Discrete Bilal Distribution in the Presence of Right-Censored Data and a Cure Fraction
Abstract
The statistical literature presents many continuous probability distributions with only one parameter, which are extensively used in the analysis of lifetime data, such as the exponential, the Lindley, and the Rayleigh distributions. Alternatively, the use of discretized versions of these distributions can provide a better fit for the data in many applications. As the novelty of this study, we present inferences for the discrete Bilal distribution (DB) with one parameter introduced by Altun et al. (2020) in the presence of right-censored data and cure fraction. We assume standard maximum likelihood methods based on asymptotic normality of the maximum likelihood estimators and also a Bayesian approach based on MCMC (Markov Chain Monte Carlo) simulation methods to get inferences for the parameters of the discrete BD distribution. The use of the proposed model was illustrated with three examples considering real medical lifetime data sets. From these applications, we concluded that the proposed model based on the discrete DB distribution has good performance even with the inclusion of a cure fraction in comparison to other existing discrete models, such as the DsFx-I, Lindley, Rayleigh, and Burr-Hatke probability distributions. Moreover, the model can be easily implemented in standard existing software, such as the R package. Under a Bayesian approach, we assumed a gamma prior distribution for the parameter of the DB discrete distribution. We also provided a brief sensitivity analysis assuming the half-normal distribution in place of the gamma distribution for the parameter of the DB distribution. From the obtained results of this study, we can conclude that the proposed methodology can be very useful for researchers dealing with medical discrete lifetime data in the presence of right-censored data and cure fraction.References
A. M. Abd-Elrahman, Utilizing ordered statistics in lifetime distributions production: A new lifetime distribution and applications, Journal of Probability and Statistical Science, vol. 11, no. 2, pp. 153–164, 2013.
A. M. Abd-Elrahman, Reliability estimation under type-II censored data from the generalized Bilal distribution, Journal of the Egyptian Mathematical Society, vol. 27, no. 1, pp. 1–15, 2019.
A. M. Abd-Elrahman, and S. F. Niazi, Approximate Bayes estimators applied to the Bilal model,
Journal of the Egyptian Mathematical Society, vol. 25, no. 1, pp. 65–70, 2017.
Z. Akhter, E. M. Almetwally, and C. Chesneau, On the generalized Bilal distribution: some properties and estimation under ranked set sampling, Axioms, vol. 11, no. 4, pp. 173, 2022.
M. H. Alamatsaz, S. Dey, T. Dey, and S. S. Harandi, Discrete generalized Rayleigh distribution, Pakistan Journal of Statistics, vol. 32, no. 1, pp. 1–20, 2016.
E. Altun, M. El-Morshedy, and M. S. Eliwa, A study on discrete Bilal distribution with properties and applications on integer-valued autoregressive process, Revstat Statistical Journal, vol. 18, pp. 70–99, 2020.
M. Amico, and I. Van Keilegom, Cure models in survival analysis, Annual Review of Statistics and Its Application, vol. 5, pp. 311–342, 2018.
M. R. P. Cardial, J. B. Fachini-Gomes, and E. Y. Nakano, Exponentiated discrete Weibull distribution for censored data, Brazilian Journal of Biometrics, vol. 38, no. 1, pp. 35–56, 2020.
S. Chakraborty, Generating discrete analogues of continuous probability distributions - A survey of methods and constructions, Journal of Statistical Distributions and Applications, vol. 2, no. 1, pp. 1–30, 2015.
S. Chib, and E. Chakraborty, Understanding the Metropolis-Hastings algorithm, The American Statistician, vol. 49, no. 4, pp. 327–335, 1995.
F. Corbière, D. Commenges, J. M. Taylor, and P. Joly, A penalized likelihood approach for mixture cure models, Statistics in Medicine, vol. 28, no. 3, pp. 510–524, 2009.
M. K. Cowles, and B. P. Carlin, Markov Chain Monte Carlo convergence diagnostics: A comparative review, Journal of American Statistical Association, vol. 91, no. 434, pp. 883–904, 1994.
D. Drevon, S. R. Fursa, and A. L. Malcolm, Intercoder reliability and validity of WebPlotDigitizer in extracting graphed data, Behavior Modification, vol. 41, no. 3, pp. 323–339, 2017.
P. K. Dunn, and G. K. Smyth, Randomized quantile residuals, Journal of Computational and Graphical Statistics, vol. 5, no. 3, pp. 236–244, 1996.
B. Efron, Censored data and the bootstrap, Journal of the American Statistical Association, vol. 76, no. 374, pp. 312–319, 1981.
M.S.Eliwa, and M.El-Morshedy, Aone-parameter discrete distribution for over-dispersed data:Statistical and reliability properties with applications, Journal of Applied Statistics, pp. 1–21, 2021.
M. El-Morshedy, M. S. Eliwa, and E. Altun, Discrete Burr-Hatke distribution with properties, estimation methods and regression model, IEEE Access, vol. 8, pp. 74359–74370, 2020.
V. T. Farewell, The use of mixture models for the analysis of survival data with long-term survivors, Biometrics, vol. 38, no. 4, pp. 1041–1046, 1982.
R. Flynn, Survival analysis, Journal of Clinical Nursing, vol. 21, pp. 2789–2797, 2012.
E. J. Freireich, E. Gehan, E. Frei-3rd, L. R. Schroeder, I. J. Wolman, R. Anbari, E. O. Burgert, S. D. Mills, D. Pinkel, O. S. Selawry, J. H. Moon, B. R. Gendel, C. L. Spurr, R. Storrs, F. Haurani, B. Hoogstraten, and S. Lee, The effect of 6-mercaptopurine on the duration of steroid induced remissions in acute leukemia - A model for evaluation of other potentially useful therapy Blood, vol. 21, no. 6, pp. 699–716, 1963.
B. C. L. Freitas, M. V. Oliveira-Peres, J. A. Achcar, and E. Z. Martinez, Classical and Bayesian inference approaches for th e exponentiated discrete Weibull model with censored data and a cure fraction, Pakistan Journal of Statistics and Operation Research, vol. 17, no. 2, pp. 467–481, 2021.
B. C. L. Freitas, M. V. Oliveira-Peres, J. A. Achcar, and E. Z. Martinez, Bayesian and maximum likelihood inference approaches for the discrete generalized Sibuya distribution with censored datan, Electronic Journal of Applied Statistical Analysis, vol. 15, no. 1, pp. 50–74, 2022.
D. I. Gallardo, M. Castro, and H. W. Gómez, An alternative promotion time cure model with overdispersed number of competing causes: an application to melanoma data, Mathematics, vol. 9, no. 15, pp. 1815, 2021.
A. E. Gelfand, and A. F. Smith, Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, vol. 85, no. 410, pp. 398–409, 1990.
A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian data analysis, 3rd ed., Chapman Hall, 2013.
J. Geweke, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, Bayesian Statistics, vol. 4, pp. 641–649, 1992.
E. Gómez-Déniz, and E. Calder´ın-Ojeda, The discrete Lindley distribution: properties and applications, Journal of Statistical Computation and Simulation, vol. 81, no. 11, pp. 1405–1416, 2011.
A. Henningsen, and O. Toomet, maxLik: A package for maximum likelihood estimation in R, Computational Statistics, vol. 26, no. 3, pp. 443–458, 2011.
C. M. Hurvich, and C. L. Tsai, Regression and time series model selection in small samples, Biometrika, vol. 76, no. 2, pp. 297–307, 1989.
P. C. Lambert, Modeling of the cure fraction in survival studies, The Stata Journal, vol. 7, no. 3, pp. 351–375, 2007.
J. Le Rademacher, and X. Wang, Time-to-event data: An overview and analysis considerations, Journal of Thoracic Oncology, vol. 16, no. 7, pp. 1067–1074, 2021.
J. Leao, M. Bourguignon, D. I. Gallardo, R. Rocha, and V. Tomazella, A new cure rate model with flexible competing causes with applications to melanoma and transplantation data, Statistics in Medicine, vol. 39, no. 24, pp. 3272–3284, 2020.
R. A. Maller, and X. Zhou, Survival analysis with long-term survivors, John Wiley & Sons, New York, 1996.
A. D. Martin, and K. M. Quinn, Applied Bayesian inference in R using MCMCpack, R News, vol. 6, pp. 2–7, 2006.
E. Z. Martinez, J. A. Achcar, A. A. Jácome, and J. S. Santos, 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, vol. 112, no. 3, pp. 343–355, 2013.
J. Myers, Survival analysis techniques in clinical research, The Journal of the Kentucky Medical Association, vol. 105, no. 11, pp. 545–550, 2007.
T. Nakagawa, and S. Osaki, The discrete Weibull distribution, IEEE Transactions on Reliability, vol. 24, no. 5, pp. 300-301, 1975.
V. Nekoukhou, and H. Bidram, The exponentiated discrete Weibull distribution, SORT Statistics and Operations Research Transactions, vol. 39, pp. 127–146, 2015.
R. P. Oliveira, M. V. Oliveira-Peres, E. Z. Martinez, and J. A. Achcar, Use of a discrete Sushila distribution in the analysis of right-censored lifetime data, Model Assisted Statistics and Applications, vol. 14, no. 3, pp. 255–2681, 2019.
M. Othus, B. Barlogie, M. L. LeBlanc, and J. J. Crowley, Cure models as a useful statistical tool for analyzing survival, Clinical Cancer Research, vol. 18, no. 14, pp. 3731–3736, 2012.
I. Paranjpe, V. Fuster, A. Lala, A. J. Russak, B. S. Glicksberg, M. A. Levin, A. W. Charney, J. Narula, Z. A. Fayad, E. Bagiella, S. Zhao, and G. N. Nadkarni, Association of treatment dose anticoagulation with in-hospital survival among hospitalized patients with COVID-19, Journal of the American College of Cardiology, vol. 76, no. 1, pp. 122–124, 2020.
M.Pedrosa-Laza,A.Lpez-Cheda,R.Cao, Cure models to estimate time until hospitalization due to COVID-19, Applied Intelligence, vol. 52, no. 1, pp. 794-807, 2022.
Y. Peng, and B. Yu, Cure models: Methods, applications, and implementation, CRC Press, 2021.
S. Rafati, M. R. Baneshi, and A. Bahrampour, Factors affecting long-survival of patients with breast cancer by non-mixture and mixture cure models using the Weibull, log-logistic and Dagum distributions: a Bayesian approach, Asian Pacific Journal of Cancer Prevention: APJCP, vol. 21, no. 2, pp. 485, 2020.
F. H. Riad, B. Alruwaili, A. M. Gemeay, and E. Hussam, Statistical modeling for COVID-19 virus spread in Kingdom of Saudi Arabia and Netherlands, Alexandria Engineering Journal, vol. 61, no. 12, pp. 9849-9866, 2022.
A. Rohatgi, WebPlotDigitizer version 4.4, Available from: https://automeris.io/WebPlotDigitizer/.
D. Roy, Discrete Rayleigh distribution, IEEE Transactions on Reliability, vol. 53, no. 2, pp. 255–260, 2004.
X. Shi, Y. Shi, and K. Zhou, Estimation for entropy and parameters of generalized Bilal distribution under adaptive type II progressive hybrid censoring scheme, Entropy, vol. 23, no. 2, pp. 206, 2021.
B. Wang, X. Xie, J. Yin, C. Zou, J. Wang, G. Huang, Y. Wang, and J. Shen, Reconstruction with modular hemipelvic endoprosthesis after pelvic tumor resection: a report of 50 consecutive cases, PLoS One, vol. 10, no. 5, pp. e0127263, 2015.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
