Mixture cure models with time-varying and multilevel frailties for recurrent event data

Abstract

Many medical studies yield data on recurrent clinical events from populations which consist of a proportion of cured patients in the presence of those who experience the event at several times (uncured). A frailty mixture cure model has recently been postulated for such data, with an assumption that the random subject effect (frailty) of each uncured patient is constant across successive gap times between recurrent events. We propose two new models in a more general setting, assuming a multivariate time-varying frailty with an AR(1) correlation structure for each uncured patient and addressing multilevel recurrent event data originated from multi-institutional (multi-centre) clinical trials, using extra random effect terms to adjust for institution effect and treatment-by-institution interaction. To solve the difficulties in parameter estimation due to these highly complex correlation structures, we develop an efficient estimation procedure via an EM-type algorithm based on residual maximum likelihood (REML) through the generalised linear mixed model (GLMM) methodology. Simulation studies are presented to assess the performances of the models. Data sets from a colorectal cancer study and rhDNase multi-institutional clinical trial were analyzed to exemplify the proposed models. The results demonstrate a large positive AR(1) correlation among frailties across successive gap times, indicating a constant frailty may not be realistic in some situations. Comparisons of findings with existing frailty models are discussed.

Keywords

AR(1) cure frailty model correlated survival data hierarchical survival data multilevel cure model random effect

Get full access to this article

View all access options for this article.

References

Kelly

Lim

LLY

. Survival analysis for recurrent event data: an application to childhood infectious diseases. Stat Med 2000; 19: 13–33.

Duchateau

Janssen

Kezic

, et al. Evolution of recurrent asthma event rate over time in frailty models. J R Stat Soc Ser C Appl Stat 2003; 52: 355–363.

Byar DP. The veterans administration study of chemoprophylaxis for recurrent stage I bladder tumors: comparisons of placebo, pyridoxine and topical thiotepa. In: Pavone-Maculuso M, Smith PH, Edsmyr F (eds). Bladder tumors and other topics in urological oncology. Plenum: New York, 1980, pp.363-370.

Gonzalez

Fernandez

Moreno

, et al. Sex differences in hospital readmission among colorectal cancer patients. J Epidemiol Commun Health 2005; 59: 506–511.

McGilchrist

Aisbett

. Regression with frailty in survival analysis. Biometrics 1991; 47: 461–466.

Therneau

Hamilton

. rhDNase as an example of recurrent event analysis. Stat Med 1997; 16: 2029–2047.

Cook RJ and Lawless JF. The statistical analysis of recurrent events. New York, NY: Springer Science & Business Media, 2007.

Box-Steffensmeier

De Boef

. Repeated events survival models: the conditional frailty model. Stat Med 2006; 25: 3518–3533.

Wei

Glidden

. An overview of statistical methods for multiple failure time data in clinical trials. Stat Med 1997; 16: 833–839.

10.

Metcalfe

Thompson

. Wei, Lin and Weissfeld's marginal analysis of multivariate failure time data: should it be applied to a recurrent events outcome? Stat Meth Med Res 2007; 16: 103–122.

11.

McGilchrist

. REML estimation for survival models with frailty. Biometrics 1993; 49: 221–225.

12.

Yau

KKW

McGilchrist

. ML and REML estimation in survival analysis with time dependent correlated frailty. Stat Med 1998; 17: 1201–1213.

13.

Othus

Barlogie

LeBlanc

, et al. Cure models as a useful statistical tool for analyzing survival. Clin Cancer Res 2012; 18: 3731–3736.

14.

. A frailty mixture cure model with application to hospital readmission data. Biom J 2008; 50: 386–394.

15.

Denaro

Merlano

Russi

. Follow-up in head and neck cancer: do more does it mean do better? a systematic review and our proposal based on our experience. Clin Exp Otorhinolaryngol 2016; 9: 287–297.

16.

Farewell

. The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 1982; 38: 1041–1046.

17.

Yau

KKW

ASK

. Long-term survivor mixture model with random effects: application to a multi-centre clinical trial of carcinoma. Stat Med 2001; 20: 1591–1607.

18.

Kuk

AYC

Chen

. A mixture model combining logistic regression with proportional hazards regression. Biometrika 1992; 79: 531–541.

19.

Peng

Dear

KBG

. A nonparametric mixture model for cure rate estimation. Biometrics 2000; 56: 237–243.

20.

Taylor

JMG

. Estimation of Cox proportional hazards cure model. Biometrics 2000; 56: 227–236.

21.

Lai

Yau

KKW

. Long-term survivor model with bivariate random effects: applications to bone marrow transplant and carcinoma study data. Stat Med 2008; 27: 5692–5708.

22.

Lin

, et al. Extension of a Cox proportional hazards cure model when cure information is partially known. Biostatistics 2014; 15: 540–554.

23.

Taylor

JMG

. Semi-parametric estimation in failure time mixture models. Biometrics 1995; 51: 899–907.

24.

Taylor

JMG

. A semi-parametric accelerated failure time cure model. Stat Med 2002; 21: 3235–3247.

25.

Zhang

. An alternative estimation method for the semiparametric accelerated failure time mixture cure model. Commun Stat Simul Comput 2009; 38: 1980–1990.

26.

Lai

Yau

KKW

. Extending the long-term survivor mixture model with random effects for clustered survival data. Comput Stat Data Anal 2010; 54: 2103–2112.

27.

Peng

Taylor

JMG

. Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study. Stat Med 2011; 30: 211–223.

28.

Rondeau

Schaffner

Corbière

, et al. Cure frailty models for survival data: application to recurrences for breast cancer and to hospital readmissions for colorectal cancer. Stat Meth Med Res 2011; 22: 243–260.

29.

Fong

DYT

Lam

Lawless

, et al. Dynamic random effects models for times between repeated events. Lifetime Data Anal 2001; 7: 345–362.

30.

McGilchrist

. Estimation in generalized mixed models. J R Stat Soc Series B Stat Methodol 1994; 56: 61–69.

31.

McGilchrist

Yau

KKW

. The derivation of BLUP, ML, REML estimation methods for generalised linear mixed models. Commun Stat Theory Meth 1995; 24: 2963–2980.

32.

Tawiah

Yau

KKW

McLachlan

, et al. Multilevel model with random effects for clustered survival data with multiple failure outcomes. Stat Med 2019; 38: 1036–1055.

33.

Lai

Yau

KKW

. Multilevel mixture cure models with random effects. Biom J 2009; 51: 456–466.

34.

Cheung

Lam

, et al. Estimation of summary protective efficacy using a frailty mixture model for recurrent event time data. Stat Med 2012; 31: 4023–4039.

35.

. Recent developments in expectation-maximization methods for analyzing complex data. Wiley Interdiscip Rev Comput Stat 2013; 5: 415–431.

36.

Abrahantes

Burzykowski

. A version of the EM algorithm for proportional hazards model with random effects. Biom J 2005; 47: 847–862.

37.

Vaida

. Proportional hazards model with random effects. Stat Med 2000; 19: 3309–3324.

38.

Peng

. Estimating baseline distribution in proportional hazards cure models. Comput Stat Data Anal 2003; 42: 187–201.

39.

Wang

Lee

Hamilton

, et al. Multilevel logistic regression modelling with correlated random effects: application to the smoking cessation for youth study. Stat Med 2006; 25: 3864–3876.

40.

Louis

. Finding the observed information matrix when using the EM algorithm. J R Stat Soc Series B Stat Methodol 1982; 44: 226–233.

41.

Kwok

West

Green

. The impact of misspecifying the within-subject covariance structure in multiwave longitudinal multilevel models: a Monte Carlo study. Multivariate Behav Res 2007; 42: 557–592.

42.

Peng

Taylor

JMG

. A marginal regression model for multivariate failure time data with a surviving fraction. Lifetime Data Anal 2007; 13: 351–369.

43.

Yamaguchi

Ohashi

. Investigating centre effects in a multi-center clinical trial of superficial bladder cancer. Stat Med 1999; 18: 1961–1971.

44.

McLachlan

Yau

KKW

, et al. Modelling the distribution of ischaemic stroke-specific survival time using an EM-based mixture approach with random effect adjustment. Stat Med 2004; 23: 2729–2744.

45.

McLachlan

. An EM-based semi-parametric mixture model approach to the regression analysis of competing-risks data. Stat Med 2003; 22: 1097–1111.

46.

McLachlan

McGiffin

. On the role of finite mixture models in survival analysis. Stat Methods Med Res 1994; 3: 211–226.

47.

Ng SK, Xiang L and Yau KKW. Mixture Modelling for Medical and Health Sciences. Boca Raton, FL: Chapman & Hall/CRC, 2019.

48.

Lee

Zhao

Yau

KKW

, et al. A computer graphical user interface for survival mixture modelling of recurrent infections. Comput Biol Med 2009; 39: 301–307.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.23 MB