Published

2017-01-16

Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring

Bandas de confianza para la función de supervivencia usando la ONU modelo de regresión de Weibull en presencia de censura arbitraria

DOI:

https://doi.org/10.15446/rce.v40n1.55807

Keywords:

Survival analysis, Biostatistical, Confidence bands, Goodness of fit, Regression models, Simulation (en)
análisis de supervivencia, bandas de confianza, bioestadística, modelos de regresión, simulación. (es)

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Authors

  • Mario César Jaramilo Elorza Statistics School, Universidad Nacional de Colombia, Medellín, Colombia
  • Juan Carlos Salazar Uribe Statistics School, Universidad Nacional de Colombia, Medellín, Colombia
Usually, the exact time at which an event occurs cannot be observed for several reasons; for instance, it is not possible to constantly monitor  a characteristic of interest. This generates a phenomenon known as censoring that can be classified as having a left censor, right censor or interval censor. When one is working with survival data in the presence of arbitrary censoring, the survival time of interest is defined as the elapsed time between an initial event and the next event that is generally unknown. This problem has been widely studied in the statistic literature and some progress has been made, toward resolving and the formulation of a bivariate likelihood to estimate parameters in a parametric regression model offers positive development opportunities. In this paper, we construct a bivariate likelihood for the Weibull regression model in the presence of interval censoring. Finally, its performance is illustrated by means of a simulation study.

Usualmente, el tiempo exacto en el que ocurre un evento no se puede
observar por diversas razones; por ejemplo, no es posible un monitoreo constante de las características de interés. Esto genera un fenómeno conocido como censura que puede ser de tres tipos: a izquierda, a derecha, o de intervalo. En datos de tiempo de vida con censura arbitraria (censura a izquierda, a derecha, o de intervalo), el tiempo de supervivencia de interés es definido como el lapso de tiempo entre un evento inicial y el evento siguiente, el cuál generalmente es desconocido. Este problema ha sido ampliamente estudiado en la literatura estadística, y se evidencian avances importantes. Sin embargo, la construcción de una verosimilitud bivariada para la estimación de los parámetros de modelos de regresión paramétricos, ofrece oportunidades de desarrollo. En este trabajo se construye una verosimilitud bivariada para el modelo de regresión Weibull, en presencia de censura arbitraria. Finalmente se ilustra su desempeño por medio de un estudio de simulación.

https://doi.org/10.15446/rce.v40n1.55807

Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring

Bandas de confianza para la función de supervivencia usando la ONU modelo de regresión de Weibull en presencia de censura arbitraria

MARIO CÉSAR JARAMILLO ELORZA1, JUAN CARLOS SALAZAR URIBE2

1Universidad Nacional de Colombia, Statistics School, Medellín, Colombia. PhD. Email: mcjarami@unal.edu.co
2Universidad Nacional de Colombia, Statistics School, Medellín, Colombia. PhD. Email: jcsalaza@unal.edu.co

Abstract

Usually, the exact time at which an event occurs cannot be observed for several reasons; for instance, it is not possible to constantly monitor a characteristic of interest. This generates a phenomenon known as censoring that can be classified as having a left censor, right censor or interval censor. When one is working with survival data in the presence of arbitrary censoring, the survival time of interest is defined as the elapsed time between an initial event and the next event that is generally unknown. This problem has been widely studied in the statistic literature and some progress has been made, toward resolving and the formulation of a bivariate likelihood to estimate parameters in a parametric regression model offers positive development opportunities. In this paper, we construct a bivariate likelihood for the Weibull regression model in the presence of interval censoring. Finally, its performance is illustrated by means of a simulation study.

Key words: Biostatistics, Confidence Bands, Goodness of Fit, Regression Models, Simulation, Survival Analysis.

Resumen

Usualmente, el tiempo exacto en el que ocurre un evento no se puede observar por diversas razones; por ejemplo, no es posible un monitoreo constante de las características de interés. Esto genera un fenómeno conocido como censura que puede ser de tres tipos: a izquierda, a derecha, o de intervalo. En datos de tiempo de vida con censura arbitraria (censura a izquierda, a derecha, o de intervalo), el tiempo de supervivencia de interés es definido como el lapso de tiempo entre un evento inicial y el evento siguiente, el cuál generalmente es desconocido. Este problema ha sido ampliamente estudiado en la literatura estadística, y se evidencian avances importantes. Sin embargo, la construcción de una verosimilitud bivariada para la estimación de los parámetros de modelos de regresión paramétricos, ofrece oportunidades de desarrollo. En este trabajo se construye una verosimilitud bivariada para el modelo de regresión Weibull, en presencia de censura arbitraria. Finalmente se ilustra su desempeño por medio de un estudio de simulación.

Palabras clave: análisis de supervivencia, bandas de confianza, bioestadística, modelos de regresión, simulación.

Texto completo disponible en PDF

References

1. Allison, P. D. (1995), Survival Analysis Using the SAS System: A Practical Guide, Springer-Verlag, New York.

2. Chang, C. H. & Weissfeld, L. A. (1999), 'Normal aproximation diagnostics for the Cox model', Biometrics 55, 1114-1119.

3. Cheng, R. & Iles, T. (1983), 'Confidence bands for cumulative distribution functions of continuous random variables', Technometrics 25(1), 77-86.

4. Cheng, R. & Iles, T. (1988), 'One-sided confidence bands for cumulative distribution functions', Technometrics 30(1), 155-159.

5. Dorey, F. J., Little, R. & Schenker, N. (1993), 'Multiple imputation for threshold-crossing data with interval censoring', Statistics in Medicine 12, 1589-1603.

6. Efron, B. (1967), The two sample problem with censored data, University of California Press.

7. Escobar, L. A., Hong, Y. & Meeker, W. Q. (2009), 'Simultaneous confidence bands and regions for log-location-scale distributions with censored data', Journal of Statistical Planning and Inference 139(9), 3231-3245.

8. Gentleman, R. & Vandal, A. C. (2001), 'Computational algorithms for censored-data problems using intersection graphs', Journal of Computational and Graphical Statistics 10, 403-421.

9. Jeng, S. & Meeker, W. Q. (2001), 'Parametric simultaneous confidence bands for cumulative distributions from censored data', Technometrics 43(4), 450-461.

10. Joly, P. & Commenges, D. (1999), 'A penalized likelihood approach for a progressive three-state model with censored and truncated data: Application to AIDS', Biometrics 55, 887-890.

11. Kaplan, E. L. & Meier, P. (1958), 'Nonparametric estimation from incomplete observations', Journal of the American statistical association 53, 457-481.

12. Lawless, J. & Babineau, D. (2006), 'Models for interval censoring and simulation-based inference for lifetime distributions', Biometrika 93, 671-686.

13. Meeker, W. & Escobar, L. (1992), 'Assessing influence in regression analysis with censored data', Biometrics 48, 507-528.

14. Nair, V. N. (1984), 'Confidence bands for survival functions with censored data: A comparative study', Technometrics 46(3), 265-275.

15. Nelder, J. & Mead, R. (1965), 'A simplex method for function minimization', Computer Journal 7, 308-313.

16. Odell, P., Anderson, K. & D'Agostinho, R. (1992), 'Maximum likelihood estimation for interval censored data using a Weibull based accelerated failure time model', Biometrics 48, 951-959.

17. Peto, R. (1973), 'Experimental survival curves for interval-censored data', Journal of the Royal Statistical Society, Series C 22, 86-91.

18. Rojas, A., Diaz, F. J., Calvo, E., Salazar, J. C., Iglesias, A., Mantilla, R. D. & Anaya, J. M. (2009), 'Familial disease, the HLA-DRB1 shared epitope and anti-CCP antibodies influence time at appearance of substantial joint damage in rheumatoid arthritis', Journal of Autoimmunity 32, 64-69.

19. Rosales, L. F. & Salazar, J. C. (2006), Estimaciones de funciones de intensidad en un modelo de 3 estados en presencia de doble censura, Master's in Statistics, Universidad Nacional De Colombia, Sede Medellín.

20. Rucker, G. & Messerer, D. (1988), 'Remission duration: an example of interval-censored observation', Statistics in Medicine 7, 1139-1145.

21. Turnbull, B. W. (1974), 'Nonparametric estimation of a survivorship function with doubly censored data', Journal of the American statistical association 69, 169-173.

22. Turnbull, B. W. (1976), 'The empirical distribution function with arbitrarily grouped censored and truncated data', Journal of the Royal Statistical Society, Series B 38, 290-295.

23. Waller, L. A. & Turnbull, B. W. (1992), 'Probability Plotting with censored data', The American Statistician 46, 5-12.

[Recibido en abril de 2015. Aceptado en febrero de 2016]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv40n1a04,
    AUTHOR  = {Jaramillo Elorza, Mario César and Salazar Uribe, Juan Carlos},
    TITLE   = {{Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2017},
    volume  = {40},
    number  = {1},
    pages   = {85-103}
}

References

Allison, P. D. (1995), Survival Analysis Using the SAS System: A Practical Guide, Springer-Verlag, New York.

Chang, C. H. & Weissfeld, L. A. (1999), ‘Normal aproximation diagnostics for the Cox model’, Biometrics 55, 1114–1119.

Cheng, R. & Iles, T. (1983), ‘Confidence bands for cumulative distribution functions of continuous random variables’, Technometrics 25(1), 77–86.

Cheng, R. & Iles, T. (1988), ‘One-sided confidence bands for cumulative distribution functions’, Technometrics 30(1), 155–159.

Dorey, F. J., Little, R. & Schenker, N. (1993), ‘Multiple imputation for thresholdcrossing data with interval censoring’, Statistics in Medicine 12, 1589–1603.

Efron, B. (1967), The two sample problem with censored data, Technical report, University of California Press.

Escobar, L. A., Hong, Y. & Meeker, W. Q. (2009), ‘Simultaneous confidence bands and regions for log-location-scale distributions with censored data’, Journal of Statistical Planning and Inference 139(9), 3231–3245.

Gentleman, R. & Vandal, A. C. (2001), ‘Computational algorithms for censoreddata problems using intersection graphs’, Journal of Computational and Graphical Statistics 10, 403–421.

Jeng, S. & Meeker, W. O. (2001), ‘Parametric simultaneous confidence bands for cumulative distributions from censored data’, Technometrics 43(4), 450–461.

Joly, P. & Commenges, D. (1999), ‘A penalized likelihood approach for a progressive three-state model with censored and truncated data: Application to AIDS’, Biometrics 55, 887–890.

Kaplan, E. L. & Meier, P. (1958), ‘Nonparametric estimation from incomplete observations’, Journal of the American statistical association 53, 457–481.

Lawless, J. & Babineau, D. (2006), ‘Models for interval censoring and simulationbased inference for lifetime distributions’, Biometrika 93, 671–686.

Meeker, W. & Escobar, L. (1992), ‘Assessing influence in regression analysis with censored data’, Biometrics 48, 507–528.

Nair, V. N. (1984), ‘Confidence bands for survival functions with censored data: A comparative study’, Technometrics 46(3), 265–275.

Nelder, J. & Mead, R. (1965), ‘A simplex method for function minimization’, Computer Journal 7, 308–313.

Odell, P., Anderson, K. & D’Agostinho, R. (1992), ‘Maximum likelihood estimation for interval censored data using a Weibull based accelerated failure time model’, Biometrics 48, 951–959.

Peto, R. (1973), ‘Experimental survival curves for interval-censored data’, Journal of the Royal Statistical Society, Series C 22, 86–91.

Rojas, A., Diaz, F. J., Calvo, E., Salazar, J. C., Iglesias, A., Mantilla, R. D. & Anaya, J. M. (2009), ‘Familial disease, the HLA-DRB1 shared epitope and anti-CCP antibodies influence time at appearance of substantial joint damage in rheumatoid arthritis’, Journal of Autoimmunity 32, 64–69.

Rosales, L. F. & Salazar, J. C. (2006), Estimaciones de funciones de intensidad en un modelo de 3 estados en presencia de doble censura, Master’s in statistics, Universidad Nacional De Colombia, Sede Medellín.

Rucker, G. & Messerer, D. (1988), ‘Remission duration: an example of intervalcensored observation’, Statistics in Medicine 7, 1139–1145.

Turnbull, B. W. (1974), ‘Nonparametric estimation of a survivorship function with doubly censored data’, Journal of the American statistical association 69, 169–173.

Turnbull, B. W. (1976), ‘The empirical distribution function with arbitrarily grouped censored and truncated data’, Journal of the Royal Statistical Society, Series B 38, 290–295.

Waller, L. A. & Turnbull, B. W. (1992), ‘Probability Plotting with censored data’, The American Statistician 46, 5–12.

How to Cite

APA

Jaramilo Elorza, M. C. and Salazar Uribe, J. C. (2017). Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring. Revista Colombiana de Estadística, 40(1), 85–103. https://doi.org/10.15446/rce.v40n1.55807

ACM

[1]
Jaramilo Elorza, M.C. and Salazar Uribe, J.C. 2017. Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring. Revista Colombiana de Estadística. 40, 1 (Jan. 2017), 85–103. DOI:https://doi.org/10.15446/rce.v40n1.55807.

ACS

(1)
Jaramilo Elorza, M. C.; Salazar Uribe, J. C. Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring. Rev. colomb. estad. 2017, 40, 85-103.

ABNT

JARAMILO ELORZA, M. C.; SALAZAR URIBE, J. C. Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring. Revista Colombiana de Estadística, [S. l.], v. 40, n. 1, p. 85–103, 2017. DOI: 10.15446/rce.v40n1.55807. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/55807. Acesso em: 28 mar. 2025.

Chicago

Jaramilo Elorza, Mario César, and Juan Carlos Salazar Uribe. 2017. “Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring”. Revista Colombiana De Estadística 40 (1):85-103. https://doi.org/10.15446/rce.v40n1.55807.

Harvard

Jaramilo Elorza, M. C. and Salazar Uribe, J. C. (2017) “Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring”, Revista Colombiana de Estadística, 40(1), pp. 85–103. doi: 10.15446/rce.v40n1.55807.

IEEE

[1]
M. C. Jaramilo Elorza and J. C. Salazar Uribe, “Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring”, Rev. colomb. estad., vol. 40, no. 1, pp. 85–103, Jan. 2017.

MLA

Jaramilo Elorza, M. C., and J. C. Salazar Uribe. “Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring”. Revista Colombiana de Estadística, vol. 40, no. 1, Jan. 2017, pp. 85-103, doi:10.15446/rce.v40n1.55807.

Turabian

Jaramilo Elorza, Mario César, and Juan Carlos Salazar Uribe. “Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring”. Revista Colombiana de Estadística 40, no. 1 (January 1, 2017): 85–103. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/55807.

Vancouver

1.
Jaramilo Elorza MC, Salazar Uribe JC. Confidence Bands for the Survival Function Using a Weibull Regression Model in Presence of Arbitrary Censoring. Rev. colomb. estad. [Internet]. 2017 Jan. 1 [cited 2025 Mar. 28];40(1):85-103. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/55807

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CrossRef citations1

1. Sundarraman Subramanian. (2022). Simultaneous confidence bands for survival functions from twice censorship. Statistics & Probability Letters, 186, p.109494. https://doi.org/10.1016/j.spl.2022.109494.

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