Joint frailty models for zero‐inflated recurrent events in the presence of a terminal event

Recurrent event data arise frequently in longitudinal medical studies. In many situations, there are a large portion of subjects without any recurrent events, manifesting the "zero-inflated" nature of the data. Some of the zero events may be "structural zeros" as patients are unsusceptible to recurrent events, while others are "random zeros" due to censoring before any recurrent events. On the other hand, there often exists a terminal event which may be correlated with the recurrent events. In this article, we propose two joint frailty models for zero-inflated recurrent events in the presence of a terminal event, combining a logistic model for "structural zero" status (Yes/No) and a joint frailty proportional hazards model for recurrent and terminal event times. The models can be fitted conveniently in SAS Proc NLMIXED. We apply the methods to model recurrent opportunistic diseases in the presence of death in an AIDS study, and tumor recurrences and a terminal event in a sarcoma study.

[1]  McGilchrist Ca,et al.  Regression with frailty in survival analysis. , 1991 .

[2]  D. Lin,et al.  Nonparametric Analysis of Recurrent Events and Death , 2000, Biometrics.

[3]  Yu Zhangsheng,et al.  A joint model of recurrent events and a terminal event with a nonparametric covariate function , 2011, Statistics in medicine.

[4]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[5]  Keunbaik Lee,et al.  Analysis of zero-inflated clustered count data: A marginalized model approach , 2011, Comput. Stat. Data Anal..

[6]  J. P. Sy,et al.  Estimation in a Cox Proportional Hazards Cure Model , 2000, Biometrics.

[7]  P. Grambsch,et al.  Penalized Survival Models and Frailty , 2003 .

[8]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[9]  Lei Liu,et al.  The use of Gaussian quadrature for estimation in frailty proportional hazards models , 2008, Statistics in medicine.

[10]  Robert L. Strawderman,et al.  Use of the Probability Integral Transformation to Fit Nonlinear Mixed-Effects Models With Nonnormal Random Effects , 2006 .

[11]  Diane Lambert,et al.  Zero-inflacted Poisson regression, with an application to defects in manufacturing , 1992 .

[12]  Richard D. Gill,et al.  A counting process approach to maximum likelihood estimation in frailty models , 1992 .

[13]  S. Keleş,et al.  Recurrent events analysis in the presence of time‐dependent covariates and dependent censoring , 2004 .

[14]  William A. Knaus,et al.  A random effects four-part model, with application to correlated medical costs , 2008, Comput. Stat. Data Anal..

[15]  Lei Liu,et al.  A likelihood reformulation method in non‐normal random effects models , 2008, Statistics in medicine.

[16]  Victor H. Lachos,et al.  On estimation and influence diagnostics for zero-inflated negative binomial regression models , 2011, Comput. Stat. Data Anal..

[17]  D. Hall.  Zero‐Inflated Poisson and Binomial Regression with Random Effects: A Case Study , 2000, Biometrics.

[18]  Lei Liu,et al.  A semiparametric recurrent events model with time‐varying coefficients , 2013, Statistics in medicine.

[19]  Lei Liu,et al.  Joint model of recurrent events and a terminal event with time‐varying coefficients , 2014, Biometrical journal. Biometrische Zeitschrift.

[20]  Joseph L Schafer,et al.  A Two-Part Random-Effects Model for Semicontinuous Longitudinal Data , 2001 .

[21]  R. Wolfe,et al.  A Frailty Model for Informative Censoring , 2002, Biometrics.

[22]  Gary K Grunwald,et al.  Analysis of repeated measures data with clumping at zero , 2002, Statistical methods in medical research.

[23]  Binbing Yu,et al.  A marginal regression model for multivariate failure time data with a surviving fraction , 2007, Lifetime data analysis.

[24]  Guang Guo,et al.  Estimating a Multivariate Proportional Hazards Model for Clustered Data Using the EM Algorithm, with an Application to Child Survival in Guatemala , 1992 .

[25]  D. Stram,et al.  Variance components testing in the longitudinal mixed effects model. , 1994, Biometrics.

[26]  Lei Liu,et al.  A Joint Frailty Model for Survival and Gap Times Between Recurrent Events , 2007, Biometrics.

[27]  Lei Liu,et al.  Analysis of Longitudinal Data in the Presence of Informative Observational Times and a Dependent Terminal Event, with Application to Medical Cost Data , 2008, Biometrics.

[28]  J P Klein,et al.  Semiparametric estimation of random effects using the Cox model based on the EM algorithm. , 1992, Biometrics.

[29]  Alan Agresti,et al.  Random effect models for repeated measures of zero-inflated count data , 2005 .

[30]  F. Vaida,et al.  Proportional hazards model with random effects. , 2000, Statistics in medicine.

[31]  Douglas E Schaubel,et al.  Semiparametric Analysis of Correlated Recurrent and Terminal Events , 2007, Biometrics.

[32]  Lei Liu,et al.  A flexible two-part random effects model for correlated medical costs. , 2010, Journal of health economics.

[33]  D. Bates,et al.  Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model , 1995 .

[34]  P. Pisters,et al.  Estimation of the Causal Effects on Survival of Two‐Stage Nonrandomized Treatment Sequences for Recurrent Diseases , 2006, Biometrics.

[35]  Daniel B. Hall,et al.  Marginal models for zero inflated clustered data , 2004 .

[36]  Lei Liu,et al.  Joint modeling longitudinal semi‐continuous data and survival, with application to longitudinal medical cost data , 2009, Statistics in medicine.

[37]  Virginie Rondeau,et al.  Cure frailty models for survival data: Application to recurrences for breast cancer and to hospital readmissions for colorectal cancer , 2013, Statistical methods in medical research.

[38]  Qing Liu,et al.  A note on Gauss—Hermite quadrature , 1994 .

[39]  P. Thall,et al.  Cohort analysis of patients with localized, high-risk, extremity soft tissue sarcoma treated at two cancer centers: chemotherapy-associated outcomes. , 2004, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[40]  T. Lancaster,et al.  Panel Data with Survival: Hospitalization of HIV-Positive Patients , 1998 .