Mazroui, Yassin; Mauguen, Audrey; Mathoulin-Pélissier, Simone; MacGrogan, Gaetan; Brouste, Véronique; Rondeau, Virginie Time-varying coefficients in a multivariate frailty model: application to breast cancer recurrences of several types and death. (English) Zbl 1356.62215 Lifetime Data Anal. 22, No. 2, 191-215 (2016). Summary: During their follow-up, patients with cancer can experience several types of recurrent events and can also die. Over the last decades, several joint models have been proposed to deal with recurrent events with dependent terminal event. Most of them require the proportional hazard assumption. In the case of long follow-up, this assumption could be violated. We propose a joint frailty model for two types of recurrent events and a dependent terminal event to account for potential dependencies between events with potentially time-varying coefficients. For that, regression splines are used to model the time-varying coefficients. Baseline hazard functions (BHF) are estimated with piecewise constant functions or with cubic M-Splines functions. The maximum likelihood estimation method provides parameter estimates. Likelihood ratio tests are performed to test the time dependency and the statistical association of the covariates. This model was driven by breast cancer data where the maximum follow-up was close to 20 years. Cited in 3 Documents MSC: 62P10 Applications of statistics to biology and medical sciences; meta analysis 62N02 Estimation in survival analysis and censored data 62G08 Nonparametric regression and quantile regression Keywords:breast cancer; frailty model; penalized likelihood; recurrent events Software:frailtypack PDFBibTeX XMLCite \textit{Y. Mazroui} et al., Lifetime Data Anal. 22, No. 2, 191--215 (2016; Zbl 1356.62215) Full Text: DOI References: [1] Abrahamowicz M, MacKenzie T (2007) Joint estimation of time-dependent and non-linear effects of continuous covariates on survival. Stat Med 26(2):392-408 [2] Abrahamowicz M, Mackenzie T, Esdaile JM (1996) Time-dependent hazard ratio: modeling and hypothesis testing with application in lupus nephritis. J Am Stat Assoc 91(436):1432-1439 · Zbl 0882.62101 [3] Bellera C, MacGrogan G, Debled M, de Lara T, Brouste V, Mathoulin-Pelissier S (2010) Variables with time-varying effects and the Cox model: some statistical concepts illustrated with a prognostic factor study in breast cancer. BMC Med Res Methodol 10(1):20 [4] Cai J, Schaubel D (2004) Marginal means/rates models for multiple type recurrent event data. Lifetime Data Anal 10(2):121-138 · Zbl 1058.62098 [5] Cai Z, Sun Y (2003) Local linear estimation for time-dependent coefficients in Cox’s regression models. Scand J Stat 30(1):93-111 · Zbl 1034.62096 [6] Chiang C, Wang M (2009) Varying-coefficient model for the occurrence rate function of recurrent events. Ann Inst Stat Math 61(1):197-213 · Zbl 1294.62079 [7] Commenges D, Rondeau V (2000) Standardized martingale residuals applied to grouped left truncated observations of dementia cases. Lifetime Data Anal 6(3):229-235 · Zbl 0955.62112 [8] Cook RJ, Lawless JF (1997) Marginal analysis of recurrent events and a terminating event. Stat Med 16:911-924 [9] De Boor C (2001) A practical guide to splines, vol 27. Springer, Berlin · Zbl 0987.65015 [10] Duchateau L, Janssen P, Kezic I, Fortpied C (2003) Evolution of recurrent asthma event rate over time in frailty models. J Royal Stat Soc 52(3):355-363 · Zbl 1111.62336 [11] Elkhuizen P, van de Vijver M, Hermans J, Zonderland H, van de Velde C, Leer J (1998) Local recurrence after breast-conserving therapy for invasive breast cancer: high incidence in young patients and association with poor survival. Int J Radiat Oncol Biol Phys 40(4):859-867 [12] Fan J, Gijbels I, King M (1997) Local likelihood and local partial likelihood in hazard regression. Ann Stat 25(4):1661-1690 · Zbl 0890.62023 [13] Ghosh D, Lin D (2000) Nonparametric analysis of recurrent events and death. Biometrics 56(2):554-562 · Zbl 1060.62614 [14] Ghosh D, Lin D (2002) Marginal regression models for recurrent and terminal events. Stat Sinica 12(3):663-688 · Zbl 1005.62084 [15] Hastie T, Tibshirani R (1993) Varying-coefficient models. J Royal Stat Soc 55(4):757-796 · Zbl 0796.62060 [16] Huang C, Wang M (2004) Joint modeling and estimation for recurrent event processes and failure time data. J Am Stat Assoc 99(468):1153-1165 · Zbl 1055.62108 [17] Huang X, Liu L (2007) A joint frailty model for survival and gap times between recurrent events. Biometrics 63(2):389-397 · Zbl 1137.62076 [18] Jemal A, Bray F, Center M, Ferlay J, Ward E, Forman D (2011) Global cancer statistics. CA 61:69-90 [19] Kauermann G, Krivobokova T, Fahrmeir L (2009) Some asymptotic results on generalized penalized spline smoothing. J Royal Stat Soc 71(2):487-503 · Zbl 1248.62055 [20] Li Q, Lagakos S (1997) Use of Wei-Lin-Weissfeld method for the analysis of a recurring and a terminating event. Stat Med 16:925-940 [21] Lin D, Wei L, Yang I, Ying Z (2000) Semiparametric regression for the mean and rate functions of recurrent events. J Royal Stat Soc 62(4):711-730 · Zbl 1074.62510 [22] Liu L, Wolfe R, Huang X (2004) Shared frailty models for recurrent events and a terminal event. Biometrics 60(3):747-756 · Zbl 1274.62827 [23] Marquardt D (1963) An algorithm for least-squares estimation of nonlinear parameters. J Soc Ind Appl Math 11(2):431-441 · Zbl 0112.10505 [24] Martinussen T, Scheike T, Skovgaard I (2002) Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Scand J Stat 29(1):57-74 · Zbl 1017.62094 [25] Mazroui Y, Mathoulin-Pelissier S, Soubeyran P, Rondeau V (2012) General joint frailty model for recurrent event data with a dependent terminal event: application to follicular lymphoma data. Stat Med 31:1162-1176 [26] Montagna E, Bagnardi V, Rotmensz N, Viale G, Renne G, Cancello G et al (2012) Breast cancer subtypes and outcome after local and regional relapse. Ann Onc 23(2):324-331 [27] Monteiro Grillo I, Jorge M, Marques Vidal P, Ortiz M, Ravasco P (2005) The effect of locoregional recurrence on survival and distant metastasis after conservative treatment for invasive breast carcinoma. Clin Oncol 17(2):111-117 [28] O’Shaughnessy J (2005) Extending survival with chemotherapy in metastatic breast cancer. Oncologist 10(3):20-29 [29] Putter H, Sasako M, Hartgrink H, Van de Velde C, Van Houwelingen J (2005) Long-term survival with non-proportional hazards: results from the dutch gastric cancer trial. Stat Med 24(18):2807-2821 [30] 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 · Zbl 1267.62110 [31] Rondeau V, Mazroui Y, Gonzalez JR (2012) frailtypack: an R package for the analysis of correlated survival data with frailty models using penalized likelihood estimation or parametrical estimation. J Stat Softw 47(4):1-28 [32] Schaubel D, Zhang M (2010) Estimating treatment effects on the marginal recurrent event mean in the presence of a terminating event. Lifetime Data Anal 16(4):451-477 · Zbl 1322.62234 [33] Spiekerman C, Lin D (1998) Marginal regression models for multivariate failure time data. J Am Stat Assoc 93(443):1164-1175 · Zbl 1064.62572 [34] Wang M, Qin J, Chiang C (2001) Analyzing recurrent event data with informative censoring. J Am Stat Assoc 96(455):1057-1065 · Zbl 1072.62646 [35] Ye Y, Kalbeisch J, Schaubel D (2007) Semiparametric analysis of correlated recurrent and terminal events. Biometrics 63(1):78-87 · Zbl 1124.62083 [36] Yu Z, Lin X (2008) Nonparametric regression using local kernel estimating equations for correlated failure time data. Biometrika 95(1):123-137 · Zbl 1437.62667 [37] Yu Z, Lin X (2010) Semiparametric regression with time-dependent coefficients for failure time data analysis. Stat Sinica 20:853-869 · Zbl 1187.62079 [38] Yu Z, Liu L, Bravata D, Williams L, Tepper R (2012) A semiparametric recurrent events model with time-varying coefficients. Stat Med 32(6):1016-1026 [39] Zeng D, Cai J (2010) A semiparametric additive rate model for recurrent events with an informative terminal event. Biometrika 97(3):699-712 · Zbl 1195.62152 [40] Zeng D, Lin D (2009) Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events. Biometrics 65(3):746-752 · Zbl 1172.62070 [41] Zhu L, Sun J, Tong X, Srivastava D (2010) Regression analysis of multivariate recurrent event data with a dependent terminal event. Lifetime Data Anal 16(4):478-490 · Zbl 1322.62297 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.