Analysis of Diabetes Data using Extended Cox Model with Frailty under Partial and Penalized partial likelihood estimation methods

Alfred A. Abidoun, Benjamin A. Oyejola, Kazeem A. Adeleke


Data on Diabetes were analyzed using partial likelihood (Pl) and penalized partial likelihood (Ppl) estimation methods in non-proportional hazards model with dichotomous time-varying covariates. Gamma and Inverse Gaussian frailty distributions were used to account for patient- specific unobserved heterogeneity. Four likelihood configurations were formed from the combinations of the two estimation methods and frailty distributions. These are Partial likelihood with Gamma frailty, Partial likelihood with Inverse Gaussian frailty, Penalized partial likelihood with Gamma frailty and Penalized partial likelihood with Gamma frailty.  The results revealed that age and body mass index of the patients significantly increased the risk of death from diabetes, while regular exercise had significant decreased risk of death. Penalized partial likelihood estimation method generally outperformed models with Partial likelihood under all scenarios for the data and Gamma frailty provided a better fit in accounting for unobserved heterogeneity among the diabetic patients.

Full Text:



Adeleke, K.A., Abiodun, A. A. & Ipinyomi, R. A. (2015). Semi-Parametric Non-Proportional Hazard Model with Time Varying Covariate. Journal of Modern Applied Statistical, Methods, 14 (2), 68-87.

Aro, S., Kangas, T., Reunanen, A., Salinto, M. & Koivist, V. (1994). Hospital use among diabetic patients and the general population. Diabetes Care, 17(11):1320–9.

Austin, P.C., Mamdani, M.M., van Walraven C. & Tu, JV. (2006). Quantifying the impact of survivor treatment bias in observational studies. Journal of Evaluation in Clinical Practice, 12(6), 601– 612.

Austin P.C. (2012). Generating survival times to simulate Cox proportional hazards models with time-varying covariates. Statistics in Medicine, 31, 3946–3958.

Australian Institute of Health and Welfare. Diabetes prevalence in Australia. An assessment of national data sources. Canberra: AIHW; 2009.

Bellera, C.A,. MacGrogan,G., Debled, M., Christine Tunon de Lara, C. T., Brouste, V. & Mathoulin-Pélissier, S. (2010). Variables with time-varying effects and the Cox model: Some statistical concepts illustrated with a prognostic factor study in breast cancer. BMC Medical Research Methodology, 10:20.

Beyersmann, J., Wolkewitz, M. & Schumacher M. (2008). The impact of time-dependent bias in Proportional hazards modelling. Statistics in Medicine, 27, 6439–6454.

Bo, S, Ciccone, G., Grassi, G., Gancia, R., Rosato, R., Merletti, F. et al. (2004). Patients with type 2 diabetes had higher rates of hospitalization than the general population. J Clin Epidemiol. 57(11):1196–201.

Carral, F., Olveira G, Salas, J., Garcia, L., Sillero, A., Aguilar, M., (2002). Care resource utilization And direct costs incurred by people with diabetes in a Spanish hospital. Diabetes Res & Clin Prac. 56(1):27–34.

Colagiuri, S., Dickinson, S., Girgis, S. & Colagiuri, R. (2009). National Evidence Based Guideline for Blood Glucose Control in Type 2 Diabetes. Canberra. Diabetes Australia and the NHMRC.

Comino, E.J., Harris, M. F., Islam, M. D. F.,Tran, D.T.,Jalaludin, B., Jorm, L., Flack, and Haas, M. (2015). Impact of diabetes on hospital admission and length of stay among a general population aged 45 year or more: a record linkage study. Health Services Research.15:12 DOI 10.1186/s12913-014-0666-2

Cortese, G., Scheike, T., Martinussen, T., (2009). Flexible survival regression modelling. Stat Methods Med Res. 00:1-24.

Cox, D.R. (1972). Regression Models and Life-tables (with discussion). J. Roy. Statist. Soc. Ser. B. 34, 187-220.

De Berardis, G., D’Ettorre, A., Graziano, G., Lucisano, G., Pellegrini, F., Cammarota, S., et al. (2012). The burden of hospitalization related to diabetes mellitus: a population-based study. Nutr Metab & Cardiovasc Dis. 22(7):605–12.

Grambsch, P. & Therneau, T. (1994). Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika, 81:515-26.

Khalid, J.M, Raluy-Callado M, Curtis BH, Boye KS, Maguire A, Reaney M. (2013). Rates and risk of hospitalisation among patients with type 2 diabetes: retrospective cohort study using the UK General Practice Research Database linked to English Hospital Episode Statistics. Int J Clin Pract., 68(1):40–80.

Kleinbaum, D.G. and Klein, M. (2012). Survival analysis: A Self- Learning Text, Third Edition. Springer New York Dordrecht Heidelberg London.

McGilchrist, C. A. & Aisbett, C.W. (1993). Regression with frailty in survival analysis. Biometrics, 47, 461-466.

Munda, M., Rotolo, F. & Legrand. C. (2012). parfm: Parametric Frailty Model in R. Journal of Statistical Software, 15(11), 1-20.

Moolgavkar, S.H., Chang, E. T., Watson, H. N. and Lau, E.C. (2018) . An Assessment of the Cox Proportional Hazards Regression Model for Epidemiologic Studies. Risk Analysis, Vol. 38, No. 4, DOI: 10.1111/risa.12865. 777-794.

Munda, M., Rotolo, F. & Legrand. C. (2012). parfm: Parametric Frailty Model in R. Journal of Statistical Software, 15(11), 1-20.

Ng’andu, N.H (1997). An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox’s model. Stat Med. 16:611-26.

Putter, H., Sasako, M., Hartgrink , H.H., van d V, van Houwelingen JC. (2005). Long-term survival With non-proportional hazards: results from the Dutch Gastric Cancer Trial. Stat Med. 24:2807-21.

Rosenthal, M.J., Fajardo, M., Gilmore, S., Morley, J.E. & Naliboff, B.D. (1998). Hospitalization and mortality of diabetes in older adults. A 3-year prospective study. Diabetes Care. 1998;21(2):231–5.

Sertkaya, D., Ata, N., & özer, M. T. (2005). Yaşam cozumlemesinde zamana bağlı acıklayıcıdeğişkenli Cox regresyon modeli. Ankara Üniversitesi ıp akültesi Mecmuası, 58, 153-58.

Therneau, T. M., Grambsch, P. M., & Pankratz, V. S.(2003). Penalized survival models and frailty. Journal of Computational and Graphical Statistics, 12, 156–175.

Suissa, S. (2007). Immortal time bias in pharmacoepidemiology. American Journal of Epidemiology, 167(4), 492–499.

Therneau, T. M., Grambsch, P. M., & Pankratz, V. S.(2003). Penalized survival models and frailty. Journal of Computational and Graphical Statistics, 12, 156–175.

Vaupel, J.W., Manton, K.G. & Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography, 16, 349-454.

Zhou M. (2001). Understanding the Cox regression models with time-change covariates. The American Statistician; 55(2):153{155.DOI:10.1198/000313001750358491.


  • There are currently no refbacks.