Machine Learning for Survival Analysis: A Survey <s> C-index <s> A method is presented for evaluating the amount of information a medical test provides about individual patients. Emphasis is placed on the role of a test in the evaluation of patients with a chronic disease. In this context, the yield of a test is best interpreted by analyzing the prognostic information it furnishes. Information from the history, physical examination, and routine procedures should be used in assessing the yield of a new test. As an example, the method is applied to the use of the treadmill exercise test in evaluating the prognosis of patients with suspected coronary artery disease. The treadmill test is shown to provide surprisingly little prognostic information beyond that obtained from basic clinical measurements. <s> BIB001 </s> Machine Learning for Survival Analysis: A Survey <s> C-index <s> Regression models such as the Cox proportional hazards model have had increasing use in modelling and estimating the prognosis of patients with a variety of diseases. Many applications involve a large number of variables to be modelled using a relatively small patient sample. Problems of overfitting and of identifying important covariates are exacerbated in analysing prognosis because the accuracy of a model is more a function of the number of events than of the sample size. We used a general index of predictive discrimination to measure the ability of a model developed on training samples of varying sizes to predict survival in an independent test sample of patients suspected of having coronary artery disease. We compared three methods of model fitting: (1) standard 'step-up' variable selection, (2) incomplete principal components regression, and (3) Cox model regression after developing clinical indices from variable clusters. We found regression using principal components to offer superior predictions in the test sample, whereas regression using indices offers easily interpretable models nearly as good as the principal components models. Standard variable selection has a number of deficiencies. <s> BIB002 </s> Machine Learning for Survival Analysis: A Survey <s> C-index <s> The assessment of the discrimination ability of a survival analysis model is a problem of considerable theoretical interest and important practical applications. This issue is, however, more complex than evaluating the performance of a linear or logistic regression. Several different measures have been proposed in the biostatistical literature. In this paper we investigate the properties of the overall C index introduced by Harrell as a natural extension of the ROC curve area to survival analysis. We develop the overall C index as a parameter describing the performance of a given model applied to the population under consideration and discuss the statistic used as its sample estimate. We discover a relationship between the overall C and the modified Kendall's τand construct a confidence interval for our measure based on the asymptotic normality of its estimate. Then we investigate via simulations the length and coverage probability of this interval. Finally, we present a real life example evaluating the performance of a Framingham Heart Study model. Copyright © 2004 John Wiley & Sons, Ltd. <s> BIB003 </s> Machine Learning for Survival Analysis: A Survey <s> C-index <s> In this paper, we show that classical survival analysis involving censored data can naturally be cast as a ranking problem. The concordance index (CI), which quantifies the quality of rankings, is the standard performance measure for model assessment in survival analysis. In contrast, the standard approach to learning the popular proportional hazard (PH) model is based on Cox's partial likelihood. We devise two bounds on CI-one of which emerges directly from the properties of PH models-and optimize them directly. Our experimental results suggest that all three methods perform about equally well, with our new approach giving slightly better results. We also explain why a method designed to maximize the Cox's partial likelihood also ends up (approximately) maximizing the CI. <s> BIB004 </s> Machine Learning for Survival Analysis: A Survey <s> C-index <s> Predicting the occurrence of a particular event of interest at future time points is the primary goal of survival analysis. The presence of incomplete observations due to time limitations or loss of data traces is known as censoring which brings unique challenges in this domain and differentiates survival analysis from other standard regression methods. The popularly used survival analysis methods such as Cox proportional hazard model and parametric survival regression suffer from some strict assumptions and hypotheses that are not realistic in most of the real-world applications. To overcome the weaknesses of these two types of methods, in this paper, we reformulate the survival analysis problem as a multi-task learning problem and propose a new multi-task learning based formulation to predict the survival time by estimating the survival status at each time interval during the study duration. We propose an indicator matrix to enable the multi-task learning algorithm to handle censored instances and incorporate some of the important characteristics of survival problems such as non-negative non-increasing list structure into our model through max-heap projection. We employ the L2,1-norm penalty which enables the model to learn a shared representation across related tasks and hence select important features and alleviate over-fitting in high-dimensional feature spaces; thus, reducing the prediction error of each task. To efficiently handle the two non-smooth constraints, in this paper, we propose an optimization method which employs Alternating Direction Method of Multipliers (ADMM) algorithm to solve the proposed multi-task learning problem. We demonstrate the performance of the proposed method using real-world microarray gene expression high-dimensional benchmark datasets and show that our method outperforms state-of-the-art methods. <s> BIB005
In survival analysis, a common way to evaluate a model is to consider the relative risk of an event occurring for different instances rather than the absolute survival times for each instance. This can be done by computing the concordance probability or the concordance index (C-index) BIB002 BIB001 BIB003 . The survival times for two instances can be ordered for two scenarios: (i) both are uncensored, and (ii) the observed event time of the uncensored instance is smaller than the censoring time of the censored instance BIB004 ). This can be visualized by the ordered graph shown in Figure 4. Figure 4 (a) and Figure 4 (b) are used to illustrate the possible ranking comparisons (denoted by edges between instances) for the survival data without and with censored instances, respectively. There are 5 2 = 10 possible pairwise comparisons for the five instances in the survival data without censored cases shown in Figure 4(a) . Due to the presence of censored instances (represented by red circles) in Figure 4(b) , only 6 out of the 10 comparisons are feasible. Note that a censored instance can only be compared with an earlier uncensored instance (for example y 2 &y 1 ). However, any censored instance cannot be compared with both censored and uncensored instances after its censored time (e.g., y 2 &y 3 and y 2 &y 4 ) since its actual event time is unknown. Consider both the observations and prediction values for two instances, (y 1 ,ŷ 1 ) and (y 2 ,ŷ 2 ), where y i andŷ i represent the actual observation time and the predicted value, respectively. The concordance probability between them can be computed as c = Pr (ŷ 1 >ŷ 2 |y 1 ≥ y 2 ). ( Applying this definition, for the binary prediction problem, the C-index will have a similar meaning to the regular area under the ROC curve (AUC), and if y i is binary, then the C-index is the AUC BIB005 ). As the preceding definition is not straightforward, in practice, there are multiple ways of calculating the C-index. (1) When the output of the model is a hazard ratio (such as the outcome obtained by Cox based models), the C-index can be computed usinĝ where i, j ∈ {1, · · · , N }, num denotes the number of all comparable pairs, I [·] is the indicator function, andβ represents the estimated parameters from the Cox based models. (2) For survival methods which aim to directly learn the survival time, the C-index should be calculated as:ĉ where S (·) corresponds to the estimated survival probabilities.