Parametric Analysis of Renal Failure Data using the Exponentiated Odd Weibull Distribution

Kahadawala Cooray; Nonhle Channon Mdziniso

doi:10.6000/1929-6029.2018.07.03.5

Authors

Kahadawala Cooray Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA
Nonhle Channon Mdziniso Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA

DOI:

https://doi.org/10.6000/1929-6029.2018.07.03.5

Keywords:

Coverage probability, hazard function, maximum likelihood, random censoring, survival function.

Abstract

In this article, we analyze renal failure data from patients with mesangioproliferative glomerulonephritis (MPGN) which was modeled by [1] non-parametrically using the Kaplan-Meier curve. In their work, they showed that the clinical variables, large increase serum creatinine (LISC) and systolic blood pressure >160 mmHg (SBP>160), and morphological variables, benign nephrosclerosis present (BNP) and interstitial score group 5-6 (IS5-6) were part of the variables which indicated progression to end-stage renal failure (ESRF). Though survival curves associated with these variables may be difficult to model by existing parametric distributions in literature. Therefore, we introduce a four-parameter Odd Weibull extension, the exponentiated Odd Weibull (EOW) distribution which is very versatile in modeling lifetime data that its hazard function exhibits ten different hazard shapes as well as various density shapes. Basic properties of the EOW distribution are presented. In the presence of random censoring, a small simulation study is conducted to assess the coverage probabilities of the estimated parameters of the EOW distribution using the maximum likelihood method. Our results show that the EOW distribution is very convenient and reliable to analyze the MPGN data since it provides an excellent fit for the variables LISC, SBP>160, BNP, and IS5-6. Furthermore, advantages of using the EOW distribution over the Kaplan-Meier curve are discussed. Comparisons of the EOW distribution with other Weibull-related distributions are also presented.

References

Vikse BE, Bostad L, Aasarod K, Lysebo DE, and Iversen BM. Prognostic factors in mesangioproliferative glomerulonephritis. Nephrol Dial Transplant 2002; 17: 1603-1613. https://doi.org/10.1093/ndt/17.9.1603 DOI: https://doi.org/10.1093/ndt/17.9.1603

Klein JP and Moeschberger ML. Survival analysis techniques for censored and truncated data. New York: Springer-Verlag; 2003. DOI: https://doi.org/10.1007/b97377

Rinne H. The Weibull distribution. Boca Raton; Chapman and Hall, CRC; 2009.

Murthy DNP, Xie M, and Jiang R. Weibull Models. Hoboken; Wiley Interscience: 2004.

Stacy EW. A generalization of the gamma distribution. Ann Math Statist 1962; 33: 1187-1192. https://doi.org/10.1214/aoms/1177704481 DOI: https://doi.org/10.1214/aoms/1177704481

Mudholkar GS, Srivastava DK, and Freimer M. The exponentiatedWeibull family: a reanalysis of the bus-motor-failure data. Technometrics 1995; 37: 436-445. https://doi.org/10.1080/00401706.1995.10484376 DOI: https://doi.org/10.1080/00401706.1995.10484376

Cooray K. Generalization of the Weibull distribution: the odd Weibull family. Stat Modelling 2006; 6: 265-277. https://doi.org/10.1191/1471082X06st116oa DOI: https://doi.org/10.1191/1471082X06st116oa

Jiang H, Xie M, and Tang LC. On the Odd Weibull Distribution. ProcInstMechEng O J Risk Reliab 2008; 222: 583-594. https://doi.org/10.1243/1748006XJRR168 DOI: https://doi.org/10.1243/1748006XJRR168

Cooray K. Analyzing grouped, censored, and truncated data using the Odd Weibull family. Commun Stat Theory Methods 2012; 41: 2661-2680. https://doi.org/10.1080/03610926.2011.556294 DOI: https://doi.org/10.1080/03610926.2011.556294

Johnson NL, Kotz S and Balakrishnan N. Continuous univariate distributions. New York: Wiley; 1994.

Choudhury A. A simple derivation of moments of the exponentiatedWeibull distribution. Metrika 2005; 62: 17-22. https://doi.org/10.1007/s001840400351 DOI: https://doi.org/10.1007/s001840400351

Cooray K. A study of moments and likelihood estimators of the odd Weibull distribution. Stat Methodol 2015; 26: 72-83. https://doi.org/10.1016/j.stamet.2015.03.003 DOI: https://doi.org/10.1016/j.stamet.2015.03.003

Cox C and Matheson M. A comparison of the generalized gamma and exponentiatedWeibull distributions. Stat Med 2014; 33: 3772-3780. https://doi.org/10.1002/sim.6159 DOI: https://doi.org/10.1002/sim.6159

Lee C, Famoye F, and Olumolade O. Beta-Weibull distribution: some properties and applications to censored data. J Mod Appl Stat Methods 2007; 6: 173-186. https://doi.org/10.22237/jmasm/1177992960 DOI: https://doi.org/10.22237/jmasm/1177992960

Marshall AW and Olkin I. A new method of adding a parameter to a family of distributions with applications to the exponential and Weibull families. Biometrika 1997; 84: 641-652. https://doi.org/10.1093/biomet/84.3.641 DOI: https://doi.org/10.1093/biomet/84.3.641

Guilbaud O. Exact Kolmogorov-type tests for left-truncated and (or) right-censored data. J Am Stat Assoc 1988; 83: 213-221. https://doi.org/10.1080/01621459.1988.10478589 DOI: https://doi.org/10.1080/01621459.1988.10478589

Rich JT, et al. A practical guide to understanding Kaplan-Meier curves. Otolaryngol Head Neck Surg 2010; 143: 331-336. https://doi.org/10.1016/j.otohns.2010.05.007 DOI: https://doi.org/10.1016/j.otohns.2010.05.007