Machine Learning for Survival Analysis: A Survey <s> TRADITIONAL STATISTICAL METHODS <s> Abstract In lifetesting, medical follow-up, and other fields the observation of the time of occurrence of the event of interest (called a death) may be prevented for some of the items of the sample by the previous occurrence of some other event (called a loss). Losses may be either accidental or controlled, the latter resulting from a decision to terminate certain observations. In either case it is usually assumed in this paper that the lifetime (age at death) is independent of the potential loss time; in practice this assumption deserves careful scrutiny. Despite the resulting incompleteness of the data, it is desired to estimate the proportion P(t) of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t). The observation for each item of a suitable initial event, marking the beginning of its lifetime, is presupposed. For random samples of size N the product-limit (PL) estimate can be defined as follows: L... <s> BIB001 </s> Machine Learning for Survival Analysis: A Survey <s> TRADITIONAL STATISTICAL METHODS <s> Measurement of patient survival is necessary for the evaluation of treatment of usually fatal chronic diseases. This is particularly true for cancer. The American College of Surgeons, recognizing this, requires the maintenance of a cancer case registration and follow-up program for approval of a hospital cancer program [1], Acceptance of survival as a criterion for measuring the effectiveness of cancer therapy is also attested to by the very large number of papers published every year reporting on the survival experience of cancer patients. <s> BIB002 </s> Machine Learning for Survival Analysis: A Survey <s> TRADITIONAL STATISTICAL METHODS <s> Modern survival analysis and more general event history analysis may be effectively handled in the mathematical framework of counting processes, stochastic integration, martingale central limit theory and product integration. This book presents this theory, which has been the subject of an intense research activity during the past one-and-a-half decades. The exposition of the theory is integrated with the careful presentation of many practical examples, based almost exlusively on the authors' experience, with detailed numerical and graphical illustrations. "Statistical Models Based on Counting Processes" may be viewed as a research monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories (demographers, econometricians, epidemiologists, actuariala mathematicians, reliability engineers, biologists). Much of the material has so far only been available in the journal literature (if at all), and a wide variety of researchers will find this an invlauable survey of the subject. <s> BIB003 </s> Machine Learning for Survival Analysis: A Survey <s> TRADITIONAL STATISTICAL METHODS <s> Introduction to Survival Analysis.- Kaplan-Meier Survival Curves and the Log-Rank Test.- The Cox Proportional Hazards Model and Its Characteristics.- Evaluating the Proportional Hazards Assumption.- The Stratified Cox Procedure.- Extension of the Cox Proportional Hazards Model for Time-Dependent Variables.- Parametric Survival Models.- Recurrent Events Survival Analysis.- Competing Risks Survival Analysis. <s> BIB004
In this section, we introduce three different types of statistical methods that are commonly used to estimate the survival/hazard functions: non-parametric, semi-parametric, and parametric methods. Table 2 shows both the advantages and disadvantages of each type based on theoretical and experimental analysis and lists the specific methods included. The detailed characteristics of each statistical survival algorithm are summarized in Supplemental Table 1 , including the assumptions made in each model, the optimization techniques used, and its computational complexity. Non-parametric methods are more efficient when there is no underlying distribution for the event time or the proportional hazard assumption does not hold. In non-parametric methods, an empirical estimate of the survival function is obtained using the Kaplan-Meier (KM) method, Nelson-Aalen (NA) estimator, or Life- Table ( LT) method. More generally, any KM estimator for the survival probability at the specified survival time will be a product of the same estimate up to the previous time and the observed survival rate for that given time. Thus, the KM method is also sometimes referred to as a product-limit method BIB001  . The NA method is an estimator based on modern counting process techniques BIB003 , while the LT method BIB002 is the application of the KM method to the interval grouped survival data. In the semi-parametric category, the Cox model is the most commonly used regression analysis approach for survival data. This differs significantly from other methods since it is built on the proportional hazards assumption and employs partial likelihood for the parameter estimation. Cox regression is described as a semi-parametric method since the distribution of the outcome remains unknown even if it is based on a parametric regression model. Several useful variants of the basic Cox model, such as the penalized Cox models (Zhang and Lu 2007), the CoxBoost algorithm (Binder have also been proposed in the literature. Parametric methods are more efficient and accurate for estimation when the time to the event of interest follows a particular distribution that can be specified in terms of certain parameters. It is relatively easy to estimate the time to the event of interest with parametric models, but it becomes awkward or even impossible to do so with the Cox model (Allison 2010). Linear regression is one of the main parametric survival methods, while the Tobit model, Buckley-James regression model, and penalized regression are the most commonly used linear models for survival analysis. Other parametric models such as Accelerated Failure Time (AFT), which models the survival time as a function of covariates BIB004 , are also widely used. These will be described in turn in this section.