Joint frailty model of recurrent and terminal events in the presence of cure fraction using a Bayesian approach
Bayesian cure joint frailty model
Abstract
Introduction: Recurrent event data are common in many longitudinal studies. Often, a terminating event such as death can be correlated with the recurrent event process. A shared frailty model applied to account for the association between recurrent and terminal events. In some situations, a fraction of subjects experience neither recurrent events nor death; these subjects are cured.
Methods: In this paper, we discussed the Bayesian approach of a joint frailty model for recurrent and terminal events in the presence of cure fraction. We compared estimates of parameters in the Frequentist and Bayesian approaches via simulation studies in various sample sizes; we applied the joint frailty model in the presence of cure fraction with Frequentist and Bayesian approaches for breast cancer.
Results: In small sample size Bayesian approach compared to Frequentist approach had a smaller standard error and mean square error, and the coverage probabilities close to nominal level of 95%. Also, in Bayesian approach, the sampling means of the estimated standard errors were close to the empirical standard error.
Conclusion: The simulation results suggested that when sample size was small, the use of Bayesian joint frailty model in the presence of cure fraction led to more efficiency in parameter estimation and statistical inference
1. Andersen PK, Gill RD. Cox's regression model for counting processes: a large sample study. The annals of statistics. 1982:1100-20.
2. Nielsen GG, Gill RD, Andersen PK, Sørensen TI. A counting process approach to maximum likelihood estimation in frailty models. Scandinavian journal of Statistics. 1992:25-43.
3. Pepe MS, Cai J. Some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates. Journal of the American statistical Association. 1993;88(423):811-20.
4. Prentice RL, Williams BJ, Peterson AV. On the regression analysis of multivariate failure time data. Biometrika. 1981;68(2):373-9.
5. Wei LJ, Lin DY, Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of the American statistical association. 1989;84(408):1065-73.
6. Zeng D, Lin D. Semiparametric transformation models with random effects for recurrent events. Journal of the American Statistical Association. 2007;102(477):167-80.
7. Choi S, Huang X, Ju H, Ning J. Semiparametric accelerated intensity models for correlated recurrent and terminal events. Statistica Sinica. 2017:625-43.
8. Huang CY, Wang MC. Joint modeling and estimation for recurrent event processes and failure time data. Journal of the American Statistical Association. 2004;99(468):1153-65.
9. Liu L, Wolfe RA, Huang X. Shared frailty models for recurrent events and a terminal event. Biometrics. 2004;60(3):747-56.
10. Ye Y, Kalbfleisch JD, Schaubel DE. Semiparametric analysis of correlated recurrent and terminal events. Biometrics. 2007;63(1):78-87.
11. Zeng D, Lin D. Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events. Biometrics. 2009;65(3):746-52.
12. Rondeau V, Schaffner E, Corbiere F, Gonzalez JR, Mathoulin-Pelissier S. Cure frailty models for survival data: Application to recurrences for breast cancer and to hospital readmissions for colorectal cancer. Statistical methods in medical research. 2013;22(3):243-60.
13. Yu B. A frailty mixture cure model with application to hospital readmission cata. Biometrical Journal: Journal of Mathematical Methods in Biosciences. 2008;50(3):386-94.
14. Sinha D, Maiti T, Ibrahim JG, Ouyang B. Current methods for recurrent events data with dependent termination: a Bayesian perspective. Journal of the American Statistical Association. 2008;103(482):866-78.
15. Paulon G, De Iorio M, Guglielmi A, Ieva F. Joint modeling of recurrent events and survival: a Bayesian non-parametric approach. Biostatistics (Oxford, England). 2018.
16. E Talebi-Ghane AB, F Zayeri, V Rondeau, A Saeedi,. A comparison of Joint frailty model for recurrent events and death using Classical and Bayesian approaches: Application to breast cancer data. JP Journal of Biostatistics 2018;16(1):71-90.
17. Liu L, Huang X, Yaroshinsky A, Cormier JN. Joint frailty models for zero‐inflated recurrent events in the presence of a terminal event. Biometrics. 2016;72(1):204-14.
18. Andersen PK, Borgan Ø, Gill RD, Keiding N. Nonparametric estimation. Statistical models based on counting processes: Springer; 1993. p. 176-331.
Table 1. Simulation Results for a generated joint frailty model with Frequentist and Bayesian approaches
Sample size Parameter Frequentist Bayesian
(Informative priors) Bayesian
(Non-informative priors)
Est SEE SEM1 MSE1 CP1 mean SD SEM2 MSE2 CP2 mean SD SEM2 MSE2 CP2
N=20
2.60 5.35 178.33 30.12 0.91 0.34 0.48 0.63 0.66 0.75 1.17 1.59 2.70 2.56 0.95
-1.28 5.85 226.70 34.90 0.94 -0.39 0.48 0.64 0.24 0.95 -0.83 1.55 2.23 2.54 0.94
1.17 4.68 14.71 22.10 0.97 0.25 0.53 0.60 0.48 0.86 0.81 1.41 1.45 2.00 0.93
1.47 1.10 0.33 1.25 0.98 1.34 0.44 0.43 0.21 0.94 1.48 0.67 0.51 0.50 0.93
1.98 4.84 0.29 23.90 0.99 1.51 0.53 0.39 0.35 0.91 1.66 0.74 0.49 0.72 0.92
1.60 10.20 25.03 1.05 0.99 0.44 0.59 0.81 0.35 0.96 0.54 0.93 1.41 0.87 0.93
-0.28 2.41 9.54 5.88 0.93 -0.51 0.61 0.67 0.37 0.93 -0.54 0.95 0.91 0.91 0.95
1.40 3.35 17.36 11.30 0.91 0.89 0.78 0.95 0.63 0.95 1.11 1.37 1.50 1.90 0.94
N=30
1.87 3.03 16.60 9.95 0.90 0.46
0.48 0.57 0.52 0.82 1.12 1.09 1.59 1.21 0.93
-0.64 3.05 5.01 9.33 0.94 -0.44 0.49 0.57 0.24 0.94 -0.72 1.03 1.24 1.11 0.95
0.59 5.56 4.43 30.90 0.99 0.30 0.49 0.54 0.40 0.87 0.66 1.02 1.07 1.05 0.95
1.36 0.51 0.24 0.27 0.94 1.28 0.33 0.34 0.11 0.94 1.37 0.42 0.44 0.19 0.94
1.67 4.00 0.23 16.20 0.99 1.40 0.39 0.32 0.11 0.92 1.49 0.47 0.50 0.28 0.91
1.12 8.02 12.87 64.7 0.99 0.42 0.50 0.66 0.26 0.95 0.53 0.73 1.42 0.54 0.95
-0.41 1.28 2.28 1.65 0.97 -0.54 0.54 0.55 0.29 0.94 -0.50 0.62 0.66 0.38 0.95
1.22 1.80 4.42 3.29 0.95 0.96 0.69 0.77 0.48 0.96 1.03 0.91 0.91 0.83 0.96
N=50
1.39 1.80 3.79 3.42 0.97 0.59 0.45 0.50 0.36 0.86 1.04 0.75 0.77 0.56 0.94
-0.61 1.25 2.017 1.53 0.99 -0.48 0.48 0.48 0.23 0.95 -0.62 0.75 0.64 0.58 0.94
0.77 0.80 0.68 0.65 0.96 0.38 0.42 0.47 0.28 0.87 0.69 0.66 0.66 0.44 0.95
1.29 0.33 0.18 0.11 0.94 1.25 0.26 0.26 0.07 0.95 1.29 0.28 0.28 0.08 0.95
1.36 0.35 0.17 0.12 0.93 1.32 0.27 0.25 0.08 0.94 1.36 0.30 0.27 0.10 0.94
0.58 1.04 4.82 1.09 0.98 0.43 0.44 0.53 0.20 0.97 0.46 0.50 0.60 0.26 0.96
-0.45 0.55 0.49 0.30 0.95 -0.51 0.44 0.53 0.20 0.94 -0.50 0.49 0.47 0.24 0.94
1.07 0.88 1.81 0.79 0.97 0.95 0.55 0.61 0.30 0.97 0.98 0.65 0.64 0.42 0.94
N=100
1.08 0.51 0.50 0.27 0.95 0.73 0.34 0.39 0.18 0.88 1.01 0.46 0.49 0.21 0.95
-0.52 0.41 0.40 0.17 0.94 -0.52 0.35 0.36 0.13 0.95 -0.59 0.41 0.40 0.18 0.93
0.74 0.51 0.45 0.26 0.96 0.47 0.36 0.37 0.18 0.91 0.67 0.44 0.45 0.21 0.95
1.25 0.19 0.13 0.03 0.95 1.22 0.18 0.18 0.03 0.95 1.24 0.19 0.19 0.03 0.95
1.29 0.19 0.13 0.04 0.93 1.27 0.18 0.18 0.03 0.95 1.28 0.20 0.18 0.04 0.94
0.49 0.33 1.56 0.11 0.94 0.43 0.30 0.37 0.09 0.97 0.44 0.32 0.40 0.10 0.96
-0.48 0.32 0.32 0.10 0.94 -0.50 0.32 0.31 0.10 0.95 -0.47 0.33 0.32 0.11 0.94
1.01 0.45 0.44 0.20 0.95 0.97 0.42 0.43 0.18 0.94 0.94 0.45 0.44 0.20 0.93
N=200
1.03 0.32 0.34 0.10 0.94 0.83 0.28 0.30 0.11 0.92 0.97 0.34 0.33 0.11 0.94
-0.51 0.29 0.27 0.08 0.95 -0.53 0.28 0.26 0.08 0.94 -0.57 0.32 0.27 0.11 0.94
0.71 0.33 0.31 0.10 0.94 0.54 0.28 0.28 0.10 0.90 0.66 0.33 0.31 0.11 0.94
1.25 0.12 0.09 0.01 0.96 1.22 0.12 0.12 0.01 0.95 1.23 0.13 0.13 0.01 0.95
1.27 0.12 0.09 0.01 0.94 1.24 0.12 0.12 0.01 0.95 1.24 0.13 0.12 0.01 0.94
0.50 0.23 0.48 0.05 0.96 0.43 0.21 0.25 0.04 0.95 0.43 0.22 0.26 0.05 0.94
-0.49 0.22 0.25 0.05 0.94 -0.50 0.24 0.22 0.06 0.94 -0.49 0.25 0.22 0.06 0.94
1.00 0.29 0.30 0.08 0.94 0.96 0.32 0.30 0.10 0.94 0.93 0.33 0.30 0.11 0.93
CP1, coverage probability of 95% confidence intervals; CP2, coverage probability of 95% equal-tail credible intervals; Est, estimates of parameters; MSE1, mean square error; MSE2, Bayesian mean square error; SD, posterior standard deviation; SEE, empirical standard error; SEM1, mean of standard error; SEM2, sample mean of the square root of posterior variances
2. Nielsen GG, Gill RD, Andersen PK, Sørensen TI. A counting process approach to maximum likelihood estimation in frailty models. Scandinavian journal of Statistics. 1992:25-43.
3. Pepe MS, Cai J. Some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates. Journal of the American statistical Association. 1993;88(423):811-20.
4. Prentice RL, Williams BJ, Peterson AV. On the regression analysis of multivariate failure time data. Biometrika. 1981;68(2):373-9.
5. Wei LJ, Lin DY, Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of the American statistical association. 1989;84(408):1065-73.
6. Zeng D, Lin D. Semiparametric transformation models with random effects for recurrent events. Journal of the American Statistical Association. 2007;102(477):167-80.
7. Choi S, Huang X, Ju H, Ning J. Semiparametric accelerated intensity models for correlated recurrent and terminal events. Statistica Sinica. 2017:625-43.
8. Huang CY, Wang MC. Joint modeling and estimation for recurrent event processes and failure time data. Journal of the American Statistical Association. 2004;99(468):1153-65.
9. Liu L, Wolfe RA, Huang X. Shared frailty models for recurrent events and a terminal event. Biometrics. 2004;60(3):747-56.
10. Ye Y, Kalbfleisch JD, Schaubel DE. Semiparametric analysis of correlated recurrent and terminal events. Biometrics. 2007;63(1):78-87.
11. Zeng D, Lin D. Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events. Biometrics. 2009;65(3):746-52.
12. Rondeau V, Schaffner E, Corbiere F, Gonzalez JR, Mathoulin-Pelissier S. Cure frailty models for survival data: Application to recurrences for breast cancer and to hospital readmissions for colorectal cancer. Statistical methods in medical research. 2013;22(3):243-60.
13. Yu B. A frailty mixture cure model with application to hospital readmission cata. Biometrical Journal: Journal of Mathematical Methods in Biosciences. 2008;50(3):386-94.
14. Sinha D, Maiti T, Ibrahim JG, Ouyang B. Current methods for recurrent events data with dependent termination: a Bayesian perspective. Journal of the American Statistical Association. 2008;103(482):866-78.
15. Paulon G, De Iorio M, Guglielmi A, Ieva F. Joint modeling of recurrent events and survival: a Bayesian non-parametric approach. Biostatistics (Oxford, England). 2018.
16. E Talebi-Ghane AB, F Zayeri, V Rondeau, A Saeedi,. A comparison of Joint frailty model for recurrent events and death using Classical and Bayesian approaches: Application to breast cancer data. JP Journal of Biostatistics 2018;16(1):71-90.
17. Liu L, Huang X, Yaroshinsky A, Cormier JN. Joint frailty models for zero‐inflated recurrent events in the presence of a terminal event. Biometrics. 2016;72(1):204-14.
18. Andersen PK, Borgan Ø, Gill RD, Keiding N. Nonparametric estimation. Statistical models based on counting processes: Springer; 1993. p. 176-331.
Table 1. Simulation Results for a generated joint frailty model with Frequentist and Bayesian approaches
Sample size Parameter Frequentist Bayesian
(Informative priors) Bayesian
(Non-informative priors)
Est SEE SEM1 MSE1 CP1 mean SD SEM2 MSE2 CP2 mean SD SEM2 MSE2 CP2
N=20
2.60 5.35 178.33 30.12 0.91 0.34 0.48 0.63 0.66 0.75 1.17 1.59 2.70 2.56 0.95
-1.28 5.85 226.70 34.90 0.94 -0.39 0.48 0.64 0.24 0.95 -0.83 1.55 2.23 2.54 0.94
1.17 4.68 14.71 22.10 0.97 0.25 0.53 0.60 0.48 0.86 0.81 1.41 1.45 2.00 0.93
1.47 1.10 0.33 1.25 0.98 1.34 0.44 0.43 0.21 0.94 1.48 0.67 0.51 0.50 0.93
1.98 4.84 0.29 23.90 0.99 1.51 0.53 0.39 0.35 0.91 1.66 0.74 0.49 0.72 0.92
1.60 10.20 25.03 1.05 0.99 0.44 0.59 0.81 0.35 0.96 0.54 0.93 1.41 0.87 0.93
-0.28 2.41 9.54 5.88 0.93 -0.51 0.61 0.67 0.37 0.93 -0.54 0.95 0.91 0.91 0.95
1.40 3.35 17.36 11.30 0.91 0.89 0.78 0.95 0.63 0.95 1.11 1.37 1.50 1.90 0.94
N=30
1.87 3.03 16.60 9.95 0.90 0.46
0.48 0.57 0.52 0.82 1.12 1.09 1.59 1.21 0.93
-0.64 3.05 5.01 9.33 0.94 -0.44 0.49 0.57 0.24 0.94 -0.72 1.03 1.24 1.11 0.95
0.59 5.56 4.43 30.90 0.99 0.30 0.49 0.54 0.40 0.87 0.66 1.02 1.07 1.05 0.95
1.36 0.51 0.24 0.27 0.94 1.28 0.33 0.34 0.11 0.94 1.37 0.42 0.44 0.19 0.94
1.67 4.00 0.23 16.20 0.99 1.40 0.39 0.32 0.11 0.92 1.49 0.47 0.50 0.28 0.91
1.12 8.02 12.87 64.7 0.99 0.42 0.50 0.66 0.26 0.95 0.53 0.73 1.42 0.54 0.95
-0.41 1.28 2.28 1.65 0.97 -0.54 0.54 0.55 0.29 0.94 -0.50 0.62 0.66 0.38 0.95
1.22 1.80 4.42 3.29 0.95 0.96 0.69 0.77 0.48 0.96 1.03 0.91 0.91 0.83 0.96
N=50
1.39 1.80 3.79 3.42 0.97 0.59 0.45 0.50 0.36 0.86 1.04 0.75 0.77 0.56 0.94
-0.61 1.25 2.017 1.53 0.99 -0.48 0.48 0.48 0.23 0.95 -0.62 0.75 0.64 0.58 0.94
0.77 0.80 0.68 0.65 0.96 0.38 0.42 0.47 0.28 0.87 0.69 0.66 0.66 0.44 0.95
1.29 0.33 0.18 0.11 0.94 1.25 0.26 0.26 0.07 0.95 1.29 0.28 0.28 0.08 0.95
1.36 0.35 0.17 0.12 0.93 1.32 0.27 0.25 0.08 0.94 1.36 0.30 0.27 0.10 0.94
0.58 1.04 4.82 1.09 0.98 0.43 0.44 0.53 0.20 0.97 0.46 0.50 0.60 0.26 0.96
-0.45 0.55 0.49 0.30 0.95 -0.51 0.44 0.53 0.20 0.94 -0.50 0.49 0.47 0.24 0.94
1.07 0.88 1.81 0.79 0.97 0.95 0.55 0.61 0.30 0.97 0.98 0.65 0.64 0.42 0.94
N=100
1.08 0.51 0.50 0.27 0.95 0.73 0.34 0.39 0.18 0.88 1.01 0.46 0.49 0.21 0.95
-0.52 0.41 0.40 0.17 0.94 -0.52 0.35 0.36 0.13 0.95 -0.59 0.41 0.40 0.18 0.93
0.74 0.51 0.45 0.26 0.96 0.47 0.36 0.37 0.18 0.91 0.67 0.44 0.45 0.21 0.95
1.25 0.19 0.13 0.03 0.95 1.22 0.18 0.18 0.03 0.95 1.24 0.19 0.19 0.03 0.95
1.29 0.19 0.13 0.04 0.93 1.27 0.18 0.18 0.03 0.95 1.28 0.20 0.18 0.04 0.94
0.49 0.33 1.56 0.11 0.94 0.43 0.30 0.37 0.09 0.97 0.44 0.32 0.40 0.10 0.96
-0.48 0.32 0.32 0.10 0.94 -0.50 0.32 0.31 0.10 0.95 -0.47 0.33 0.32 0.11 0.94
1.01 0.45 0.44 0.20 0.95 0.97 0.42 0.43 0.18 0.94 0.94 0.45 0.44 0.20 0.93
N=200
1.03 0.32 0.34 0.10 0.94 0.83 0.28 0.30 0.11 0.92 0.97 0.34 0.33 0.11 0.94
-0.51 0.29 0.27 0.08 0.95 -0.53 0.28 0.26 0.08 0.94 -0.57 0.32 0.27 0.11 0.94
0.71 0.33 0.31 0.10 0.94 0.54 0.28 0.28 0.10 0.90 0.66 0.33 0.31 0.11 0.94
1.25 0.12 0.09 0.01 0.96 1.22 0.12 0.12 0.01 0.95 1.23 0.13 0.13 0.01 0.95
1.27 0.12 0.09 0.01 0.94 1.24 0.12 0.12 0.01 0.95 1.24 0.13 0.12 0.01 0.94
0.50 0.23 0.48 0.05 0.96 0.43 0.21 0.25 0.04 0.95 0.43 0.22 0.26 0.05 0.94
-0.49 0.22 0.25 0.05 0.94 -0.50 0.24 0.22 0.06 0.94 -0.49 0.25 0.22 0.06 0.94
1.00 0.29 0.30 0.08 0.94 0.96 0.32 0.30 0.10 0.94 0.93 0.33 0.30 0.11 0.93
CP1, coverage probability of 95% confidence intervals; CP2, coverage probability of 95% equal-tail credible intervals; Est, estimates of parameters; MSE1, mean square error; MSE2, Bayesian mean square error; SD, posterior standard deviation; SEE, empirical standard error; SEM1, mean of standard error; SEM2, sample mean of the square root of posterior variances
| Files | ||
| Issue | Vol 8 No 3 (2022) | |
| Section | Original Article(s) | |
| DOI | https://doi.org/10.18502/jbe.v8i3.12306 | |
| Keywords | ||
| Bayesian approach Breast cancer Cure fraction Joint frailty model Recurrent event. | ||
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How to Cite
1.
Baghestani AR, Arab Borzu zahra, Talebi Ghane E, Khadem maboudi AA, Saeedi A, Akhavan A. Joint frailty model of recurrent and terminal events in the presence of cure fraction using a Bayesian approach. JBE. 2023;8(3):304-313.
