The Weibull-Exponentiated Exponential Cure Fraction Model for Right Censored Survival Data with Applications to Cancer Data
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10.58578/amjsai.v1i2.3855
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Abstract
The cure fraction model also known as the long-term survival model is used in fitting data from a population with two different types of individuals: individuals who experienced the event of interest (susceptible) and individuals who will never experience the event of interest (non-susceptible). The present paper introduced a cure fraction model considering the Weibull exponentiated exponential distribution that will be used in modeling such type of information. The parameters of the model were estimated via the maximum likelihood procedure (MLE) under the assumption of right censoring. Furthermore, the statistical properties of the model were studied comprehensively. Simulation studies and medical data sets were used to demonstrate the applicability of the proposed methodology. Bias and standard error were used as discrimination criteria in the simulation study while Akaike Information criteria (AIC), Bayesian Information Criteria (BIC), and Consistent Akaike Information criteria (CAIC) were used as discrimination criteria in real-life applications. Results from the applications showed that the Weibull exponentiated exponential non-mixture cure fraction model is a strong competitor.
Keywords:
Survival Analysis; Mixture Cure Fraction Model; Non-Mixture Cure Fraction Model; Weibull Cure Fraction Model; Right censoring
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Yakubu, A., Muhammad, N. I., Falgore, J. Y., & Rabiu, A. (2024). The Weibull-Exponentiated Exponential Cure Fraction Model for Right Censored Survival Data with Applications to Cancer Data. African Multidisciplinary Journal of Sciences and Artificial Intelligence, 1(2), 711-736. https://doi.org/10.58578/amjsai.v1i2.3855
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
References
Achcar, J. A., Coelho-Barros, E. A., & Mazucheli, J. (2012). Cure fraction models using mixture and non-mixture models. Tatra Mountains Mathematical Publications, 51(1), 1 – 9.
Andrei, Y. Y., Asselain, B., et al. (1996). Stochastic models of tumor latency and their biostatistical applications (Vol. 1). World Scientific.
Berkson, J., & Gage, R. P. (1952). Survival curve for cancer patients following treatment. Journal of the American Statistical Association, 47(259), 501–515.
Boag, J. W. (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society. Series B (Methodological), 11(1), 15–53.
Cancho, V. G., & Bolfarine, H. (2001). Modeling the presence of immunes by using the exponentiated-Weibull model. Journal of Applied Statistics, 28(6), 659–671.
Cancho, V. G., Rodrigues, J., & Castro, M. de. (2011). A flexible model for survival data with a cure rate: a Bayesian approach. Journal of Applied Statistics, 38(1), 57–70.
Carrasco, J. M., Ortega, E. M., & Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics & Data Analysis, 53(2), 450–462.
Chen, M.-H., Ibrahim, J. G., & Sinha, D. (1999). A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association, 94(447), 909–919.
Coelho-Barros, E. A., Achcar, J. A., & Mazucheli, J. (2017). Cure rate models considering the burr xii distribution in the presence of covariate. Journal of Statistical Theory and Applications, 16(2), 150–164.
Corbi`ere, F., Commenges, D., Taylor, J. M., & Joly, P. (2009). A penalized likelihood approach for mixture cure models. Statistics in medicine, 28(3), 510–524.
Farewell, V. T. (1982). The use of mixture models for the analysis of survival data with long-term survivors. Biometrics, 1041–1046.
Farewell, V. T. (1986). Mixture models in survival analysis: Are they worth the risk? Canadian Journal of Statistics, 14(3), 257–262.
Gamel, J. W., McLean, I. W., & Rosenberg, S. H. (1990). Proportion cured and mean log survival time as functions of tumor size. Statistics in Medicine, 9(8), 999–1006.
Ghazali, A. K. (2018). Modeling of survival and incidence for colorectal cancer in Malaysia. Unpublished doctoral dissertation, Lancaster University.
Ghitany, M., & Maller, R. A. (1992). Asymptotic results for exponential mixture models with long-term survivors. Statistics: A Journal of Theoretical and Applied Statistics, 23(4), 321–336.
Goldman, A. I. (1984). Survivorship analysis when cure is a possibility: a Monte Carlo study. Statistics in Medicine, 3(2), 153–163.
Hassan, M. R. A., Suan, M. A. M., Soelar, S. A., Mohammed, N. S., Ismail, I., & Ahmad, F. (2016). Survival analysis and prognostic factors for colorectal cancer patients in Malaysia. Asian Pacific Journal of Cancer Prevention, 17(7), 3575 – 3581.
Herring, A. H., & Ibrahim, J. G. (2002). Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates. Biostatistics, 3(3), 387–405.
Ishaq, A., Usman, A., Tasiaˆu, M., Aliyu, Y., & Idris, F. (2017). A new Weibull Kumaraswamy distribution: Theory and applications. Nigerian Journal of Scientific Research, 16(2), 158–166.
Jones, D., Powles, R., Machin, D., & Sylvester, R. (1981). On estimating the proportion of cured patients in clinical studies. Biometrie-Praximetrie, 21, 1–11.
Kannan, N., Kundu, D., Nair, P., & Tripathi, R. C. (2010). The generalized exponential cure rate model with covariates. Journal of Applied Statistics, 37(10), 1625 aˆ 1636.
Kittrongsiri, K., Wanitsuwan, W., Prechawittayakul, P., Sangroongruangsri, S., Cairns, J., & Chaikledkaew, U. (2020). Survival analysis of colorectal cancer patients in a Thai hospital-based cancer registry. Expert review of gastroenterology & hepatology, 14(4), 291–300.
Kutal, D., & Qian, L. (2018). A non-mixture cure model for right-censored data with Fr´echet distribution. Stats, 1(1), 176–188.
Liu, H., & Shen, Y. (2009). A semiparametric regression cure model for interval-censored data. Journal of the American Statistical Association, 104(487), 1168 – 1178.
Lopes, C. M. C., & Bolfarine, H. (2012). Random effects in promotion time cure rate models. Computational statistics & data analysis, 56(1), 75–87.
Magaji, B. A., Moy, F. M., Roslani, A. C., & Law, C. W. (2014). Descriptive epidemiology of colorectal cancer in University Malaya Medical Centre, 2001 to 2010. Asian Pacific Journal of Cancer Prevention, 15(15), 6059–6064.
Magaji, B. A., Moy, F. M., Roslani, A. C., & Law, C. W. (2017). Survival rates and predictors of survival among colorectal cancer patients in a Malaysian tertiary hospital. BMC cancer, 17(1), 1 – 8.
Maller, R. A., & Zhou, X. (1996). Survival analysis with long-term survivors. Wiley New York.
Martinez, E. Z., Achcar, J. A., Ja´come, A. A., & Santos, J. S. (2013). Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data. Computer methods and programs in biomedicine, 112(3), 343–355.
Mazucheli, J., Coelho-Barros, E. A., & Achcar, J. A. (2013). The exponentiated exponential mixture and non-mixture cure rate model in the presence of covariates. Computer methods and programs in biomedicine, 112(1), 114–124.
Meeker, W. Q. (1987). Limited failure population life tests: application to integrated circuit reliability. Technometrics, 29(1), 51–65.
Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE transactions on reliability, 42(2), 299–302.
Naishadham, D., Lansdorp-Vogelaar, I., Siegel, R., Cokkinides, V., & Jemal, A. (2011). State disparities in colorectal cancer mortality patterns in the United States. Cancer Epidemiology and Prevention Biomarkers, 20(7), 1296–1302.
Ng, S., & McLachlan, G. (1998). On modifications to the long-term survival mixture model in the presence of competing risks. Journal of Statistical Computation and Simulation, 61(1-2), 77–96.
Ortega, E. M., Cordeiro, G. M., & Kattan, M. W. (2013). The log-beta Weibull regression model with application to predict recurrence of prostate cancer. Statistical Papers, 54(1), 113–132.
Pal, M., Ali, M. M., & Woo, J. (2006). Exponentiated Weibull distribution. Statistica, 66(2), 139–147.
Peng, Y., Dear, K. B., & Denham, J. (1998). A generalized f mixture model for cure rate estimation. Statistics in medicine, 17(8), 813–830.
Peto, R., Lee, P., & Paige, W. (1972). Statistical analysis of the bioassay of continuous carcinogens. British journal of cancer, 26(4), 258–261.
Ramos, P. L., Nascimento, D., & Louzada, F. (2017). The long term fr\’echet distribution: Estimation, properties and its application. arXiv preprint arXiv:1709.07593.
Salem, H. M., & Selim, M. (2014). The generalized Weibull-exponential distribution: properties and applications. International Journal of Statistics and Applications, 4(2), 102–112.
Shao, Q., & Zhou, X. (2004). A new parametric model for survival data with long-term survivors. Statistics in medicine, 23(22), 3525–3543.
Sy, J. P., & Taylor, J. M. (2000). Estimation in a Cox proportional hazards cure model. Biometrics, 56(1), 227–236.
Tsodikov, A., Ibrahim, J., & Yakovlev, A. (2003). Estimating cure rates from survival data: an alternative to two-component mixture models. Journal of the American Statistical Association, 98(464), 1063–1078.
Uddin, M., Islam, M., & Ibrahim, Q. (2006). An analytical approach to cure rate estimation based on uncensored data. Journal of Applied Sciences, 6(3), 548 –552.
Uddin, M., Sen, A., Noor, M., Islam, M., & Chowdhury, Z. (2006). An analytical approach on non-parametric estimation of cure rate based on uncensored data. Journal of Applied Sciences, 6(6), 1258–1264.
Usman, U., Shamsuddeen, S., Arkilla, B. M., & Aliyu, Y. (2020). Inferences on the Weibull exponentiated exponential distribution and applications. International Journal of Statistical Distributions and Applications, 6(1), 1-10.
Usman, U., Shamsuddeen, S., Arkilla, B. M., & Yakubu, A. (2022). Mixture cure model for right censored survival data with Weibull exponentiated exponential distribution. Pakistan Journal of Statistics, 38(4).
Usman, U., Suleiman, S., Arkilla, B. M., & Aliyu, Y. (2021). Nadarajah-Haghighi model for survival data with long-term survivors in the presence of right censored data. Pakistan Journal of Statistics and Operation Research, 695–709.
Yakovlev, A. Y., Asselain, B., Bardou, V., Fourquet, A., Hoang, T., Rochefediere, A., et al. (1993). A simple stochastic model of tumor recurrence and its application to data on premenopausal breast cancer. Biometrie et analyse de donnees spatiotemporelles, 12, 66–82.
Yakubu, A., & Doguwa, S. I. (2017). On the properties of the Weibull-Burr iii distribution and its application to uncensored and censored survival data. CBN Journal of Applied Statistics, 8(2), 91–116.
Andrei, Y. Y., Asselain, B., et al. (1996). Stochastic models of tumor latency and their biostatistical applications (Vol. 1). World Scientific.
Berkson, J., & Gage, R. P. (1952). Survival curve for cancer patients following treatment. Journal of the American Statistical Association, 47(259), 501–515.
Boag, J. W. (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society. Series B (Methodological), 11(1), 15–53.
Cancho, V. G., & Bolfarine, H. (2001). Modeling the presence of immunes by using the exponentiated-Weibull model. Journal of Applied Statistics, 28(6), 659–671.
Cancho, V. G., Rodrigues, J., & Castro, M. de. (2011). A flexible model for survival data with a cure rate: a Bayesian approach. Journal of Applied Statistics, 38(1), 57–70.
Carrasco, J. M., Ortega, E. M., & Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics & Data Analysis, 53(2), 450–462.
Chen, M.-H., Ibrahim, J. G., & Sinha, D. (1999). A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association, 94(447), 909–919.
Coelho-Barros, E. A., Achcar, J. A., & Mazucheli, J. (2017). Cure rate models considering the burr xii distribution in the presence of covariate. Journal of Statistical Theory and Applications, 16(2), 150–164.
Corbi`ere, F., Commenges, D., Taylor, J. M., & Joly, P. (2009). A penalized likelihood approach for mixture cure models. Statistics in medicine, 28(3), 510–524.
Farewell, V. T. (1982). The use of mixture models for the analysis of survival data with long-term survivors. Biometrics, 1041–1046.
Farewell, V. T. (1986). Mixture models in survival analysis: Are they worth the risk? Canadian Journal of Statistics, 14(3), 257–262.
Gamel, J. W., McLean, I. W., & Rosenberg, S. H. (1990). Proportion cured and mean log survival time as functions of tumor size. Statistics in Medicine, 9(8), 999–1006.
Ghazali, A. K. (2018). Modeling of survival and incidence for colorectal cancer in Malaysia. Unpublished doctoral dissertation, Lancaster University.
Ghitany, M., & Maller, R. A. (1992). Asymptotic results for exponential mixture models with long-term survivors. Statistics: A Journal of Theoretical and Applied Statistics, 23(4), 321–336.
Goldman, A. I. (1984). Survivorship analysis when cure is a possibility: a Monte Carlo study. Statistics in Medicine, 3(2), 153–163.
Hassan, M. R. A., Suan, M. A. M., Soelar, S. A., Mohammed, N. S., Ismail, I., & Ahmad, F. (2016). Survival analysis and prognostic factors for colorectal cancer patients in Malaysia. Asian Pacific Journal of Cancer Prevention, 17(7), 3575 – 3581.
Herring, A. H., & Ibrahim, J. G. (2002). Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates. Biostatistics, 3(3), 387–405.
Ishaq, A., Usman, A., Tasiaˆu, M., Aliyu, Y., & Idris, F. (2017). A new Weibull Kumaraswamy distribution: Theory and applications. Nigerian Journal of Scientific Research, 16(2), 158–166.
Jones, D., Powles, R., Machin, D., & Sylvester, R. (1981). On estimating the proportion of cured patients in clinical studies. Biometrie-Praximetrie, 21, 1–11.
Kannan, N., Kundu, D., Nair, P., & Tripathi, R. C. (2010). The generalized exponential cure rate model with covariates. Journal of Applied Statistics, 37(10), 1625 aˆ 1636.
Kittrongsiri, K., Wanitsuwan, W., Prechawittayakul, P., Sangroongruangsri, S., Cairns, J., & Chaikledkaew, U. (2020). Survival analysis of colorectal cancer patients in a Thai hospital-based cancer registry. Expert review of gastroenterology & hepatology, 14(4), 291–300.
Kutal, D., & Qian, L. (2018). A non-mixture cure model for right-censored data with Fr´echet distribution. Stats, 1(1), 176–188.
Liu, H., & Shen, Y. (2009). A semiparametric regression cure model for interval-censored data. Journal of the American Statistical Association, 104(487), 1168 – 1178.
Lopes, C. M. C., & Bolfarine, H. (2012). Random effects in promotion time cure rate models. Computational statistics & data analysis, 56(1), 75–87.
Magaji, B. A., Moy, F. M., Roslani, A. C., & Law, C. W. (2014). Descriptive epidemiology of colorectal cancer in University Malaya Medical Centre, 2001 to 2010. Asian Pacific Journal of Cancer Prevention, 15(15), 6059–6064.
Magaji, B. A., Moy, F. M., Roslani, A. C., & Law, C. W. (2017). Survival rates and predictors of survival among colorectal cancer patients in a Malaysian tertiary hospital. BMC cancer, 17(1), 1 – 8.
Maller, R. A., & Zhou, X. (1996). Survival analysis with long-term survivors. Wiley New York.
Martinez, E. Z., Achcar, J. A., Ja´come, A. A., & Santos, J. S. (2013). Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data. Computer methods and programs in biomedicine, 112(3), 343–355.
Mazucheli, J., Coelho-Barros, E. A., & Achcar, J. A. (2013). The exponentiated exponential mixture and non-mixture cure rate model in the presence of covariates. Computer methods and programs in biomedicine, 112(1), 114–124.
Meeker, W. Q. (1987). Limited failure population life tests: application to integrated circuit reliability. Technometrics, 29(1), 51–65.
Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE transactions on reliability, 42(2), 299–302.
Naishadham, D., Lansdorp-Vogelaar, I., Siegel, R., Cokkinides, V., & Jemal, A. (2011). State disparities in colorectal cancer mortality patterns in the United States. Cancer Epidemiology and Prevention Biomarkers, 20(7), 1296–1302.
Ng, S., & McLachlan, G. (1998). On modifications to the long-term survival mixture model in the presence of competing risks. Journal of Statistical Computation and Simulation, 61(1-2), 77–96.
Ortega, E. M., Cordeiro, G. M., & Kattan, M. W. (2013). The log-beta Weibull regression model with application to predict recurrence of prostate cancer. Statistical Papers, 54(1), 113–132.
Pal, M., Ali, M. M., & Woo, J. (2006). Exponentiated Weibull distribution. Statistica, 66(2), 139–147.
Peng, Y., Dear, K. B., & Denham, J. (1998). A generalized f mixture model for cure rate estimation. Statistics in medicine, 17(8), 813–830.
Peto, R., Lee, P., & Paige, W. (1972). Statistical analysis of the bioassay of continuous carcinogens. British journal of cancer, 26(4), 258–261.
Ramos, P. L., Nascimento, D., & Louzada, F. (2017). The long term fr\’echet distribution: Estimation, properties and its application. arXiv preprint arXiv:1709.07593.
Salem, H. M., & Selim, M. (2014). The generalized Weibull-exponential distribution: properties and applications. International Journal of Statistics and Applications, 4(2), 102–112.
Shao, Q., & Zhou, X. (2004). A new parametric model for survival data with long-term survivors. Statistics in medicine, 23(22), 3525–3543.
Sy, J. P., & Taylor, J. M. (2000). Estimation in a Cox proportional hazards cure model. Biometrics, 56(1), 227–236.
Tsodikov, A., Ibrahim, J., & Yakovlev, A. (2003). Estimating cure rates from survival data: an alternative to two-component mixture models. Journal of the American Statistical Association, 98(464), 1063–1078.
Uddin, M., Islam, M., & Ibrahim, Q. (2006). An analytical approach to cure rate estimation based on uncensored data. Journal of Applied Sciences, 6(3), 548 –552.
Uddin, M., Sen, A., Noor, M., Islam, M., & Chowdhury, Z. (2006). An analytical approach on non-parametric estimation of cure rate based on uncensored data. Journal of Applied Sciences, 6(6), 1258–1264.
Usman, U., Shamsuddeen, S., Arkilla, B. M., & Aliyu, Y. (2020). Inferences on the Weibull exponentiated exponential distribution and applications. International Journal of Statistical Distributions and Applications, 6(1), 1-10.
Usman, U., Shamsuddeen, S., Arkilla, B. M., & Yakubu, A. (2022). Mixture cure model for right censored survival data with Weibull exponentiated exponential distribution. Pakistan Journal of Statistics, 38(4).
Usman, U., Suleiman, S., Arkilla, B. M., & Aliyu, Y. (2021). Nadarajah-Haghighi model for survival data with long-term survivors in the presence of right censored data. Pakistan Journal of Statistics and Operation Research, 695–709.
Yakovlev, A. Y., Asselain, B., Bardou, V., Fourquet, A., Hoang, T., Rochefediere, A., et al. (1993). A simple stochastic model of tumor recurrence and its application to data on premenopausal breast cancer. Biometrie et analyse de donnees spatiotemporelles, 12, 66–82.
Yakubu, A., & Doguwa, S. I. (2017). On the properties of the Weibull-Burr iii distribution and its application to uncensored and censored survival data. CBN Journal of Applied Statistics, 8(2), 91–116.
