Some Useful Properties of Log-Logistic Random Variables for Health Care Simulations
DOI:
https://doi.org/10.6000/1929-6029.2015.04.01.9Keywords:
Log-logistic distribution, Log-normal distribution, Mean residual time, Median residual time, Simulation.Abstract
A log-logistic (LL) random variable is one whose logarithm has a logistic distribution. Since the logistic distribution is similar to the normal distribution, log-logistic random variables are similar to log-normal (LN) random variables. However, many of the important properties of LN random variables can only be described using integrals, while the corresponding properties of LL random variables can be described using simple algebra. LL random variables may therefore be a useful alternative to LN random variable for computer simulation of operating room processes or other health care applications, especially when they fit the data more closely. We review the properties of LL random variables, and derive some relationships of the mean residual time to the median residual time. We describe methods of fitting LL distributions to observed data, and discuss potential advantages of using them for simulation of operating room utilization.
References
Denton B, Viapiano J, Vogl A. Optimization of surgery sequencing and scheduling decisions under uncertainty. Health Care Manag Sci 2007; 10(1): 13-24. http://dx.doi.org/10.1007/s10729-006-9005-4 DOI: https://doi.org/10.1007/s10729-006-9005-4
Cardoen B, Demeulemeester E, Belien J. Operating room planning and scheduling: A literature review. Eur J Operational Res 2010; 201(3): 921-932. http://dx.doi.org/10.1016/j.ejor.2009.04.011 DOI: https://doi.org/10.1016/j.ejor.2009.04.011
Dexter F, Traub RD. How to schedule elective surgical cases into specific operating rooms to maximize the efficiency of use of operating room time. Anesth Analg 2002; 94(4): 933-942. http://dx.doi.org/10.1097/00000539-200204000-00030 DOI: https://doi.org/10.1097/00000539-200204000-00030
Ledolter J, Dexter F, Epstein RH. Analysis of variance of communication latencies in anesthesia: Comparing means of multiple log-normal distributions. Anesth Analg 2011; 113(4): 888-896. http://dx.doi.org/10.1213/ANE.0b013e318227518f DOI: https://doi.org/10.1213/ANE.0b013e318227518f
Strum DP, May JH, Vargas LG. Modeling the uncertainty of surgical procedure times: Comparison of log-normal and normal models. Anesthesiology 2000; 92(4): 1160-1167. http://dx.doi.org/10.1097/00000542-200004000-00035 DOI: https://doi.org/10.1097/00000542-200004000-00035
Fisk PR. The graduation of income distributions. Econometrica 1961; 29(2): 171-185. http://dx.doi.org/10.2307/1909287 DOI: https://doi.org/10.2307/1909287
Kleiber C, Kotz S. Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken NJ: Wiley-Interscience; 2003. DOI: https://doi.org/10.1002/0471457175
Balakrishnan N (ed.). Handbook of the Logistic Distribution. New York: Marcel Dekker, Inc.; 1992. DOI: https://doi.org/10.1201/9781482277098
Hosmer DW, Jr., Lemeshow S. Applied Logistic Regression, 2nd Edition. New York: John Wiley & Sons; 2000. DOI: https://doi.org/10.1002/0471722146
Ashkar F, Mahdi S. Fitting the log-logistic distribution by generalized moments. J Hydrol 2006; 328: 694-703. http://dx.doi.org/10.1016/j.jhydrol.2006.01.014 DOI: https://doi.org/10.1016/j.jhydrol.2006.01.014
Ragab A, Green J. Estimation of the parameters of the loglogistic distribution based on order statistics. Am J Mathematical and Management Sci 1987; 7: 307-323. DOI: https://doi.org/10.1080/01966324.1987.10737224
Cleves MA, Gould WW, Gutierrez RG, Marchenko YU. An Introduction to Survival Analysis using Stata. College Station, TX: StataCorp; 2008.
Collett D. Modelling Survival Data in Medical Research. London: Chapman & Hall; 1994. DOI: https://doi.org/10.1007/978-1-4899-3115-3
Dexter F, Epstein RH, Lee JD, Ledolter J. Automatic updating of times remaining in surgical cases using Bayesian analysis of historical case duration data and "instant messaging" updates from anesthesia providers. Anesth Analg 2009; 108(3): 929-940. http://dx.doi.org/10.1213/ane.0b013e3181921c37 DOI: https://doi.org/10.1213/ane.0b013e3181921c37
Finkelstein MS. On the shape of the mean residual lifetime function. Appl Stochastic Models in Business and Industry 2002; 18: 135-146. http://dx.doi.org/10.1002/asmb.461 DOI: https://doi.org/10.1002/asmb.461
Lillo RE. On the median residual lifetime and its aging properties: A characterization theorem and applications. Naval Res Logistics 2005; 52: 370-380. http://dx.doi.org/10.1002/nav.20082 DOI: https://doi.org/10.1002/nav.20082
Gupta RC, Akman O, Lvin S. A study of log-logistic model in survival analysis. Biometrical J 1999; 41(4): 431-443. http://dx.doi.org/10.1002/(SICI)1521-4036(199907)41:4<431::AID-BIMJ431>3.0.CO;2-U DOI: https://doi.org/10.1002/(SICI)1521-4036(199907)41:4<431::AID-BIMJ431>3.0.CO;2-U
Abramowitz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover; 1965.
Ross S. A First Course in Probability. Englewood Cliffs NJ: Prentice Hall; 1994.
Schmittlein DC, Morrison DG. The median residual lifetime: A characterization theorem and an application. Operations Res 1981; 29(2): 392-399. http://dx.doi.org/10.1287/opre.29.2.392 DOI: https://doi.org/10.1287/opre.29.2.392
Gupta RC, Akman O. Mean residual life functions for certain types of nonmonotonic ageing. Commun Statist - Stochastic Models 1995; 11: 219-225. http://dx.doi.org/10.1080/15326349508807340 DOI: https://doi.org/10.1080/15326349508807340
Bennett S. Log-logistic regression models for survival data. Appl Statist 1983; 32(2): 165-171. http://dx.doi.org/10.2307/2347295 DOI: https://doi.org/10.2307/2347295
von Hippel PT. Mean, median, and skew: Correcting a textbook rule. J Stat Educ 2005; 13(2): 1-16. DOI: https://doi.org/10.1080/10691898.2005.11910556
Dey AK, Kundu D. Discriminating between the log-normal and log-logistic distributions. Commun Statist - Theor Meth 2010; 39: 280-292. DOI: https://doi.org/10.1080/03610920902737100
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 David E. Clark, Muhammad El-Taha
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Policy for Journals/Articles with Open Access
Authors who publish with this journal agree to the following terms:
- 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 permitted and encouraged to post links to 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
Policy for Journals / Manuscript with Paid Access
Authors who publish with this journal agree to the following terms:
- Publisher retain copyright .
- Authors are permitted and encouraged to post links to 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 .
