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Sun J, Cook T. 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. Stat Methods Med Res 2024; 33:765-793. [PMID: 38625756 DOI: 10.1177/09622802241236934] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/18/2024]
Abstract
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.
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Affiliation(s)
- Jiren Sun
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI, USA
| | - Thomas Cook
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI, USA
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2
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Erdmann A, Beyersmann J, Bluhmki E. Comparison of nonparametric estimators of the expected number of recurrent events. Pharm Stat 2024; 23:339-369. [PMID: 38153191 DOI: 10.1002/pst.2356] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2022] [Revised: 09/21/2023] [Accepted: 11/29/2023] [Indexed: 12/29/2023]
Abstract
We compare the performance of nonparametric estimators for the mean number of recurrent events and provide a systematic overview for different recurrent event settings. The mean number of recurrent events is an easily interpreted marginal feature often used for treatment comparisons in clinical trials. Incomplete observations, dependencies between successive events, terminating events acting as competing risk, or gaps between at risk periods complicate the estimation. We use survival multistate models to represent different complex recurrent event situations, profiting from recent advances in nonparametric estimation for non-Markov multistate models, and explain several estimators by using multistate intensity processes, including the common Nelson-Aalen-type estimators with and without competing mortality. In addition to building on estimation of state occupation probabilities in non-Markov models, we consider a simple extension of the Nelson-Aalen estimator by allowing for dependence on the number of prior recurrent events. We pay particular attention to the assumptions required for the censoring mechanism, one issue being that some settings require the censoring process to be entirely unrelated while others allow for state-dependent or event-driven censoring. We conducted extensive simulation studies to compare the estimators in various complex situations with recurrent events. Our practical example deals with recurrent chronic obstructive pulmonary disease exacerbations in a clinical study, which will also be used to illustrate two-sample-inference using resampling.
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Affiliation(s)
| | | | - Erich Bluhmki
- Boehringer Ingelheim Pharma GmbH & Go. KG, Biberach, Germany
- Biberach University of Applied Sciences, Biberach, Germany
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3
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LeMasters K, Renson A, Zalla L, Martin CL, Edwards JK. Understanding the Accumulation of Health-Related Inequities Over the Life Course Using the Mean Cumulative Count. Am J Epidemiol 2023; 192:1425-1431. [PMID: 37218618 PMCID: PMC10666969 DOI: 10.1093/aje/kwad123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 04/26/2023] [Accepted: 05/17/2023] [Indexed: 05/24/2023] Open
Abstract
Understanding how health inequities develop over time is necessary to inform interventions, but methods for doing so are underutilized. We provide an example of the accumulation of stressful life events using the mean cumulative count (MCC), which estimates the expected number of events per person as a function of time, allowing for censoring and competing events. Data came from the National Longitudinal Survey of Youth 1997, a nationally representative data set. To compare the MCC with standard practice, we present the proportions of persons experiencing 1, 2, and ≥3 stressful events and the cumulative probability of experiencing at least 1 event by the end of follow-up. Our sample included 6,522 individuals aged 18-33 years who were followed for a median of 14 years. Using the MCC, by age 20 years the expected number of encounters was 56 events per 100 participants for Black non-Hispanic persons, 47 per 100 for White non-Hispanic persons, and 50 per 100 for Hispanic persons. By age 33 years, inequities grew to 117, 99, and 108 events per 100 persons, respectively. The MCC revealed that inequities in stressful events accumulate over the course of early adulthood, partially driven by repeat events; this information was not evident from conventional approaches. This method can be used to identify intervention points for disrupting the accumulation of repeat events to improve health equity.
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Affiliation(s)
- Katherine LeMasters
- Correspondence to Dr. Katherine LeMasters, Division of General Internal Medicine, School of Medicine, Anschutz Medical Campus, University of Colorado, Academic Office 1, 12631 East 17th Avenue, Aurora, CO 80045 (e-mail: )
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4
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Mao L. Nonparametric inference of general while-alive estimands for recurrent events. Biometrics 2023; 79:1749-1760. [PMID: 35731993 PMCID: PMC9772359 DOI: 10.1111/biom.13709] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 06/16/2022] [Indexed: 12/24/2022]
Abstract
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).
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Affiliation(s)
- Lu Mao
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI, 53792, USA
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5
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Mao L. On restricted mean time in favor of treatment. Biometrics 2023; 79:61-72. [PMID: 34562019 PMCID: PMC8948098 DOI: 10.1111/biom.13570] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Revised: 06/27/2021] [Accepted: 09/03/2021] [Indexed: 12/29/2022]
Abstract
The restricted mean time in favor (RMT-IF) of treatment is a nonparametric effect size for complex life history data. It is defined as the net average time the treated spend in a more favorable state than the untreated over a prespecified time window. It generalizes the familiar restricted mean survival time (RMST) from the two-state life-death model to account for intermediate stages in disease progression. The overall estimand can be additively decomposed into stage-wise effects, with the standard RMST as a component. Alternate expressions of the overall and stage-wise estimands as integrals of the marginal survival functions for a sequence of landmark transitioning events allow them to be easily estimated by plug-in Kaplan-Meier estimators. The dynamic profile of the estimated treatment effects as a function of follow-up time can be visualized using a multilayer, cone-shaped "bouquet plot." Simulation studies under realistic settings show that the RMT-IF meaningfully and accurately quantifies the treatment effect and outperforms traditional tests on time to the first event in statistical efficiency thanks to its fuller utilization of patient data. The new methods are illustrated on a colon cancer trial with relapse and death as outcomes and a cardiovascular trial with recurrent hospitalizations and death as outcomes. The R-package rmt implements the proposed methodology and is publicly available from the Comprehensive R Archive Network (CRAN).
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Affiliation(s)
- Lu Mao
- Department of Biostatistics and Medical Informatics, School of Medicine and Public Health, University of Wisconsin, Madison, Wisconsin 53792, U.S.A
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6
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Cortese G, Scheike TH. Efficient estimation of the marginal mean of recurrent events. J R Stat Soc Ser C Appl Stat 2022. [DOI: 10.1111/rssc.12586] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Affiliation(s)
- Giuliana Cortese
- Department of Statistical Sciences University of Padova Padova Italy
| | - Thomas H. Scheike
- Section of Biostatistics, Department of Public Health University of Copenhagen Copenhagen N Denmark
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7
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Jin J, Song X, Sun L. Dynamic semiparametric transformation models for recurrent event data with a terminal event. Stat Med 2022; 41:5432-5447. [PMID: 36121319 DOI: 10.1002/sim.9577] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2021] [Revised: 08/01/2022] [Accepted: 09/05/2022] [Indexed: 11/06/2022]
Abstract
Recurrent event data with a terminal event commonly arise in many longitudinal follow-up studies. This article proposes a class of dynamic semiparametric transformation models for the marginal mean functions of the recurrent events with a terminal event, where some covariate effects may be time-varying. An estimation procedure is developed for the model parameters, and the asymptotic properties of the resulting estimators are established. In addition, relevant significance tests are suggested for examining whether or not covariate effects vary with time, and a model checking procedure is presented for assessing the adequacy of the proposed models. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is provided.
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Affiliation(s)
- Jin Jin
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, China
| | - Xinyuan Song
- Department of Statistics, The Chinese University of Hong Kong, Hong Kong, China
| | - Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China.,School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
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8
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Cheung LC, Albert PS, Das S, Cook RJ. Multistate models for the natural history of cancer progression. Br J Cancer 2022; 127:1279-1288. [PMID: 35821296 DOI: 10.1038/s41416-022-01904-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2022] [Revised: 06/21/2022] [Accepted: 06/28/2022] [Indexed: 11/09/2022] Open
Abstract
BACKGROUND Multistate models can be effectively used to characterise the natural history of cancer. Inference from such models has previously been useful for setting screening policies. METHODS We introduce the basic elements of multistate models and the challenges of applying these models to cancer data. Through simulation studies, we examine (1) the impact of assuming time-homogeneous Markov transition intensities when the intensities depend on the time since entry to the current state (i.e., the process is time-inhomogenous semi-Markov) and (2) the effect on precancer risk estimation when observation times depend on an unmodelled intermediate disease state. RESULTS In the settings we examined, we found that misspecifying a time-inhomogenous semi-Markov process as a time-homogeneous Markov process resulted in biased estimates of the mean sojourn times. When screen-detection of the intermediate disease leads to more frequent future screening assessments, there was minimal bias induced compared to when screen-detection of the intermediate disease leads to less frequent screening. CONCLUSIONS Multistate models are useful for estimating parameters governing the process dynamics in cancer such as transition rates, sojourn time distributions, and absolute and relative risks. As with most statistical models, to avoid incorrect inference, care should be given to use the appropriate specifications and assumptions.
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Affiliation(s)
- Li C Cheung
- Biostatistics Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, National Institutes of Health, Rockville, MD, USA.
| | - Paul S Albert
- Biostatistics Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, National Institutes of Health, Rockville, MD, USA
| | - Shrutikona Das
- Biostatistics Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, National Institutes of Health, Rockville, MD, USA
| | - Richard J Cook
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada
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Su CL, Steele RJ, Shrier I. The semiparametric accelerated trend-renewal process for recurrent event data. LIFETIME DATA ANALYSIS 2021; 27:357-387. [PMID: 33768490 DOI: 10.1007/s10985-021-09519-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2019] [Accepted: 02/23/2021] [Indexed: 06/12/2023]
Abstract
Recurrent event data arise in many biomedical longitudinal studies when health-related events can occur repeatedly for each subject during the follow-up time. In this article, we examine the gap times between recurrent events. We propose a new semiparametric accelerated gap time model based on the trend-renewal process which contains trend and renewal components that allow for the intensity function to vary between successive events. We use the Buckley-James imputation approach to deal with censored transformed gap times. The proposed estimators are shown to be consistent and asymptotically normal. Model diagnostic plots of residuals and a method for predicting number of recurrent events given specified covariates and follow-up time are also presented. Simulation studies are conducted to assess finite sample performance of the proposed method. The proposed technique is demonstrated through an application to two real data sets.
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Affiliation(s)
- Chien-Lin Su
- Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada.
- Department of Epidemiology, Biostatistics and Occupational Health, McGill University, Montréal, QC, Canada.
| | - Russell J Steele
- Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada
| | - Ian Shrier
- Centre for Clinical Epidemiology, Lady Davis Institute, Jewish General Hospital, McGill University, Montréal, QC, Canada
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10
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Zhong Y, Cook RJ. Semiparametric recurrent event vs time-to-first-event analyses in randomized trials: Estimands and model misspecification. Stat Med 2021; 40:3823-3842. [PMID: 33880781 DOI: 10.1002/sim.9002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2020] [Revised: 02/27/2021] [Accepted: 04/07/2021] [Indexed: 12/18/2022]
Abstract
Insights regarding the merits of recurrent event and time-to-first-event analyses are needed to provide guidance on strategies for analyzing intervention effects in randomized trials involving recurrent event responses. Using established asymptotic results we introduce a framework for studying the large sample properties of estimators arising from semiparametric proportional rate function models and Cox regression under model misspecification. The asymptotic biases and power implications are investigated for different data generating models, and we study the impact of dependent censoring on these findings. Illustrative applications are given involving data from a cystic fibrosis trial and a carcinogenicity experiment, following which we summarize findings and discuss implications for clinical trial design.
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Affiliation(s)
- Yujie Zhong
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, P.R. China
| | - Richard J Cook
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada
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11
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Osmani F, Hajizadeh E, Akbari ME. Prognostic Factors Associated with Curing in Patients with Breast Cancer: A Joint Frailty Model. Int J Prev Med 2021; 12:9. [PMID: 34084306 PMCID: PMC8106266 DOI: 10.4103/ijpvm.ijpvm_89_19] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2019] [Accepted: 01/17/2020] [Indexed: 11/30/2022] Open
Abstract
Background: Recurrent event data arise frequently in longitudinal medical studies. In many situations, there are a large portion of patients without any recurrences, manifesting the “zero-inflated” nature of the data. Moreover, there often exists a terminal event which may be correlated with the recurrent events. The goal of this study is to extend the application of joint frailty model to identify the prognostic factors associated with curing in patients with breast cancer. Methods: As a prospective study, medical records of women who had been attended to Cancer Research Center, Shahid Beheshti University of Medical Sciences from January 1998 to February 2016 were reviewed. Finally, after an initial review of medical records, 711 patients were included in the study and analyzed. A checklist that included items drawn from the demographic background of patients was provided in the study. Two joint frailty models for zero-inflated recurrent events, combining a logistic model for “structural zero” status (Yes/No) and a joint frailty proportional hazards model for recurrent and terminal event times were performed to identify factors associated with BCS. Results: The mean age of patients was 38.2 years. The numbers of subjects with 1, 2, 3, and 4 recurrent events were 392, 207, 97, and 15, respectively. The median follow-up time was 6.87 years. There were 137 (19.2%) deaths from cancer during the follow-up. Among the 574 patients who were censored, 418 had no tumor recurrence. Thus, there may exist a large portion of “cured” subjects. We can see that the radiation (OR = 6.02, CI = (3.87, 8.61)) and tumor size interaction with radiation (OR = 1.065, CI = (1.002–1.26)) were significant in the cure model (P < 0.05) which means that patients with smaller tumor sizes were more likely to be cured by radiation therapy. Conclusions: Our proposed models can help investigators to evaluate which treatment will result in a higher fraction of cured subjects. This is usually an important research question in biomedical studies.
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Affiliation(s)
- Freshteh Osmani
- Infectious Diseases Research Center, Birjand University of Medical Sciences, Birjand, Iran
| | - Ebrahim Hajizadeh
- Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, Iran
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12
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Sun X, Ding J, Sun L. A semiparametric additive rates model for the weighted composite endpoint of recurrent and terminal events. LIFETIME DATA ANALYSIS 2020; 26:471-492. [PMID: 31549283 DOI: 10.1007/s10985-019-09486-w] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2018] [Accepted: 09/17/2019] [Indexed: 06/10/2023]
Abstract
Recurrent event data with a terminal event commonly arise in longitudinal follow-up studies. We use a weighted composite endpoint of all recurrent and terminal events to assess the overall effects of covariates on the two types of events. A semiparametric additive rates model is proposed to analyze the weighted composite event process and the dependence structure among recurrent and terminal events is left unspecified. An estimating equation approach is developed for inference, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is illustrated.
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Affiliation(s)
- Xiaowei Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
- University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Jieli Ding
- School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, China
| | - Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China.
- University of Chinese Academy of Sciences, Beijing, 100049, China.
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13
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Maitra P, Amorim LDAF, Cai J. Multiplicative rates model for recurrent events in case-cohort studies. LIFETIME DATA ANALYSIS 2020; 26:134-157. [PMID: 30734884 PMCID: PMC6687570 DOI: 10.1007/s10985-019-09466-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2018] [Accepted: 01/29/2019] [Indexed: 06/09/2023]
Abstract
In large prospective cohort studies, accumulation of covariate information and follow-up data make up the majority of the cost involved in the study. This might lead to the study being infeasible when there are some expensive variables and/or the event is rare. Prentice (Biometrika 73(1):1-11, 1986) proposed the case-cohort study for time to event data to tackle this problem. There has been extensive research on the analysis of univariate and clustered failure time data, where the clusters are formed among different individuals under case-cohort sampling scheme. However, recurrent event data are quite common in biomedical and public health research. In this paper, we propose case-cohort sampling schemes for recurrent events. We consider a multiplicative rates model for the recurrent events and propose a weighted estimating equations approach for parameter estimation. We show that the estimators are consistent and asymptotically normally distributed. The proposed estimator performed well in finite samples in our simulation studies. For illustration purposes, we examined the association between prior occurrence of measles on acute lower respiratory tract infections (ALRI) among young children in Brazil.
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Affiliation(s)
- Poulami Maitra
- Department of Biostatistics, University of North Carolina at Chapel Hill,
| | - Leila DAF Amorim
- Department of Statistics, Institute of Mathematics, Federal University of Bahia, Brazil,
| | - Jianwen Cai
- Department of Biostatistics, University of North Carolina at Chapel Hill,
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Andersen PK, Angst J, Ravn H. Modeling marginal features in studies of recurrent events in the presence of a terminal event. LIFETIME DATA ANALYSIS 2019; 25:681-695. [PMID: 30697652 DOI: 10.1007/s10985-019-09462-4] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Accepted: 01/21/2019] [Indexed: 06/09/2023]
Abstract
We study models for recurrent events with special emphasis on the situation where a terminal event acts as a competing risk for the recurrent events process and where there may be gaps between periods during which subjects are at risk for the recurrent event. We focus on marginal analysis of the expected number of events and show that an Aalen-Johansen type estimator proposed by Cook and Lawless is applicable in this situation. A motivating example deals with psychiatric hospital admissions where we supplement with analyses of the marginal distribution of time to the competing event and the marginal distribution of the time spent in hospital. Pseudo-observations are used for the latter purpose.
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Affiliation(s)
- Per Kragh Andersen
- Section of Biostatistics, University of Copenhagen, Ø. Farimagsgade 5, PB 2099, 1014, Copenhagen K, Denmark.
| | - Jules Angst
- Department of Psychiatry, Psychotherapy and Psychosomatics, Psychiatric Hospital, University of Zürich, Zurich, Switzerland
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15
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Jazić I, Haneuse S, French B, MacGrogan G, Rondeau V. Design and analysis of nested case-control studies for recurrent events subject to a terminal event. Stat Med 2019; 38:4348-4362. [PMID: 31290191 DOI: 10.1002/sim.8302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2018] [Revised: 06/06/2019] [Accepted: 06/06/2019] [Indexed: 11/08/2022]
Abstract
The process by which patients experience a series of recurrent events, such as hospitalizations, may be subject to death. In cohort studies, one strategy for analyzing such data is to fit a joint frailty model for the intensities of the recurrent event and death, which estimates covariate effects on the two event types while accounting for their dependence. When certain covariates are difficult to obtain, however, researchers may only have the resources to subsample patients on whom to collect complete data: one way is using the nested case-control (NCC) design, in which risk set sampling is performed based on a single outcome. We develop a general framework for the design of NCC studies in the presence of recurrent and terminal events and propose estimation and inference for a joint frailty model for recurrence and death using data arising from such studies. We propose a maximum weighted penalized likelihood approach using flexible spline models for the baseline intensity functions. Two standard error estimators are proposed: a sandwich estimator and a perturbation resampling procedure. We investigate operating characteristics of our estimators as well as design considerations via a simulation study and illustrate our methods using two studies: one on recurrent cardiac hospitalizations in patients with heart failure and the other on local recurrence and metastasis in patients with breast cancer.
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Affiliation(s)
- Ina Jazić
- Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, Massachusetts
| | - Sebastien Haneuse
- Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, Massachusetts
| | - Benjamin French
- Department of Statistics, Radiation Effects Research Foundation, Hiroshima, Japan
| | | | - Virginie Rondeau
- Centre de recherche INSERM U1219, Université de Bordeaux-ISPED, Bordeaux, France
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16
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Osmani F, Hajizadeh E, Rasekhi A, Akbari ME. Prognostic factors associated with locoronal relapses, metastatic relapses, and death among women with breast cancer. Population-based cohort study. Breast 2019; 48:82-88. [PMID: 31568898 DOI: 10.1016/j.breast.2019.09.010] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2019] [Revised: 09/17/2019] [Accepted: 09/18/2019] [Indexed: 10/26/2022] Open
Abstract
BACKGROUND This study was designed to identify the prognostic factors associated with two types of relapses of breast neoplasms. The aim of this study was to evaluate the association between two possible relapses for the increased incidence of distant metastases observed in patients with local relapses injuries, using multivariate statistical models. DESIGN And Setting: A population-based cohort study that was designed as a single center: the cancer research center, Shahid Beheshti University of medical sciences. METHODS This study was conducted on 1815 patients with breast cancer having age of 22 or more. This study considers the analysis of recurrence and survival by joint modeling of three correlated outcomes: local recurrence, distant recurrence (metastasis) and death. The goals are to find out the effects of treatments on recurrences and death, the effects of relapses on death and the correlation between local and distant recurrences. RESULTS According to obtained results of the fitted models, the risk of local and metastatic relapses or death increased for patients with at least one positive lymph node (N+) or for patients with a grade greater than I. Also, the variable HR+ was significantly associated with the hazards of locoronal, metastatic recurrence and the death for both reduced and proposed models (P < 0.05). CONCLUSIONS We concluded that if the association between these outcomes are not taken into account, we may lose important information. Given the small number of recurrent events, these results should be considered with caution.
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Affiliation(s)
- Freshteh Osmani
- Infectious Diseases Research Center, Faculty of Medicine, Birjand University of Medical Sciences, Birjand, Iran.
| | - Ebrahim Hajizadeh
- Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University (TMU), Tehran, Iran.
| | - Aliakbar Rasekhi
- Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University (TMU), Tehran, Iran.
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Lee J, Cook RJ. On estimands arising from misspecified semiparametric rate‐based analysis of recurrent episodic conditions. Stat Med 2019; 38:4977-4998. [DOI: 10.1002/sim.8345] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2018] [Revised: 07/18/2019] [Accepted: 07/22/2019] [Indexed: 11/10/2022]
Affiliation(s)
- Jooyoung Lee
- Department of Statistics and Actuarial Science University of Waterloo Waterloo Ontario Canada
| | - Richard J. Cook
- Department of Statistics and Actuarial Science University of Waterloo Waterloo Ontario Canada
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Lawless JF, Cook RJ. A new perspective on loss to follow‐up in failure time and life history studies. Stat Med 2019; 38:4583-4610. [DOI: 10.1002/sim.8318] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2018] [Revised: 05/07/2019] [Accepted: 06/20/2019] [Indexed: 11/11/2022]
Affiliation(s)
- Jerald F. Lawless
- Department of Statistics and Actuarial ScienceUniversity of Waterloo Waterloo Ontario Canada
| | - Richard J. Cook
- Department of Statistics and Actuarial ScienceUniversity of Waterloo Waterloo Ontario Canada
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19
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Zhan T, Schaubel DE. Semiparametric temporal process regression of survival-out-of-hospital. LIFETIME DATA ANALYSIS 2019; 25:322-340. [PMID: 29796979 PMCID: PMC6251773 DOI: 10.1007/s10985-018-9433-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2017] [Accepted: 05/09/2018] [Indexed: 06/08/2023]
Abstract
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.
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Affiliation(s)
- Tianyu Zhan
- Department of Biostatistics, University of Michigan, 1415 Washington Hts., Ann Arbor, MI, 48109-2029, USA
| | - Douglas E Schaubel
- Department of Biostatistics, University of Michigan, 1415 Washington Hts., Ann Arbor, MI, 48109-2029, USA.
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20
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Zhong Y, Cook RJ. The effect of omitted covariates in marginal and partially conditional recurrent event analyses. LIFETIME DATA ANALYSIS 2019; 25:280-300. [PMID: 29767377 PMCID: PMC6423006 DOI: 10.1007/s10985-018-9430-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/24/2017] [Accepted: 03/19/2018] [Indexed: 06/08/2023]
Abstract
There have been many advances in statistical methodology for the analysis of recurrent event data in recent years. Multiplicative semiparametric rate-based models are widely used in clinical trials, as are more general partially conditional rate-based models involving event-based stratification. The partially conditional model provides protection against extra-Poisson variation as well as event-dependent censoring, but conditioning on outcomes post-randomization can induce confounding and compromise causal inference. The purpose of this article is to examine the consequences of model misspecification in semiparametric marginal and partially conditional rate-based analysis through omission of prognostic variables. We do so using estimating function theory and empirical studies.
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Affiliation(s)
- Yujie Zhong
- MRC Biostatistics Unit, School of Clinical Medicine, University of Cambridge, Cambridge Institute of Public Health, Forvie Site, Robinson Way, Cambridge, CB2 0SR UK
| | - Richard J. Cook
- Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1 Canada
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21
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Analyzing Relationship Between Local and Metastasis Relapses with Survival of Patients with Breast Cancer: A Study Using Joint Frailty Model. INTERNATIONAL JOURNAL OF CANCER MANAGEMENT 2018. [DOI: 10.5812/ijcm.81783] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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22
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Qu L, Sun L. The Cox–Aalen model for recurrent‐event data with a dependent terminal event. STAT NEERL 2018. [DOI: 10.1111/stan.12167] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Affiliation(s)
- Lianqiang Qu
- School of Mathematics and StatisticsCentral China Normal University 430079 Wuhan China
| | - Liuquan Sun
- Institute of Applied MathematicsAcademy of Mathematics and Systems Science, Chinese Academy of Sciences 100190 Beijing China
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23
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Chen X, Ding J, Sun L. A semiparametric additive rate model for a modulated renewal process. LIFETIME DATA ANALYSIS 2018; 24:675-698. [PMID: 29185212 DOI: 10.1007/s10985-017-9413-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2016] [Accepted: 11/22/2017] [Indexed: 06/07/2023]
Abstract
Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.
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Affiliation(s)
- Xin Chen
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People's Republic of China
| | - Jieli Ding
- School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People's Republic of China
| | - Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People's Republic of China.
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24
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Vidula H, Kutyifa V, Johnson BA, Strawderman RL, Harrington D, Polonsky B, Papernov A, Alexis JD. Readmission Patterns During Long-Term Follow-Up After Left Ventricular Assist Device Implantation. Am J Cardiol 2018; 122:1021-1027. [PMID: 30064855 DOI: 10.1016/j.amjcard.2018.05.037] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 02/11/2018] [Revised: 05/21/2018] [Accepted: 05/21/2018] [Indexed: 01/06/2023]
Abstract
As more patients are supported for longer periods by a left ventricular assist device (LVAD), hospital readmission is becoming a growing problem. However, data about temporal changes in readmission rates and causes for patients with prolonged LVAD support are limited. We aimed to evaluate rates, causes, and predictors of any and long-term readmission after LVAD placement at our institution. We followed 177 HeartMate II LVAD patients for a mean of 1.90 ± 1.33 years post initial discharge after implantation. A marginal rate model was used to evaluate readmission rates, accounting for mortality. During the first year, the readmission rate was 1.79 (95% confidence interval 1.51 to 2.10) readmissions per year. The readmission rate was 1.54 (95% confidence interval 1.07 to 1.93) 2 to 3 years after initial discharge. There was a further decrease in readmission rate in the 3- to 4-year interval. The most common causes of readmission during the first year and even after 3 to 4 years of LVAD support were bleeding (excluding intracranial bleeding) and infection. Female gender was associated with an increased risk of readmission in multivariable analyses, while blood urea nitrogen was predictive of long-term readmissions. In conclusion, readmission after LVAD implantation is common, but readmission rates decrease during long-term follow-up. Bleeding and infection remain leading causes of readmission during longer follow-up and strategies to decrease these complications may reduce readmission rates. Female patients and patients with renal dysfunction have increased risk of readmission and further studies are needed to improve outcomes in these groups.
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25
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Smith AR, Zhu D, Goodrich NP, Merion RM, Schaubel DE. Estimating the effect of a rare time-dependent treatment on the recurrent event rate. Stat Med 2018; 37:1986-1996. [PMID: 29479838 PMCID: PMC5943190 DOI: 10.1002/sim.7626] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2016] [Revised: 10/31/2017] [Accepted: 01/05/2018] [Indexed: 11/05/2022]
Abstract
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.
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Affiliation(s)
- Abigail R Smith
- Department of Biostatistics, University of Michigan, 1415 Washington Heights, Ann Arbor, Michigan 48109-2029, USA
- Arbor Research Collaborative for Health, 340 E. Huron St, Suite 300, Ann Arbor, Michigan 48104, USA
| | - Danting Zhu
- Department of Biostatistics, University of Michigan, 1415 Washington Heights, Ann Arbor, Michigan 48109-2029, USA
| | - Nathan P Goodrich
- Arbor Research Collaborative for Health, 340 E. Huron St, Suite 300, Ann Arbor, Michigan 48104, USA
| | - Robert M Merion
- Arbor Research Collaborative for Health, 340 E. Huron St, Suite 300, Ann Arbor, Michigan 48104, USA
| | - Douglas E Schaubel
- Department of Biostatistics, University of Michigan, 1415 Washington Heights, Ann Arbor, Michigan 48109-2029, USA
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26
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A novel approach for analyzing data on recurrent events with duration to estimate the combined cumulative rate of both variables over time. Contemp Clin Trials Commun 2018; 10:50-56. [PMID: 29696158 PMCID: PMC5898580 DOI: 10.1016/j.conctc.2018.03.002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2017] [Revised: 02/23/2018] [Accepted: 03/08/2018] [Indexed: 11/24/2022] Open
Abstract
Recurrent adverse events, once occur often continue for some duration of time in clinical trials; and the number of events along with their durations is clinically considered as a measure of severity of a disease under study. While there are methods available for analyzing recurrent events or durations or for analyzing both side by side, no effort has been made so far to combine them and present as a single measure. However, this single-valued combined measure may help clinicians assess the wholesome effect of recurrence of incident comprising events and durations. Non-parametric approach is adapted here to develop an estimator for estimating the combined rate of both, the recurrence of events as well as the event-continuation, that is the duration per event. The proposed estimator produces a single numerical value, the interpretation and meaningfulness of which are discussed through the analysis of a real-life clinical dataset. The algebraic expression of variance is derived, asymptotic normality of the estimator is noted, and demonstration is provided on how the estimator can be used in the setup of testing of statistical hypothesis. Further possible development of the estimator is also noted, to adjust for the dependence of event occurrences on the history of the process generating recurrent events through covariates and for the case of dependent censoring.
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27
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Joint analysis of recurrent event data with additive–multiplicative hazards model for the terminal event time. METRIKA 2018. [DOI: 10.1007/s00184-018-0654-3] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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28
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Han M, Han D, Sun L. A class of partially linear transformation models for recurrent gap times. COMMUN STAT-THEOR M 2018. [DOI: 10.1080/03610926.2017.1313986] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Miao Han
- Department of Applied Statistics, School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, P. R. China
| | - Dongxiao Han
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P. R. China
| | - Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P. R. China
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29
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Ding J, Sun L. Additive mixed effect model for recurrent gap time data. LIFETIME DATA ANALYSIS 2017; 23:223-253. [PMID: 26296808 DOI: 10.1007/s10985-015-9341-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2014] [Accepted: 08/17/2015] [Indexed: 06/04/2023]
Abstract
Gap times between recurrent events are often of primary interest in medical and observational studies. The additive hazards model, focusing on risk differences rather than risk ratios, has been widely used in practice. However, the marginal additive hazards model does not take the dependence among gap times into account. In this paper, we propose an additive mixed effect model to analyze gap time data, and the proposed model includes a subject-specific random effect to account for the dependence among the gap times. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite sample behavior of the proposed methods is evaluated through simulation studies, and an application to a data set from a clinic study on chronic granulomatous disease is provided.
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Affiliation(s)
- Jieli Ding
- School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People's Republic of China
| | - Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People's Republic of China.
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30
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Mao L, Lin DY. Semiparametric regression for the weighted composite endpoint of recurrent and terminal events. Biostatistics 2016; 17:390-403. [PMID: 26668069 PMCID: PMC4804115 DOI: 10.1093/biostatistics/kxv050] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2015] [Revised: 09/23/2015] [Accepted: 11/02/2015] [Indexed: 11/12/2022] Open
Abstract
Recurrent event data are commonly encountered in clinical and epidemiological studies. A major complication arises when recurrent events are terminated by death. To assess the overall effects of covariates on the two types of events, we define a weighted composite endpoint as the cumulative number of recurrent and terminal events properly weighted by the relative severity of each event. We propose a semiparametric proportional rates model which specifies that the (possibly time-varying) covariates have multiplicative effects on the rate function of the weighted composite endpoint while leaving the form of the rate function and the dependence among recurrent and terminal events completely unspecified. We construct appropriate estimators for the regression parameters and the cumulative frequency function. We show that the estimators are consistent and asymptotically normal with variances that can be consistently estimated. We also develop graphical and numerical procedures for checking the adequacy of the model. We then demonstrate the usefulness of the proposed methods in simulation studies. Finally, we provide an application to a major cardiovascular clinical trial.
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Affiliation(s)
- Lu Mao
- Department of Biostatistics, University of North Carolina, Chapel Hill, NC 27599-7420, USA
| | - D Y Lin
- Department of Biostatistics, University of North Carolina, Chapel Hill, NC 27599-7420, USA
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31
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Smith AR, Schaubel DE. Time-dependent prognostic score matching for recurrent event analysis to evaluate a treatment assigned during follow-up. Biometrics 2015; 71:950-9. [PMID: 26295563 DOI: 10.1111/biom.12361] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2014] [Revised: 04/01/2015] [Accepted: 05/01/2015] [Indexed: 11/30/2022]
Abstract
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.
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Affiliation(s)
- Abigail R Smith
- Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, USA
| | - Douglas E Schaubel
- Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, USA
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32
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Lawless JF, Nazeri Rad N. Estimation and assessment of markov multistate models with intermittent observations on individuals. LIFETIME DATA ANALYSIS 2015; 21:160-179. [PMID: 25332076 DOI: 10.1007/s10985-014-9310-z] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2014] [Accepted: 10/06/2014] [Indexed: 06/04/2023]
Abstract
Multistate models provide important methods of analysis for many life history processes, and this is an area where John Klein made numerous contributions. When individuals in a study group are observed continuously so that all transitions between states, and their times, are known, estimation and model checking is fairly straightforward. However, individuals in many studies are observed intermittently, and only the states occupied at the observation times are known. We review methods of estimation and assessment for Markov models in this situation. Numerical studies that show the effects of inter-observation times are provided, and new methods for assessing fit are given. An illustration involving viral load dynamics for HIV-positive persons is presented.
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Affiliation(s)
- J F Lawless
- Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada,
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33
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Bailey L, Weaver FM, Chin AS, Carbone LD. Estimation of a recurrent event gap time distribution: an application to morbidity outcomes following lower extremity fracture in Veterans with spinal cord injury. HEALTH SERVICES AND OUTCOMES RESEARCH METHODOLOGY 2015. [DOI: 10.1007/s10742-014-0128-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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34
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Chen CM, Chuang YW, Shen PS. Two-stage estimation for multivariate recurrent event data with a dependent terminal event. Biom J 2014; 57:215-33. [DOI: 10.1002/bimj.201400001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/02/2014] [Revised: 09/14/2014] [Accepted: 10/01/2014] [Indexed: 11/11/2022]
Affiliation(s)
- Chyong-Mei Chen
- Department of Statistics and Informatics Science; Providence University; Taichung 43301 Taiwan Republic of China
- Department of Financial and Computational Mathematics; Providence University; Taichung 43301 Taiwan Republic of China
| | - Ya-Wen Chuang
- Division of Nephrology; Department of Internal Medicine, Taichung Veterans General Hospital; Taichung 40705 Taiwan Republic of China
| | - Pao-Sheng Shen
- Department of Statistics; Tunghai University; Taichung 40704 Taiwan Republic of China
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35
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LIN FENGCHANG, CAI JIANWEN, FINE JASONP, LAI HUICHUANJ. Nonparametric estimation of the mean function for recurrent event data with missing event category. Biometrika 2014; 100:10.1093/biomet/ast016. [PMID: 24415792 PMCID: PMC3887139 DOI: 10.1093/biomet/ast016] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Recurrent event data frequently arise in longitudinal studies when study subjects possibly experience more than one event during the observation period. Often, such recurrent events can be categorized. However, part of the categorization may be missing due to technical difficulties. If the event types are missing completely at random, then a complete case analysis may provide consistent estimates of regression parameters in certain regression models, but estimates of the baseline event rates are generally biased. Previous work on nonparametric estimation of these rates has utilized parametric missingness models. In this paper, we develop fully nonparametric methods in which the missingness mechanism is completely unspecified. Consistency and asymptotic normality of the nonparametric estimators of the mean event functions accommodate nonparametric estimators of the event category probabilities, which converge more slowly than the parametric rate. Plug-in variance estimators are provided and perform well in simulation studies, where complete case estimators may exhibit large biases and parametric estimators generally have a larger mean squared error when the model is misspecified. The proposed methods are applied to data from a cystic fibrosis registry.
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Affiliation(s)
- FENG-CHANG LIN
- Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina 27599, U.S.A
| | - JIANWEN CAI
- Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina 27599, U.S.A
| | - JASON P. FINE
- Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina 27599, U.S.A
| | - HUICHUAN J. LAI
- Department of Nutritional Sciences, University of Wisconsin, Madison, Wisconsin 53706, U.S.A
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36
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Cook RJ, Lawless JF. Statistical Issues in Modeling Chronic Disease in Cohort Studies. STATISTICS IN BIOSCIENCES 2013. [DOI: 10.1007/s12561-013-9087-8] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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37
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Wu L, Cook RJ. The Design of Intervention Trials Involving Recurrent and Terminal Events. STATISTICS IN BIOSCIENCES 2013. [DOI: 10.1007/s12561-013-9083-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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38
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Lawless JF. Armitage Lecture 2011: the design and analysis of life history studies. Stat Med 2013; 32:2155-72. [DOI: 10.1002/sim.5754] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2012] [Accepted: 01/17/2013] [Indexed: 11/08/2022]
Affiliation(s)
- Jerald F. Lawless
- Department of Statistics and Actuarial Science; University of Waterloo; 200 University Avenue West Waterloo Ontario Canada
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39
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Sun L, Kang F. An additive-multiplicative rates model for recurrent event data with informative terminal event. LIFETIME DATA ANALYSIS 2013; 19:117-137. [PMID: 22965680 DOI: 10.1007/s10985-012-9228-2] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2011] [Accepted: 08/29/2012] [Indexed: 06/01/2023]
Abstract
In this article, we propose an additive-multiplicative rates model for recurrent event data in the presence of a terminal event such as death. The association between recurrent and terminal events is nonparametric. For inference on the model parameters, estimating equation approaches are developed, and the asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is provided.
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Affiliation(s)
- Liuquan Sun
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China.
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40
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Wang J, Quartey G. Nonparametric estimation for cumulative duration of adverse events. Biom J 2011; 54:61-74. [DOI: 10.1002/bimj.201000256] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2010] [Revised: 09/15/2011] [Accepted: 09/17/2011] [Indexed: 11/06/2022]
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41
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Zhangsheng Y, Liu L. A joint model of recurrent events and a terminal event with a nonparametric covariate function. Stat Med 2011; 30:2683-95. [PMID: 21751230 DOI: 10.1002/sim.4297] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2010] [Revised: 03/28/2011] [Accepted: 05/02/2011] [Indexed: 11/09/2022]
Abstract
We extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function. We include a Gaussian random effect (frailty) in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes. We employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model. We use Laplace approximation to evaluate the marginal penalized partial likelihood without a closed form. We also propose the variance estimates for regression coefficients. Numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components. We apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death.
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Affiliation(s)
- Yu Zhangsheng
- Division of Biostatistics, Indiana University School of Medicine, Indianapolis, IN, USA.
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42
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Ghosh S, Majumdar SK. Reliability modeling and prediction using classical and Bayesian approach. INTERNATIONAL JOURNAL OF QUALITY & RELIABILITY MANAGEMENT 2011. [DOI: 10.1108/02656711111132580] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
PurposeThe purpose of this paper is to provide the maintenance personnel with a methodology for modeling and estimating the reliability of critical machine systems using the historical data of their inter‐failure times.Design/methodology/approachThe failure patterns of five different machine systems were modeled with NHPP‐log linear process and HPP belonging to stochastic point process for predicting their reliability in future time frames. Besides the classical approach, Bayesian approach was also used involving Jeffreys's invariant non‐informative independent priors to derive the posterior densities of the model parameters of NHPP‐LLP and HPP with a view to estimating the reliability of the machine systems in future time intervals.FindingsFor at least three machine systems, Bayesian approach gave lower reliability estimates and a larger number of (expected) failures than those obtained by the classical approach. Again, Bayesian estimates of the probability that “ROCOF (rate of occurrence of failures) would exceed its upper threshold limit” in future time frames were uniformly higher for these machine systems than those obtained with the classical approach.Practical implicationsThis study indicated that, the Bayesian approach would give more realistic estimates of reliability (in future time frames) of the machine systems, which had dependent inter‐failure times. Such information would be helpful to the maintenance team for deciding on appropriate maintenance strategy.Originality/valueWith the help of Bayesian approach, the posterior densities of the model parameters were found analytically by considering Jeffreys's invariant non‐informative independent prior. The case study would serve to motivate the maintenance teams to model the failure patterns of the repairable systems making use of the historical data on inter‐failure times and estimating their reliability in future time frames.
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Luo X, Huang CY. Analysis of recurrent gap time data using the weighted risk-set method and the modified within-cluster resampling method. Stat Med 2011; 30:301-11. [PMID: 20963733 DOI: 10.1002/sim.4074] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
The gap times between recurrent events are often of primary interest in medical and epidemiology studies. The observed gap times cannot be naively treated as clustered survival data in analysis because of the sequential structure of recurrent events. This paper introduces two important building blocks, the averaged counting process and the averaged at-risk process, for the development of the weighted risk-set (WRS) estimation methods. We demonstrate that with the use of these two empirical processes, existing risk-set based methods for univariate survival time data can be easily extended to analyze recurrent gap times. Additionally, we propose a modified within-cluster resampling (MWCR) method that can be easily implemented in standard software. We show that the MWCR estimators are asymptotically equivalent to the WRS estimators. An analysis of hospitalization data from the Danish Psychiatric Central Register is presented to illustrate the proposed methods.
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Affiliation(s)
- Xianghua Luo
- Division of Biostatistics, School of Public Health, University of Minnesota, 420 Delaware Street SE, Minneapolis, MN 55455, U.S.A.
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Cook RJ, Lawless JF, Lee KA. A copula-based mixed Poisson model for bivariate recurrent events under event-dependent censoring. Stat Med 2010; 29:694-707. [PMID: 20146240 DOI: 10.1002/sim.3830] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
In many chronic disease processes subjects are at risk of two or more types of events. We describe a bivariate mixed Poisson model in which a copula function is used to model the association between two gamma distributed random effects. The resulting model is a bivariate negative binomial process in which each type of event arises from a negative binomial process. Methods for parameter estimation are described for parametric and semiparametric models based on an EM algorithm. We also consider the issue of event-dependent censoring based on one type of event, which arises when one event is sufficiently serious that its occurence may influence the decision of whether to withdraw a patient from a study. The asymptotic biases of estimators of rate and mean functions from naive marginal analyses are discussed, as well as associated treatment effects. Because the joint model is fit based on a likelihood, consistent estimates are obtained. Simulation studies are carried out to evaluate the empirical performance of the proposed estimators with independent and event-dependent censoring and applications to a trial of breast cancer patients with skeletal metastases and a study of patients with chronic obstructive pulmonary disease illustrate the approach.
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Affiliation(s)
- Richard J Cook
- Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1.
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