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 novel estimation techniques for estimating these parameters under a general semiparametric shared frailty model. We also present a simple, easily computed, and non-iterative nonparametric estimator for the cumulative baseline hazard function. A rigorous large sample theory for the proposed estimators of these parameters is given along with simulations and a real data example illustrate the utility of the proposed method.
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…
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…
We consider estimation in a class of semiparametric transformation models for right--censored data. These models gained much attention in survival analysis; however, most authors consider only regression models derived from frailty distributions whose hazards are decreasing. This paper considers estimation in a more flexible class of models and proposes conditional rank M-estimators for estimation of the Euclidean component of the model.
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…
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.
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…
Current status data abounds in the field of epidemiology and public health, where the only observable data for a subject is the random inspection time and the event status at inspection. Motivated by such a current status data from a periodontal study where data are inherently clustered, we propose a unified methodology to analyze such complex data. We allow the time-to-event to follow the semiparametric GOR model with a cure fraction, and develop a unified estimation scheme powered by the EM algorithm. The within-subject correlation is accounted for by a random (frailty) effect, and the non-p…
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）
One of Pranab K. Sen's major research areas is sequential nonparametrics and semiparametrics and their applications to clinical trials, to which he has made many important contributions. Herein we review a number of these contributions and related developments. We also describe some recent work on nonparametric and semiparametric inference and the associated computational methods in time-sequential clinical trials with survival endpoints.
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…
The infinite-dimensional information operator for the nuisance parameter plays a key role in semiparametric inference, as it is closely related to the regular estimability of the target parameter. Calculation of information operators has traditionally proceeded in a case-by-case manner and has easily entailed lengthy derivations with complicated arguments. We develop a unified framework for this task by exploiting commonality in the form of semiparametric likelihoods. The general formula allows one to derive information operators with simple calculus and, if necessary at all, a minimal amount …
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…
We consider the efficient estimation of the semiparametric additive transformation model with current status data. A wide range of survival models and econometric models can be incorporated into this general transformation framework. We apply the B-spline approach to simultaneously estimate the linear regression vector, the nondecreasing transformation function, and a set of nonparametric regression functions. We show that the parametric estimate is semiparametric efficient in the presence of multiple nonparametric nuisance functions. An explicit consistent B-spline estimate of the asymptotic …
We propose a general index model for survival data, which generalizes many commonly used semiparametric survival models and belongs to the framework of dimension reduction. Using a combination of geometric approach in semiparametrics and martingale treatment in survival data analysis, we devise estimation procedures that are feasible and do not require covariate-independent censoring as assumed in many dimension reduction methods for censored survival data. We establish the root-$n$ consistency and asymptotic normality of the proposed estimators and derive the most efficient estimator in this …
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…
Palliative medicine is an interdisciplinary specialty focusing on improving quality of life (QOL) for patients with serious illness and their families. Palliative care programs are available or under development at over 80% of large US hospitals (300+ beds). Palliative care clinical trials present unique analytic challenges relative to evaluating the palliative care treatment efficacy which is to improve patients diminishing QOL as disease progresses towards end of life (EOL). A unique feature of palliative care clinical trials is that patients will experience decreasing QOL during the trial d…
The proportional hazards (PH), proportional odds (PO) and accelerated failure time (AFT) models have been widely used in different applications of survival analysis. Despite their popularity, these models are not suitable to handle lifetime data with crossing survival curves. In 2005, Yang and Prentice proposed a semiparametric two-sample strategy (YP model), including the PH and PO frameworks as particular cases, to deal with this type of data. Assuming a general regression setting, the present paper proposes an unified approach to fit the YP model by employing Bernstein polynomials to manage…
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…
We consider a class of semiparametric regression models which are one-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] model for right-censored univariate failure times. These models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family of distributions. Inference is based on a nonparametric likelihood. The behavior of the likelihood maximizer is studied under general conditions where the fitt…
One goal in survival analysis of right-censored data is to estimate the marginal survival function in the presence of dependent censoring. When many auxiliary covariates are sufficient to explain the dependent censoring, estimation based on either a semiparametric model or a nonparametric model of the conditional survival function can be problematic due to the high dimensionality of the auxiliary information. In this paper, we use two working models to condense these high-dimensional covariates in dimension reduction; then an estimate of the marginal survival function can be derived nonparamet…
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 …
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…
Proportional hazards (PH), proportional odds (PO) and accelerated failure time (AFT) models have been widely used to deal with survival data in different fields of knowledge. Despite their popularity, such models are not suitable to handle survival data with crossing survival curves. Yang and Prentice (2005) proposed a semiparametric two-sample approach, denoted here as the YP model, allowing the analysis of crossing survival curves and including the PH and PO configurations as particular cases. In a general regression setting, the present work proposes a fully likelihood-based approach to fit…
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…
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 this paper, we mainly discuss the cure model with survival data. Different from the usual survival data with right-censoring, we incorporate the features of left-truncation and measurement error in covariates. Generally speaking, left-truncation causes a biased sample in survival analysis; measurement error in covariates may incur a tremendous bias if we do not deal with it properly. To deal with these challenges, we propose a flexible way to analyze left-truncated survival data and correct measurement error in covariates. The theoretical results are also established in this paper.
Estimating individualized treatment rules is a central task for personalized medicine. [zhao2012estimating] and [zhang2012robust] proposed outcome weighted learning to estimate individualized treatment rules directly through maximizing the expected outcome without modeling the response directly. In this paper, we extend the outcome weighted learning to right censored survival data without requiring either an inverse probability of censoring weighting or a semiparametric modeling of the censoring and failure times as done in [zhao2015doubly]. To accomplish this, we take advantage of the tree ba…
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…
