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**|**Brief Bioinform**|**PMC3030811

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- Abstract
- INTRODUCTION
- RANKING PROGNOSIS MARKERS
- SIMULATION
- DATA ANALYSIS
- RESULTS
- DISCUSSION
- SUPPLEMENTARY DATA
- FUNDING
- References

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Brief Bioinform. 2011 January; 12(1): 33–40.

Published online 2010 November 18. doi: 10.1093/bib/bbq069

PMCID: PMC3030811

Corresponding author. Shuangge Ma, 60 College ST, LEPH 209, School of Public Health, Yale University, New Haven, CT 06520, USA. Tel: Phone: +1 203 785 3119; Fax: +1 203 785 6912; E-mail: ude.elay@am.eggnauhs

Received 2010 August 23; Accepted 2010 October 18.

Copyright © The Author 2010. Published by Oxford University Press. For Permissions, please email: journals.permissions@oup.com

This article has been cited by other articles in PMC.

In cancer research, high-throughput genomic studies have been extensively conducted, searching for markers associated with cancer diagnosis, prognosis and variation in response to treatment. In this article, we analyze cancer prognosis studies and investigate ranking markers based on their marginal prognosis power. To avoid ambiguity, we focus on microarray gene expression studies where genes are the markers, but note that the methodology and results are applicable to other high-throughput studies. The objectives of this study are 2-fold. First, we investigate ranking markers under three commonly adopted semiparametric models, namely the Cox, accelerated failure time and additive risk models. Data analysis shows that the ranking may vary significantly under different models. Second, we describe a nonparametric concordance measure, which has roots in the time-dependent ROC (receiver operating characteristic) framework and relies on much weaker assumptions than the semiparametric models. In simulation, it is shown that ranking using the concordance measure is not sensitive to model specification whereas ranking under the semiparametric models is. In data analysis, the concordance measure generates rankings significantly different from those under the semiparametric models.

Cancer research has entered the -omics era. High-throughput profiling studies have been extensively conducted, searching for markers associated with cancer diagnosis, prognosis and variation in response to treatment [1]. In this article, we analyze cancer prognosis studies with survival outcomes. To avoid ambiguity, we focus on microarray gene expression studies, where genes are the markers, but note that the methods and results are applicable to other high-throughput -omics studies.

Denote *T* as the survival time, which can be progression-free, overall, or other types of survival. Denote *Z* = (*Z*^{1},…,*Z*^{d}) as the length *d* microarray gene expression measurements. Denote *C*as the censoring time. Under right censoring, one observation consists of where *I* is the indicator function. In a typical microarray study, *d* is of the order 10^{3‐4}. In other genomic (for example, genome wide association) studies, *d* can be even larger. In practice, it is not feasible to investigate all *d* genes in detail. More importantly, among the *d* genes, only a subset is cancer associated, whereas the rest are noises. Thus, it is of great interest to rank the *d* genes, and only the top-ranked genes are investigated in downstream analysis.

Ranking the marginal prognosis power of markers consists of the following steps.

- For
*j*= 1,…,*d*,- describe the relationship between marker
*j*and event time*T*using the model where β^{j}is the regression coefficient, and ϕ is the link function; - a statistic measuring the prognosis power of marker
*j*is computed. Examples of the statistic include the magnitude of the estimate of β^{j}, significance level (*P*-value) of the estimate, likelihood of the model and others. In this study, we use the*P*-value of the estimate as the ranking statistic, and refer to studies such as [2] for other ranking statistics;

- All
*d*markers are ranked based on the magnitudes of ranking statistics. When the*P*-values are used for ranking, the FDR (false discovery rate) approach can be used to determine a cutoff [3]. Only top-ranked markers with ranking statistics below the cutoff are studied in downstream analysis.

In published studies, the most extensively adopted prognosis model is the Cox proportional hazards model [4, 5]. Alternatives, including the accelerated failure time model, additive risk model and others have also been used. Our literature review suggests that quite often only a single prognosis model is used to rank markers, and there is insufficient model justification or diagnostics. As the first goal of this study, we show that rankings under different prognosis models can be significantly different. We note that, most cancer genomic studies are still at the marker identification stage, and the identified markers will not be used in clinical practice before they are validated in independent studies. However, failing to acknowledge the difference of rankings (under different prognosis models) and adopting inappropriate ranking methods can lead to the wrong sets of markers for downstream validation studies. With multiple candidate models, the ideal approach is to select the most appropriate model using data-adaptive approaches. However, available approaches (for example those in [4, 6]) are developed for data with a small number of markers. To the best of our knowledge, there is still no approach that can select the appropriate survival model for high dimensional cancer genomic data. ‘As an alternative and the second goal of this study, we introduce a nonparametric concordance measure for ranking markers'. This measure depends on weaker assumptions and hence can provide a more robust ranking.

In a recent study, Ma *et al*. [7] also investigated multiple prognosis models. Unlike in this study, in [7], a joint mode is assumed. Denote *R*_{n}(β^{1},…,β^{d}) as the log-likelihood function constructed using *n* iid observations. Ma *et al*. [7] investigated marker selection based on the Lasso penalized estimate and showed that under different prognosis models, the Lasso approach selected significantly different sets of markers. Although this study and [7] both investigate multiple prognosis models, they differ significantly from each other. More specifically, Ma *et al*. [7] investigated the joint effects of multiple markers, whereas this study focuses on the marginal prognosis power of markers. Both analyses of marginal and joint effects have been extensively conducted in the literature. They describe markers from different prospective and complement each other. In Ma *et al*. [7], marker selection is achieved using a penalization approach, whereas in this study, ranking statistics are computed for individual markers and marker selection is achieved using the FDR approach. In addition in this study, beyond the semiparametric prognosis models, we also introduce a nonparametric concordance measure, which can provide a more robust ranking of markers. Thus this study is warranted beyond [7].

When computing the ranking statistics, the first approach assumes a semiparametric model with an unknown regression coefficient for each marker. An estimate of the regression coefficient is then computed based on the likelihood or estimating equations, and its significance level is taken as the ranking statistic. An alternative approach assumes a nonparametric model. A nonparametric concordance measure can be computed, and its significance level is taken as the ranking statistic.

As parametric models are usually too restricted, we focus on semiparametric prognosis models. Among the many available models, the following have attracted the most attention.

For marker under the Cox model, the conditional hazard function is

Here λ_{0}(*t*) is the unknown baseline hazard function, and β^{j} is the regression coefficient. The Cox model has been extensively employed in gene expression studies [5, 8–10].

Assume *n* iid observations {(*Y*_{i},δ_{i},*Z*_{i}),*i* = 1…n}. For marker *j*, the log-partial likelihood function is , where is the at-risk set at time *Y*_{i}. Consider the maximum (partial) likelihood estimate . Its significance level can be computed using the martingale-based approach in [6]. Many software packages (for example the *coxph* function in R) can be used to compute the *P*-value.

The AFT (accelerated failure time) model shares a similar spirit with the linear regression model [11]. For marker *j* under the AFT model,

where, α^{j} is the intercept and ε^{j} is the random error with an unknown distribution. Here the logarithm transformation can be replaced by other known monotone transformations. Examples of the AFT model in gene expression studies include [12–15].

For subject *i*, define

Here and is the KM (Kaplan–Meier) estimate of *F*, the distribution function of ε^{j}, based on the transformed data That is

Define The Buckley-James estimator of is defined as . Calculation of the significance level of is described in [16].

For marker *j*, the additive risk model assumes that the conditional hazard function is

where notations have similar interpretations as under the Cox model [11]. The additive risk model has also been used extensively in gene expression studies [16, 17–19].

Define For the *i*^{th} subject, denote and as the observed event process and the at-risk process, respectively. β^{j} can be estimated by solving Here is the estimate of Λ_{0} satisfying . The resulting estimate of β^{j} satisfies the estimating equation

Inference for is described in [20].

There are other semiparametric prognosis models, including for example the proportional odds model, the accelerated hazard model and others. They are less extensively used and will not be discussed. Among the above three models, the Cox and AFT models belong to the family of transformation models. The Cox and additive risk models describe the conditional hazard function, whereas the AFT model describes the event time directly. A common advantage of the three models is that, although they are semiparametric, the regression coefficients can be estimated without estimating the nonparametric parameters, which significantly reduces the computational complexity. There are several software packages that can be used to compute the estimates and their significance level.

Under mild conditions, each β^{j} can be consistently estimated. When as , the estimates of β^{j}s are uniformly consistent. Similar results hold for the *P*-values. Thus, ranking the genes based on the *P*-values and the FDR approach is asymptotically valid [21].

A drawback of ranking markers using semiparametric models is that the validity of ranking depends on the validity of model assumptions. With cancer genomic data, it is difficult to determine whether the model assumptions are appropriate. Below we describe a concordance measure, which was proposed in [X. Song *et al*., submitted for publication]. It relies on much weaker assumptions and hence provides a more robust way of ranking markers.

For marker *j*, assume that *E*(*T*|Z^{j}) = η^{j}(*Z*^{j}), where η^{j} is an unknown monotone function. Note that this model is generic and includes many existing parametric and semiparametric models as special cases. Without loss of generality, assume that η^{j} is an increasing function (a recoding can be conducted if necessary). Intuitively, if marker *j* has prognosis power, the order of {*T*_{1},…,*T*_{n}} should be similar to that of . Thus, the prognosis power of a marker can be evaluated using the concordance between the ranking of event times and the ranking of marker values. Particularly, the nonparametric concordance measure is defined as

This measure has roots in the time-dependent receiver operating characteristic (ROC) techniques [X. Song *et al*., submitted for publication, 22]. The ROC approaches have been extensively used in evaluating the diagnosis and prognosis power of markers. With ROC, the diagnosis/prognosis power can be summarized with the area under curve (AUC). For marker *j* at time *t*, the AUC for the incident ROC curve is The concordance measure τ^{j} is related to the AUC through the formula where *w*(*t*) = 2*f*(*t*)*S*(*t*), and *f*(*t*) and *S*(*t*) are the density and survival functions of the survival time *T* [23]. Thus the concordance measure can be viewed as a weighted average of the AUC over time. Unlike *AUC*^{j}(*t*) which is a function of time, τ^{j} is time-independent. It thus can better summarize the prognosis performance of a marker and facilitate the comparison of markers.

Note that τ^{j} can be rewritten as

If Here *S*_{C} is the survival function of the censoring time *C*. In addition, if *C* is independent of *Z*, Thus, τ^{j} can be estimated with

Here is the KM estimate of *S*_{C}. In [X. Song *et al*., submitted for publication], it is shown that

Hence is *n*^{1/2} consistent and asymptotically normal. Inference can be based on the asymptotic normality result. For integrity of this article, we provide the definition of and outline of the proof of asymptotic properties in Supplementary Appendix I and refer to [X. Song *et al*., submitted for publication] for more details. Calculation of and its significance level can be realized using existing software. Sample R code is available from the authors.

We conduct simulation study to better understand properties of different ranking approaches. We simulate expressions of 1000 genes for 100 subjects. Gene expressions have marginally standard normal distributions, and expressions of genes *j* and *k* have correlation coefficient 0.3^{|j‐k|}. We intentionally set the distributions of gene expressions simpler than those observed in practice. Usually performance of marker identification methods decreases as the correlation structure and distributions of gene expressions become more complicated. Simpler distributions may better reveal the difference caused by ranking approaches. Among the 1000 genes, 30 have small, moderate and large prognosis power and the rest 970 are noises. The true underlying models are set as the Cox, AFT and additive risk models, respectively. We simulate and analyze 500 replicates and present the summary results in Supplementary Appendix II.

Simulation suggests that when the ranking approach matches the true data generating model, it can rank the genes properly. For example, when the Cox model is the true data generating model, ranking under the Cox model puts 23 true positives in the top 30. However, ranking under other semiparametric models is less satisfactory. Ranking under the AFT and additive risk models puts 13 and 15 true positives in the top 30, respectively. Under all three different data generating models, performance of the nonparametric concordance measure is satisfactory. It ranks 21, 22 and 20 true positives in the top 30, respectively. We also compute the Kendall tau rank correlation coefficient [24] between the following two rankings. The first is the ranking of the 30 true positives under the approach that matches the data generating model (i.e. ranking under the optimal ranking approach). The second is the ranking of the true positives under an alternative approach. The Kendall tau correlation is adopted as the ranks are discrete. Supplementary Appendix II suggests that ranking using the nonparametric concordance has a high correlation with ranking under the optimal approach.

Simulation shows that ranking using semiparametric models may be subject to model misspecification. In contrast, ranking using the nonparametric concordance measure is more robust. Even though it is asymptotically consistent, ranking using the concordance measure still differs from ranking under the optimal approach. This finding is not surprising. The concordance measure is nonparametric, which makes it less efficient particularly under small to moderate sample sizes. This explains why the nonparametric approach ranks slightly fewer true positives in the top 30.

Rosenwald *et al*. [25] reported a study using microarray gene expression analysis in mantle cell lymphoma (MCL). Among 101 untreated patients with no history of previous lymphoma, 92 were classified as having MCL based on established morphologic and immunophenotypic criteria. Survival times of 64 patients were available and 28 patients were censored. The median survival time was 2.8 years (range 0.02–14.05 years). Lymphochip DNA microarrays were used to quantify mRNA expressions in the lymphoma samples from the 92 patients. Data that contains expression values of 8810 cDNA elements is available at http://llmpp.nih.gov/MCL.

Rosenwald *et al*. [26] reported a diffuse large B-cell lymphoma (DLBCL) prognosis study. This study retrospectively collected tumor biopsy specimens and clinical data for 240 patients with untreated DLBCL. The median follow up is 2.8 years, with 138 observed deaths. Lymphochip cDNA microarray is used to measure the expressions of 7399 genes. Raw data and detailed experiment protocol are available at http://llmpp.nih.gov/DLBCL/.

A study was conducted to determine whether the survival risks of patients with follicular lymphoma (FL) can be predicted by gene expression profiles of the tumors [27]. Fresh-frozen tumor-biopsy specimens from 191 untreated patients who had received a diagnosis of FL were obtained. The median age at diagnosis was 51 years, and the median follow-up time was 6.6 years. Affymetrix U133A and U133B microarray genechips were used to measure expressions of 44928 probes. Detailed experimental setup and the raw data can be accessed at http://llmpp.nih.gov/FL/.

Breast cancer is the second leading cause of death from cancer among women in the United States. Despite major progress in breast cancer treatment, the ability to predict metastasis of the tumor remains limited. van't Veer *et al*. [28] reported a breast cancer prognosis study investigating the time to distant metastasis. About 97 lymph node-negative breast cancer patients, 55-years old or younger, participated in this study. Among them, 46 developed distant metastases within 5 years. Expression levels for 24481 gene probes were collected. Data is available at http://www.rii.com/publications/2002/vantveer.html.

The original goal of the breast cancer study conducted by Sorlie *et al*. [29] was to classify breast tumors based on the gene expression patterns derived from cDNA microarrays. A total of 85 cDNA microarray experiments representing 78 cancers were conducted. Survival data is available for 76 patients. Four of these patients were excluded, because the microarray data was from samples obtained after treatment.

The goal of the study reported in Huang *et al*. [30] was to predict metastasis of breast tumor using gene expressions. Affymetrix genechips were used for the profiling of 71 samples. Expression measurements on 12625 probes are available.

Sotiriou *et al*. [31] reported a study correlating gene expression measurements generated using cDNA with clinico-pathological characteristics and clinical outcomes in an unselected group of 99 breast cancer patients. In the original analysis, the Cox model was used to identify genes that were marginally significantly associated with relapse-free survival. In this study, we analyze the 98 patients with complete survival information.

For each data set, we fill in missing expressions using medians across samples. Since usually researchers are more interested in genes with a higher level of variations of expressions, we conduct an unsupervised screening and select the 2000 genes with the largest variances of expressions. We normalize the gene expressions to have median zero and variance one.

We rank markers under the three specific semiparametric models and using the nonparametric concordance measure. Analysis results are presented in Supplementary Appendix III.

In the first set of analysis, we construct the complete ranking of all the 2000 markers. We evaluate the (dis)similarity among the rankings using the Kendall tau rank correlation coefficient [24]. Our data analysis suggests that different rankings are moderately to strongly correlate. Consider for example the MCL data, the rank correlation coefficient ranges from 0.546 to 0.902. With the seven data sets, the smallest rank correlation coefficient is only 0.293 (DLBCL data, the rankings under the AFT model and using the concordance measure). We conclude that the ranking of markers highly depends on the ranking methods and underlying prognosis model.

As the *P*-value is adopted as the ranking statistic, it is possible to apply the FDR approach and identify markers with significant prognosis power. We adopt the approach in [3] and set the target FDR=0.1. Results on the numbers of markers identified are provided in Supplementary Appendix III. We find that the numbers of markers identified also highly depend on the ranking methods and prognosis models. For example, with the breast cancer data in [26], the four ranking methods identify 18 (nonparametric concordance), 3 (Cox), 51 (AFT) and 1 (additive) markers. The sets of significant markers identified using different approaches can be significantly different. For example with the breast cancer data in [29], using the nonparametric concordance measure, 20 genes are identified. Under the AFT model, 5 genes are identified. However, the two sets only have 1 gene in common.

We examine the top 100 markers ranked using different methods and find that they have overlaps but can be significantly different. For example, with the DLBCL data, the set of top 100 markers ranked using the nonparametric concordance measure has 56 (Cox), 26 (AFT), 47 (additive) overlapped markers with those ranked under semiparametric models.

In cancer genomic studies, it is of interest to rank markers based on their marginal prognosis power. Top ranked markers will be investigated in detail in downstream analysis. When ranking markers, the most commonly adopted approach assumes a specific (semi-, non-)parametric model, computes a statistic from the model, and uses that statistic for ranking. In this article, we review the three most commonly adopted semiparametric prognosis models. Analysis of multiple cancer microarray data sets shows that the ranking may vary significantly under different prognosis models. Under a very weak conditional expectation assumption, we propose using a nonparametric concordance measure to rank markers. As much weaker assumptions are made, the concordance measure provides a more robust way of ranking markers. Another advantage of ranking based on τ^{j} is that the ranking is invariant under monotone increasing transformations of *Z*^{j}. Thus it is capable of accommodating nonlinear effects of markers. Because of the nonparametric nature, a drawback of the concordance measure is its lack of efficiency. Simulation suggests the satisfactory performance of the concordance measure. Data analysis suggests that ranking using the concordance measure can be significantly different from those under specific semiparametric models. Considering the robustness and slight lack of efficiency of the concordance measure, we suggest the following approach in practical data analysis. Ranking using the concordance measure is first conducted. Then multiple parametric and semiparametric models can be used to rank the genes. The Kendall tau rank correlations between the concordance measure ranking and the parametric or semiparametric model rankings are computed. If there is a parametric or semiparametric model rankings close enough to the concordance measure ranking, the corresponding model can be used in downstream analysis. If there is no such parametric or semiparametric model ranking, then the nonparametric concordance measure ranking should be used.

In this study, our conclusions are drawn from both simulation and analysis of real data. The seven data sets focus on lymphoma and breast cancer. We expect similar results to hold with other types of cancers. All four breast cancer studies investigate prognosis free survival. However, we note that all four studies are retrospective, with considerable differences among the demographic characteristics of patients and experimental settings. Thus, it is unclear whether it is appropriate to compare results of the four data sets. It is nontrivial to conduct meta-analysis that can accommodate multiple data sets or extend the nonparametric concordance measure to multi-data sets cases.

In this article, our presentation is in the context of cancer microarray studies. We expect similar conclusions to hold for other profiling platforms and other diseases. We adopt the significance level as the ranking statistic, which can be naturally coupled with the FDR approach to identify significant markers. We have experimented with other ranking statistics (for example the magnitude of the estimated regression coefficients) and reached similar conclusions. Ranking markers is also of great importance in diagnosis and treatment selection studies. In other types of studies and with other types of outcomes/phenotypes, similar ranking problems are also worth investigation. However, it is beyond the scope of this article and hence will not be pursued.

National Institutes of Health (grant CA142774 to S.M.); National Science Foundation (NSF) (grant DMS 0904181 to S.M.).

Supplementary data are available online at http://bib.oxfordjournals.org/.

- In cancer genomic studies, it is of interest to rank the marginal prognosis power of genomic markers.
- There are multiple ways of ranking markers. Numerical studies suggest that different methods may lead to significantly different rankings.
- Ranking can also be based on a nonparametric concordance measure, which depends on weaker data and model assumptions and hence can be more robust.
- In practical data analysis, more attention needs to be paid to the ranking methods and underlying prognosis models.

The authors thank the editor and three referees for their careful review and insightful comments, which have led to significant improvement of the article.

•

**Shuangge Ma** obtained his Ph.D. in Statistics from University of Wisconsin, Madison. He is an Assistant Professor in School of Public Health, Yale University.

•

**Xiao Song** obtained her Ph.D. in Statistics from North Carolina State University. She is an Assistant Professor in Department of Epidemiology and Biostatistics, College of Public Health, University of Georgia.

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