Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> SUMMARY We give a method of estimating parameters in the linear regression model which allows the dependent variable to be censored and the residual distribution to be unspecified. The method differs from that of Miller (1976) in that the normal equations rather than the sum of squares of residuals are modified and this appears to overcome the inconsistency problems in Miller's approach. Large sample properties of the estimator of slope are derived heuristically and substantiated by simulations. Some of the heart transplant data reported and analysed by Miller are reanalysed using the present method. <s> BIB001 </s> Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> SUMMARY There are four regression techniques currently available for use with censored data which do not assume particular parametric families of survival distributions. They are due to (i) Cox (1972), (ii) Miller (1976), (iii) Buckley & James (1979), and (iv) Koul, Susarla & Van Ryzin (1981). These four methods are described, and their performances compared on the updated Stanford heart transplant data. Conclusions on the usefulness of the four procedures are drawn. <s> BIB002 </s> Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> Recent interest in cancer research focuses on predicting patients' survival by investigating gene Recent interest in cancer research focuses on predicting patients' survival by investigating gene expression profiles based on microarray analysis. We propose a doubly penalized Buckley-James method for the semiparametric accelerated failure time model to relate high-dimensional genomic data to censored survival outcomes, which uses the elastic-net penalty that is a mixture of L 1 - and L 2 -norm penalties. Similar to the elastic-net method for a linear regression model with uncensored data, the proposed method performs automatic gene selection and parameter estimation, where highly correlated genes are able to be selected (or removed) together. The two-dimensional tuning parameter is determined by generalized crossvalidation. The proposed method is evaluated by simulations and applied to the Michigan squamous cell lung carcinoma study. <s> BIB003 </s> Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> Penalized regression methods for simultaneous variable selection and coe-cient estimation, especially those based on the lasso of Tibshirani (1996), have received a great deal of attention in recent years, mostly through frequen- tist models. Properties such as consistency have been studied, and are achieved by difierent lasso variations. Here we look at a fully Bayesian formulation of the problem, which is ∞exible enough to encompass most versions of the lasso that have been previously considered. The advantages of the hierarchical Bayesian for- mulations are many. In addition to the usual ease-of-interpretation of hierarchical models, the Bayesian formulation produces valid standard errors (which can be problematic for the frequentist lasso), and is based on a geometrically ergodic Markov chain. We compare the performance of the Bayesian lassos to their fre- quentist counterparts using simulations, data sets that previous lasso papers have used, and a di-cult modeling problem for predicting the collapse of governments around the world. In terms of prediction mean squared error, the Bayesian lasso performance is similar to and, in some cases, better than, the frequentist lasso. <s> BIB004 </s> Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the $\ell_1$-norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the $\ell_1$-norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of non-linear variable selection. <s> BIB005 </s> Machine Learning for Survival Analysis: A Survey <s> Linear Regression Models. <s> Censoring is a common phenomenon that arises in many longitudinal studies where an event of interest could not be recorded within the given time frame. Censoring causes missing time-to-event labels, and this effect is compounded when dealing with datasets which have high amounts of censored instances. In addition, dependent censoring in the data, where censoring is dependent on the covariates in the data leads to bias in standard survival estimators. This motivates us to develop an approach for pre-processing censored data which calibrates the right censored (RC) times in an attempt to reduce the bias in the survival estimators. This calibration is done using an imputation method which estimates the sparse inverse covariance matrix over the dataset in an iterative convergence framework. During estimation, we apply row and column-based regularization to account for both row and column-wise correlations between different instances while imputing them. This is followed by comparing these imputed censored times with the original RC times to obtain the final calibrated RC times. These calibrated RC times can now be used in the survival dataset in place of the original RC times for more effective prediction. One of the major benefits of our calibration approach is that it is a pre-processing method for censored data which can be used in conjunction with any survival prediction algorithm and improve its performance. We evaluate the goodness of our approach using a wide array of survival prediction algorithms which are applied over crowdfunding data, electronic health records (EHRs), and synthetic censored datasets. Experimental results indicate that our calibration method improves the AUC values of survival prediction algorithms, compared to applying them directly on the original survival data. <s> BIB006
In data analysis, the linear regression model, together with the least squares estimation method, represents one of the most commonly used approaches. However, it cannot be applied directly to solve survival analysis problems since the actual event times are missing for censored instances. A number of linear models BIB002 BIB001 BIB003  , including Tobit and Buckley-James (BJ) regressions, have been proposed to handle censored instances in survival analysis. Strictly speaking, although a linear regression is a specific parametric censored regression, this method is fundamental for many different types of data analysis, and hence linear regression methods for censored data will be discussed se parately here. Tobit Regression: The Tobit model  ) is one of the earliest attempts to extend a linear regression with a Gaussian distribution for data analysis with censored observations. In this model, a latent variable y * is introduced and the assumption made is that this depends linearly on X via the parameter β as y * = X β + ϵ, ϵ ∼ N (0, σ 2 ), where ϵ is a normally distributed error term. Then, for the ith instance, the observable variable y i will be y * i if y * i > 0; otherwise, it will be 0. This means that if the latent variable is above zero, the observed variable equals the latent variable and is zero otherwise. Based on the latent variable, the parameters in the model can be estimated using an MLE method. Buckley-James Regression: The Buckley-James (BJ) regression (Buckley and James 1979) estimates the survival time of censored instances in terms of the response value based on the KM estimation method, and then fits a linear (AFT) model by simultaneously considering the survival times of uncensored instances and the approximate survival times of the censored instances. To handle high-dimensional survival data, Wang et al. BIB003 applied the elastic net regularizer in the BJ regression (EN-BJ). Penalized Regression: Penalized regression methods BIB004 ) are best known for their useful property of simultaneous variable selection and coefficient estimation. A penalized regression method can provide better prediction results in the presence of either multicollinear covariates or high dimensionality. Recently, these methods have received a great deal of attention in survival analysis. For example, a weighted linear regression model with different regularizers for high-dimensional censored data is an efficient way to handle censored data by assigning different weights to different instances  ). In addition, a structured regularization-based linear regression algorithm BIB005 BIB006 ) for right-censored data can be used to infer the underlying structure of the survival data. -Weighted Regression: The weighted regression method  ) can be used when the constant variance assumption about the errors in the ordinary least squares regression methods is violated (i.e., heteroscedasticity), which is different from a constant variance in the errors (i.e., homoscedasticity) in ordinary least squares regression methods. Instead of minimizing the residual sum of squares, the weighted regression method minimizes the weighted sum of squares n i=1 w i (y i − X i β ) 2 ; an ordinary least squares method is a special case of this where all the weights w i = 1. A weighted regression method can be solved in the same manner as an ordinary linear least squares problem with a time complexity of O (N P ). The weighted regression method enables us to assign higher weights to instances that we want to emphasize or those where mistakes are especially costly. If we give selected samples high weights, the model will be pulled toward matching the data for those samples. This is very helpful for survival analysis as it allows us to place more emphasis on the instances whose information may contribute most to the model. -Structured Regularization: The ability to effectively infer latent knowledge through treebased hierarchies and graph-based relationships is crucial in survival analysis. This is also supported by the effectiveness of structured sparsity-based regularization methods in regression BIB005 . The Structured regularization based LInear REgression algorithm for right Censored data (SLIREC) BIB006 infers the underlying structure of the survival data directly using a Sparse Inverse Covariance Estimation (SICE) method and then applies this structural knowledge to guide the base linear regression model. The structured approach is more robust than standard statistical or Cox-based methods since it can automatically adapt to different distributions of events and censored instances.