BACKWARD ESTIMATION OF STOCHASTIC PROCESSES WITH FAILURE EVENTS AS TIME ORIGINS

Conditional modeling of recurrent event data with terminal event

Recurrent event data with a terminal event arise in follow-up studies. The current literature has primarily focused on the effect of covariates on the recurrent event process using marginal estimating equation approaches or joint modeling approaches via frailties. In this article, we propose a conditional model for recurrent event data with a terminal event, which provides an intuitive interpretation of the effect of the terminal event: at an early time, the rate of recurrent events is nearly independent of the terminal event, but the dependence gets stronger as time goes close to the terminal event time. A two-stage likelihood-based approach is proposed to estimate parameters of interest. Asymptotic properties of the estimators are established. The finite-sample behavior of the proposed method is examined through simulation studies. A real data of colorectal cancer is analyzed by the proposed method for illustration.

Joint modeling in presence of informative censoring on the retrospective time scale with application to palliative care research

Joint modeling of longitudinal data such as quality of life data and survival data is important for palliative care researchers to draw efficient inferences because it can account for the associations between those two types of data. Modeling quality of life on a retrospective from death time scale is useful for investigators to interpret the analysis results of palliative care studies which have relatively short life expectancies. However, informative censoring remains a complex challenge for modeling quality of life on the retrospective time scale although it has been addressed for joint models on the prospective time scale. To fill this gap, we develop a novel joint modeling approach that can address the challenge by allowing informative censoring events to be dependent on patients' quality of life and survival through a random effect. There are two sub-models in our approach: a linear mixed effect model for the longitudinal quality of life and a competing-risk model for the death time and dropout time that share the same random effect as the longitudinal model. Our approach can provide unbiased estimates for parameters of interest by appropriately modeling the informative censoring time. Model performance is assessed with a simulation study and compared with existing approaches. A real-world study is presented to illustrate the application of the new approach.

Semiparametric estimation and testing for panel count data with informative interval-censored failure event

Panel count data and interval-censored data are two types of incomplete data that often occur in event history studies. Almost all existing statistical methods are developed for their separate analysis. In this paper, we investigate a more general situation where a recurrent event process and an interval-censored failure event occur together. To intuitively and clearly explain the relationship between the recurrent current process and failure event, we propose a failure time-dependent mean model through a completely unspecified link function. To overcome the challenges arising from the blending of nonparametric components and parametric regression coefficients, we develop a two-stage conditional expected likelihood-based estimation procedure. We establish the consistency, the convergence rate and the asymptotic normality of the proposed two-stage estimator. Furthermore, we construct a class of two-sample tests for comparison of mean functions from different groups. The proposed methods are evaluated by extensive simulation studies and are illustrated with the skin cancer data that motivated this study.

Kernel Estimation of Bivariate Time-varying Coefficient Model for Longitudinal Data with Terminal Event

We propose a nonparametric bivariate time-varying coefficient model for longitudinal measurements with the occurrence of a terminal event that is subject to right censoring. The time-varying coefficients capture the longitudinal trajectories of covariate effects along with both the followup time and the residual lifetime. The proposed model extends the parametric conditional approach given terminal event time in recent literature, and thus avoids potential model misspecification. We consider a kernel smoothing method for estimating regression coefficients in our model and use cross-validation for bandwidth selection, applying undersmoothing in the final analysis to eliminate the asymptotic bias of the kernel estimator. We show that the kernel estimates follow a finite-dimensional normal distribution asymptotically under mild regularity conditions, and provide an easily computed sandwich covariance matrix estimator. We conduct extensive simulations that show desirable performance of the proposed approach, and apply the method to analyzing the medical cost data for patients with end-stage renal disease.

Premature termination of the unified protocol for the transdiagnostic treatment of emotional disorders: The role of ambivalence towards change

Ambivalence towards change is an expected, recurrent process in psychological change. However, the prolonged experience of ambivalence in psychotherapy contributes to client disengagement, which could result in treatment dropout. Considering the negative effects of premature termination of therapy and the convenience of the identification of clients who are at risk of dropping out before achieving good-outcome, the current study explored the predictive power of ambivalence for premature therapy termination using a multilevel time-backwards model (i.e., considering the session of the dropout as session zero and then modelling what occurred from the dropout until session 1). Participants included a total of 96 psychotherapy clients (38 dropouts) treated in a university-based clinic following the Unified Protocol for Transdiagnostic Treatment of Emotional Disorders. Multilevel modelling using a time-backwards model to analyse dropout data provided evidence of the predictive power of ambivalence evolution throughout treatment on the decision to prematurely discontinue treatment (p < .0001; R² _adj  = .29). Specifically, good-outcome dropouts presented a decreasing ambivalence trend throughout treatment, whereas poor-outcome dropouts tended to experience the same levels of ambivalence before deciding to drop out (time × dropout; β₁₁  = .64, p = .014). Additionally, poor-outcome dropouts presented higher levels of ambivalence (β₀₁  = 9.92, p < .0001) in the last session. The results suggest that the pattern of client ambivalence towards change is a predictor of premature termination of therapy. Implications for clinical and research contexts are discussed.

Joint modeling of longitudinal data with informative cluster size adjusted for zero-inflation and a dependent terminal event

Repeated measures are often collected in longitudinal follow-up from clinical trials and observational studies. In many situations, these measures are adherent to some specific event and are only available when it occurs; an example is serum creatinine from laboratory tests for hospitalized acute kidney injuries. The frequency of event recurrences is potentially correlated with overall health condition and hence may influence the distribution of the outcome measure of interest, leading to informative cluster size. In particular, there may be a large portion of subjects without any events, thus no longitudinal measures are available, which may be due to insusceptibility to such events or censoring before any events, and this zero-inflation nature of the data needs to be taken into account. On the other hand, there often exists a terminal event that may be correlated with the recurrent events. Previous work in this area suffered from the limitation that not all these issues were handled simultaneously. To address this deficiency, we propose a novel joint modeling approach for longitudinal data adjusting for zero-inflated and informative cluster size as well as a terminal event. A three-stage semiparametric likelihood-based approach is applied for parameter estimation and inference. Extensive simulations are conducted to evaluate the performance of our proposal. Finally, we utilize the Assessment, Serial Evaluation, and Subsequent Sequelae of Acute Kidney Injury (ASSESS-AKI) study for illustration.

Trajectories of longitudinal biomarkers for mortality in severely burned patients

This study aimed to investigate the differences in the trajectory of blood biomarkers routinely assessed through forward- and backward-looking approaches among burn patients. This cohort study included patients above 18 years of age from February 2007 to December 2018. All the biomarkers were estimated from admission to discharge from the intensive care unit. Significant differences were observed in the platelet count at 40 days, prothrombin time (PT) at 32 days, white blood cell count at 26 days, creatinine levels at 22 days, and lactate and total bilirubin levels at 19 days before death. In reverse order, significant differences were observed in the fitted model in platelet count at 44 days and in the platelet count and PT at 33 days. We obtained more valuable information from the longitudinal biomarker trajectory using the backward-looking method than using the forward-looking method. The platelet count served as the earliest predictor of mortality among burn patients.

Statistical modeling of longitudinal medical cost trajectory: renal cell cancer care cost analyses

Estimating the current cost of cancer care is important to health policy makers. An indispensable step in cost projection is to estimate cost trajectories from an incident cohort of cancer patients using longitudinal medical cost data, accounting for terminal events such as death, and right censoring due to loss of follow-up. Since the cost of cancer care and survival are correlated, a scientifically meaningful quantity for inference in this context is the mean cost trajectory conditional on survival. We propose a two-stage semiparametric approach to estimate the longitudinal cost trajectories from a joint model of longitudinal medical costs and survival. The longitudinal cost trajectories corresponding to various survival times form a bivariate surface in a triangular area. The cost trajectories are estimated using the tensor products of discretized measurement time and survival, as well as effective ridge penalties for data in 2D arrays. The proposed approach balances the practical considerations of model flexibility, statistical efficiency, and computational tractability. We used the proposed method to estimate the cost trajectories of renal cell cancer patients using the Surveillance, Epidemiology, and End Results-Medicare linked database.

Estimation of the distribution of longitudinal biomarker trajectories prior to disease progression

Most studies characterize longitudinal biomarker trajectories by looking forward at them from a commonly used time origin, such as the initial treatment time. For a better understanding of the relationship between biomarkers and disease progression, we propose to align all subjects by using their disease progression time as the origin and then looking backward at the biomarker distributions prior to that event. We demonstrate that such backward-looking plots are much more informative than forward-looking plots when the research goal is to understand the shape of the trajectory leading up to the event of interest. Such backward-looking plotting is an easy task if disease progression is observed for all the subjects. However, when these events are censored for a significant proportion of subjects in the study cohort, their time origins cannot be identified, and the task of aligning them cannot be performed. We propose a new method to tackle this problem by considering the distributions of longitudinal biomarker data conditional on the failure time. We use landmark analysis models to estimate these distributions. Compared to a naïve method, our new method greatly reduces estimation bias. We apply our method to a study for chronic myeloid leukemia patients whose BCR-ABL transcript expression levels after treatment are good indicators of residual disease. Our proposed method provides a good visualization tool for longitudinal biomarker studies for the early detection of disease.

Survival models and health sequences

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique-reverse alignment-for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.

Semiparametric Estimation of Longitudinal Medical Cost Trajectory

Estimating the average monthly medical costs from disease diagnosis to a terminal event such as death for an incident cohort of patients is a topic of immense interest to researchers in health policy and health economics because patterns of average monthly costs over time reveal how medical costs vary across phases of care. The statistical challenges to estimating monthly medical costs longitudinally are multifold; the longitudinal cost trajectory (formed by plotting the average monthly costs from diagnosis to the terminal event) is likely to be nonlinear, with its shape depending on the time of the terminal event, which can be subject to right censoring. The goal of this paper is to tackle this statistically challenging topic by estimating the conditional mean cost at any month t given the time of the terminal event s. The longitudinal cost trajectories with different terminal event times form a bivariate surface of t and s, under the constraint t ≤ s. We propose to estimate this surface using bivariate penalized splines in an Expectation-Maximization algorithm that treats the censored terminal event times as missing data. We evaluate the proposed model and estimation method in simulations and apply the method to the medical cost data of an incident cohort of stage IV breast cancer patients from the Surveillance, Epidemiology and End Results-Medicare Linked Database.

Conditional modeling of longitudinal data with terminal event

We consider a random effects model for longitudinal data with the occurrence of an informative terminal event that is subject to right censoring. Existing methods for analyzing such data include the joint modeling approach using latent frailty and the marginal estimating equation approach using inverse probability weighting; in both cases the effect of the terminal event on the response variable is not explicit and thus not easily interpreted. In contrast, we treat the terminal event time as a covariate in a conditional model for the longitudinal data, which provides a straight-forward interpretation while keeping the usual relationship of interest between the longitudinally measured response variable and covariates for times that are far from the terminal event. A two-stage semiparametric likelihood-based approach is proposed for estimating the regression parameters; first, the conditional distribution of the right-censored terminal event time given other covariates is estimated and then the likelihood function for the longitudinal event given the terminal event and other regression parameters is maximized. The method is illustrated by numerical simulations and by analyzing medical cost data for patients with end-stage renal disease. Desirable asymptotic properties are provided.

A unified framework of demographic time

Demographic thought and practice is largely conditioned by the Lexis diagram, a two-dimensional graphical representation of the identity between age, period, and birth cohort. This relationship does not account for remaining years of life, total length of life, or time of death, whose use in demographic research is both underrepresented and incompletely situated. We describe an identity between these six demographic time measures and describe the sub-identities and diagrams that pertain to this identity. We provide an application of this framework to the measurement of late-life morbidity prevalence. We generalize these relationships to higher order identities derived from an arbitrary number of events in calendar time. Our examples are based on classic human demography, but the concepts we present can reveal patterns and relationships in any event history data, and contribute to the study of human or non-human population dynamics measured on any scale of calendar time.

A semiparametric joint model for terminal trend of quality of life and survival in palliative care research

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 despite potentially beneficial treatment. Often longitudinal QOL and survival data are highly correlated which, in the face of censoring, makes it challenging to properly analyze and interpret terminal QOL trend. To address these issues, we propose a novel semiparametric statistical approach to jointly model the terminal trend of QOL and survival data. There are two sub-models in our approach: a semiparametric mixed effects model for longitudinal QOL and a Cox model for survival. We use regression splines method to estimate the nonparametric curves and AIC to select knots. We assess the model performance through simulation to establish a novel modeling approach that could be used in future palliative care research trials. Application of our approach in a recently completed palliative care clinical trial is also presented.

Semiparametric modeling and estimation of the terminal behavior of recurrent marker processes before failure events

Recurrent event processes with marker measurements are mostly and largely studied with forward time models starting from an initial event. Interestingly, the processes could exhibit important terminal behavior during a time period before occurrence of the failure event. A natural and direct way to study recurrent events prior to a failure event is to align the processes using the failure event as the time origin and to examine the terminal behavior by a backward time model. This paper studies regression models for backward recurrent marker processes by counting time backward from the failure event. A three-level semiparametric regression model is proposed for jointly modeling the time to a failure event, the backward recurrent event process, and the marker observed at the time of each backward recurrent event. The first level is a proportional hazards model for the failure time, the second level is a proportional rate model for the recurrent events occurring before the failure event, and the third level is a proportional mean model for the marker given the occurrence of a recurrent event backward in time. By jointly modeling the three components, estimating equations can be constructed for marked counting processes to estimate the target parameters in the three-level regression models. Large sample properties of the proposed estimators are studied and established. The proposed models and methods are illustrated by a community-based AIDS clinical trial to examine the terminal behavior of frequencies and severities of opportunistic infections among HIV infected individuals in the last six months of life.

Nonparametric inference for the joint distribution of recurrent marked variables and recurrent survival time

Time between recurrent medical events may be correlated with the cost incurred at each event. As a result, it may be of interest to describe the relationship between recurrent events and recurrent medical costs by estimating a joint distribution. In this paper, we propose a nonparametric estimator for the joint distribution of recurrent events and recurrent medical costs in right-censored data. We also derive the asymptotic variance of our estimator, a test for equality of recurrent marker distributions, and present simulation studies to demonstrate the performance of our point and variance estimators. Our estimator is shown to perform well for a wide range of levels of correlation, demonstrating that our estimators can be employed in a variety of situations when the correlation structure may be unknown in advance. We apply our methods to hospitalization events and their corresponding costs in the second Multicenter Automatic Defibrillator Implantation Trial (MADIT-II), which was a randomized clinical trial studying the effect of implantable cardioverter-defibrillators in preventing ventricular arrhythmia.

Joint modeling of longitudinal, recurrent events and failure time data for survivor's population

Recurrent events together with longitudinal measurements are commonly observed in follow-up studies where the observation is terminated by censoring or a primary failure event. In this article, we developed a joint model where the dependence of longitudinal measurements, recurrent event process and time to failure event is modeled through rescaling the time index. The general idea is that the trajectories of all biology processes of subjects in the survivors' population are elongated or shortened by the rate identified from a model for the failure event. To avoid making disputing assumptions on recurrent events or biomarkers after the failure event (such as death), the model is constructed on the basis of survivors' population. The model also possesses a specific feature that, by aligning failure events as time origins, the backward-in-time model of recurrent events and longitudinal measurements shares the same parameter values with the forward time model. The statistical properties, simulation studies and real data examples are conducted. The proposed method can be generalized to analyze left-truncated data.

Interdialytic weight gain, systolic blood pressure, serum albumin, and C-reactive protein levels change in chronic dialysis patients prior to death

Reports from a United States cohort of chronic hemodialysis patients suggested that weight loss, a decline in pre-dialysis systolic blood pressure, and decreased serum albumin may precede death. However, no comparative studies have been reported in such patients from other countries. Here we analyzed dynamic changes in these parameters in hemodialysis patients and included 3593 individuals from 5 Asian countries; 35,146 from 18 European countries; 8649 from Argentina; and 4742 from the United States. In surviving prevalent patients, these variables appeared to have notably different dynamics than in patients who died. While in all populations the interdialytic weight gain, systolic blood pressure, and serum albumin levels were stable in surviving patients, these indicators declined starting more than a year ahead in those who died with the dynamics similar irrespective of gender and geographic region. In European patients, C-reactive protein levels were available on a routine basis and indicated that levels of this acute-phase protein were low and stable in surviving patients but rose sharply before death. Thus, relevant fundamental biological processes start many months before death in the majority of chronic hemodialysis patients. Longitudinal monitoring of these dynamics may help to identify patients at risk and aid the development of an alert system to initiate timely interventions to improve outcomes.