Survival data is encountered in a range of disciplines, most notably health and medical research. Although Bayesian approaches to the analysis of survival data can provide a number of benefits, they are less widely used than classical (e.g. likelihood-based) approaches. This may be in part due to a relative absence of user-friendly implementations of Bayesian survival models. In this article we describe how the rstanarm R package can be used to fit a wide range of Bayesian survival models. The rstanarm package facilitates Bayesian regression modelling by providing a user-friendly interface (users specify their model using customary R formula syntax and data frames) and using the Stan software (a C++ library for Bayesian inference) for the back-end estimation. The suite of models that can be estimated using rstanarm is broad and includes generalised linear models (GLMs), generalised linear mixed models (GLMMs), generalised additive models (GAMs) and more. In this article we focus only on the survival modelling functionality. This includes standard parametric (exponential, Weibull, Gompertz) and flexible parametric (spline-based) hazard models, as well as standard parametric accelerated failure time (AFT) models. All types of censoring (left, right, interval) are allowed, as is delayed entry (left truncation), time-varying covariates, time-varying effects, and frailty effects. We demonstrate the functionality through worked examples. We anticipate these implementations will increase the uptake of Bayesian survival analysis in applied research.
Applications of machine learning in healthcare often require working with time-to-event prediction tasks including prognostication of an adverse event, re-hospitalization or death. Such outcomes are typically subject to censoring due to loss of follow up. Standard machine learning methods cannot be applied in a straightforward manner to datasets with censored outcomes. In this paper, we present auton-survival, an open-source repository of tools to streamline working with censored time-to-event or survival data. auton-survival includes tools for survival regression, adjustment in the presence o…
PyDTS: A Python Package for Discrete-Time Survival Analysis with Competing Risks and Optional Penalization
Faculty of Data and Decision Sciences, Technion - Israel Institute of Technology（数据与决策科学学院，技术离子-以色列理工学院）；Department of Statistics and Operations Research, Tel Aviv University（统计与运筹学系，特拉维夫大学）
Spatial survival analysis has received a great deal of attention over the last 20 years due to the important role that geographical information can play in predicting survival. This paper provides an introduction to a set of programs for implementing some Bayesian spatial survival models in R using the package spBayesSurv. The function survregbayes includes the three most commonly-used semiparametric models: proportional hazards, proportional odds, and accelerated failure time. All manner of censored survival times are simultaneously accommodated including uncensored, interval censored, curren…
survex: an R package for explaining machine learning survival models
MI2.AI；Warsaw University of Technology（华沙理工大学）；Leibniz Institute for Prevention Research and Epidemiology – BIPS（莱布尼茨预防研究与流行病学研究所——BIPS）；Faculty of Mathematics and Computer Science, University of Bremen（不来梅大学数学与计算机科学学院）；University of Warsaw（华沙大学）；University of Copenhagen（哥本哈根大学）
针对机器学习生存模型缺乏易用解释工具的问题，本文提出 survex R 包，用可解释人工智能方法统一解释任意生存模型，支持模型诊断、可靠性评估和偏差检测。
In comparative research on time-to-event data for two groups, when two survival curves cross each other, it may be difficult to use the log-rank test and hazard ratio (HR) to properly assess the treatment benefit. Our aim was to identify a method for evaluating the treatment benefits for two groups in the above situation. We quantified treatment benefits based on an intuitive measure called the area between two survival curves (ABS), which is a robust measure of treatment benefits in clinical trials regardless of whether the proportional hazards assumption is violated or two survival curves cr…
orthoDr is a package in R that solves dimension reduction problems using orthogonality constrained optimization approach. The package serves as a unified framework for many regression and survival analysis dimension reduction models that utilize semiparametric estimating equations. The main computational machinery of orthoDr is a first-order algorithm developed by \cite{wen2013feasible} for optimization within the Stiefel manifold. We implement the algorithm through Rcpp and OpenMP for fast computation. In addition, we developed a general-purpose solver for such constrained problems with user-…
In epidemiological studies of time-to-event data, a quantity of interest to the clinician and the patient is the risk of an event given a covariate profile. However, methods relying on time matching or risk-set sampling (including Cox regression) eliminate the baseline hazard from the likelihood expression or the estimating function. The baseline hazard then needs to be estimated separately using a non-parametric approach. This leads to step-wise estimates of the cumulative incidence that are difficult to interpret. Using case-base sampling, Hanley & Miettinen (2009) explained how the parametr…
Missing data occur in many types of studies and typically complicate the analysis. Multiple imputation, either using joint modelling or the more flexible fully conditional specification approach, are popular and work well in standard settings. In settings involving non-linear associations or interactions, however, incompatibility of the imputation model with the analysis model is an issue often resulting in bias. Similarly, complex outcomes such as longitudinal or survival outcomes cannot be adequately handled by standard implementations. In this paper, we introduce the R package JointAI, whic…
Methods for the evaluation of the predictive accuracy of biomarkers with respect to survival outcomes subject to right censoring have been discussed extensively in the literature. In cancer and other diseases, survival outcomes are commonly subject to interval censoring by design or due to the follow up schema. In this paper, we present an estimator for the area under the time-dependent receiver operating characteristic ROC curve for interval censored data based on a nonparametric sieve maximum likelihood approach. We establish the asymptotic properties of the proposed estimator, and illustrat…
Factorial analyses offer a powerful nonparametric means to detect main or interaction effects among multiple treatments. For survival outcomes, e.g. from clinical trials, such techniques can be adopted for comparing reasonable quantifications of treatment effects. The key difficulty to solve in survival analysis concerns the proper handling of censoring. So far, all existing factorial analyses for survival data were developed under the independent censoring assumption, which is too strong for many applications. As a solution, the central aim of this article is to develop new methods in factori…
While analysing time-to-event data, it is possible that a certain fraction of subjects will never experience the event of interest and they are said to be cured. When this feature of survival models is taken into account, the models are commonly referred to as cure models. In the presence of covariates, the conditional survival function of the population can be modelled by using cure model which depends on the probability of being uncured (incidence) and the conditional survival function of the uncured subjects (latency), and a combination of logistic regression and Cox proportional hazards (P…
Survival regression is used to estimate the relation between time-to-event and feature variables, and is important in application domains such as medicine, marketing, risk management and sales management. Nonlinear tree based machine learning algorithms as implemented in libraries such as XGBoost, scikit-learn, LightGBM, and CatBoost are often more accurate in practice than linear models. However, existing state-of-the-art implementations of tree-based models have offered limited support for survival regression. In this work, we implement loss functions for learning accelerated failure time (A…
Survival analysis is one of the most important fields of statistics in medicine and the biological sciences. In addition, the computational advances in the last decades have favoured the use of Bayesian methods in this context, providing a flexible and powerful alternative to the traditional frequentist approach. The objective of this paper is to summarise some of the most popular Bayesian survival models, such as accelerated failure time, proportional hazards, mixture cure, competing risks, frailty, and joint models of longitudinal and survival data. Moreover, an implementation of each presen…
Pathologic complete response (pCR) is a common primary endpoint for a phase II trial or even accelerated approval of neoadjuvant cancer therapy. If granted, a two-arm confirmatory trial is often required to demonstrate the efficacy with a time-to-event outcome such as overall survival. However, the design of a subsequent phase III trial based on prior information on the pCR effect is not straightforward. Aiming at designing such phase III trials with overall survival as primary endpoint using pCR information from previous trials, we consider a mixture model that incorporates both the survival …
We introduce a framework to build a survival/risk bump hunting model with a censored time-to-event response. Our Survival Bump Hunting (SBH) method is based on a recursive peeling procedure that uses a specific survival peeling criterion derived from non/semi-parametric statistics such as the hazards-ratio, the log-rank test or the Nelson-Aalen estimator. To optimize the tuning parameter of the model and validate it, we introduce an objective function based on survival or prediction-error statistics, such as the log-rank test and the concordance error rate. We also describe two alternative cro…
Longitudinal and high-dimensional measurements have become increasingly common in biomedical research. However, methods to predict survival outcomes using covariates that are both longitudinal and high-dimensional are currently missing. In this article, we propose penalized regression calibration (PRC), a method that can be employed to predict survival in such situations. PRC comprises three modeling steps: First, the trajectories described by the longitudinal predictors are flexibly modeled through the specification of multivariate mixed effects models. Second, subject-specific summaries of t…
Survival analysis of right censored data arises often in many areas of research including medical research. Effect of covariates (and their interactions) on survival distribution can be studied through existing methods which requires to pre-specify the functional form of the covariates including their interactions. Survival trees offer relatively flexible approach when the form of covariates' effects is unknown. Most of the currently available survival tree construction techniques are not based on a formal test of significance; however, recently proposed ctree algorithm (Hothorn et al., 2006) …
We propose the interval censored recursive forests (ICRF) which is an iterative tree ensemble method for interval censored survival data. This nonparametric regression estimator makes the best use of censored information by iteratively updating the survival estimate, and can be viewed as a self-consistent estimator with convergence monitored using out-of-bag samples. Splitting rules optimized for interval censored data are developed and kernel-smoothing is applied. The ICRF displays the highest prediction accuracy among competing nonparametric methods in most of the simulations and in an appli…
We propose a novel frailty model with change points applying random effects to a Cox proportional hazard model to adjust the heterogeneity between clusters. Because the frailty model includes random effects, the parameters are estimated using the expectation-maximization (EM) algorithm. Additionally, our model needs to estimate change points; we thus propose a new algorithm extending the conventional estimation algorithm to the frailty model with change points to solve the problem. We show a practical example to demonstrate how to estimate the change point and random effect. Our proposed model…
We overview Bayesian estimation, hypothesis testing, and model-averaging and illustrate how they benefit parametric survival analysis. We contrast the Bayesian framework to the currently dominant frequentist approach and highlight advantages, such as seamless incorporation of historical data, continuous monitoring of evidence, and incorporating uncertainty about the true data generating process. We illustrate the application of the Bayesian approaches on an example data set from a colon cancer trial. We compare the Bayesian parametric survival analysis and frequentist models with AIC/BIC model…
We propose a class of two-sample statistics for testing the equality of proportions and the equality of survival functions. We build our proposal on a weighted combination of a score test for the difference in proportions and a Weighted Kaplan-Meier statistic-based test for the difference of survival functions. The proposed statistics are fully non-parametric and do not rely on the proportional hazards assumption for the survival outcome. We present the asymptotic distribution of these statistics, propose a variance estimator and show their asymptotic properties under fixed and local alternati…
In survival analysis, the hazard function often depends on a set of covariates. Martingale and deviance residual are most widely used for examining the validity of the function form of covariates by checking whether there is a discernible trend in their scatterplot against continuous covariates. However, visual inspection of martingale and deviance residuals is often subjective. In addition, these residuals lack a reference distribution due to censoring. It is therefore challenging to derive numerical statistical tests based on martingale or deviance residuals. In this paper, we extend the ide…
In medical research, it is often needed to obtain subgroups with heterogeneous survivals, which have been predicted from a prognostic factor. For this purpose, a binary split has often been used once or recursively; however, binary partitioning may not provide an optimal set of well separated subgroups. We propose a multi-way partitioning algorithm, which divides the data into K heterogeneous subgroups based on the information from a prognostic factor. The resulting subgroups show significant differences in survival. Such a multi-way partition is found by maximizing the minimum of the subgroup…
We present a new method for analyzing stochastic epidemic models under minimal assumptions. The method, dubbed DSA, is based on a simple yet powerful observation, namely that population-level mean-field trajectories described by a system of PDE may also approximate individual-level times of infection and recovery. This idea gives rise to a certain non-Markovian agent-based model and provides an agent-level likelihood function for a random sample of infection and/or recovery times. Extensive numerical analyses on both synthetic and real epidemic data from the FMD in the United Kingdom and the C…
University of Warwick（华威大学）；Nanjing University（南京大学）；London School of Economics and Political Science（伦敦政治经济学院）；University of Cambridge（剑桥大学）；Yale School of Public Health（耶鲁公共卫生学院）
Survival analysis is an essential tool for the study of health data. An inherent component of such data is the presence of missing values. In recent years, researchers proposed new learning algorithms for survival tasks based on neural networks. Here, we studied the predictive performance of such algorithms coupled with different methods for handling missing values on simulated data that reflect a realistic situation, i.e., when individuals belong to unobserved clusters. We investigated different patterns of missing data. The results show that, without further feature engineering, no single im…
Many medical decisions involve the use of dynamic information collected on individual patients toward predicting likely transitions in their future health status. If accurate predictions are developed, then a prognostic mode can identify patients at greatest risk for future adverse events, and may be used clinically to define populations appropriate for targeted intervention. In practice, a prognostic model is often used to guide decisions at multiple time points over the course of disease, and classification performance, i.e. sensitivity and specificity, for distinguishing high-risk versus lo…
Randomized clinical trials are often designed to assess whether a test treatment prolongs survival relative to a control treatment. Increased patient heterogeneity, while desirable for generalizability of results, can weaken the ability of common statistical approaches to detect treatment differences, potentially hampering the regulatory approval of safe and efficacious therapies. A novel solution to this problem is proposed. A list of baseline covariates that have the potential to be prognostic for survival under either treatment is pre-specified in the analysis plan. At the analysis stage, u…
A bivariate cure copula model with zero-inflated gamma frailty for right-censored data
Graduate Institute for Advanced Studies, SOKENDAI（SOKENDAI 高级研究研究生院）；Institute of Statistical Mathematics（统计数学研究所）；School of Informatics and Data Science, Hiroshima University（广岛大学信息科学与数据科学学院）
Non-proportional hazards (NPH) have been observed recently in many immuno-oncology clinical trials. Weighted log-rank tests (WLRT) with suitably chosen weights can be used to improve the power of detecting the difference of the two survival curves in the presence of NPH. However, it is not easy to choose a proper WLRT in practice when both robustness and efficiency are considered. A versatile maxcombo test was proposed to achieve the balance of robustness and efficiency and has received increasing attentions in both methodology development and application. However, survival trials often warran…
Predictive modeling from high-dimensional genomic data is often preceded by a dimension reduction step, such as principal components analysis (PCA). However, the application of PCA is not straightforward for multi-source data, wherein multiple sources of 'omics data measure different but related biological components. In this article we utilize recent advances in the dimension reduction of multi-source data for predictive modeling. In particular, we apply exploratory results from Joint and Individual Variation Explained (JIVE), an extension of PCA for multi-source data, for prediction of diffe…
