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Multivariate frailty models for two types of recurrent events with a dependent terminal event: application to breast cancer data. (English) Zbl 1441.62435
Summary: Individuals may experience more than one type of recurrent event and a terminal event during the life course of a disease. Follow-up may be interrupted for several reasons, including the end of a study, or patients lost to follow-up, which are noninformative censoring events. Death could also stop the follow-up, hence, it is considered as a dependent terminal event. We propose a multivariate frailty model that jointly analyzes two types of recurrent events with a dependent terminal event. Two estimation methods are proposed: a semiparametrical approach using penalized likelihood estimation where baseline hazard functions are approximated by M-splines, and another one with piecewise constant baseline hazard functions. Finally, we derived martingale residuals to check the goodness-of-fit. We illustrate our proposals with a real dataset on breast cancer. The main objective was to model the dependency between the two types of recurrent events (locoregional and metastatic) and the terminal event (death) after a breast cancer.

62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] Broët, Analyzing prognostic factors in breast cancer using a multistate model, Breast Cancer Research and Treatment 54 pp 83– (1999) · doi:10.1023/A:1006197524405
[2] Burzykowski, Validation of surrogate end points in multiple randomized clinical trials with failure time end points, Journal of the Royal Statistical Society: Series C (Applied Statistics) 50 pp 405– (2001) · Zbl 1112.62336 · doi:10.1111/1467-9876.00244
[3] Cai, Marginal means/rates models for multiple type recurrent event data, Lifetime Data Analysis 10 pp 121– (2004) · Zbl 1058.62098 · doi:10.1023/B:LIDA.0000030199.23383.45
[4] Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika 65 pp 141– (1978) · Zbl 0394.92021 · doi:10.1093/biomet/65.1.141
[5] Commenges, Choice between semi-parametric estimators of Markov and non-Markov multistate models from generally coarsened observations, Scandinavian Journal of Statistics 34 pp 33– (2007) · Zbl 1142.62054 · doi:10.1111/j.1467-9469.2006.00536.x
[6] Commenges, Standardized martingale residuals applied to grouped left truncated observations of dementia cases, Lifetime Data Analysis 6 pp 229– (2000) · Zbl 0955.62112 · doi:10.1023/A:1009637608473
[7] Cook, The Statistical Analysis of Recurrent Events (2007) · Zbl 1159.62061
[8] Cox, Regression models and life-tables, Journal of the Royal Statistical Society. Series B (Methodological) 34 pp 187– (1972) · Zbl 0243.62041
[9] De Bock, The impact of loco-regional recurrences on metastatic progression in early-stage breast cancer: a multistate model, Breast Cancer Research and Treatment 117 pp 401– (2009) · doi:10.1007/s10549-008-0300-2
[10] Dignam, Hazard of recurrence and adjuvant treatment effects over time in lymph node-negative breast cancer, Breast Cancer Research and Treatment 116 pp 595– (2009) · doi:10.1007/s10549-008-0200-5
[11] Duchateau, The Frailty Model (2008) · Zbl 1210.62153
[12] Elkhuizen, Local recurrence after breast-conserving therapy for invasive breast cancer: high incidence in young patients and association with poor survival, International Journal of Radiation Oncology Biology Physics 40 pp 859– (1998) · doi:10.1016/S0360-3016(97)00917-6
[13] Fisher, Significance of ipsilateral breast tumour recurrence after lumpectomy, The Lancet 338 pp 327– (1991) · doi:10.1016/0140-6736(91)90475-5
[14] Fortin, Local failure is responsible for the decrease in survival for patients with breast cancer treated with conservative surgery and postoperative radiotherapy, Journal of Clinical Oncology 17 pp 101– (1999)
[15] Hilsenbeck, Time-dependence of hazard ratios for prognostic factors in primary breast cancer, Breast Cancer Research and Treatment 52 pp 227– (1998) · doi:10.1023/A:1006133418245
[16] Huang, A joint frailty model for survival and gap times between recurrent events, Biometrics 63 pp 389– (2007) · Zbl 1137.62076 · doi:10.1111/j.1541-0420.2006.00719.x
[17] Huang, A frailty model for informative censoring, Biometrics 58 pp 510– (2002) · Zbl 1210.62129 · doi:10.1111/j.0006-341X.2002.00510.x
[18] Jemal, Global cancer statistics, CA: A Cancer Journal for Clinicians 61 pp 69– (2011) · doi:10.3322/caac.20107
[19] Joly, Estimation de la fonction de risque dans un contexte général de troncature et de censure: application à l’estimation de l’incidence de la démence (1996)
[20] Knight, Mathematical Statistics, Texts in Statistical Science (2000)
[21] Lancaster, Panel data with survival: hospitalization of HIV-positive patients, Journal of the American Statistical Association 93 pp 46– (1998) · Zbl 0915.62090 · doi:10.1080/01621459.1998.10474086
[22] Liu, Shared frailty models for recurrent events and a terminal event, Biometrics 60 pp 747– (2004) · Zbl 1274.62827 · doi:10.1111/j.0006-341X.2004.00225.x
[23] Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 pp 431– (1963) · Zbl 0112.10505 · doi:10.1137/0111030
[24] Mazroui, General joint frailty model for recurrent event data with a dependent terminal event: application to follicular lymphoma data, Statistics in Medicine 31 pp 1162– (2012) · doi:10.1002/sim.4479
[25] Montagna, Breast cancer subtypes and outcome after local and regional relapse, Annals of Oncology 23 pp 324– (2011) · doi:10.1093/annonc/mdr129
[26] Grillo, The effect of locoregional recurrence on survival and distant metastasis after conservative treatment for invasive breast carcinoma, Clinical Oncology 17 pp 111– (2005) · doi:10.1016/j.clon.2004.10.005
[27] O’Shaughnessy, Extending survival with chemotherapy in metastatic breast cancer, The Oncologist 10 pp 20– (2005) · doi:10.1634/theoncologist.10-90003-20
[28] Putter, Estimation and prediction in a multistate model for breast cancer, Biometrical Journal 48 pp 366– (2006) · doi:10.1002/bimj.200510218
[29] Putter, Frailties in multistate models: are they identifiable? Do we need them?, Statistical Methods in Medical Research (2011) · doi:10.1177/0962280211424665
[30] Ramsay, Monotone regression splines in action, Statistical Science 3 pp 425– (1988) · doi:10.1214/ss/1177012761
[31] Rondeau, frailtypack: a computer program for the analysis of correlated failure time data using penalized likelihood estimation, Computer Methods and Programs in Biomedicine 80 pp 154– (2005) · Zbl 05462535 · doi:10.1016/j.cmpb.2005.06.010
[32] Rondeau, Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events, Biostatistics 8 pp 708– (2007) · Zbl 1267.62110 · doi:10.1093/biostatistics/kxl043
[33] Rondeau, Frailtypack: an r package for the analysis of correlated survival data with frailty models using penalized likelihood estimation or parametrical estimation, Journal of Statistical Software 47 (2012) · doi:10.18637/jss.v047.i04
[34] Rondeau, A joint model for the dependence between clustered times to tumour progression and deaths: a meta-analysis of chemotherapy in head and neck cancer, Statistical Methods in Medical Research (2011) · doi:10.1177/0962280211425578
[35] Schaubel, Analysis of clustered recurrent event data with application to hospitalization rates among renal failure patients, Biostatistics 6 pp 404– (2005) · Zbl 1070.62110 · doi:10.1093/biostatistics/kxi018
[36] Schmoor, Role of isolated locoregional recurrence of breast cancer: results of four prospective studies, Journal of Clinical Oncology 18 pp 1696– (2000)
[37] Tanis, Locoregional recurrence after breast-conserving therapy remains an independent prognostic factor even after an event free interval of 10 years in early stage breast cancer, European Journal of Cancer 48 pp 1751– (2012) · doi:10.1016/j.ejca.2012.02.051
[38] Vaupel, The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography 16 pp 439– (1979) · doi:10.2307/2061224
[39] Vicini, Does local recurrence affect the rate of distant metastases and survival in patients with early-stage breast carcinoma treated with breast-conserving therapy?, Cancer 97 pp 910– (2003) · doi:10.1002/cncr.11143
[40] Wang, Analyzing recurrent event data with informative censoring, Journal of the American Statistical Association 96 pp 1057– (2001) · Zbl 1072.62646 · doi:10.1198/016214501753209031
[41] Wapnir, Prognosis after ipsilateral breast tumor recurrence and locoregional recurrences in five national surgical adjuvant breast and bowel project node-positive adjuvant breast cancer trials, Journal of Clinical Oncology 24 pp 2028– (2006) · doi:10.1200/JCO.2005.04.3273
[42] Zeng, A semiparametric additive rate model for recurrent events with an informative terminal event, Biometrika 97 pp 699– (2010) · Zbl 1195.62152 · doi:10.1093/biomet/asq039
[43] Zhu, Regression analysis of multivariate recurrent event data with a dependent terminal event, Lifetime Data Analysis 16 pp 478– (2010) · Zbl 1322.62297 · doi:10.1007/s10985-010-9158-9
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