PROS Nariza Wanti WS, Santi WP Survival analysis fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. BC.51–54
SC.51
ISBN: 978-602-1047-21-7
Survival analysis for recurrent event data
with Andersen–Gill approach
Nariza Wanti Wulan Sari and Santi Wulan Purnami
Department of Statistics,Sepuluh Nopember Institute of Technology, Indonesia
Email: *nariza.wanti90@gmail.com
Abstract
Cervical cancer is the second most common cancer among women worldwide. In
Indonesia, cervical cancer ranked for the second highest cancer next only to breast cancer
and is the leading cause of cancer-related death. Survival analysis can also be used to
analyze the recurrent failure event. Recurrent event in this study was the return of a
patient to the hospital for examination or treatment. Recurrent event data in this study
analysis with Andersen–Gill approach. Andersen–Gill Model is a generalization of the Cox
proportional hazard regression for analysing independent increment models of the Cox
type on the basis of the theory of counting process for survival analysis, the risk of an
event is assumed not to be affected by previous events.Data in this study were patients
with cervical cancer who had undergone treatment in RSUD Dr. Soetomo in 2014 and
variable used is age, education, domicile, stadium, medication, comorbidities,
complication, weight, and parity. The result show that there exist 3 variables that affect
the recurrent data of patients with cervical cancer is stadium, complication and weight.
Keywords Andersen–Gill, recurrent, servix, survival
1. Introduction
Survival analysis is the phrase used to describe the analysis of data that corresponden
to the time from a well-defined time origin until the occurence of some particular event or
end-point (Collect, 2003). Survival analysis can also be used to analyze the recurrent failure
event. Recurents events, that is, events that occur several times for the same subject
(Eimermacher, 2008). There are several approaches used to analyze the recurrent data
derived from Cox proportional hazard is counting process approach dan stratified Cox
approach. The stratified Cox approach there are three approaches that can be used is
stratified counting process, gap time, dan marginal. In addition, recurrent data can also be
analyzed with parametric approach using a frailty model (Kleinbaum & Klein, 2012).
Kelly & Lim (2000) researching survival analysis for recurrent event data an application
to childhood infectious diseases by comparing five models of Cox for recurrent event:
Andersen–Gill (AG), Prentice Williams and Peterson–gap time (PWP-GT) dan total time (PWPCP), Lee Wei and Amato (LWA), and Wei Lin and Weissfeld (WLW). This study concludes that
five models give different result. however, have not been able to conclude which model is
suitable for recurrent event data.
Ullah et al. (2012) also researching survival analysis for recurrent event in sports injuries
using five models: Cox proportional hazard, Andersen–Gill (AG), frailty, Wei Lin and Weissfeld
total time (WLW-TT) marginal, and Pretince Williams and Peterson gap time (PWP-GP)
condition models. This study concludes that Andersen–Gill (AG) and frailty models performed
best and provided better data fits to the recurrent sports injuries data, there was no statistical
SWUP
Survival analysis for recurrent event data with Andersen–Gill approach
SC.52
different between the Andersen–Gill (AG) dan frailty models in terms of model selection,
goodness of fit or occuracy. Accordingly, Recurrent event data in this study analysis with
Andersen–Gill approach. Andersen–Gill Model is a generalization of the Cox proportional
hazard regression for analysing independent increment models of the Cox type on the basis
of the theory of counting process for survival analysis, the risk of an event is assumed not to
be affected by previous events (Eimermacher, 2008).
Cervical cancer is the second most common cancer among women worldwide, with an
estimated 471.000 new cases (and 233.000 deaths) in the year 2000. Almost 80% of the cases
occur in developing countries, where, in many regions, it is the most common cancer among
women. The highest incidence rates are observed in Latin America and the Caribbean, subSaharan Africa and south and south-east Asia (WHO, 2005). Cervical cancer continues to be a
widespread public health problem in women throught the world, especially in developing
country like Indonesia (Nuranna, 2012). In this study will examine the implementation model
Andersen–Gill in cervical cancer patients data.
2.
Materials and methods
2.1 Data and method
Data in this study case are patients with cervical cancer who had undergone treatment
at the Hospital Dr. Soetomo 2014. Data obtained from cervical cancer patient medical record
data cervical cancer patients Obstetrics Gynecology section. Variables Research in this study,
there are several variables where each variable has a classification.
2.2 Proportional Hazard test
Testing the proportional hazards assumption on categorical variables, include
education, domicile, staging, treatment, accompanying diseases, and complication variable.
lls
2
lls
2
1
0
1
0
-1
-2
-1
-3
-4
-2
-5
-6
-3
-4
-7
-8
-5
0
100
200
300
0
400
100
200
Pendidikan
1
300
400
300
400
stop
stop
2
3
4
Domisili
5
lls
2
1
2
lls
2
1
1
0
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-6
-5
-6
-7
-8
-7
0
100
200
300
400
0
100
200
stop
Stadium
1
stop
2
3
4
5
Pengobatan
1
2
3
4
lls
2
lls
2
1
0
1
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-6
-5
-6
-7
-8
-7
-8
0
100
200
300
400
0
100
200
Penyerta
1
300
400
stop
stop
2
Komplikasi
1
2
Figure 1. Survival curve proportional hazard test graphically.
SWUP
N.W.W. Sari, S.W. Purnami
SC.53
Based on Figure 1, known plot for the variable: (a) education, (b) the domicile, (c) stage,
(d) treatment, (e) accompanying diseases, and (f) the complications shows that the line
between categories parallel the proportional hazards assumption for all categorical variables
have been met. Because the proportional hazards assumption test with graphical approach
is subjective, then the approach to the goodness of fit test, the results are as follows.
Table 1. PH Assumption testing Results with GOF.
Variable
Correlation
p-value
Education
0,0271
0,3749
Domicile
-0,0006
0,9829
Staging
-0,0420
0,1699
Treatment
0,0224
0,4653
Accompanying diseasess
0,0501
0,8682
Complication
0,0458
0,1346
Based on Table 1, we conclude that all variables have been fulfilled assumptions for
PH, so that, the analysis with Andersen–Gill may be continued.
3.
Results and discussion
Data was analyzed by Andersen–Gill approach , obtained, there are several variable is
not significant. So that variables removed from analysis, performed repeated analysis, thus
obtained the result as follows.
Variable
Staging
Complication
Weight
Table 2. Estimation Andersen–Gill result.
Hazard Ratio
Chi-Square
ä
–0.1788
0.836
24.2793
–0.6269
0.534
68.4363
0.0057
1.006
7.1221
Likelihood Ratio
77.9905
AIC = 12562.376
p-value
< 0.0001
< 0.0001
0.0076
< 0.0001
Based on Table 2, it was concluded that the variable stage, complications, and partial
body weight and overall had an effect on cervical cancer recurrences. Thus obtained the
following models
ℎãIa 1(Ia , å2 = ℎãÅ 1(Ia 2 exp[−0,1788 !é − 0,6269 !ë + 0,0057 !ì o.
Based on Table 2, it can be seen that the hazard ratio for the variable stage is 0.836.
This means that patients with stage 0 cervical cancer have an increased risk 0.836 times to
experience the event or recurrence than stage I, II, III, and IV, or in other words, compared
with patients with stage 0, the risk of cervical cancer patients with stage I, II, III, and IV
experienced recurrence event or amounted to 1,208 times.
Value hazard ratio for complications variable is 0.534. This means that cervical cancer
patients that there are no complications have 0.534 times risk to experience an event than
there are complications or recurrences, or in other words, compared with patients that there
are no complications, the risk of patients who have experienced complications or recurrence
event is by 2 times. Hazard ratio value for the variable weight is 1.006. This means that each
additional kilogram of body weight of cervical cancer patients, the risk of experiencing an
event or recurrence will also increase the time.
SWUP
Survival analysis for recurrent event data with Andersen–Gill approach
4.
SC.54
Conclusion and remarks
Survival analysis for recurrent event data with Andersen–Gill approach with study case
patients cervical cancer data was studied and major conclusions are as follows:
1. In this case study showed various ways to check the PH assumption graphically and
analytically, both give the same result, all categorical variables have fulfilled proportional
hazards assumption.
2. This study showed there are three variables that affect the recurrent data of patients with
cervical cancer, complication have greatest risk, where the risk is two times compared
with no complications.
It is important to check model assumptions before carrying out analysis using any
statistical model. The proportional hazard assumption is especially important for the use of
the Cox model, and is commonly or intentionally ignored.
References
Collett, D. (2003). Modelling survival data in medical research (2nd ed.). London: Chapman and Hall.
Eimermacher, A.J. (2008). Comparison of the Andersen-Gill model with poisson and negatif binomial
regression on recurrent event data. Computational Statistics and Data Analysis, 52, 4989–4997.
Kleinbaum, D.G., & Klein, M. (2012). Survival analysis a self-learning text (3rd ed.). London: Springer.
Kelly, P. J., & Lim, L. L.-Y. (2000). Survival analysis for recurrent event data: An application to childhood
infectious siseases. Statistics in Medicine, 19, 13–33.
Ullah, S., Gabbett, T. J., & Finch, C. F. (2012). Statistical modelling for recurrent events: An application
to sports injuries. Br J Sport Med., 0, 1–8.
WHO, I.A. (2005). Cervical Cancer Sreening. Lyon: IARCH Press.
Nuranna, L. (2012). Cervical cancer prevention program in Jakarta, Indonesia: See and treat model in
developing country. Journal of Gynecology Oncology, 23(3), 147–152.
SWUP
SC.51
ISBN: 978-602-1047-21-7
Survival analysis for recurrent event data
with Andersen–Gill approach
Nariza Wanti Wulan Sari and Santi Wulan Purnami
Department of Statistics,Sepuluh Nopember Institute of Technology, Indonesia
Email: *nariza.wanti90@gmail.com
Abstract
Cervical cancer is the second most common cancer among women worldwide. In
Indonesia, cervical cancer ranked for the second highest cancer next only to breast cancer
and is the leading cause of cancer-related death. Survival analysis can also be used to
analyze the recurrent failure event. Recurrent event in this study was the return of a
patient to the hospital for examination or treatment. Recurrent event data in this study
analysis with Andersen–Gill approach. Andersen–Gill Model is a generalization of the Cox
proportional hazard regression for analysing independent increment models of the Cox
type on the basis of the theory of counting process for survival analysis, the risk of an
event is assumed not to be affected by previous events.Data in this study were patients
with cervical cancer who had undergone treatment in RSUD Dr. Soetomo in 2014 and
variable used is age, education, domicile, stadium, medication, comorbidities,
complication, weight, and parity. The result show that there exist 3 variables that affect
the recurrent data of patients with cervical cancer is stadium, complication and weight.
Keywords Andersen–Gill, recurrent, servix, survival
1. Introduction
Survival analysis is the phrase used to describe the analysis of data that corresponden
to the time from a well-defined time origin until the occurence of some particular event or
end-point (Collect, 2003). Survival analysis can also be used to analyze the recurrent failure
event. Recurents events, that is, events that occur several times for the same subject
(Eimermacher, 2008). There are several approaches used to analyze the recurrent data
derived from Cox proportional hazard is counting process approach dan stratified Cox
approach. The stratified Cox approach there are three approaches that can be used is
stratified counting process, gap time, dan marginal. In addition, recurrent data can also be
analyzed with parametric approach using a frailty model (Kleinbaum & Klein, 2012).
Kelly & Lim (2000) researching survival analysis for recurrent event data an application
to childhood infectious diseases by comparing five models of Cox for recurrent event:
Andersen–Gill (AG), Prentice Williams and Peterson–gap time (PWP-GT) dan total time (PWPCP), Lee Wei and Amato (LWA), and Wei Lin and Weissfeld (WLW). This study concludes that
five models give different result. however, have not been able to conclude which model is
suitable for recurrent event data.
Ullah et al. (2012) also researching survival analysis for recurrent event in sports injuries
using five models: Cox proportional hazard, Andersen–Gill (AG), frailty, Wei Lin and Weissfeld
total time (WLW-TT) marginal, and Pretince Williams and Peterson gap time (PWP-GP)
condition models. This study concludes that Andersen–Gill (AG) and frailty models performed
best and provided better data fits to the recurrent sports injuries data, there was no statistical
SWUP
Survival analysis for recurrent event data with Andersen–Gill approach
SC.52
different between the Andersen–Gill (AG) dan frailty models in terms of model selection,
goodness of fit or occuracy. Accordingly, Recurrent event data in this study analysis with
Andersen–Gill approach. Andersen–Gill Model is a generalization of the Cox proportional
hazard regression for analysing independent increment models of the Cox type on the basis
of the theory of counting process for survival analysis, the risk of an event is assumed not to
be affected by previous events (Eimermacher, 2008).
Cervical cancer is the second most common cancer among women worldwide, with an
estimated 471.000 new cases (and 233.000 deaths) in the year 2000. Almost 80% of the cases
occur in developing countries, where, in many regions, it is the most common cancer among
women. The highest incidence rates are observed in Latin America and the Caribbean, subSaharan Africa and south and south-east Asia (WHO, 2005). Cervical cancer continues to be a
widespread public health problem in women throught the world, especially in developing
country like Indonesia (Nuranna, 2012). In this study will examine the implementation model
Andersen–Gill in cervical cancer patients data.
2.
Materials and methods
2.1 Data and method
Data in this study case are patients with cervical cancer who had undergone treatment
at the Hospital Dr. Soetomo 2014. Data obtained from cervical cancer patient medical record
data cervical cancer patients Obstetrics Gynecology section. Variables Research in this study,
there are several variables where each variable has a classification.
2.2 Proportional Hazard test
Testing the proportional hazards assumption on categorical variables, include
education, domicile, staging, treatment, accompanying diseases, and complication variable.
lls
2
lls
2
1
0
1
0
-1
-2
-1
-3
-4
-2
-5
-6
-3
-4
-7
-8
-5
0
100
200
300
0
400
100
200
Pendidikan
1
300
400
300
400
stop
stop
2
3
4
Domisili
5
lls
2
1
2
lls
2
1
1
0
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-6
-5
-6
-7
-8
-7
0
100
200
300
400
0
100
200
stop
Stadium
1
stop
2
3
4
5
Pengobatan
1
2
3
4
lls
2
lls
2
1
0
1
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-6
-5
-6
-7
-8
-7
-8
0
100
200
300
400
0
100
200
Penyerta
1
300
400
stop
stop
2
Komplikasi
1
2
Figure 1. Survival curve proportional hazard test graphically.
SWUP
N.W.W. Sari, S.W. Purnami
SC.53
Based on Figure 1, known plot for the variable: (a) education, (b) the domicile, (c) stage,
(d) treatment, (e) accompanying diseases, and (f) the complications shows that the line
between categories parallel the proportional hazards assumption for all categorical variables
have been met. Because the proportional hazards assumption test with graphical approach
is subjective, then the approach to the goodness of fit test, the results are as follows.
Table 1. PH Assumption testing Results with GOF.
Variable
Correlation
p-value
Education
0,0271
0,3749
Domicile
-0,0006
0,9829
Staging
-0,0420
0,1699
Treatment
0,0224
0,4653
Accompanying diseasess
0,0501
0,8682
Complication
0,0458
0,1346
Based on Table 1, we conclude that all variables have been fulfilled assumptions for
PH, so that, the analysis with Andersen–Gill may be continued.
3.
Results and discussion
Data was analyzed by Andersen–Gill approach , obtained, there are several variable is
not significant. So that variables removed from analysis, performed repeated analysis, thus
obtained the result as follows.
Variable
Staging
Complication
Weight
Table 2. Estimation Andersen–Gill result.
Hazard Ratio
Chi-Square
ä
–0.1788
0.836
24.2793
–0.6269
0.534
68.4363
0.0057
1.006
7.1221
Likelihood Ratio
77.9905
AIC = 12562.376
p-value
< 0.0001
< 0.0001
0.0076
< 0.0001
Based on Table 2, it was concluded that the variable stage, complications, and partial
body weight and overall had an effect on cervical cancer recurrences. Thus obtained the
following models
ℎãIa 1(Ia , å2 = ℎãÅ 1(Ia 2 exp[−0,1788 !é − 0,6269 !ë + 0,0057 !ì o.
Based on Table 2, it can be seen that the hazard ratio for the variable stage is 0.836.
This means that patients with stage 0 cervical cancer have an increased risk 0.836 times to
experience the event or recurrence than stage I, II, III, and IV, or in other words, compared
with patients with stage 0, the risk of cervical cancer patients with stage I, II, III, and IV
experienced recurrence event or amounted to 1,208 times.
Value hazard ratio for complications variable is 0.534. This means that cervical cancer
patients that there are no complications have 0.534 times risk to experience an event than
there are complications or recurrences, or in other words, compared with patients that there
are no complications, the risk of patients who have experienced complications or recurrence
event is by 2 times. Hazard ratio value for the variable weight is 1.006. This means that each
additional kilogram of body weight of cervical cancer patients, the risk of experiencing an
event or recurrence will also increase the time.
SWUP
Survival analysis for recurrent event data with Andersen–Gill approach
4.
SC.54
Conclusion and remarks
Survival analysis for recurrent event data with Andersen–Gill approach with study case
patients cervical cancer data was studied and major conclusions are as follows:
1. In this case study showed various ways to check the PH assumption graphically and
analytically, both give the same result, all categorical variables have fulfilled proportional
hazards assumption.
2. This study showed there are three variables that affect the recurrent data of patients with
cervical cancer, complication have greatest risk, where the risk is two times compared
with no complications.
It is important to check model assumptions before carrying out analysis using any
statistical model. The proportional hazard assumption is especially important for the use of
the Cox model, and is commonly or intentionally ignored.
References
Collett, D. (2003). Modelling survival data in medical research (2nd ed.). London: Chapman and Hall.
Eimermacher, A.J. (2008). Comparison of the Andersen-Gill model with poisson and negatif binomial
regression on recurrent event data. Computational Statistics and Data Analysis, 52, 4989–4997.
Kleinbaum, D.G., & Klein, M. (2012). Survival analysis a self-learning text (3rd ed.). London: Springer.
Kelly, P. J., & Lim, L. L.-Y. (2000). Survival analysis for recurrent event data: An application to childhood
infectious siseases. Statistics in Medicine, 19, 13–33.
Ullah, S., Gabbett, T. J., & Finch, C. F. (2012). Statistical modelling for recurrent events: An application
to sports injuries. Br J Sport Med., 0, 1–8.
WHO, I.A. (2005). Cervical Cancer Sreening. Lyon: IARCH Press.
Nuranna, L. (2012). Cervical cancer prevention program in Jakarta, Indonesia: See and treat model in
developing country. Journal of Gynecology Oncology, 23(3), 147–152.
SWUP