In the field of cardio-thoracic surgery, valve function is monitored over time after surgery. The motivation for our research comes from a study which includes patients who received a human tissue valve in the aortic position. These patients are followed prospectively over time by standardized echocardiographic assessment of valve function. Loss of follow-up could be caused by valve intervention or the death of the patient. One of the main characteristics of the human valve is that its durability is limited. Therefore, it is of interest to obtain a prognostic model in order for the physicians to scan trends in valve function over time and plan their next intervention, accounting for the characteristics of the data.
Several authors have focused on deriving predictions under the standard joint modeling of longitudinal and survival data framework that assumes a constant effect for the coefficient that links the longitudinal and survival outcomes. However, in our case this may be a restrictive assumption. Since the valve degenerates, the association between the biomarker with survival may change over time.
To improve dynamic predictions we propose a Bayesian joint model that allows a time-varying coefficient to link the longitudinal and the survival processes, using P-splines. We evaluate the performance of the model in terms of discrimination and calibration, while accounting for censoring.
Cystic fibrosis is a chronic lung disease which requires frequent patient monitoring to maintain lung function over time and minimize onset of acute respiratory events known as pulmonary exacerbations. From the clinical point of view it is important to characterize the association between key biomarkers such as $FEV_1$ and time-to first exacerbation. Progression of the disease is heterogeneous, yielding different sub-groups in the population exhibiting distinct longitudinal profiles. It is desirable to categorize these unobserved sub-groups (latent classes) according to their distinctive traje…
Dynamic predictions of survival outcomes are of great interest to physicians and patients, since such predictions are useful elements of clinical decision-making. Joint modelling of longitudinal and survival data has been increasingly used to obtain dynamic predictions. A common assumption of joint modelling is that random-effects and error terms in the longitudinal sub-model are Gaussian. However, this assumption may be too restrictive, e.g. in the presence of outliers as commonly encountered in many real-life applications. A natural extension is to robustify the joint models by assuming more…
Bayesian joint modeling of longitudinal patient-reported outcomes and survival: an application to chronic obstructive pulmonary disease
Basque Center for Applied Mathematics BCAM（巴斯克应用数学中心）；University of the Basque Country UPV/EHU（巴斯克大学）；MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计学单位）；IE University（IE大学）
The objective is to model longitudinal and survival data jointly taking into account the dependence between the two responses in a real HIV/AIDS dataset using a shared parameter approach inside a Bayesian framework. We propose a linear mixed effects dispersion model to adjust the CD4 longitudinal biomarker data with a between-individual heterogeneity in the mean and variance. In doing so we are relaxing the usual assumption of a common variance for the longitudinal residuals. A hazard regression model is considered in addition to model the time since HIV/AIDS diagnostic until failure, being th…
This paper takes a quick look at Bayesian joint models (BJM) for longitudinal and survival data. A general formulation for BJM is examined in terms of the sampling distribution of the longitudinal and survival processes, the conditional distribution of the random effects and the prior distribution. Next a basic BJM defined in terms of a mixed linear model and a Cox survival regression models is discussed and some extensions and other Bayesian topics are briefly outlined.
The mixture cure model for analyzing survival data is characterized by the assumption that the population under study is divided into a group of subjects who will experience the event of interest over some finite time horizon and another group of cured subjects who will never experience the event irrespective of the duration of follow-up. When using the Bayesian paradigm for inference in survival models with a cure fraction, it is common practice to rely on Markov chain Monte Carlo (MCMC) methods to sample from posterior distributions. Although computationally feasible, the iterative nature of…
We propose a Bayesian approach using improper priors for hierarchical linear mixed models with flexible random effects and residual error distributions. The error distribution is modelled using scale mixtures of normals, which can capture tails heavier than those of the normal distribution. This generalisation is useful to produce models that are robust to the presence of outliers. The case of asymmetric residual errors is also studied. We present general results for the propriety of the posterior that also cover cases with censored observations, allowing for the use of these models in the con…
Longitudinal Outcomes Truncated by Death: Causal Estimands and Bayesian Estimators
Amsterdam UMC（阿姆斯特丹大学医学中心）；Singapore University of Technology and Design（新加坡科技设计大学）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）
We introduce a numerically tractable formulation of Bayesian joint models for longitudinal and survival data. The longitudinal process is modelled using generalised linear mixed models, while the survival process is modelled using a parametric general hazard structure. The two processes are linked by sharing fixed and random effects, separating the effects that play a role at the time scale from those that affect the hazard scale. This strategy allows for the inclusion of non-linear and time-dependent effects while avoiding the need for numerical integration, which facilitates the implementati…
The COVID-19 pandemic has caused severe public health consequences in the United States. In this study, we use a hierarchical Bayesian model to estimate the age-specific COVID-19 attributable deaths over time in the United States. The model is specified by a novel non-parametric spatial approach, a low-rank Gaussian Process (GP) projected by regularised B-splines. We show that this projection defines a new GP with attractive smoothness and computational efficiency properties, derive its kernel function, and discuss the penalty terms induced by the projected GP. Simulation analyses and benchmar…
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 …
Bayesian joint modelling of longitudinal biomarkers to enable extrapolation of overall survival: an application using larotrectinib trial clinical data
Joint latent class modelling has been developed considerably in the past two decades. In some instances, the models are linked by the latent class k (i.e. the number of subgroups), in others they are joined by shared random effects or a heterogeneous random covariance matrix. We propose an extension to the joint latent class model (JLCM) in which probabilities of subjects being in latent class k can be set to vary with time. This can be a more flexible way to analyse the effect of treatments to patients. For example, a patient may be in period I at the first visit time and may move to period I…
We propose a flexible joint longitudinal-survival framework to examine the association between longitudinally collected biomarkers and a time-to-event endpoint. More specifically, we use our method for analyzing the survival outcome of end-stage renal disease patients with time-varying serum albumin measurements. Our proposed method is robust to common parametric assumptions in that it avoids explicit distributional assumptions on longitudinal measures and allows for subject-specific baseline hazard in the survival component. Fully joint estimation is performed to account for the uncertainty i…
Repeated measures of biomarkers have the potential of explaining hazards of survival outcomes. In practice, these measurements are intermittently measured and are known to be subject to substantial measurement error. Joint modelling of longitudinal and survival data enables us to associate intermittently measured error-prone biomarkers with risks of survival outcomes. Most of the joint models available in the literature have been built on the Gaussian assumption. This makes them sensitive to outliers. In this work, we study a range of robust models to address this issue. For medical data, it h…
An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall's tau association measure that covers the whole range $(-1,1)$. Inference on the model is done under a Bayes…
Joint models have received increasing attention during recent years with extensions into various directions; numerous hazard functions, different association structures, linear and non-linear longitudinal trajectories amongst others. Many of these resulted in new R packages and new formulations of the joint model. However, a joint model with a linear bivariate Gaussian association structure is still a latent Gaussian model (LGM) and thus can be implemented using most existing packages for LGM's. In this paper, we will show that these joint models can be implemented from a LGM viewpoint using t…
This paper considers the problem of estimation in the generalized semiparametric model for longitudinal data when the number of parameters diverges with the sample size. A penalization type of generalized estimating equation method is proposed, while we use the regression spline to approximate the nonparametric component. The proposed procedure involves the specification of the posterior distribution of the random effects, which cannot be evaluated in a closed-form. However, it is possible to approximate this posterior distribution by producing random draws from the distribution using a Metrop…
In medical and biological research, longitudinal data and survival data types are commonly seen. Traditional statistical models mostly consider to deal with either of the data types, such as linear mixed models for longitudinal data, and the Cox models for survival data, while they do not adjust the association between these two different data types. It is desirable to have a joint modeling approach which accomadates both data types and the dependency between them. In this paper, we extend traditional single-index models to a new joint modeling approach, by replacing the single-index component…
Recent evidence has shown that structural magnetic resonance imaging (MRI) is an effective tool for Alzheimer's disease (AD) prediction and diagnosis. While traditional MRI-based diagnosis uses images acquired at a single time point, a longitudinal study is more sensitive and accurate in detecting early pathological changes of the AD. Two main difficulties arise in longitudinal MRI-based diagnosis: (1) the inconsistent longitudinal scans among subjects (i.e., different scanning time and different total number of scans); (2) the heterogeneous progressions of high-dimensional regions of interest…
Joint models of longitudinal and survival data have become an important tool for modeling associations between longitudinal biomarkers and event processes. The association between marker and log-hazard is assumed to be linear in existing shared random effects models, with this assumption usually remaining unchecked. We present an extended framework of flexible additive joint models that allows the estimation of nonlinear, covariate specific associations by making use of Bayesian P-splines. Our joint models are estimated in a Bayesian framework using structured additive predictors for all model…
Bayesian joint modelling using semiparametric accelerated failure time approaches
The University of Queensland’s Clinical Trials Capability (ULTRA)（昆士兰大学临床试验能力中心（ULTRA））；NHMRC Clinical Trial Centre, University of Sydney（悉尼大学国家健康与医学研究委员会临床试验中心）；Australian and New Zealand Urogenital and Prostate (ANZUP) Cancer Trials Group（澳大利亚和新西兰泌尿生殖和前列腺癌症试验组）
Often in follow-up studies intermediate events occur in some patients, such as reinterventions or adverse events. These intermediate events directly affect the shapes of their longitudinal profiles. Our work is motivated by two studies in which such intermediate events have been recorded during follow-up. The first study concerns Congenital Heart Diseased patients who were followed-up echocardiographically, with several patients undergoing reintervention. The second study concerns patients who participated in the SPRINT study and experienced adverse events during follow-up. We are interested i…
Joint modeling of geometric features of longitudinal process and discrete survival time measured on nested timescales: an application to fecundity studies
In animal behavior studies, a common goal is to investigate the causal pathways between an exposure and outcome, and a mediator that lies in between. Causal mediation analysis provides a principled approach for such studies. Although many applications involve longitudinal data, the existing causal mediation models are not directly applicable to settings where the mediators are measured on irregular time grids. In this paper, we propose a causal mediation model that accommodates longitudinal mediators on arbitrary time grids and survival outcomes simultaneously. We take a functional data analys…
A key question in clinical practice is accurate prediction of patient prognosis. To this end, nowadays, physicians have at their disposal a variety of tests and biomarkers to aid them in optimizing medical care. These tests are often performed on a regular basis in order to closely follow the progression of the disease. In this setting it is of medical interest to optimally utilize the recorded information and provide medically-relevant summary measures, such as survival probabilities, that will aid in decision making. In this work we present and compare two statistical techniques that provide…
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…
When Method Choice Changes Statistical Inference: A Comparison of a Baseline Two-Stage Approach and Bayesian Joint Modeling for Longitudinal and Survival Data in an HIV Clinical Trial
This research examines the simultaneous influences of height and weight on longitudinally measured systolic and diastolic blood pressure in children. Previous studies have shown that both height and weight are positively associated with blood pressure. In children, however, the concurrent increases of height and weight have made it all but impossible to discern the effect of height from that of weight. To better understand these influences, we propose to examine the joint effect of height and weight on blood pressure. Bivariate thin plate spline surfaces are used to accommodate the potentially…
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…
Evaluating treatment effects on longitudinal outcomes with attrition due to death: Methods for a two-dimentional estimand with a case study in Quality of Life
Ghent University（根特大学）；Leiden University Medical Center（莱顿大学医学中心）；Pfizer Inc（辉瑞公司）；Oslo University Hospital（奥斯陆大学医院）；European Organisation for Research and Treatment of Cancer (EORTC) Headquarters（欧洲癌症研究与治疗组织（EORTC）总部）；University of Freiburg（弗赖堡大学）
Mixed model repeated measures (MMRM) is the most common analysis approach used in clinical trials for Alzheimer's disease and other progressive diseases measured with continuous outcomes measured over time. The model treats time as a categorical variable, which allows an unconstrained estimate of the mean for each study visit in each randomized group. Categorizing time in this way can be problematic when assessments occur off-schedule, as including off-schedule visits can induce bias, and excluding them ignores valuable information and violates the intention to treat principle. This problem ha…
FastJM: An R Package for Efficient Implementation of Semiparametric Joint Models for Longitudinal and Survival Data
Beijing Normal–Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；University of California at Riverside（加州大学河滨分校）；University of California at Los Angeles（加州大学洛杉矶分校）
Due to the heterogeneity of the randomized controlled trial (RCT) and external target populations, the estimated treatment effect from the RCT is not directly applicable to the target population. For example, the patient characteristics of the ACTG 175 HIV trial are significantly different from that of the three external target populations of interest: US early-stage HIV patients, Thailand HIV patients, and southern Ethiopia HIV patients. This paper considers several methods to transport the treatment effect from the ACTG 175 HIV trial to the target populations beyond the trial population. Mos…
Studies of Alzheimer's disease (AD) often collect multiple longitudinal clinical outcomes, which are correlated and predictive of AD progression. It is of great scientific interest to investigate the association between the outcomes and time to AD onset. We model the multiple longitudinal outcomes as multivariate sparse functional data and propose a functional joint model linking multivariate functional data to event time data. In particular, we propose a multivariate functional mixed model (MFMM) to identify the shared progression pattern and outcome-specific progression patterns of the outco…
Conventional approaches to statistical inference preclude structures that facilitate incorporation of supplemental information acquired from similar circumstances. For example, the analysis of data obtained using perfusion computed tomography to characterize functional imaging biomarkers in cancerous regions of the liver can benefit from partially informative data collected concurrently in noncancerous regions. This paper presents a hierarchical model structure that leverages all available information about a curve, using penalized splines, while accommodating important between-source features…
