Notation

We first introduce some basic notation that will be used throughout the article. Let be the survival time of interest and a vector of predictor variables. In addition to the gene expression levels, may contain the measurements of some clinical predictor variables. Denote the conditional survival function of given by . The probability density and survival functions of are denoted by and , respectively. Further let be a random censoring time and denote the observed survival time by . The random variable indicates whether is right-censored () or not ().

A prediction rule for will be formed as a linear combination(1)where is an unknown vector of coefficients. We generally assume that the estimation of is based on an i.i.d. learning sample . In case of the Cox regression model, for example, is related to by the equation(2)where is the cumulative baseline hazard function. Because there is a one-to-one relationship between and the expected survival time , the linear combination can be used as a biomarker to predict the survival of individual patients.

Concordance index

Our proposed framework to derive and evaluate biomarker combinations is based on the concordance index (“-index”) which is a general discrimination measure for the evaluation of prediction models [16], [17]. It can be applied to continuous, ordinal and dichotomous outcomes [25]. For time-to-event outcomes, the -index is defined as(3)where , and , are the event times and the predicted marker values, respectively, of two observations in an i.i.d. test sample . By definition, the -index for survival data measures whether large values of are associated with short survival times and vice versa. Moreover, it can be shown that the -index is equivalent to the area under the time-dependent ROC curve, which is a measure of the discriminative ability of at each time point under consideration (see [26], p. 95 for a formal proof).

During the last decades, the -index has gained enormous popularity in biomedical research; for example, searching for the terms “concordance index” and “c-index” in PubMed [27] resulted in 1156 articles by the time of writing this article. Generally, a value of close to 1 indicates that the marker is close to a perfect discriminatory power, while a marker that does not perform better than chance results in a value of 0.5. For example, the famous Gail model [28] for the prediction of breast cancer is estimated to yield a value of [29].

Being a flexible discrimination measure, the -index is especially useful for selecting and ranking genes from a pre-processed set of high-dimensional gene expression data (Task 1 described in the Introduction). In other words, Task 1 can be addressed by computing the -index (and hence the marginal discriminatory power) for each individual gene or biomarker, where only those genes with the highest -index are incorporated into the yet-to-derive optimal combination (Task 2). Although there exist various other ways to rank genes and select the most influential ones, the -index has been demonstrated to be especially advantageous for this task [9].

An estimator of the -index for survival data is given by(4)

with (“Harrell's for survival data”, [30]). Generally, is a consistent estimator of the -index in situations where no censoring is present. However, because pairs of observations where the smaller observed survival time is censored are ignored in formula (4), is known to show a notable upward bias in the presence of censoring. This bias tends to be correlated with the censoring rate [15], [30].

To overcome the censoring bias of , Uno et al. [18] proposed a modified version of , which is defined as(5)

where denotes the Kaplan-Meier estimator of the unconditional survival function of (estimated from the learning data). In the following, we will assume that the censoring times are independent of . Under this assumption, is a consistent and asymptotically normal estimator of (see [18], pp. 1113–1115). Consistency is ensured by applying inverse probability weighting (using the weights , which account for the inverse probability that an observation in the test data is censored [31]). Numerical studies suggest that is remarkably robust against violations of the random censoring assumption [32].

Apart from the estimators and , there exist various other approaches to estimate the probability in (3) (see, e.g., [15] for an overview). Most of these approaches are based on the assumptions of a Cox proportional hazards model, so that they are not valid in case these assumptions are violated. Because is model-free and because the consistency of is guaranteed even in situations where censoring rates are high (in contrast to the estimator ), we will base our boosting method on .

Boosting the concordance index

The core of our proposed framework to address Tasks 1 – 3 is the derivation of a prediction-optimized linear combination of genes that is optimal w.r.t. to the -index for time-to-event data. Our approach will be based on a component-wise gradient boosting algorithm [24] that uses the -index as optimization criterion.

Gradient boosting algorithms [33] are generally based on a loss function that is assumed to be differentiable with respect to the predictor . The aim is then to estimate the “optimal” prediction function(6)by using gradient descent techniques. As the theoretical mean in (6) is usually unknown in practice, gradient boosting algorithms minimize the empirical risk over instead.

When considering the -index for survival data, the aim is to determine the optimal predictor that maximizes the concordance measure – which is essentially the solution to Task 2. Hence a natural choice for the empirical risk function would be the negative concordance index estimator(7)

Setting , however, is unfeasible because is not differentiable with respect to and can therefore not be used in a gradient boosting algorithm. To solve this problem, we follow the approach of Ma and Huang [34] and approximate the indicator function in (7) by the sigmoid function . Here, is a tuning parameter that controls the smoothness of the approximation (details on the choice of will be given in the Numerical Results section). Replacing the indicator function in (7) by its smoothed version results in the smoothed empirical risk function(8)

with weights(9)

By definition, the smoothed empirical risk is differentiable with respect to the predictor . Its derivative is given by(10)

In the proposed gradient boosting algorithm, the derivative in (10) is iteratively fitted to a set of base-learners. Typically, an individual base-learner (simple regression tool, e.g., a tree or a simple linear model) is specified for each marker. To ensure that the estimate of the optimal predictor is a linear combination of the components of , we will apply simple linear models as base-learners (cf. [35]). In other words, each base-learner is a simple linear model with one component of as input variable. Consequently, there are base-learners, which will be denoted by , . Each base-learner refers to one component of and therefore to one marker (or gene).

The component-wise gradient boosting algorithm for the optimization of the smoothed -index is then given as follows:

1. Initialize the estimate of the marker combination with offset values. For example, set , leading to for all components . Choose a sufficiently large maximum number of iterations and set the iteration counter to 1.
2. Compute the negative gradient vector by using formula (10) and evaluate it at the marker combination of the previous iteration:
3. Fit the negative gradient vector separately to each of the components of via the base-learners :
4. Select the component that best fits the negative gradient vector according to the least squares criterion, i.e., select the base-learner defined by
5. Update the marker combination for this component:where sl is a small step length . For example, if , only 10% of the fit of the base-learner is added to the current marker. This procedure shrinks the effect estimates towards zero, effectively increasing the numerical stability of the update step [23], [25].As only the base learner was selected, only the effect of component is updated () while all other effects stay constant ( for ).
6. Stop if . Else increase by one and go back to step (2).

By construction, the proposed boosting algorithm automatically estimates the optimal linear biomarker combination that maximizes the smoothed -index. The principle behind the proposed algorithm is to minimize the empirical risk by using gradient descent in function space, where the function space is spanned by the base-learners , . In other words, the algorithm iteratively descents the empirical risk by updating via the best fitting base-learner. Because the base-learners are simple linear models (each containing only one possible biomarker as predictor variable) and because the update in step (5) of the algorithm is additive, the final solution effectively becomes a linear combination of these markers.

The two main tuning parameters of gradient boosting algorithms are the stopping iteration and the step length sl. In the literature it has been argued that the choice of the step length is of minor importance for the performance of boosting algorithms [36]. Generally, a larger step length leads to faster convergence of the algorithm. However, it also increases the risk of overshooting near the minimum of . In the following sections we will use a fixed step-length of , which is a common recommendation in the literature on gradient boosting and which is also the default value in the R package mboost [37].

The stopping iteration is considered to be the most important tuning parameter of boosting algorithms [38]. The optimal value of is usually determined by using cross-validation techniques [24]. Small values of reduce the complexity of the resulting linear combination and avoid overfitting via shrinking the effect estimates. In case of boosting the -index, however, overfitting is less problematic as the predictive performance of is not related to the actual size of the coefficients but to the concordance of the rankings between marker values and the observed survival times. As a result, the stopping iteration in this specific case is less relevant and can be also specified by a fixed large value (e.g., ).

Regarding the boosting algorithm for the smoothed -index, an additional tuning parameter is given by the smoothing parameter . While too large values of will lead to a poor approximation of the indicator functions in (7), too small values of might overfit the data (and might therefore result in a decreased prediction accuracy). Details on how to best choose the value of will be given in the Numerical Results section.

The boosting algorithm presented above is implemented in the add-on software package mboost of the open source statistical programming environment R [39]. The specification of the new Cindex() family and a short description of how to apply the algorithm in practice are given in Text S1.

Evaluation

After having applied the -index to select the most influential genes (Task 1), and after having used the proposed boosting algorithm to combine the selected genes (Task 2), a final challenge is to evaluate the prediction accuracy of the resulting gene combination (Task 3). Since the -index was used for Tasks 1 and 2, it is also a natural criterion to evaluate the derived marker combination in Task 3. As argued before, it is advantageous from both a methodological perspective as well as from a practical one to use the same criterion for estimation and evaluation of a biomarker combination.

To avoid over-optimistic estimates of prediction accuracy, it is crucial to use different observations for the optimization and evaluation of the marker signature [25], [40]. This can be achieved either by using two completely different data sets (external evaluation) or by splitting one data-set into a learning sample and a test sample . The learning sample is used to optimize the marker combination while the “external” test sample serves only for evaluation purposes. A more elaborate strategy is subsampling (such as five-fold or ten-fold cross-validation), which is based on multiple random splits of the data. In our numerical analysis, we will use subsampling techniques in combination with stratification to divide the underlying data sets into learning and test samples (for details, see the next section).

When it comes to the -index, two additional points have to be taken into consideration: First, as the task is to obtain the most precise estimation for the discriminatory power, it is no longer necessary to use the smoothed version (which was included for numerical reasons in the boosting algorithm) for evaluation. Consequently, we propose to apply the original estimator for evaluating biomarker combinations in Task 3. Second, when applying the estimator to the observations in a test sample, a natural question is how to calculate the Kaplan-Meier estimator of the unconditional survival function of . In principle, there are three possibilities for the calculation of : The Kaplan-Meier estimator can be computed from either the test or from the training data, or, alternatively, from the combined data set containing all observations in the learning and test samples. Following the principle that all estimation steps should be carried out prior to Task 3, we will base computation of the Kaplan-Meier estimator on the learning data.

Numerical results

Simulation study.

We first investigated the performance of our approach based on simulated data. The aim of our simulation study was:

1. To analyze if the proposed framework is able to select a small amount of informative markers from a much larger set of candidate variables.
2. To check if gradient boosting is able to derive the optimal combination of the selected markers, and to compare its performance to competing Cox-based estimation schemes.
3. To investigate the effect of the smoothing parameter that controls the smoothness inside the sigmoid function, as well as potential effects of the sample size and the censoring rate on the performance of our approach.

The simulated survival times are generated via a log-logistic distribution for accelerated failure time (AFT) models [41]. They are based on the model equation , where is the survival time, and are location and scale parameters, and is a noise variable, following a standard logistic distribution. As a result, the density function for realizations can be written as(11)

with and . The possible markers were drawn from a multivariate normal distribution with pairwise correlation (). The true underlying combinations of the predictors were given by

Note that only four out of possible markers have an actual effect on the survival time. Furthermore, those four markers do not only contribute to the location parameter but also to the scale parameter (cf. [42]) – a setting where standard survival analysis clearly becomes problematic. Additionally to the survival times , we generated an independent sample of censoring times and defined the observed survival time by leading to independent censoring of 50% of the observations.

In order to answer the above questions, we investigated the performance of our framework in all three tasks that are necessary to develop new gene signatures in practice (Tasks 1–3 described in the Introduction). To avoid over-optimism and biased results, we simulated separate data sets for the different tasks. In simulation runs, we first simulated a data set with observations to pre-select the most influential predictors based on the -index (Task 1). The optimal combination of those predictors was later estimated (Task 2) by our boosting algorithm based on smaller training samples of size . For the final external evaluation of the prediction accuracy (Task 3) we simulated a separate test data set with .

For Task 1, we first pre-selected a subset of predictors from the available markers. We ranked the predictors based on their individual values of and included only the best-performing markers in the boosting algorithm. The results suggest that the -index is clearly able to identify markers that are truly related to the outcome: Although all predictors had a relatively high pairwise correlation (), the four informative markers had a selection probability of 98.5% for (99% for and 99.5% for ).

To find the optimal combination of the pre-selected markers (Task 2), we applied the proposed boosting approach on training samples with size . The resulting coefficients for and smoothing parameter are presented in Figure 1. The boosting algorithm seems to be able to derive the optimal combination of the pre-selected markers, as the structure displayed by the coefficients is essentially the same as the one of the underlying true combination . The discriminatory power of the resulting biomarker does not depend on the absolute size of the coefficients: As the -index is based solely on the concordance between biomarker and survival time, what matters in practice is the relative size of the coefficients. As can be seen from Figure 1, the estimated positive effect for is larger than the one for . On the other hand, the negative effect of is correctly estimated to be more pronounced than the the one of . The coefficient of the falsely selected marker is on average close to zero.

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Figure 1. Coefficient estimates for pre-selected markers obtained from 100 simulation runs.

The marker combinations were optimized via gradient boosting based on training samples of size (left) and (right). Boxplots represent the empirical distribution of the resulting coefficients. Only markers to had an actual effect on the survival time.

https://doi.org/10.1371/journal.pone.0084483.g001

In a third step, we evaluated the performance of the resulting optimized marker combinations (Task 3) on separate test samples. The resulting estimates for different simulation settings are presented in Table 1. The highest discriminatory power (median , range  =  0.559–0.819) can be observed for , which is closest to the true number of informative markers. We compared the performance of our proposed algorithm to penalized Cox regression approaches such as Cox-Lasso [11], [12] and Cox regression with ridge-penalization [13], [14] – see Figure 2. The proposed boosting approach clearly outperforms the competing estimation schemes, supporting our view that applying traditional Cox regression might be suboptimal if the discriminatory power is the performance criterion of interest. We additionally computed the optimal -index resulting from the true marker combination with known coefficients. The values of the true -index on the test samples are on average only slightly better than the ones of boosting the concordance index (median – see Table 1).

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Figure 2. Simulation results for the discriminatory power obtained via the proposed -index boosting approach and competing Cox-based estimation schemes.

The marker combinations were optimized via the different approaches based on training samples of size (left) and (right). Boxplots represent the empirical distribution of the resulting on corresponding test samples. The dotted line corresponds to the discriminatory power resulting from the true combination of predictors with known coefficients.

https://doi.org/10.1371/journal.pone.0084483.g002

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Table 1. Results of the simulation study.

https://doi.org/10.1371/journal.pone.0084483.t001

To evaluate the possible effects of different sample sizes and censoring rates we modified the mean censoring time leading to approximate censoring rates of 30% and 70% and generated training samples of size . Results are included in Table 1. As expected, the -index resulting from our framework increases as censoring rates become small (median , range  =  0.736–0.858) and decreases in settings with a large proportion of censored observations (median , range  =  0.421–0.776). However, the same effect can be observed for the true -index resulting from the true marker combination (30% censoring , 70% censoring ). For larger training samples, the variance of the coefficient estimates decreases (see Figure 1). As a result, the discriminatory power resulting from our boosting algorithm improves (for , median , range  =  0.614–0.818) and gets nearly as good as the true -index (). This finding further confirms the ability of our approach to find the most optimal marker combination possible – see Figure 2. Note that also the true -index differs slightly between the different sample sizes, as the training sample enters in via the Kaplan-Meier estimator .

To investigate the effect of the smoothing parameter inside the sigmoid function, we additionally applied our boosting procedure for every simulation setting with different values of . The resulting estimates are presented in Table 2. Compared to the effects of the sample size or the number of pre-selected markers , the smoothing parameter only seems to have a minor effect on the performance of our algorithm. In light of these empirical results, we recommend to apply a fixed small value (e.g., , which is also the default value in the Cindex() family for the mboost package [37] – see Text S1).

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Table 2. Simulation results for different values of the smoothing parameter.

https://doi.org/10.1371/journal.pone.0084483.t002

For both approaches to fit penalized Cox regression (Cox lasso, Cox ridge), we applied the R add-on package penalized [43]. In order to evaluate , we used the UnoC() function implemented in the survAUC package [44].

Applications to predict the time to distant metastases

In the next step, we further analyzed the performance of our gradient boosting algorithm in two applications to estimate and evaluate the optimal combination of pre-selected biomarkers. All markers are used to predict the time to distant metastases in breast cancer patients. As in the simulation study, we compared the results of our proposed algorithm to Cox regression with lasso and ridge penalization. Additionally, we considered four competing boosting approaches for survival analysis, which do not directly optimize the -index. The first is classical Cox regression, estimated via gradient boosting, while the other three are parametric accelerated failure-time (AFT) models assuming a Weibull, log-normal or log-logistic distribution [45], [46]. For all boosting approaches (Weibull AFT boosting, loglog-AFT boosting and Cox boosting) we used the corresponding pre-implemented functions of the mboost package. To ensure comparability, we used the same linear base-learners as described above for all boosting approaches.

To ensure that the combined predictor did not only work on the data it was derived from but also on “external” validation data, we carried out a subsampling procedure for both data sets to generate different learning samples and test samples , respectively. Consequently, we randomly split the corresponding data sets to use of the observations as learning sample in order to optimize the biomarker combination . To ensure an equal distribution of patients with and without an observed development of distant metastases we applied stratified sampling. Correspondingly, the of the observations not included in the learning sample were used to evaluate the resulting predictions via the -index. As a result, for every data set and every method we computed 100 different combinations and 100 corresponding values of .

Breast cancer data by Desmedt et al

Desmedt et al. [1] collected a data set of 196 node-negative breast cancer patients to validate a 76-gene expression signature developed by Wang et al. [10]. The signature, which is based on Affymetrix microarrays, was developed separately for estrogen-receptor (ER) positive patients (60 genes) and ER-negative patients (16 genes). In addition to the expression levels of the 76 genes, four clinical predictor variables were considered (tumor size, estrogen receptor (ER) status, grade of the tumor and patient age). The data are publicly available on GEO (http://www.ncbi.nlm.nih.gov/geo, accession number GSE 7390).

Similar to Wang et al. [10], we used the time from diagnosis to distant metastases as primary outcome and considered the 76 genes together with the four clinical predictors. Observed metastasis-free survival ranged from 125 days to 3652 days, with 79.08% of the survival times being censored.

The main results of our analysis are presented in Figure 3. As expected, the unified framework to estimate and evaluate the optimal marker signature based on the -index is not only methodologically more consistent than the Cox and AFT approaches, but also leads to to marker signatures that show a higher discriminatory power on external or future data (median , range  =  0.467–0.854). As discussed in the methodological section, it is crucial to evaluate the discriminatory power on external data: the estimated -index on the training sample was more than 35% higher (median ) and hence extremely over-optimistic [25], [40].

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Figure 3. Comparing the discriminatory power of biomarker combinations to predict the time to distant metastases resulting from the proposed -index boosting approach with competing estimation schemes.

The plot on the left refers to the Desmedt et al. data, whereas the plot on the right presents results from the van de Vijver et al. data. All biomarker combinations were optimized via the corresponding algorithms based on the same 100 learning samples. Boxplots represent the empirical distribution of the resulting on corresponding test samples. The dotted line corresponds to the median -index resulting from the new approach.

https://doi.org/10.1371/journal.pone.0084483.g003

Considering the interpretation of the resulting coefficient estimates for the clinical predictors, it is crucial to note that boosting methods for the -index and the AFT models yield biomarker combinations where larger values indicate longer predicted survival times. On the other hand, classical Cox regression models rely on the hazard; higher values are hence associated with smaller survival times. If this is taken into account, results from the different approaches were in fact very similar. Both age of the patients and size of the tumor had a negative effect on the time to recurrence for all seven approaches. The same holds true for the tumor grade poor differentiation which resulted in a negative effect compared to good differentiation and intermediate differentiation. A positive ER status, on the other hand, was associated with a larger metastasis-free survival in all approaches. Regarding the coefficients of the 76 genes, results from our approach to boost the -index were highly correlated to the ones of the other four boosting approaches (which rely on the same base-learners) – absolute correlation coefficients computed from the 100 subsamples ranged from 0.77 to 0.99. Also coefficients resulting from the popular ridge-penalized Cox regression were highly correlated with the ones from our approach – absolute correlation coefficients ranged from 0.47 to 0.84.

Breast cancer data by van de Vijver et al

The second data set consists of lymph node positive breast cancer patients that was collected by the Netherlands Cancer Institute [2]. The data set, which is publicly available as part of the R add-on package penalized [43], was used by van de Vijver et al. [2] to validate a 70-gene signature for metastasis-free survival after surgery developed by van't Veer et al. [47]. In addition to the expression levels of the 70 genes, the data set contains five clinical predictor variables (tumor diameter, number of affected lymph nodes, ER status, grade of the tumor and patient age). Observed metastasis-free survival times ranged from months to months, with of the survival times being censored.

Resulting values of the -index of the new approach and the six considered competitors are presented in Figure 3. The improvement from applying the proposed unified framework compared to boosting the Cox proportional hazard model or applying ridge-penalized Cox regression was much less pronounced than in the previous data set. However, on average, boosting the -index still led to the best combination of markers regarding the discriminatory power (median , range  =  0.257–0.836). Interestingly, as in the previous data set, the lasso penalized Cox regression was clearly outperformed by the ridge-penalized competitor (which has been suggested for this specific data set by van Houwelingen et al. [48]). Furthermore, the ridge-penalized approach performed at least as good as the considered boosting approaches (except the new approach to boost the -index). As in the previous data set, we again additionally evaluated the -index on the training sample in order to assess the resulting over-optimism. As expected, the estimated -index on the training sample was extremely biased (median ).

The resulting coefficients for the clinical predictors were again comparable for the seven different approaches. A positive ER status was associated with a larger metastasis-free survival for all seven approaches, the same also holds true for the age of the patient. On the other hand, the size of the tumor, the number of affected lymph nodes and a poor tumor grade resulted for all approaches in a negative effect on the survival time. Regarding the coefficients of the 79 genes, the highest correlation could again be observed for the boosting algorithms: Absolute correlation coefficients obtained from the 100 subsamples ranged from 0.64 to 0.95. Correlation between coefficients resulting from our approach to boost the -index and the ones from ridge-penalized Cox regression was slightly lower, it ranged from 0.30 to 0.82.