Frailty models are essential tools in survival analysis for addressing unobserved heterogeneity and random effects in the data. These models incorporate a random effect, the frailty, which is assumed to impact the hazard rate multiplicatively. In this paper, we introduce a novel class of frailty models in both univariate and multivariate settings, using phase-type distributions as the underlying frailty specification. We investigate the properties of these phase-type frailty models and develop expectation-maximization algorithms for their maximum-likelihood estimation. In particular, we show that the resulting model shares similarities with the Gamma frailty model, has closed-form expressions for its functionals, and can approximate any other frailty model. Through a series of simulated and real-life numerical examples, we demonstrate the effectiveness and versatility of the proposed models in addressing unobserved heterogeneity in survival analysis.
The present article studies survival analytic aspects of semiparametric copula dependence models with arbitrary univariate marginals. The underlying survival functions admit a representation via exponent measures which have an interpretation within the context of hazard functions. In particular, correlated frailty survival models are linked to copulas. Additionally, the relation to exponent measures of minumum-infinitely divisible distributions as well as to the Lévy measure of the Lévy-Khintchine formula is pointed out. The semiparametric character of the current analyses and the construction…
In this work we provide a simple estimation procedure for a general frailty model for analysis of prospective correlated failure times. Rigorous large-sample theory for the proposed estimators of both the regression coefficient vector and the dependence parameter is given, including consistent variance estimators. In a simulation study under the widely used gamma frailty model, our proposed approach was found to have essentially the same efficiency as the EM-based estimator considered by other authors, with negligible difference between the standard errors of the two estimators. The proposed a…
In this work we deal with correlated failure time (age at onset) data arising from population-based case-control studies, where case and control probands are selected by population-based sampling and an array of risk factor measures is collected for both cases and controls and their relatives. Parameters of interest are effects of risk factors on the hazard function of failure times and within-family dependencies of failure times after adjusting for the risk factors. Due to the retrospective nature of sampling, a large sample theory for existing methods has not been established. We develop a n…
This paper studies Cox's regression hazard model with an unobservable random frailty where no specific distribution is postulated for the frailty variable, and the marginal lifetime distribution allows both parametric and non-parametric models. Laplace's approximation method and gradient search on smooth manifolds embedded in Euclidean space are applied, and a non-iterative profile likelihood optimization method is proposed for estimating the regression coefficients. The proposed method is compared with the Expected-Maximization method developed based on a gamma frailty assumption, and also in…
Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modelled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (e.g. Gamma or log-Normal). Despite that, in the context of parametric frailty models little is known about the impact of misspecifying the baseline hazard, the frailty distribution, or both. Therefore, our ai…
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…
In statistical models for the analysis of time-to-event data, individual heterogeneity is usually accounted for by means of one or more random effects, also known as frailties. In the vast majority of the literature, the random effect is assumed to follow a continuous probability distribution. However, in some areas of application, a discrete frailty distribution may be more appropriate. We investigate and compare various existing families of discrete univariate and shared frailty models by taking as our focus the variance of the relative frailty distribution in survivors. The relative frailty…
The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterise the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. We call this model the double-Cox model. The R programs for fitting the double-Cox model are available on Github. We formally introduce the double-Cox model with shared frailty and investigat…
Understanding the factors that explain differences in survival times is an important issue for establishing policies to improve national health systems. Motivated by breast cancer data arising from the Surveillance Epidemiology and End Results program, we propose a covariate-adjusted proportional hazards frailty model for the analysis of clustered right-censored data. Rather than incorporating exchangeable frailties in the linear predictor of commonly-used survival models, we allow the frailty distribution to flexibly change with both continuous and categorical cluster-level covariates and mod…
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…
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…
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）
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…
Multiple indications of disease progression found in a cancer patient by loco-regional relapse, distant metastasis and death. Early identification of these indications is necessary to change the treatment strategy. Biomarkers play an essential role in this aspect. The survival chance of a patient is dependent on the biomarker, and the treatment strategy also differs accordingly, e.g., the survival prediction of breast cancer patients diagnosed with HER2 positive status is different from the same with HER2 negative status. This results in a different treatment strategy. So, the heterogeneity of…
A common practice in clinical trials is to evaluate a treatment effect on an intermediate endpoint when the true outcome of interest would be difficult or costly to measure. We consider how to validate intermediate endpoints in a causally-valid way when the trial outcomes are time-to-event. Using counterfactual outcomes, those that would be observed if the counterfactual treatment had been given, the causal association paradigm assesses the relationship of the treatment effect on the surrogate $S$ with the treatment effect on the true endpoint $T$. In particular, we propose illness death model…
We are concerned with the flexible parametric analysis of bivariate survival data. Elsewhere, we have extolled the virtues of the "power generalized Weibull" (PGW) distribution as an attractive vehicle for univariate parametric survival analysis: it is a tractable, parsimonious, model which interpretably allows for a wide variety of hazard shapes and, when adapted (to give an adapted PGW, or APGW, distribution), covers a wide variety of important special/limiting cases. Here, we additionally observe a frailty relationship between a PGW distribution with one value of the parameter which control…
In this work we present a simple estimation procedure for a general frailty model for analysis of prospective correlated failure times. Earlier work showed this method to perform well in a simulation study. Here we provide rigorous large-sample theory for the proposed estimators of both the regression coefficient vector and the dependence parameter, including consistent variance estimators.
Correlated frailty model for analysis of genetic association in family studies
Lunenfeld-Tanenbaum Research Institute, Mount Sinai Hospital（伦内费尔德-塔纳鲍姆研究所，西奈山医院）；Bordeaux Population Health Research Center U1219, Inserm（波尔多人口健康研究中心U1219，法国国家健康与医学研究院）；Department of Epidemiology and Biostatistics, Western University（韦仕敦大学流行病学与生物统计学系）
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…
Frailty models are often the model of choice for heterogeneous survival data. A frailty model contains both random effects and fixed effects, with the random effects accommodating for the correlation in the data. Different estimation procedures have been proposed for the fixed effects and the variances of and covariances between the random effects. Especially with an unspecified baseline hazard, i.e., the Cox model, the few available methods deal only with a specific correlation structure. In this paper, an estimation procedure, based on the integrated partial likelihood, is introduced, which …
Hazard ratios are often used to evaluate time to event outcomes, but they may be hard to interpret. A particular issue arise because hazards are typically estimated conditional on survival, i.e.\ on left truncated samples. Then, hazard ratios from conventional models cannot be interpreted as counterfactual hazard ratios that are immediately relevant to individual patients. This article explores how the hazard ratios from Cox models may differ from hazard ratios with a causal interpretation. Using summary data from twin studies, I suggest an approach to learn about the unmeasured heterogeneity …
Laplacian-P-splines for shared Gamma frailty models applied to clustered right-censored time-to-event data
Department of Neurodegeneration Diagnostics, University Hospital Bialystok, Poland（比亚韦斯托克医院神经退行性疾病诊断系）；Interuniversity Institute for Biostatistics and statistical Bioinformatics (I-BioStat), Data Science Institute, Hasselt University, Belgium（哈瑟尔特大学生物统计与统计生物信息学跨大学研究所（I-BioStat））；Global Health Institute, Family Medicine and Population Health, University of Antwerp, Belgium（安特卫普大学全球健康学院）
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…
Frailty and resilience models provide a way to introduce random effects in hazard and reversed hazard rate modeling by random variables, called frailty and resilience random variables, respectively, to account for unobserved or unexplained heterogeneity among experimental units. This paper investigates the effects of frailty and resilience random variables on the baseline random variables using some shifted stochastic orders based on some ageing properties of the baseline random variables. Relevant examples are provided to illustrate the results. Some results are illustrated with real-world da…
In this paper we apply survival techniques to soccer data, treating a goal scoring as the event of interest. It specifically concerns the relationship between the time of the first goal in the game and the time of the second goal. In order to do so, the relevant survival analysis concepts are readjusted to fit the problem and a Cox model is developed for the hazard function. Attributes such as time dependent covariates and a frailty term are also being considered. We also use a reliable propensity score to summarize the pre-game covariates. The conclusions derived from the results are that a f…
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…
In heterogeneous cohorts and those where censoring by non-primary risks is informative many conventional survival analysis methods are not applicable; the proportional hazards assumption is usually violated at population level and the observed crude hazard rates are no longer estimators of what they would have been in the absence of other risks. In this paper, we develop a fully Bayesian survival analysis to determine the probabilistically optimal description of a heterogeneous cohort and we propose a novel means of recovering hazard rates and survival functions `decontaminated' of the effects…
Most papers implicitly assume competing risks to be induced by residual cohort heterogeneity, i.e. heterogeneity that is not captured by the recorded covariates. Based on this observation we develop a generic statistical description of competing risks that unifies the main schools of thought. Assuming heterogeneity-induced competing risks is much weaker than assuming risk independence. However, we show that it still imposes sufficient constraints to solve the competing risk problem, and derive exact formulae for decontaminated primary risk hazard rates and cause-specific survival functions. Th…
A class of multivariate mixed survival models for continuous and discrete time with a complex covariance structure is introduced in a context of quantitative genetic applications. The methods introduced can be used in many applications in quantitative genetics although the discussion presented concentrates on longevity studies. The framework presented allows to combine models based on continuous time with models based on discrete time in a joint analysis. The continuous time models are approximations of the frailty model in which the hazard function will be assumed to be piece-wise constant. T…
Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Department of Data Sciences and Operations（数据科学与运营系）；Department of Computer Science, University of Southern California（计算机科学系，南加州大学）；Merck & Co., Inc.（默克公司）
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…
This article analyzes the problem of estimating the time until an event occurs, also known as survival modeling. We observe through substantial experiments on large real-world datasets and use-cases that populations are largely heterogeneous. Sub-populations have different mean and variance in their survival rates requiring flexible models that capture heterogeneity. We leverage a classical extension of the logistic function into the survival setting to characterize unobserved heterogeneity using the beta distribution. This yields insights into the geometry of the problem as well as efficient …
Oslo University Hospital（奥斯陆大学医院）；University of Oslo（奥斯陆大学）；Leiden University Medical Center（莱顿大学医学中心）；Leiden University（莱顿大学）；Erasmus Medical Center, University Medical Center Rotterdam（伊拉斯姆斯医学中心，鹿特丹大学医学中心）
SurvHTE-Bench: A Benchmark for Heterogeneous Treatment Effect Estimation in Survival Analysis
Machine Learning Department & Heinz College Carnegie Mellon University（机器学习系及卡内基梅隆大学海因兹学院）；Heinz College Carnegie Mellon University（卡内基梅隆大学海因兹学院）；National Library of Medicine National Institutes of Health（国家医学图书馆国立卫生研究院）
It is known that the hazard ratio lacks a useful causal interpretation. Even for data from a randomized controlled trial, the hazard ratio suffers from built-in selection bias as, over time, the individuals at risk in the exposed and unexposed are no longer exchangeable. In this work, we formalize how the observed hazard ratio evolves and deviates from the causal hazard ratio of interest in the presence of heterogeneity of the hazard of unexposed individuals (frailty) and heterogeneity in effect (individual modification). For the case of effect heterogeneity, we define the causal hazard ratio.…
Antibodies, an essential part of our immune system, develop through an intricate process to bind a wide array of pathogens. This process involves randomly mutating DNA sequences encoding these antibodies to find variants with improved binding, though mutations are not distributed uniformly across sequence sites. Immunologists observe this nonuniformity to be consistent with "mutation motifs", which are short DNA subsequences that affect how likely a given site is to experience a mutation. Quantifying the effect of motifs on mutation rates is challenging: a large number of possible motifs makes…
Increasingly, medical research is dependent on data collected for non-research purposes, such as electronic health records data (EHR). EHR data and other large databases can be prone to measurement error in key exposures, and unadjusted analyses of error-prone data can bias study results. Validating a subset of records is a cost-effective way of gaining information on the error structure, which in turn can be used to adjust analyses for this error and improve inference. We extend the mean score method for the two-phase analysis of discrete-time survival models, which uses the unvalidated covar…
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…
