Survival models are a popular tool for the analysis of time to event data with applications in medicine, engineering, economics, and many more. Advances like the Cox proportional hazard model have enabled researchers to better describe hazard rates for the occurrence of single fatal events, but are unable to accurately model competing events and transitions. Common phenomena are often better described through multiple states, for example: the progress of a disease modeled as healthy, sick and dead instead of healthy and dead, where the competing nature of death and disease has to be taken into account. Moreover, Cox models are limited by modeling assumptions, like proportionality of hazard rates and linear effects. Individual characteristics can vary significantly between observational units, like patients, resulting in idiosyncratic hazard rates and different disease trajectories. These considerations require flexible modeling assumptions. To overcome these issues, we propose the use of neural ordinary differential equations as a flexible and general method for estimating multi-state survival models by directly solving the Kolmogorov forward equations. To quantify the uncertainty in the resulting individual cause-specific hazard rates, we further introduce a variational latent variable model and show that this enables meaningful clustering with respect to multi-state outcomes as well as interpretability regarding covariate values. We show that our model exhibits state-of-the-art performance on popular survival data sets and demonstrate its efficacy in a multi-state setting
Survival analysis concerns the study of timeline data where the event of interest may remain unobserved (i.e., censored). Studies commonly record more than one type of event, but conventional survival techniques focus on a single event type. We set out to integrate both multiple independently censored time-to-event variables as well as missing observations. An energy-based approach is taken with a bi-partite structure between latent and visible states, known as harmoniums (or restricted Boltzmann machines). The present harmonium is shown, both theoretically and experimentally, to capture non-l…
University of Michigan（密歇根大学）；Merck Research Laboratories（默克研究实验室）
There has been increasing interest in modelling survival data using deep learning methods in medical research. Current approaches have focused on designing special cost functions to handle censored survival data. We propose a very different method with two steps. In the first step, we transform each subject's survival time into a series of jackknife pseudo conditional survival probabilities and then use these pseudo probabilities as a quantitative response variable in the deep neural network model. By using the pseudo values, we reduce a complex survival analysis to a standard regression probl…
We consider a parametric modelling approach for survival data where covariates are allowed to enter the model through multiple distributional parameters, i.e., scale and shape. This is in contrast with the standard convention of having a single covariate-dependent parameter, typically the scale. Taking what is referred to as a multi-parameter regression (MPR) approach to modelling has been shown to produce flexible and robust models with relatively low model complexity cost. However, it is very common to have clustered data arising from survival analysis studies, and this is something that is …
Modelling customer churn for the retail industry in a deep learning based sequential framework
University of Essex（埃塞克斯大学）；Profusion Media Ltd.（Profusion传媒有限公司）；Essex Business School（埃塞克斯商学院）；Friedrich-Alexander University Erlangen-Nuremberg（埃尔朗根-纽伦堡弗里德里希-亚历山大大学）
We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards…
We discuss causal mediation analyses for survival data and propose a new approach based on the additive hazards model. The emphasis is on a dynamic point of view, that is, understanding how the direct and indirect effects develop over time. Hence, importantly, we allow for a time varying mediator. To define direct and indirect effects in such a longitudinal survival setting we take an interventional approach (Didelez (2018)) where treatment is separated into one aspect affecting the mediator and a different aspect affecting survival. In general, this leads to a version of the non-parametric g-…
Survival analysis is a critical tool for the modelling of time-to-event data, such as life expectancy after a cancer diagnosis or optimal maintenance scheduling for complex machinery. However, current neural network models provide an imperfect solution for survival analysis as they either restrict the shape of the target probability distribution or restrict the estimation to pre-determined times. As a consequence, current survival neural networks lack the ability to estimate a generic function without prior knowledge of its structure. In this article, we present the metaparametric neural netwo…
Survival analysis, or time-to-event modelling, is a classical statistical problem that has garnered a lot of interest for its practical use in epidemiology, demographics or actuarial sciences. Recent advances on the subject from the point of view of machine learning have been concerned with precise per-individual predictions instead of population studies, driven by the rise of individualized medicine. We introduce here a conditional normalizing flow based estimate of the time-to-event density as a way to model highly flexible and individualized conditional survival distributions. We use a nove…
Joint modelling of longitudinal and time-to-event data is usually described by a joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the random effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas to introduce non-linear correlation between the marginal distributions of the longitudinal and time…
Neural Fine-Gray: Monotonic neural networks for competing risks
The Alan Turing Institute（艾伦·图灵研究所）；MRC Biostatistics Unit, University of Cambridge（剑桥大学MRC生物统计学组）；Big Data Institute, University of Oxford（牛津大学大数据研究所）
Datasets containing large samples of time-to-event data arising from several small heterogeneous groups are commonly encountered in statistics. This presents problems as they cannot be pooled directly due to their heterogeneity or analyzed individually because of their small sample size. Bayesian nonparametric modelling approaches can be used to model such datasets given their ability to flexibly share information across groups. In this paper, we will compare three popular Bayesian nonparametric methods for modelling the survival functions of heterogeneous groups. Specifically, we will first c…
merlin can do a lot of things. From simple stuff, like fitting a linear regression or a Weibull survival model, to a three-level logistic mixed effects model, or a multivariate joint model of multiple longitudinal outcomes (of different types) and a recurrent event and survival with non-linear effects...the list is rather endless. merlin can do things I haven't even thought of yet. I'll take a single dataset, and attempt to show you the full range of capabilities of merlin, and discuss some future directions for the implementation in Stata.
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 (us…
Tick is a statistical learning library for Python~3, with a particular emphasis on time-dependent models, such as point processes, and tools for generalized linear models and survival analysis. The core of the library is an optimization module providing model computational classes, solvers and proximal operators for regularization. tick relies on a C++ implementation and state-of-the-art optimization algorithms to provide very fast computations in a single node multi-core setting. Source code and documentation can be downloaded from https://github.com/X-DataInitiative/tick
Survival analysis/time-to-event models are extremely useful as they can help companies predict when a customer will buy a product, churn or default on a loan, and therefore help them improve their ROI. In this paper, we introduce a new method to calculate survival functions using the Multi-Task Logistic Regression (MTLR) model as its base and a deep learning architecture as its core. Based on the Concordance index (C-index) and Brier score, this method outperforms the MTLR in all the experiments disclosed in this paper as well as the Cox Proportional Hazard (CoxPH) model when nonlinear depende…
Deep Learning-Based Survival Analysis with Copula-Based Activation Functions for Multivariate Response Prediction
Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris（明尼苏达州立大学莫里斯分校统计学系）；EGADE Business School, Tecnológico de Monterrey（墨西哥蒙特雷技术学院EGADE商学院）；Department of Statistics and Data Science, Pukyong National University（浦项国立大学统计学与数据科学系）；Department of Management Information Systems, Dong-A University（东阿大学管理信息系）
Patient-level health economic data collected alongside clinical trials are an important component of the process of technology appraisal, with a view to informing resource allocation decisions. For end of life treatments, such as cancer treatments, modelling of cost-effectiveness/utility data may involve some form of partitioned survival analysis, where measures of health-related quality of life and survival time for both pre- and post-progression periods are combined to generate some aggregate measure of clinical benefits (e.g. quality-adjusted survival). In addition, resource use data are of…
When modelling competing risks survival data, several techniques have been proposed in both the statistical and machine learning literature. State-of-the-art methods have extended classical approaches with more flexible assumptions that can improve predictive performance, allow high dimensional data and missing values, among others. Despite this, modern approaches have not been widely employed in applied settings. This article aims to aid the uptake of such methods by providing a condensed compendium of competing risks survival methods with a unified notation and interpretation across approach…
Often in Phase 3 clinical trials measuring a long-term time-to-event endpoint, such as overall survival or progression-free survival, investigators also collect repeated measures on biomarkers which may be predictive of the primary endpoint. Although these data may not be leveraged directly to support early stopping decisions, can we make greater use of these data to increase efficiency and improve interim decision making? We present a joint model for longitudinal and time-to-event data and a method which establishes the distribution of successive estimates of parameters in the joint model acr…
Survival analysis is the branch of statistics that studies the relation between the characteristics of living entities and their respective survival times, taking into account the partial information held by censored cases. A good analysis can, for example, determine whether one medical treatment for a group of patients is better than another. With the rise of machine learning, survival analysis can be modeled as learning a function that maps studied patients to their survival times. To succeed with that, there are three crucial issues to be tackled. First, some patient data is censored: we do…
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 medicine, survival analysis studies the time duration to events of interest such as mortality. One major challenge is how to deal with multiple competing events (e.g., multiple disease diagnoses). In this work, we propose a transformer-based model that does not make the assumption for the underlying survival distribution and is capable of handling competing events, namely SurvTRACE. We account for the implicit \emph{confounders} in the observational setting in multi-events scenarios, which causes selection bias as the predicted survival probability is influenced by irrelevant factors. To su…
Health economic evaluations often require predictions of survival rates beyond the follow-up period. Parametric survival models can be more convenient for economic modelling than the Cox model. The generalized gamma (GG) and generalized F (GF) distributions are extensive families that contain almost all commonly used distributions with various hazard shapes and arbitrary complexity. In this study, we present a new SAS macro for implementing a wide variety of flexible parametric models including the GG and GF distributions and their special cases, as well as the Gompertz distribution. Proper cu…
In decision modelling with time to event data, there are a variety of parametric models which could be used to extrapolate the survivor function. Each of these implies a different hazard function and in situations where there is moderate censoring, they can result in quite different extrapolations. Expert opinion on the long-term survival or other quantities could reduce model uncertainty. We present a general and easily implementable approach for including a variety of types of expert opinions. Expert opinion is incorporated by penalizing the likelihood function. Inference is performed in a B…
Joint modelling of time-dependent biomarker variability and time-to-event outcomes, a two-step approach
Erasmus University Medical Center（伊拉斯姆斯大学医学中心）；European Organisation for Research and Treatment of Cancer（欧洲癌症研究与治疗组织）；Julius Center for Health Sciences and Primary Care, University Medical Center UtrechtUtrecht University（乌得勒支大学医学中心朱利叶斯健康科学与初级保健中心）
This paper describes the use of survival analysis and simulation to model the lifetime of high voltage instrument transformers in the Dutch transmission sys-tem. To represent asset aging, the non-parametric Kaplan-Meier method is used to enable the fitting of Weibull distribution. Such an approach is implemented on three different voltage levels, namely 110kV, 150kV, and 220/380kV. Real failure and inspection data is used to achieve a realistic failure model of the instrument trans-formers. Failure and maintenance data occurring between 1989 and 2021 have been used for this study. In spite of …
Data-driven approaches which aim to identify and predict player engagement are becoming increasingly popular in games industry contexts. This is due to the growing practice of tracking and storing large volumes of in-game telemetries coupled with a desire to tailor the gaming experience to the end-user's needs. These approaches are particularly useful not just for companies adopting Game-as-a-Service (GaaS) models (e.g. for re-engagement strategies) but also for those working under persistent content-delivery regimes (e.g. for better audience targeting). A major challenge for the latter is to …
Background Survival extrapolation is essential in the cost-effectiveness analysis to quantify the lifetime survival benefit associated with a new intervention, due to the restricted duration of randomized controlled trials (RCTs). Current approaches of extrapolation often assume that the treatment effect observed in the trial can continue indefinitely, which is unrealistic and may have a huge impact on decisions for resource allocation. Objective We introduce a novel methodology as a possible solution to alleviate the problem of performing survival extrapolation with heavily censored data from…
On a penalised likelihood approach for joint modelling of longitudinal covariates and partly interval-censored data -- an application to the Anti-PD1 brain collaboration trial
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…
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）
PISA: An AI Pipeline for Interpretable-by-design Survival Analysis Providing Multiple Complexity-Accuracy Trade-off Models
a Leiden University Medical Center, Leiden, The Netherlands；b Amsterdam University Medical Centers location Vrije Universiteit Amsterdam, Amsterdam, The Netherlands；c Cancer Center Amsterdam, Cancer Treatment；Quality of Life, Amsterdam, The Netherlands；d Centrum Wiskunde \& Informatica, Amsterdam, The Netherlands
We develop a functional proportional hazards mixture cure (FPHMC) model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the EM algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness …
Kidney transplantation can significantly enhance living standards for people suffering from end-stage renal disease. A significant factor that affects graft survival time (the time until the transplant fails and the patient requires another transplant) for kidney transplantation is the compatibility of the Human Leukocyte Antigens (HLAs) between the donor and recipient. In this paper, we propose 4 new biologically-relevant feature representations for incorporating HLA information into machine learning-based survival analysis algorithms. We evaluate our proposed HLA feature representations on a…
This article concerns the statistical modelling of emergency service response times. We apply advanced methods from spatial survival analysis to deliver inference for data collected by the London Fire Brigade on response times to reported dwelling fires. Existing approaches to the analysis of these data have been mainly descriptive; we describe and demonstrate the advantages of a more sophisticated approach. Our final parametric proportional hazards model includes harmonic regression terms to describe how response time varies with time-of-day and shared spatially correlated frailties on an aux…
We are interested in survival analysis of hemodialysis patients for whom several biomarkers are recorded over time. Motivated by this challenging problem, we propose a general framework for multivariate joint longitudinal-survival modeling that can be used to examine the association between several longitudinally recorded covariates and a time-to-event endpoint. Our method allows for simultaneous modeling of longitudinal covariates by taking their correlation into account. This leads to a more efficient method for modeling their trajectories over time, and hence, it can better capture their re…
We develop flexible multi-parameter regression survival models for interval censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multi-parameter Weibull regression survival model, which is wholly parametric, and has non-proportional hazards, is the main focus of the paper. We describe the basic model, develop the interval-censored likelihood and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed re-analysis of data from the Signal Tandmobie…
Predicting patient survival probabilities based on observed covariates is an important assessment in clinical practice. These patient-specific covariates are often measured over multiple follow-up appointments. It is then of interest to predict survival based on the history of these longitudinal measurements, and to update predictions as more observations become available. The standard approaches to these so-called `dynamic prediction' assessments are joint models and landmark analysis. Joint models involve high-dimensional parametrisations, and their computational complexity often prohibits i…
In this article, I present the user written stmixed command for the fitting of multilevel survival models, which serves as both an alternative to Stata's official mestreg, and a complimentary program with substantial extensions. stmixed can fit multilevel survival models with any number of levels and random effects at each level, including flexible spline-based approaches (such as Royston-Parmar and the log hazard equivalent) or user-defined hazard models. Simple or complex time-dependent effects can be included, as well as the addition of expected mortality for a relative survival model. Left…
We study the non-parametric estimation of an unknown survival function S with support on R+ based on a sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the survival function S and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is dealt with by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin-Sobolev spaces which characterize the regularity of the unknown survival function S throu…
Unobserved individual heterogeneity is a common challenge in population cancer survival studies. This heterogeneity is usually associated with the combination of model misspecification and the failure to record truly relevant variables. We investigate the effects of unobserved individual heterogeneity in the context of excess hazard models, one of the main tools in cancer epidemiology. We propose an individual excess hazard frailty model to account for individual heterogeneity. This represents an extension of frailty modelling to the relative survival framework. In order to facilitate the infe…
Technical University of Denmark（丹麦技术大学）；University of Leeds（利兹大学）；TU Dortmund（多特蒙德工业大学）
We provide a novel approach and an exploratory study for modelling life event choices and occurrence from a probabilistic perspective through causal discovery and survival analysis. Our approach is formulated as a bi-level problem. In the upper level, we build the life events graph, using causal discovery tools. In the lower level, for the pairs of life events, time-to-event modelling through survival analysis is applied to model time-dependent transition probabilities. Several life events were analysed, such as getting married, buying a new car, child birth, home relocation and divorce, toget…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of California Los Angeles（加州大学洛杉矶分校）
Analyzing electronic health records (EHR) poses significant challenges because often few samples are available describing a patient's health and, when available, their information content is highly diverse. The problem we consider is how to integrate sparsely sampled longitudinal data, missing measurements informative of the underlying health status and fixed demographic information to produce estimated survival distributions updated through a patient's follow up. We propose a nonparametric probabilistic model that generates survival trajectories from an ensemble of Bayesian trees that learns …
