t From a theoretical point of view, the Prentice-Williams-Peterson models are the only models that take the order of the events into account in the definition of the risk sets.
recurrent time-to-event are implemented to identify and to determine the e ects of the clinical and the genetic risk factors of tumor recurrence.
Biometrika. Barreto
Mean number of recurrences is 1.5, varying from 0 to 9. A
Butler
Cai
Vaccarino
J Neurolog Sci. They differ regarding the age group and presence of a toilet at home. Schaubel
New York: Springer; 2007. Chang PS, Nallamothu BK, Hayward RA. So, recurrent event models were used in addition to time to first event models, to explore the treatment effect on the number of occurrences of events over time. GY
S
The most well-known approach for analysis of survival data is the Cox proportional hazards model.2 Due to the independence assumption, the original Cox model is only appropriate for modelling the time to the first event,2 which is an inefficient use of data because data from the later events are discarded.
. A Regression Model for Dependent Gap Times A Regression Model for Dependent Gap Times Strawderman, Robert L 2006-01-13 00:00:00 A natural choice of time scale for analyzing recurrent event data is the ``gap" (or soujourn) time between successive events. Many diseases and clinical outcomes may recur in the same patient. Extensions of the original Cox model have been proposed for analyses of recurrent event data such as Andersen-Gill (AG),3 Prentice, Williams and Peterson (PWP) (total and gap times),4 Wei, Lin and Weissfeld (WLW)5 and frailty models.6 Another analysis strategy is through modelling the mean number of events or their occurrence rate.7,8,13 More recently, multi-state models (MSM) have been extended for recurrent events,14,15 but their application for analysis of epidemiological data is still limited. 2006; 48(4):850.
G
2016; 35(13):2195–205. In this case, the Andersen-Gill model remains more influenced by the recurrent event process.
Since each of the models has distinct assumptions, their results should not be directly compared. Of particular note are five Cox-based models for recurrent event data: Andersen and Gill (AG); Wei, Lin and Weissfeld (WLW); Prentice, Williams and Peterson, total time (PWP-CP) and gap time (PWP-GT); and Lee, Wei and Amato (LWA). Subjects 2 and 4 had 7 tumours at baseline and 4 cm was the size of their largest initial tumour. Unlike the AG model, the effect of covariates may vary from event to event in the stratified PWP models. Williams
J Am Coll Cardiol.
Fit of frailty models and MSM, however, is less accessible. Keiding
Choice of the appropriate approach for analysis of recurrent event data is determined by many factors, including number of events, relationship between consecutive events, effects that may or may not vary across recurrences, biological process, dependence structure and research question.
The frailty models are indicated when a subject-specific random effect can explain the unmeasured heterogeneity that cannot be explained by covariates alone, which leads to a person-specific interpretation of the estimates in a similar way as that for mixed models for analysis of longitudinal data. infected by Xanthomonas citri subsp. Negative Binomial Pharm Stat.
Subject 3 is the one with smallest probabilities of going from healthy to disease status (tumour recurrences) during the follow-up period. A
Nevertheless, we were also able to use full data for analysis using the AG model, marginal rates model and frailty model. Yip et al. The Statistical Analysis of Recurrent Events. Caballero
We assume that, conditional on the covariates, the event and censoring times are independent (independent censoring assumption). . M
Theprodlim package implements a fast algorithm and some features not included insurvival. Jahn-Eimermacher A.
Lawless
Cook RJ, Lawless JF. . Many models assume that future events depend only on the immediate past (AG, PWP, MSM), also known as Markov process, whereas others assume dependency upon shared random effects (frailty models). Another important factor is the presence of a toilet at home, which reduces by about 40% the risk of ALRI (e.g. Comput Stat Data Anal. PubMed Survival analysis for recurrent event data: an application to childhood infectious diseases. AT
citri at different infection stages. For MSM we considered only one type of transition, which means that the individual returns to the initial condition just after the occurrence of the event, that is, the event is immediately reversible.13 In this case, we are interested in the transition from healthy to disease status, assuming the probability of recovery is 1. Insititue of Medical Biometry and Epidemiology, University Medical Center Hamburg-Eppendorf, Martinistraße 52, Hamburg, 20246, Germany, Institute of Medical Biometry and Informatics, Universtiy Medical Center Ruprecht-Karls Universtiy Heidelberg, Im Neuenheimer Feld 130.3, Heidelberg, 69120, Germany, Ann-Kathrin Ozga, Meinhard Kieser & Geraldine Rauch, Charité - Universitätsmedizin Berlin, corporate member of Freie Universität Berlin, Humboldt-Universität zu Berlin, Berlin Institute of Health, Institute of Biometry and Clinical Epidemiology, Charitéplatz 1, Berlin, 10117, Germany, You can also search for this author in E-mail: Search for other works by this author on: Repeated occurrence of basal cell carcinoma of the skin and multifailure survival analysis: follow-up data from the Nambour Skin Cancer Prevention Trial, Regression models and life-tables (with Discussion), Cox’s regression model for counting processes: a large sample study, On the regression analysis of multivariate failure time data, Regression analysis of multivariate incomplete failure time data by modelling marginal distributions, Modelling Survival Data: Extending The Cox Model, Some graphical displays and marginal regression analysis for recurrent failure times and time dependent covariates, Semiparametric regression for the mean and rate functions of recurrent events, Repeated hospitalizations and self-rated health among the elderly: a multivariate failure time analysis, Comparison of regression models for the analysis of fall risk factors in older veterans, Multivariate time-to-event models for studies of recurrent childhood diseases, Survival analysis for recurrent event data: an application to childhood infectious diseases, The Statistical Analysis of Recurrent Events, A SAS macro for estimating transition probabilities in semiparametric models for recurrent events, Appraisal of several methods to model time to multiple events per subject: modelling time to hospitalizations and death, Efficient estimation of semiparametric transformation models for counting processes, Effect of vitamin A supplementation on diarrhoea and acute lower-respiratory-tract infections in young children in Brazil, The analysis of recurrent events for multiple subjects, Handbook of Statistics: Advances in Survival Analysis, Regression splines in the time-dependent coefficient rates model for recurrent event data, Multi-state models for event history analysis, Multi-state models for the analysis of time-to-event data. Paper 237, Analysis of multiple failure-time data with Stata, Modelling recurrence in colorectal cancer, Recurrent transient ischaemic attack and early risk of stroke: data from the PROMAPA Study, Recurrent events analysis in the presence of time-dependent covariates and dependent censoring, Proportional rate models for recurrent time event data under dependent censoring: a comparative study, Recent Advances in Biostatistics: False Discovery Rates, Survival Analysis and Related Topics, © The Author 2014; all rights reserved.
Advantages of an AG model include the ability to accommodate time-varying covariates and discontinuous intervals of risk.15.
The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Such states may be considered to be of the same type of recurrent events (Figure 2). A
Castañeda
D
Many statistical challenges arise when performing analyses of repeated time-to-event data and the researcher should be careful to address them adequately.
. In summary, the choice of the approach for analysis of recurrent event data will be determined by many factors, including: number of the events; relationship between subsequent events; effects varying or not across recurrences; biological process; and dependence structure.
Keles
C
Stat Med. This work was supported by the German Research Foundation (Grant RA 2347/1-2). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. An Approach to Nonparametric Regression for Life History Data Using Local Linear Fitting Li, Gang and Doss, Hani, Annals of Statistics, 1995; Cox's Periodic Regression Model Pons, O. and de Turckheim, E., Annals of Statistics, 1988; Robust inference for univariate proportional hazards frailty regression models Kosorok, Michael R., Lee, Bee Leng, and Fine, Jason P., Annals of Statistics, 2004
PubMed
Results of five analytical approaches for recurrent events: hazard and rate ratios of tumour recurrences in bladder cancer patients. In particular, we consider Andersen-Gill (A-G), Wei-Lin-Weissfeld (WLW), Prentice-Williams-Peterson Total Time (PWP-TT), Prentice-Williams-Peterson Gap Time (PWP-GT) and Frailty models. BS
Several approaches have been proposed to account for intra-subject correlation that rises from multiple events settings in survival analysis. Examples of applications for this approach include analysis of accumulated cost of medical care, and multiple infections in patients with chronic granulomatous disease. This approach has been used to evaluate repeated occurrence of basal cell carcinoma2 and hospitalizations due to all causes and to cardiovascular diseases in the elderly,9 for instance. Andersen-Gill I Extension of Cox proportional-hazards model I Analyses gap times I Each gap time contributes to the likelihood I Gives a hazard ratio for recurrent events I Assumes that events are independent I Robust standard errors accommodates heterogeneity. In Section 3 the pros and cons of these modelling strategies are illustrated using the HF-dataset, analyzing the time to death and/or … N
Article These models differ in assumptions and the data layout for analysis (Appendix 1, available as Supplementary data at IJE online). For example, separate strata for the different event types could be defined. is incorporated. Cai
This work was supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)/Ministry of Education-Brazil (grant 0617/11-3 to L.D.A. Log hazards ratio and 95% confidence intervals for (A) demographic variables, (B) cardiovascular risk factors, and (C) novel risk factors stratified by first versus subsequent CHF events from the PWP total time and gap time … Article
Mazroui Y, Mathoulin-Pelissier S, MacGrogan G, Brouste V, Rondeau V. Multivariate frailty models for two types or recurrent events with a dependent terminal event: application to breast cancer data. Ying
et al. The predicted transition probabilities through MSM are presented in Figure 4 and were estimated using the AG model.
From our results, no general recommendation regarding the choice between the total time or the gap time approach can be derived.
In conclusion, apart from the general interpretation difficulty of an overall mixed effect, the conditional models from Prentice, Williams, and Peterson [5] could be recommended to analyze clinical trials with a composite endpoint which is justified from a theoretical point of view as well as from the results of our simulation study. Stat Med. However, more work has to be done to consider the situation of more than two correlated event processes, e.g. M
Subjects 1 and 3 had both only 1 tumour at baseline of size 1 cm. However, they consider the total number of events per a fixed period of time, ignoring the time between repeated occurrences. J
Nonetheless, the power values of the Wei-Lin-Weissfeld model are usually smaller which is due to the considerably higher standard deviations of the estimated hazard ratios.
LH
from Z on multiple events. Statistical modelling for recurrent events: an application to sports injuries. Schematic plot for recurrent time-to-event data for five hypothetical subjects. Departmento de Estatística, Instituto de Matemática, Universidade Federal da Bahia, Av.Adhemar de Barros s/n, Campus de Ondina, Salvador-Bahia, Brazil CEP 40170-115.
Article 2002; 11(2):91–115.
Several approaches have been proposed in the literature to account for intra-subject correlation that arises from recurrent events in survival analysis. However, the majority of analyses focus only on time to the first event, ignoring the subsequent events. Supported data formats include clustered failures with left truncation and recurrent events in gap-time or Andersen-Gill format. STSET is used to input information on the survival times, censoring time and identification variables. Lin
Jahn-Eimermacher A, Ingel K, Ozga A, Preussler S, Binder H. Simulating recurrent event data with hazard functions defined on a total time scale. According to all models, younger children (≤12 months) are more likely to have recurrent episodes of ALRI than older children (e.g. An episode of ALRI was defined as cough plus a respiratory rate of 50 breaths per min or higher for children under 12 months of age, and 40 breaths per min or higher for older children.17 Censoring occurred when children were lost to follow-up or the study reached the end. Lifetime Data Anal. Literature review indicates that most statistical models used for such data are often based on time to the first event or consider events within a subject as independent. Analysis based only on the first event time cannot be used to examine the effect of the risk factors on the number of recurrences over time.1,28 Many researchers continue to use logistic regression for such analysis, despite known limitations and the increasing availability of analytical approaches that handle recurrent events.10,29 In cohort studies, there is little justification for fitting logistic regression once there are other available approaches for estimating risk.10 The count data models, such as Poisson and negative binomial, are the simplest ways to analyse repeated events. Moreover, the inclusion of prior event history may attenuate estimates of covariate effects compared with the marginal effects. Miloslavsky
Manage cookies/Do not sell my data we use in the preference centre. Gaps between events are often useful with infrequent events, when a renewal occurs after an event or when the interest lies on prediction of a next event. All these models can be extended with a frailty term to account for heterogeneity between individuals [7–9]. The so called Andersen-Gil counting process model, two variants of the conditional models of Prentice, Williams, and Peterson (gap time model, conditional probability model), and the so called frailty model were applied to a dataset of 17’415 patients observed during a 12 years period starting from 1996 and leading to 37’697 psychiatric hospitalisations. Contains functions for fitting shared frailty models with a semi-parametric baseline hazard with the Expectation-Maximization algorithm. Note that there is no difference on the effects for treatment and gender on the two transitions (Table 3). The extended Cox models were: 1) the Andersen-Gill counting process (CP), 2) the Prentice-Williams-Peterson (PWP-CP/Total time),3) PWP – Gap time (PWP-GT) model, 4) Marginal (Wei, Lin and Weissfeld)Model and 5) Cox frailty model. As stated above, the results from the Andersen-Gill model differ barely from the Prentice-Williams-Peterson models because the risk sets are similar for both approaches as long as only a few strata are considered in the Prentice-Williams-Peterson model. 3; complete data in 162 PEx). Hence, it is very important to consider the use of as much data as possible and to conduct analysis that can enhance a comprehensive understanding of the role of the risk factors in the disease process. Comparison of the andersen-gill model with poisson and negative binomial regression on recurrent event data.
On the other hand, when the investigators are interested in modelling the expected number of events or the rate of event occurrence, conditional on covariates, the means/rates marginal model should be used. Barreto
Data from a double-blinded randomized clinical trial with 1207 children followed for 1 year to evaluate the impact of high doses of vitamin A on diarrhoea and acute lower respiratory tract infections (ALRI) was used.17 Daily information on respiratory rates was available and measured twice for those children who reported cough.
1989; 84(408):1074–8. 1972; 34(2):187–220.
If it is reasonable to assume that the occurrence of the first event increases the likelihood of a recurrence, then PWP is recommended. PubMed de Lima
Speechley
I include time-varying covariates in this model as per the 1982 paper from Andersen and Gill - for example I use the "dynamic" covariate recurrent outcome history to model the within-subject dependence in the recurrent events. Quantin
The robust inference for the cox proportional hazards model. DY
Another major difference among them is the way the repeated events are modelled.
Andersen
Various confidence intervals and confidence bands for the Kaplan-Meier estimator are implemented in thekm.ci package.plot.Surv of packageeha plots the …
Assuming that there is unequal risk for the different transitions, the analysis can be stratified by transition (event) using PWP-TT model (data not shown). B
A change of state is called a transition (or an event) and is central in this framework, which is fully characterized through estimation of transition probabilities between states and transition intensities that are defined as instantaneous hazards of progression to one state, conditional on occupying another state.22 Both of them depend on the process over time, the history up to time t. Graphically, MSM are illustrated using diagrams with boxes (the states) and arrows between the states (the transitions).15 In Figure 2 we represent an MSM for k recurrent events. D
Covariates include treatment group, number of initial tumours (found at baseline) and size of the largest initial tumour (in centimeters); 55% of the patients had at least one recurrence, resulting in 130 recurrences. There has been a considerable amount of discussion on methods of analysis for recurrent or repeated events in biostatistics, epidemiological and medical literature.1,3–13 Nevertheless, inefficient or inappropriate statistical approaches are still used to analyse such type of data. https://doi.org/10.1186/s12874-017-0462-x, DOI: https://doi.org/10.1186/s12874-017-0462-x. Competing Risks and Multistate Models with R. Heidelberg: Springer; 2012. . When there is heterogeneous susceptibility to the risk of recurrent events, the frailty model can be applied. events: Andersen & Gill (1982), Wei et al. . Schematic for full multi-state model for recurrent events. The differences described before between the Andersen-Gill or Prentice-Williams-Peterson models to the approach of Wei, Lin, and Weissfeld were already shown in previous works [3, 6, 16]. HR = 0.60; 95% CI: 0.46, 0.77, PWP-TT model). .
Analysis of survival data with clustered events. The simplest multi-state model (MSM) is defined for two states: alive (a transient state) and dead (an absorbing state).21 A special case of MSM occurs when an individual moves from one state to another through time, and intermediate states are identified. CAS For the gap time model all starting times are set to zero and the stopping time denotes the time since the previous Meira-Machado
R: A language and environment for statistical computing. Thus, we estimated the probability of recovery. Andersen and Gill 1 – Analyzes time between events (gap time) independently – Time-varying covariates to account for correlations and clustering on patient Kelly
Due to lack of software developments for fitting MSM, this approach has been rarely applied to analysis of recurrent event data to date.14,15. Conditional on the unmeasured heterogeneity and covariates, the frailty model indicates that each additional tumour at baseline is associated with a 26% increase in the recurrence risk (HR = 1.26; 95% CI: 1.05, 1.51).
DY
account for heterogeneity is to model the capture process via covariates. However, most of these previous findings are not exactly comparable to ours as the authors considered only one recurrent event process which, in contrast to most of our results, leads to greater differences between the Andersen-Gill and Prentice-Williams-Peterson approaches. Bender R, Augustin T, Blettner M. Generating survival times to simulate cox proportional hazards models. AO implemented the simulations, produced the results and wrote the first draft of the manuscript. An alternative model is the marginal means/rates model,8,13,18–20 which can be interpreted in terms of the mean number of events when there are no time-dependent covariates. Another example of application with recurrent event data is in the evaluation of factors on the risk of catheter loss in patients with chronic renal failure, when the event is reversible and the interest in on the estimation of transition probabilities. .
Pepe
BJ
[11] could thereby be of interest. . . Subjects were followed for up to 64 months. previous PEx). Santos
STCOX is used to fit the Cox model and its extensions.12,27 Currently, there are no specific options to fit MSM for recurrent event data in Stata.
For the Prentice-Williams-Peterson total time approach the same data structure as in the Andersen-Gill model is required but with an additional stratum variable which counts the number of events for each individual. Cook
Andersen
Conversely, in simulation studies often a gap time scale is used [3,4,6,16], sometimes by defining constant hazards. Cox
Pandeya
However, the MSM allows us to additionally quantify the magnitude of the effect of the covariates on the transition ALRI-healthy. On the other hand, models that account for correlation between recurrent events using robust covariance matrix, time-varying covariates or frailties (marginal means/rates, AG and frailties models) are indicated for frequent events with constant hazard between recurrences.
Patient 1 had the largest number of events, 6, at times 4, 6, 9, 12, 15 and 28 months, wheras patient 3 had only two events at times 12 and 47 months. $$, $$\begin{array}{@{}rcl@{}} R^{PWP}_{j}(t):=\left\{l,\ l=1,{\ldots},n:T_{l(j-1)}

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