gastric cancer  <cit> .
CONCLUSIONS
we have proposed a prediction method for a survival endpoint using snps.
the paper also proposes a maxtest to screen out nonprognostic snps and identify genetic model of prognostic snps.
the simulation results indicate substantial prognostic power for the proposed prediction method.
noteworthy is that, in conjunction with the maxtest, the proposed method provides more parsimonious prediction models with more prognostic snps than those prediction methods ignoring the true genetic model of prognostic snps.
we apply real gwas data to patients with acute myeloid leukemia and find that the proposed method provides a prediction model that can efficiently classify the patients into high- and low-risk groups by using a small number of snps that are known to be biologically informative.
although the proposed method is limited to the prediction of time-to-event traits, it can be easily extended to a wide range of traits, including dichotomous or continuous ones.
competing interests
the authors declare no conflict of interest
author contributions
jk and is performed the statistical analysis and wrote the manuscript.
ds and ta supported the research.
dhk provided a biological interpretation of the statistical analysis.
sj proposed the research project.
all authors read and approved the final manuscript.

 a genome-wide association study involves an examination of the entire genome, typically single-nucleotide polymorphisms, of different individuals to determine whether any variant is associated with a particular clinical outcome. many researchers have considered the design and analysis of gwass with respect to binary clinical outcomes such as case/control or response/non-response ones. in clinical cancer research, the primary endpoint of interest is usually a time-to-event trait subject to censoring. in calgb for example, germline dnas are collected, together with time to progression and overall survival data, from advanced pancreatic cancer patients. one objective of an snp correlative study is to discover snp markers that are correlated with such time-to-event endpoints. one of the first objectives of a statistical analysis in a gwas is the discovery of snp markers that are associated with a particular trait. the major statistical issue in marker discovery is multiple testing to avoid enlarged type i error probability due to the large number of univariate tests. each prognostic snp has two or three possible outcomes depending on its genetic model, and the efficiency of a statistical method in associating it with a trait is maximized when the true genetic model is known. for most snps, however, the true genetic model is unknown. to identify the true genetic model of each snp and optimize the association analysis, many researchers have considered some candidate genetic models for a given trait and derived a null distribution of the maximum of test statistics specific to individual genetic models. this test is referred to as the maxtest. these methods have been developed for binary traits such as case/control or response/non-response ones. we develop a maxtest to identify the genetic model of each snp when the trait is a survival endpoint, e.g., the time to tumor progression or death. another major objective of a gwas is to predict a trait of interest by using snps. prediction methods using microarray data have been widely investigated, but cannot be directly applied to snp-based predictions. the number of snp markers in genome-wide snp data far exceeds that of gene markers in microarray data, e.g., 1m vs. 20k. in addition, although gene expression data in microarray studies are continuous, genome-wide snp data are discrete, taking only three different values at most and showing different values depending on the genetic model. this paper presents a method for predicting a survival outcome that uses genome-wide snp data but can be easily modified for any type of trait, including binary or continuous outcomes. the proposed method uses the gradient lasso method, which has been developed for microarray data. some investigators fit a prediction model while ignoring the genetic model of each snp. we also propose a maxtest associating between a time-to-event trait and a snp accounting for its possible genetic model and identifies the genetic model of each candidate prognostic snp by using the proposed maxtest before fitting a prediction model. the simulation results show that this procedure improves prediction efficiency and prognostic power. for computational efficiency, nonsignificant snps are excluded using the maxtest before starting the gradient lasso. for the facilitation of the proposed maxtest and prediction method, glcoxphsnp r packages are provided. methods genetic models of snps suppose that the genotype for an snp is encoded as aa, ab, or bb. let g denote the number of copies of the b allele. that is, g= or if the genotype is aa, ab, or bb, respectively. let λg denote the hazard function of genotype g. without loss of generality, assume that b is the risk allele in the sense that having b increases the risk of an event. more specifically, assume that λ0≤λ1≤λ for all t≥ we now consider the following three popular genetic models: •recessive model: λ0=λ1<λ •dominant model: λ0<λ1=λ •multiplicative model: λ2/λ1=λ1/λ or equivalently λ1=γλ and λ2=γ2λ for γ> for a chosen score cg, we consider a proportional hazard model, λg=λ exp. then cox’s partial maximum likelihood test has optimal power with = for a recessive model, for a dominant model, and for a multiplicative model. note that the phm is invariant to the linear transformation of the covariate. maxtest suppose that we want to test whether an snp is associated with a given clinical outcome. the test statistic is dependent on the true genetic model of the snp. at the time of testing, however, we usually have no knowledge of the true genetic model. in this case, a naive approach is to conduct all statistical tests by assuming different genetic models and choose the lowest p-value as the measurement of the association. this approach can lead to an enlarged type i error because of multiple tests. to adjust for multiple tests, investigators have proposed a method considering the maximum of test statistics with respect to all candidate genetic models under consideration, namely the maxtest. many studies have considered the maxtest for binary clinical outcomes. zheng et al. propose a robust ranking method when the underlying genetic model is unknown, namely the max-rank test. conneely and boehnke propose a method for computing p-values that adjusts for correlated tests and show that the method can improve the accuracy of permutation tests with greater computational efficiency. li et al. propose a method for approximating the p-value for the maxtest with or without covariates adjusted for, namely the p-rank test. li et al. compare the results of the max-rank and p-rank tests. hoggart et al. formulate the problem as variable selection in a logistic regression analysis including a covariate for each snp and find the posterior mode for shrinkage priors based on a stochastic search on a penalized likelihood function. we propose a maxtest for survival endpoints. here we assign numeric scores to three genotypes based on their genetic model: = for a recessive model, = for a dominant model, and = for a multiplicative model. for a given score cg assigned to the genotype g of an snp, we consider the cox propotional hazard model, λg=λ0exp. for patient i, let ti and ci denote the survival and censoring times, respectively. we observe that xi= min and δi=i, where i indicates the indicator function. in addition, for t≥ let yi=i and ni=δii denote the at-risk and event processes, respectively. for a given score, we set zi=cg if patient i has genotype g. let sk=∑i=1nzikyi, k=, then, the partial score test statistic by cox, w=n−1/2∑i=1n∫0∞{zi−s1/s0}dni is asymptotically normal with mean and variance that can be consistently estimated by σ^2=n−1∑i=1n∫0∞s2s0−s12s02dni under h0:λ0=λ1=λ. by combining the statistics with respect to the three candidate genetic models, we can derive a maxtest statistic. let wl and σ^l denote the test statistic and the variance estimator with respect to genetic model l. then we can define the proposed maxtest as q= max, where tl=wl/σ^l. under h σll′=cov is consistently estimated by σ^ll′=n−1∑i=1n∫0∞zli−sl1sl0zl′i−sl′1sl′0yiydn where zli and slk denote zi and sk, respectively, for genetic model l; y=∑i=1nyi, n=∑i=1nni. let Σ^=1≤l,l′≤ where ρ^ll′=σ^ll′/σ^lσ^l′. then we can obtain the critical value of q by a numerical method or a simulation method from the n distribution. this is a survival trait counterpart for the maxtest with a binary trait, as discussed in several studies. we can construct an alternative test based on the quadratic form w2=stΣ^−1s, where s=t. in addition to recessive, dominant, and multiplicative genetic models, we can consider other models to develop a test statistic to measure the relationship between an snp and a survival trait. for example, we may consider the long-rank test based on the one-way anova in or the test based on the wilcoxon rank-sum test in, which require no specific genetic model assumptions. in particular, the anova-type test is a reasonable option if the monotone trend in genotypes g = and is doubtful. cox model with a lasso penalty in an analysis using snp data, we may face a problem in which the number of snps exceeds the size of data, that is, m≫p, which frequently occurs even when a smaller number of snps are selected through a prescreening step. this may lead to a serious collinearity problem when directly applying the partial likelihood estimation to the cox model. to address this problem, tibshirani estimates the parameters of the cox model under the l constraint as follows: β^=argmaxβ,sl,subject to∑j=1m|βj|≤s, where l is the partial likelihood function. the above optimization problem is suitable for reducing the dimension of covariates but is computationally difficult because the l objective function is not differentiable. to address this computational problem, previous studies have proposed many algorithms. tibshirani proposes an algorithm using quadratic programming within an iterative procedure. gui and li propose an las-cox procedure applying the choleski decomposition and the lars procedure. however, these algorithms can be computationally burdensome and sometimes fail to converge to an optimum because they involve quadratic programming and/or matrix inversions. sohn et al. propose glcoxph for the cox model by using the gradient lasso algorithm in. this glcoxph employs a coordinate-wise gradient decent with a deletion step and requires only univariate optimization in each iteration. its convergence speed is almost independent of the number of input variables, and it does not require a matrix inversion, which makes it scalable to high-dimensional data and allows it to converge to a global optimum faster. glmpath provides the entire penalization path for the cox model in a forward stagewise manner. because it requires matrix inversions only for active sets, it is faster and more stable than other methods. sohn et al. provided a comparative analysis using real and simulated data and show that the gradient lasso algorithm outperforms glmpath in analyzing high-dimensional survival data in terms of the sparsity, predictability, and computational efficiency of the final prediction model. therefore, the following gradient lasso algorithm can be a useful alternative for fitting the cox model to predict the survival time of patients based on high-dimensional snp data: initialize: β= and k= do until convergence addition: compute the gradient ∇l. find the γj maximizing |∂l/∂βj| for j= …,p and γ^=−s×sign/∂βĵ). let v be a p-dimensional vector such that its γj-th element γ^ and other elements are zeros. find α^=argminα∈ lβ+αv. update β=β+α^v. deletion: calculate hσ=−∇l+θσ∇ltθσ/|σ|, where σ={j:βj≠0}. find δ^=argminδ∈l, where h=hσ and u= mink∈σ{−βk/hk:βkhk<0}. update β=β+δ^h. set m=m+ returnβ. proposed algorithm for predicting a survival trait we propose a new algorithm for fitting a cox regression model using snp data. the proposed algorithm consists of the following four steps: we select significant snps by the maxtest, as in section “example using real data”, convert these snps into corresponding scores by genetic models identified by the maxtest, standardize these scores, and apply the gradient lasso algorithm to selected snps. we summarize the algorithm in greater detail as follows:  read in the clinical data {,i=...,n} and snp data {,i=...,n}, where sij denotes the number of b alleles for snp j.  for snp j, calculate the variance and covarince matrix Σ^j, and generate the null distribution of the maxtest as follows. for b=...,b, generate,t2j,t3j) from n. let qj=max|,|t2j|,|t3j|) for b=...,b.  for snp j, using original data, calculate the test statistics, the maxtest statistic qj= max, and two-sided p-values p1j,p2j,p3j from the marginal test for respective genetic models. approximate the p-value of the maxtest by pj=b−1∑b=1bi≥qj)  snp screening: select j snps with pj<α for specified a α.  for selected snps j, identify the genetic model by the lowest marginal p-value from p1j,p2j,p3j or the largest test statistic from t1j,t2j,t3j.  for patient i, define covariates by the identified genetic model and the corresponding score.  standardize the covariates: zij′=zij−z¯jsj, where z¯j=n−1∑i=1nzij and sj2=n−1∑i=1n  apply the gradient lasso to the cox regression model with response data {,i=...,n) and standardized covariates {,i=...,n}.  simulation study we provide a simulation study. the data generation scheme is as follows: we generate snp data z...,zm from n random variables with an ar correlation structure with the autocorrelation coefficient ρ, x...,xm. due to linkage disequilibrium, snps which lay in close vicinity within chromosomes tend to have a stronger association. in this sense, an ar correlation structure is a reasonable correlation structure for the continuous random variables generating snp data. let x1=ε and xj=ρxj−1+1−ρ2εj for j=...,m, where ε...,εm∼iidn random numbers. the cutoff values for xj for generating zj determine allele frequency. for each snp, let f f f denote the frequency of aa, ab, and bb genotypes, respectively, where b denotes the risk allele. note that marginally xj∼n. the true model for the survival times is given as Λ=Λ0exp∑j=1dβjzij, where d denotes the number of prognostic snps. for the experiment, we set m= n= d= ρ= or, βj=, and a uniform censoring distribution for 15% or 30% of censoring. all six prognostic snps have =. snp and snp have a dominant model; snp and snp a recessive model; and snp and snp a multiplicative model. each of the remaining snps has with =. to evaluate the performance of the proposed method, we generate random samples and divide them into a training set and a test set. we calculate the maxtest p-value of each snp by using b= permutations from the training set and identify the genetic model for each snp. we select snps with p-values less than α= and convert selected snps into corresponding scores by their genetic models. we apply the gradient lasso to the selected snps to fit the prediction model. let snps j be included in the fitted prediction model with corresponding regression estimates β^...,β^k. then we can define the risk score for sample i as ri=β^1zi1+⋯+β^kzik. using the median risk score from the test set as a cutoff value, we divide the patients in the test set into high- and low-risk groups. we apply a two-sample log-rank test to compare the survival distribution between these two risk groups. we repeat this procedure times and count the number of snps and that of prognostic snps included in each fitted prediction model by the gradient lasso. we summarize the distribution of log-rank p-values from the test set, and for comparison purposes, we consider the prediction methods by assuming that all m snps have the same genetic model. example using real data we apply the proposed method to the gwas data in choi et al., who provide a gwas of patients with normal karyotype acute myeloid leukemia by using affymetric genome-wide human snp arrays. we exclude those snps with missing genotype data for any patient. we also exclude those snps with only one genotype for the patients. the final data set for the analysis includes m = autosomal snps from n = patients. the primary endpoint in this analysis is event-free survival, which is defined as the interval between the registration and the end of induction chemotherapy for patients showing no complete response, a relapse after achieving a cr to induction chemotherapy, or death. we employ the leave-one-out cross-validation procedure to evaluate the predictive performance of the proposed method for the data set. from a training set of size n− = we calculate the maxtest p-value of each snp based on b= permutations, select j candidate snps with p-values less than α= by maxtest, and apply the gradient lasso to candidate snps to obtain a prediction model. using the median risk score for the patients in the training set, we allocate those patients who are left out for the validation to the high- or low-risk group. we repeat this procedure n times and calculate the log-rank p-value to compare the efs between the two risk groups. figure shows the kaplan-meier curves for the high- and low-risk groups classified by the loocv procedure. the five-year efs rate for the low-risk group is much higher than that for the high-risk group with the estimated hazard ratio of, and the log-rank p-value is. a standard approach may be to fit a prediction model assuming a multiplicative genetic models for all snps, e.g. tan et al.. we analyzed this data set using the same method as above except that all snps were assumed to have a multiplicative model. figure displays the loocv results. note that the fitted prediction models do not significantly partition the test set into high- and low-risk groups by ignoring the possible genetic models. we also apply our prediction procedure to the whole data set with n= using α=, we select j= candidate snps, among which are shown to have a recessive model, a dominant model, and a multiplicative model. by applying the gradient lasso to the selected snps, we obtain the final prediction model including k= snps. table lists the rs ids, the chromosome numbers, the base-pair position and the gene name of the snps included in the final prediction model. for each of the snps, we report the genetic model identified by the maxtest, the marginal maxtest p-value and number of times that each snp is included in the prediction models during the loocv. note that the first four snps in table are included in all prediction models during loocv. the rgs gene is associated with treatment outcomes in aml-nk patients. rgs a regulator of g-protein signaling modulates the g-protein function in the signaling pathway by activating the intrinsic gtpase activity of alpha subunits. an snp on rgs has been found to modulate the risk of bladder cancer. in addition, it is known that rgs induces apoptosis through a mitochondrial-dependent pathway, which implies that rgs may be involved in cancer progression. further, membrane drug transporters, including slc25a and slc24a, are known to be associated with event-free survival. slc25a solute carrier family, member is known to interact with the bcl-2-associated x protein, which is involved in the apoptosis pathway. the rs snp on family with sequence similarity member c, fam5c, is selected by the proposed model. a loss of hypermethylated fam5c is known to be associated with the development of tongue squamous cell carcinoma or gastric cancer.  we have proposed a prediction method for a survival endpoint using snps. the paper also proposes a maxtest to screen out nonprognostic snps and identify genetic model of prognostic snps. the simulation results indicate substantial prognostic power for the proposed prediction method. noteworthy is that, in conjunction with the maxtest, the proposed method provides more parsimonious prediction models with more prognostic snps than those prediction methods ignoring the true genetic model of prognostic snps. we apply real gwas data to patients with acute myeloid leukemia and find that the proposed method provides a prediction model that can efficiently classify the patients into high- and low-risk groups by using a small number of snps that are known to be biologically informative. although the proposed method is limited to the prediction of time-to-event traits, it can be easily extended to a wide range of traits, including dichotomous or continuous ones. competing interests the authors declare no conflict of interest author contributions jk and is performed the statistical analysis and wrote the manuscript. ds and ta supported the research. dhk provided a biological interpretation of the statistical analysis. sj proposed the research project. all authors read and approved the final manuscript. 
in this paper, we present a prediction method for time-to-event traits using genome-wide single-nucleotide polymorphisms. the proposed maxtest can help screen out nonprognostic snps and identify genetic models of prognostic snps. in conjunction with the maxtest, the proposed method provides more parsimonious prediction models but includes more prognostic snps than some naive prediction methods. the proposed method is demonstrated with real gwas data. we also propose a maxtest associating between a time-to-event trait and a snp accounting for its possible genetic models.
 a popular objective of many high-throughput genome projects is to discover various genomic markers associated with traits and develop statistical models to predict traits of future patients based on marker values.  in this paper, we present a prediction method for time-to-event traits using genome-wide single-nucleotide polymorphisms. we also propose a maxtest associating between a time-to-event trait and a snp accounting for its possible genetic models. the proposed maxtest can help screen out nonprognostic snps and identify genetic models of prognostic snps. the performance of the proposed method is evaluated through simulations.  in conjunction with the maxtest, the proposed method provides more parsimonious prediction models but includes more prognostic snps than some naive prediction methods. the proposed method is demonstrated with real gwas data. 