Mixture method to estimate baseline hazard for non-arbitrary function of the Cox proportional model

  • Farag Hamad Department of Statistic, Faculty of Art & Science, University of Benghazi, Al Marj City, Libya.
  • Salem Abdulkarim Department of Physics, Faculty of Art& Science, University of Benghazi, Al Marj City, Libya.
  • Ayman Hamad Department of Physics, Faculty of Art& Science, University of Benghazi, Al Marj City, Libya.
Keywords: Kaplan Meier, Cox proportional hazard model, maximum likelihood, baseline hazard and partial likelihood, Gompertz distribution, Akaike information criterion, Bayesian information criterion


This study is extended to the work that was published by (Hamad & Kachouie, 2019). In this paper, a mixture method was used to estimate the Gompertz distribution parameters using hazard and cumulative hazard functions. This mixture method depends on the two different models (semi-parametric model, and nonparametric model). Cox proportional hazard model and Kaplan Meier used to estimate the Gompertz distribution parameters. The parameters of the logistic function (RHS) in the Cox proportional hazard can be estimated by the partial likelihood method. The hazard function (LHS) in the Cox model can be estimated by Kaplan Meier. The estimated parametric of the logistic function combined with the nonparametric estimate of the survival function by Kaplan Meier in order to get an estimate of the baseline hazard for Gompertz distribution. Improvement was archived in the estimated parameters of the baseline hazard using the mixture method compared to the use of the Cox method. The improvement of the mixture method was measured based on the estimated parameters for the baseline hazard as well as by the model goodness of fit. Different data types (simulation and real data) were used to measure the improvement of the mixture method. Monte Carlo simulations were carried out for evaluating the proposed method’s performance. The results showed that the mixture method provided a better estimate value of the baseline and the model parameters compared to the estimated values using the Cox model.


D. Collett, Modelling survival data in medical research. CRC press, 2015.

Z. Zhang, “Parametric regression model for survival data: Weibull regression model as an example,” Ann. Transl. Med., vol. 4, no. 24, 2016.

J. Wieczorek, “Finite sample properties of minimum Kolmogorov-Smirnov estimator and maximum likelihood estimator for right-censored data,” Available ReseachGate, 2009.

F. E. Harrell, Regression modeling strategies: with applications to linear models, logistic regression, and survival analysis, vol. 608. Springer, 2001.

P. Hu, A. A. Tsiatis, and M. Davidian, “Estimating the parameters in the Cox model when covariate variables are measured with error,” Biometrics, pp. 1407–1419, 1998.

E. T. Lee and J. Wang, Statistical methods for survival data analysis, vol. 476. John Wiley & Sons, 2003.

P. Wang, Y. Li, and C. K. Reddy, “Machine learning for survival analysis: A survey,” ACM Comput. Surv. CSUR, vol. 51, no. 6, pp. 1–36, 2019.

D. G. Kleinbaum and M. Klein, Survival analysis, vol. 3. Springer, 2010.

G. Whalen, “A proportional hazards model of bank failure: an examination of its usefulness as an early warning tool,” Econ. Rev., vol. 27, no. 1, pp. 21–31, 1991.

I. Langner, R. Bender, R. Lenz-Tönjes, H. Küchenhoff, and M. Blettner, “Bias of maximum-likelihood estimates in logistic and Cox regression models: a comparative simulation study,” Discussion Paper, 2003.

M. Franco, N. Balakrishnan, D. Kundu, and J.-M. Vivo, “Generalized mixtures of Weibull components,” Test, vol. 23, no. 3, pp. 515–535, 2014.

P. Royston, “Estimating a smooth baseline hazard function for the Cox model,” Lond. Dep. Stat. Sci. Univ. Coll. Lond., 2011.

V. Bewick, L. Cheek, and J. Ball, “Statistics review 12: survival analysis,” Crit. Care, vol. 8, no. 5, pp. 1–6, 2004.

S. J. Wang, J. Kalpathy-Cramer, J. S. Kim, C. D. Fuller, and C. R. Thomas Jr, “Parametric survival models for predicting the benefit of adjuvant chemoradiotherapy in gallbladder cancer,” 2010, vol. 2010, p. 847.

F. Hamad and N. N. Kachouie, “A hybrid method to estimate the full parametric hazard model,” Commun. Stat.-Theory Methods, vol. 48, no. 22, pp. 5477–5491, 2019.

A. A. Jafari, S. Tahmasebi, and M. Alizadeh, “The beta-Gompertz distribution,” Rev. Colomb. Estad., vol. 37, no. 1, pp. 141–158, 2014.

G. M. Cordeiro, M. Alizadeh, A. D. Nascimento, and M. Rasekhi, “The Exponentiated Gompertz Generated Family of Distributions: Properties and Applications.,” Chil. J. Stat. ChJS, vol. 7, no. 2, 2016.

A. Lenart, “The Gompertz distribution and maximum likelihood estimation of its parameters: a revision,” MPDIR Work Pap, vol. 49, pp. 0–19, 2012.

world data, “https://data.world/deviramanan2016/nki-breast-cancer-data.” 2016.

How to Cite
Hamad, F., Salem Abdulkarim, & Ayman Hamad. (2022). Mixture method to estimate baseline hazard for non-arbitrary function of the Cox proportional model. International Journal of Sciences: Basic and Applied Research (IJSBAR), 62(2), 235-248. Retrieved from https://gssrr.org/index.php/JournalOfBasicAndApplied/article/view/14023