Estimating treatment effects on the marginal recurrent event mean in the presence of a terminating event

A simple and robust parametric shared frailty model for recurrent events with the competing risk of death: An application to the Carvedilol Prospective Randomized Cumulative Survival trial

Many non-fatal events can be considered recurrent in that they can occur repeatedly over time, and some researchers may be interested in the trajectory and relative risk of non-fatal events. With the competing risk of death, the treatment effect on the mean number of recurrent events is non-identifiable since the observed mean is a function of both the recurrent event and terminal event processes. In this paper, we assume independence between the non-fatal and the terminal event process, conditional on the shared frailty, to fit a parametric model that recovers the trajectory of, and identifies the effect of treatment on, the non-fatal event process in the presence of the competing risk of death. Simulation studies are conducted to verify the reliability of our estimators. We illustrate the method and perform model diagnostics using the Carvedilol Prospective Randomized Cumulative Survival trial which involves heart-failure events.

Study design for restricted mean time analysis of recurrent events and death

The restricted mean time in favor (RMT-IF) of treatment has just been added to the analytic toolbox for composite endpoints of recurrent events and death. To help practitioners design new trials based on this method, we develop tools to calculate the sample size and power. Specifically, we formulate the outcomes as a multistate Markov process with a sequence of transient states for recurrent events and an absorbing state for death. The transition intensities, in this case the instantaneous risks of another nonfatal event or death, are assumed to be time-homogeneous but nonetheless allowed to depend on the number of past events. Using the properties of Coxian distributions, we derive the RMT-IF effect size under the alternative hypothesis as a function of the treatment-to-control intensity ratios along with the baseline intensities, the latter of which can be easily estimated from historical data. We also reduce the variance of the nonparametric RMT-IF estimator to calculable terms under a standard set-up for censoring. Simulation studies show that the resulting formulas provide accurate approximation to the sample size and power in realistic settings. For illustration, a past cardiovascular trial with recurrent-hospitalization and mortality outcomes is analyzed to generate the parameters needed to design a future trial. The procedures are incorporated into the rmt package along with the original methodology on the Comprehensive R Archive Network (CRAN).

Nonparametric inference of general while-alive estimands for recurrent events

Measuring the treatment effect on recurrent events like hospitalization in the presence of death has long challenged statisticians and clinicians alike. Traditional inference on the cumulative frequency unjustly penalizes survivorship as longer survivors also tend to experience more adverse events. Expanding a recently suggested idea of the "while-alive" event rate, we consider a general class of such estimands that adjust for the length of survival without losing causal interpretation. Given a user-specified loss function that allows for arbitrary weighting, we define as estimand the average loss experienced per unit time alive within a target period and use the ratio of this loss rate to measure the effect size. Scaling the loss rate by the width of the corresponding time window gives us an alternative, and sometimes more photogenic, way of showing the data. To make inferences, we construct a nonparametric estimator for the loss rate through the cumulative loss and the restricted mean survival time and derive its influence function in closed form for variance estimation and testing. As simulations and analysis of real data from a heart failure trial both show, the while-alive approach corrects for the false attenuation of treatment effect due to patients living longer under treatment, with increased statistical power as a result. The proposed methods are implemented in the R-package WA, which is publicly available from the Comprehensive R Archive Network (CRAN).

Doubly robust estimation and causal inference for recurrent event data

Many longitudinal databases record the occurrence of recurrent events over time. In this article, we propose a new method to estimate the average causal effect of a binary treatment for recurrent event data in the presence of confounders. We propose a doubly robust semiparametric estimator based on a weighted version of the Nelson-Aalen estimator and a conditional regression estimator under an assumed semiparametric multiplicative rate model for recurrent event data. We show that the proposed doubly robust estimator is consistent and asymptotically normal. In addition, a model diagnostic plot of residuals is presented to assess the adequacy of our proposed semiparametric model. We then evaluate the finite sample behavior of the proposed estimators under a number of simulation scenarios. Finally, we illustrate the proposed methodology via a database of circus artist injuries.

Semiparametric temporal process regression of survival-out-of-hospital

The recurrent/terminal event data structure has undergone considerable methodological development in the last 10-15 years. An example of the data structure that has arisen with increasing frequency involves the recurrent event being hospitalization and the terminal event being death. We consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-Out-of-Hospital is a useful alternative strategy for the analysis of hospitalization/survival in the chronic disease setting, with the response variate representing a refinement to survival time through the incorporation of an objective quality-of-life component. The semiparametric model we consider assumes multiplicative covariate effects and leaves unspecified the baseline probability of being alive-and-out-of-hospital. Using zero-mean estimating equations, the proposed regression parameter estimator can be computed without estimating the unspecified baseline probability process, although baseline probabilities can subsequently be estimated for any time point within the support of the censoring distribution. We demonstrate that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulation studies are performed to show that our estimating procedures have satisfactory finite sample performances. The proposed methods are applied to the Dialysis Outcomes and Practice Patterns Study (DOPPS), an international end-stage renal disease study.

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.

Estimating the effect of a rare time-dependent treatment on the recurrent event rate

In many observational studies, the objective is to estimate the effect of treatment or state-change on the recurrent event rate. If treatment is assigned after the start of follow-up, traditional methods (eg, adjustment for baseline-only covariates or fully conditional adjustment for time-dependent covariates) may give biased results. We propose a two-stage modeling approach using the method of sequential stratification to accurately estimate the effect of a time-dependent treatment on the recurrent event rate. At the first stage, we estimate the pretreatment recurrent event trajectory using a proportional rates model censored at the time of treatment. Prognostic scores are estimated from the linear predictor of this model and used to match treated patients to as yet untreated controls based on prognostic score at the time of treatment for the index patient. The final model is stratified on matched sets and compares the posttreatment recurrent event rate to the recurrent event rate of the matched controls. We demonstrate through simulation that bias due to dependent censoring is negligible, provided the treatment frequency is low, and we investigate a threshold at which correction for dependent censoring is needed. The method is applied to liver transplant (LT), where we estimate the effect of development of post-LT End Stage Renal Disease (ESRD) on rate of days hospitalized.

Evaluating Utility Measurement from Recurrent Marker Processes in the Presence of Competing Terminal Events

In follow-up studies, utility marker measurements are usually collected upon the occurrence of recurrent events until a terminal event such as death takes place. In this article, we define the recurrent marker process to characterize utility accumulation over time. For example, with medical cost and repeated hospitalizations being treated as marker and recurrent events respectively, the recurrent marker process is the trajectory of cumulative cost, which stops to increase after death. In many applications, competing risks arise as subjects are at risk of more than one mutually exclusive terminal event, such as death from different causes, and modeling the recurrent marker process for each failure type is often of interest. However, censoring creates challenges in the methodological development, because for censored subjects, both failure type and recurrent marker process after censoring are unobserved. To circumvent this problem, we propose a nonparametric framework for recurrent marker process with competing terminal events. In the presence of competing risks, we start with an estimator by using marker information from uncensored subjects. As a result, the estimator can be inefficient under heavy censoring. To improve efficiency, we propose a second estimator by combining the first estimator with auxiliary information from the estimate under non-competing risks model. The large sample properties and optimality of the second estimator is established. Simulation studies and an application to the SEER-Medicare linked data are presented to illustrate the proposed methods. Supplemental materials are available online.

Time-varying coefficients in a multivariate frailty model: Application to breast cancer recurrences of several types and death

During their follow-up, patients with cancer can experience several types of recurrent events and can also die. Over the last decades, several joint models have been proposed to deal with recurrent events with dependent terminal event. Most of them require the proportional hazard assumption. In the case of long follow-up, this assumption could be violated. We propose a joint frailty model for two types of recurrent events and a dependent terminal event to account for potential dependencies between events with potentially time-varying coefficients. For that, regression splines are used to model the time-varying coefficients. Baseline hazard functions (BHF) are estimated with piecewise constant functions or with cubic M-Splines functions. The maximum likelihood estimation method provides parameter estimates. Likelihood ratio tests are performed to test the time dependency and the statistical association of the covariates. This model was driven by breast cancer data where the maximum follow-up was close to 20 years.

Time-dependent prognostic score matching for recurrent event analysis to evaluate a treatment assigned during follow-up

Recurrent events often serve as the outcome in epidemiologic studies. In some observational studies, the goal is to estimate the effect of a new or "experimental" (i.e., less established) treatment of interest on the recurrent event rate. The incentive for accepting the new treatment may be that it is more available than the standard treatment. Given that the patient can choose between the experimental treatment and conventional therapy, it is of clinical importance to compare the treatment of interest versus the setting where the experimental treatment did not exist, in which case patients could only receive no treatment or the standard treatment. Many methods exist for the analysis of recurrent events and for the evaluation of treatment effects. However, methodology for the intersection of these two areas is sparse. Moreover, care must be taken in setting up the comparison groups in our setting; use of existing methods featuring time-dependent treatment indicators will generally lead to a biased treatment effect since the comparison group construction will not properly account for the timing of treatment initiation. We propose a sequential stratification method featuring time-dependent prognostic score matching to estimate the effect of a time-dependent treatment on the recurrent event rate. The performance of the method in moderate-sized samples is assessed through simulation. The proposed methods are applied to a prospective clinical study in order to evaluate the effect of living donor liver transplantation on hospitalization rates; in this setting, conventional therapy involves remaining on the wait list or receiving a deceased donor transplant.

Semiparametric Transformation Rate Model for Recurrent Event Data

In this article, we propose a class of semiparametric transformation rate models for recurrent event data subject to right-censoring and potentially stopped by a terminating event (e.g., death). These transformation models include both additive rates model and proportional rates model as special cases. Respecting the property that no recurrent events can occur after the terminating event, we model the conditional recurrent event rate given survival. Weighted estimating equations are constructed to estimate the regression coefficients and baseline rate function. In particular, the baseline rate function is approximated by wavelet function. Asymptotic properties of the proposed estimators are derived and a data-dependent criterion is proposed for selecting the most suitable transformation. Simulation studies show that the proposed estimators perform well for practical sample sizes. The proposed methods are used in two real-data examples: a randomized trial of rhDNase and a community trial of Vitamin A.

Computationally efficient marginal models for clustered recurrent event data

Large observational databases derived from disease registries and retrospective cohort studies have proven very useful for the study of health services utilization. However, the use of large databases may introduce computational difficulties, particularly when the event of interest is recurrent. In such settings, grouping the recurrent event data into prespecified intervals leads to a flexible event rate model and a data reduction that remedies the computational issues. We propose a possibly stratified marginal proportional rates model with a piecewise-constant baseline event rate for recurrent event data. Both the absence and the presence of a terminal event are considered. Large-sample distributions are derived for the proposed estimators. Simulation studies are conducted under various data configurations, including settings in which the model is misspecified. Guidelines for interval selection are provided and assessed using numerical studies. We then show that the proposed procedures can be carried out using standard statistical software (e.g., SAS, R). An application based on national hospitalization data for end-stage renal disease patients is provided.