Term Survival Function Research Articles

A Framework for Leveraging Machine Learning Tools to Estimate Personalized Survival Curves

The conditional survival function of a time-to-event outcome subject to censoring and truncation is a common target of estimation in survival analysis. This parameter may be of scientific interest and also often appears as a nuisance in nonparametric and semiparametric problems. In addition to classical parametric and semiparametric methods (e.g., based on the Cox proportional hazards model), flexible machine learning approaches have been developed to estimate the conditional survival function. However, many of these methods are either implicitly or explicitly targeted toward risk stratification rather than overall survival function estimation. Others apply only to discrete-time settings or require inverse probability of censoring weights, which can be as difficult to estimate as the outcome survival function itself. Here, we employ a decomposition of the conditional survival function in terms of observable regression models in which censoring and truncation play no role. This allows application of an array of flexible regression and classification methods rather than only approaches that explicitly handle the complexities inherent to survival data. We outline estimation procedures based on this decomposition, empirically assess their performance, and demonstrate their use on data from an HIV vaccine trial. Supplementary materials for this article are available online.

Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution

Considering the impact of the heterogeneous conditions of the mixture baseline distribution on the parameter estimation of a composite dynamical system (CDS), we propose an approach to infer the model parameters and baseline survival function of CDS using the maximum likelihood estimation and Bayesian estimation methods. The power-trend hazard rate function and Burr type XII mixture distribution as the baseline distribution are used to characterize the changes of the residual lifetime distribution of surviving components. The Markov chain Monte Carlo approach via using a new Metropolis–Hastings within the Gibbs sampling algorithm is proposed to overcome the computation complexity when obtaining the Bayes estimates of model parameters. A numerical example is generated from the proposed CDS to analyze the proposed procedure. Monte Carlo simulations are conducted to investigate the performance of the proposed methods, and results show that the proposed Bayesian estimation method outperforms the maximum likelihood estimation method to obtain reliable estimates of the model parameters and baseline survival function in terms of the bias and mean square error.

Regression Models and Multivariate Life Tables

Semiparametric, multiplicative-form regression models are specified for marginal single and double failure hazard rates for the regression analysis of multivariate failure time data. Cox-type estimating functions are specified for single and double failure hazard ratio parameter estimation, and corresponding Aalen–Breslow estimators are specified for baseline hazard rates. Generalization to allow classification of failure times into a smaller set of failure types, with failures of the same type having common baseline hazard functions, is also included. Asymptotic distribution theory arises by generalization of the marginal single failure hazard rate estimation results of Lin et al. The Péano series representation for the bivariate survival function in terms of corresponding marginal single and double failure hazard rates leads to novel estimators for pairwise bivariate survival functions and pairwise dependency functions, at specified covariate history. Related asymptotic distribution theory follows from that for the marginal single and double failure hazard rates and the continuity, compact differentiability of the Péano series transformation and bootstrap applicability. Simulation evaluation of the proposed estimation procedures is presented, and an application to multiple clinical outcomes in the Women’s Health Initiative Dietary Modification Trial is provided. Higher dimensional marginal hazard rate regression modeling is briefly mentioned. Supplementary materials for this article are available online.

Risk factors and outcome analysis after surgical management of ventricular septal rupture complicating acute myocardial infarction: a retrospective analysis.

BackgroundVentricular septal rupture (VSR) is an uncommon but well-recognized mechanical complication of acute myocardial infarction (AMI). The outcome of VSR remains poor even in the era of reperfusion therapy. We reviewed our experience with surgical repair of post-infarction VSR and analyzed outcomes in an attempt to identify prognostic factors.MethodsFrom October 1995 to December 2013, data from 47 consecutive patients (mean age, 68 ± 9.5 years) with post-infarction VSR who underwent surgical repair at our institute were retrospectively reviewed. The preoperative conditions, morbidity and surgical mortality were analyzed. Multivariate analysis was subsequently carried out by constructing a logistic regression model in order to identify independent predictors of postoperative mortality. Long term survival function were estimated using the Kaplan-Meier method and compared using the log-rank test.ResultsPercutaneous coronary intervention was performed in 17 (36.2%) patients, intra-aortic balloon pump (IABP) was used in 34 (72.3%), and six (12.8%) were supported with extracorporeal membrane oxygenation (ECMO) preoperatively. Forty-one (87.2%) patients received emergent surgical treatment. Concomitant coronary artery bypass grafting was performed in 27 (57.4%) patients.Operative mortality was 36.2% (17 of 47). The survival rate was 59.3% with concomitant CABG and 70% without concomitant CABG (p = 14). Multivariate analysis revealed that the survivors had higher preoperative left ventricular ejection fractions (LVEFs) compared with those who died (51 ± 13.7% vs. 36.6 ± 6.4% , respectively; p < 0.001) and lower European system for cardiac operative risk evaluation II (EuroSCORE II) (22.9 ± 14.9 vs. 38.3 ± 13.9, respectively; p < 0.001). The patients receiving total revascularization has long term survival benefit (p = 0.028).ConclusionsPost-infarction VSR remains a serious and challenging complication of AMI in the modern surgical era. The EuroSCORE II can be used for an approximate prediction of operative mortality. Preserved LVEF was associated with better prognosis, while the need for postoperative RRT was associated with higher early and late mortality. Besides, the strategy of total revascularization should be applied to ensure long-term survival benefit.

Finite-time survival probability and credit default swaps pricing under geometric Lévy markets

We study the first-passage time over a fixed threshold for a pure-jump subordinator with negative drift. We obtain a closed-form formula for its survival function in terms of marginal density functions of the subordinator. We then use this formula to calculate finite-time survival probabilities in a structural model for credit risk, and thus obtain a closed-form pricing formula for a single-name credit default swap (CDS). This pricing formula is well calibrated on market CDS quotes. In particular, it explains why the par CDS credit spread is not negligible when the maturity becomes short.

The second-order version of Karamata’s theorem with applications

Karamata’s theorem is well known, which examines the integral properties of regular variation functions. In this paper, we obtain the second-order version of Karamata’s theorem, and give its one application in characterizing the second-order regular variation property of a survival function in terms of conditional moments.

Assessing effects on long-term survival after early termination of randomized trials.

In this paper we present an approach to using historical control data to augment information from a randomized controlled clinical trial, when it is not possible to continue the control regimen to obtain the most reliable and valid assessment of long term treatment effects. Using an adjustment procedure to the historical control data, we investigate a method of estimating the long term survival function for the clinical trial control group and for evaluating the long term treatment effect. The suggested method is simple to interpret, and particularly motivated in clinical trial settings when ethical considerations preclude the long term follow-up of placebo controls. A simulation study reveals that the bias in parameter estimates that arises in the setting of group sequential monitoring will be attenuated when long term historical control information is used in the proposed manner. Data from the first and second National Wilms' Tumor studies are used to illustrate the method.

General characterization theorems via the mean absolute deviation

The Chernoff–Borovkov–Utev inequality resulted owing to earlier inequalities established by Chernoff (1981) and Borovkov and Utev (1983), respectively, giving bounds for the variance of functions of normal r.v.’s and leading to characterizations of normality. Subsequently, several analytic properties of variance bounds and other relevant results were established by others. Defining the mean absolute deviation (about a median) as E| X−med( X)| where med( X) is a median of the distribution of the random variable X, Freimer and Mudholkar (1991) gave a bound for the mean absolute deviation of a certain real-valued function of an absolutely continuous random variable (w.r.t. Lebesgue measure) and Korwar (1991) presented an analogue of this in the discrete case; these authors, also, characterized the Laplace and a mixture of two Waring distributions via the respective bounds. We extend these latter results theorems to the case where the distributions are not necessarily purely discrete or absolutely continuous, via the approach of Alharbi and Shanbhag (1996). The results in Freimer and Mudholkar (1991) and Korwar (1991) are now corollaries to our findings. Also, following Alharbi and Shanbhag (1996), we relate these results to the modified version of Cox’s representation for a survival function in terms of the hazard measure, given in Kotz and Shanbhag (1980). (The original version of the representation mentioned had appeared in Cox (1972).)

Expressing the Kaplan-Meier Estimator as a Function of Empirical Subsurvival Functions

Abstract The Kaplan-Meier estimator for the survival function in the censored data problem can be expressed for finite samples as an explicit function of two empirical subsurvival functions. This function is the natural one that expresses the survival function in terms of the sub-survival functions. As an illustration of the usefulness of expressing the Kaplan-Meier estimator in this way, the strong consistency of the Kaplan-Meier estimator is easily proved.