Abstract
In studies of recurrent events, joint modeling approaches are often needed to allow for potential dependent censoring by a terminal event such as death. Joint frailty models for recurrent events and death with an additional dependence parameter have been studied for cases in which individuals are observed from the start of the event processes. However, samples are often selected at a later time, which results in delayed entry so that only individuals who have not yet experienced the terminal event will be included. In joint frailty models such left truncation has effects on the frailty distribution that need to be accounted for in both the recurrence process and the terminal event process, if the two are associated. We demonstrate, in a comprehensive simulation study, the effects that not adjusting for late entry can have and derive the correctly adjusted marginal likelihood, which can be expressed as a ratio of two integrals over the frailty distribution. We extend the estimation method of Liu and Huang (Stat Med 27:2665–2683, 2008. https://doi.org/10.1002/sim.3077) to include potential left truncation. Numerical integration is performed by Gaussian quadrature, the baseline intensities are specified as piecewise constant functions, potential covariates are assumed to have multiplicative effects on the intensities. We apply the method to estimate age-specific intensities of recurrent urinary tract infections and mortality in an older population.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The data set on recurrent urinary tract infections analyzed in the present manuscript cannot be made available for reasons of confidentiality.
Code availability
R-code to reproduce the results of Section 3 is available from the corresponding author upon request.
References
Abbring JH, van den Berg GJ (2007) The unobserved heterogeneity distribution in duration analysis. Biometrika 94(1):87–99
Balan TA, Jonker MA, Johannesma PC, Putter H (2016) Ascertainment correction in frailty models for recurrent events data. Stat Med 35:4183–4201. https://doi.org/10.1002/sim.6968
Cai Q, Wang M-C, Chan KCG (2017) Joint modeling of longitudinal, recurrent events and failure time data for survivor’s population. Biometrics 73:1150–1160. https://doi.org/10.1111/biom.12693
Caljouw MAA, van den Hout WB, Putter H, Achterberg WP, Cools HJM, Gussekloo J (2014) Effectiveness of cranberry capsules to prevent urinary tract infections in vulnerable older persons: a double-blind randomized placebo-controlled trial in long-term care facilities. J Am Geriatr Soc 62:103–110. https://doi.org/10.1111/jgs.12593
Cook RJ, Lawless JF (1997) Marginal analysis of recurrent events and a terminating event. Stat Med 16:911–924
Cook RJ, Lawless JF (2007) The statistical analysis of recurrent events. Statistics for biology and health. Springer, New York
Crowther MJ, Andersson TM-L, Lambert PC, Abrams KR, Humphreys K (2016) Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification. Stat Med 35:1193–1209. https://doi.org/10.1002/sim.6779
Duchateau L, Janssen P (2008) The frailty model. Springer, Berlin
Emura T, Nakatochi M, Murotani K, Rondeau V (2017) A joint frailty-copula model between tumour progression and death for meta-analysis. Stat Methods Med Res 26:2649–2666. https://doi.org/10.1177/0962280215604510
Eriksson F, Martinussen T, Scheike TH (2015) Clustered survival data with left-truncation. Scand J Stat 42:1149–1166. https://doi.org/10.1111/sjos.12157
Ghosh D, Lin DY (2003) Semiparametric analysis of recurrent events data in the presence of dependent censoring. Biometrics 59:877–885. https://doi.org/10.1111/j.0006-341X.2003.00102.x
Huang C-Y, Wang M-C (2004) Joint modeling and estimation for recurrent event processes and failure time data. J Am Stat Assoc 99:1153–1165. https://doi.org/10.1198/016214504000001033
Jazić I, Haneuse S, French B, MacGrogan G, Rondeau V (2019) Design and analysis of nested case-control studies for recurrent events subject to a terminal event. Stat Med 38(22):4348–4362. https://doi.org/10.1002/sim.8302
Jensen H, Brookmeyer R, Aaby P, Andersen PK (2004) Shared frailty model for left-truncated multivariate survival data. Museum Tusculanum, Biostatistisk Afdeling
Kalbfleisch JD, Schaubel DE, Ye Y, Gong Q (2013) An estimating function approach to the analysis of recurrent and terminal events. Biometrics 69:366–374. https://doi.org/10.1111/biom.12025
Klein JP, Moeschberger ML (2003) Survival analysis. Techniques for censored and truncated data, 2nd edn. Springer, Berlin
Lawless JF, Fong DYT (1999) State duration models in clinical and observational studies. Stat Med 18:2365–2376
Liu D, Schaubel DE, Kalbfleisch JD (2012) Computationally efficient marginal models for clustered recurrent event data. Biometrics 68:637–647. https://doi.org/10.1111/j.1541-0420.2011.01676.x
Liu L, Huang X (2008) The use of Gaussian quadrature for estimation in frailty proportional hazards models. Stat Med 27:2665–2683. https://doi.org/10.1002/sim.3077
Liu L, Wolfe RA, Huang X (2004) Shared frailty models for recurrent events and a terminal event. Biometrics 60:747–756. https://doi.org/10.1111/j.0006-341X.2004.00225.x
Piulachs X, Andrinopoulou E-R, Guillén M, Rizopoulos D (2021) A Bayesian joint model for zero-inflated integers and left-truncated event times with a time-varying association: applications to senior health care. Stat Med 40(1):147–166. https://doi.org/10.1002/sim.8767
R Core Team (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria
Rogers JK, Yaroshinsky A, Pocock SJ, Stokar D, Pogoda J (2016) Analysis of recurrent events with an associated informative dropout time: application of the joint frailty model. Stat Med 35:2195–2205. https://doi.org/10.1002/sim.6853
Rondeau V, Mathoulin-Pelissier S, Jacqmin-Gadda H, Brouste V, Soubeyran P (2007) Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events. Biostatistics 8(4):708–721. https://doi.org/10.1093/biostatistics/kxl043
Rondeau V, Mauguen A, Laurent A, Berr C, Helmer C (2017) Dynamic prediction models for clustered and interval-censored outcomes: investigating the intra-couple correlation in the risk of dementia. Stat Methods Med Res 26(5):2168–2183
Rondeau V, Gonzalez JR, Mazroui Y, Mauguen A, Diakite A, Laurent A, Lopez M, Król A, Sofeu CL, Dumerc J, Rustand D, Chauvet J, Coent, Le Q (2021) frailtypack: general frailty models: shared, joint and nested frailty models with prediction; evaluation of failure-time surrogate endpoints. R package version 3.5.0. https://CRAN.R-project.org/package=frailtypack
van den Berg GJ, Drepper B (2016) Inference for shared-frailty survival models with left-truncated data. Econ Rev 35:1075–1098. https://doi.org/10.1080/07474938.2014.975640
van den Hout A, Muniz-Terrera G (2016) Joint models for discrete longitudinal outcomes in aging research. J R Stat Soc Ser C 65:167–186. https://doi.org/10.1111/rssc.12114
Wienke A (2011) Frailty models in survival analysis. Chapman & Hall/CRC Biostatistics Series, Boca Raton
Funding
The authors received no specific funding for this work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Böhnstedt, M., Gampe, J., Caljouw, M.A.A. et al. Incorporating delayed entry into the joint frailty model for recurrent events and a terminal event. Lifetime Data Anal 29, 585–607 (2023). https://doi.org/10.1007/s10985-022-09587-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-022-09587-z