Abstract
We propose the Binary Geometric Process (BGP) model for longitudinal binary data with trends. The Geometric Process (GP) model contains two components to capture the dynamics on a trend: the mean of an underlying renewal process and the ratio which measures the direction and strength of the trend. The GP model is extended to binary data using a latent GP. The statistical inference for the BGP models is conducted using the least-square, maximum likelihood (ML) and Bayesian methods. The model is demonstrated through simulation studies and real data analyzes. Results reveal that all estimators perform satisfactorily and that the ML estimator performs the best. Moreover the BGP model is better than the ordinary logistic regression model.
Article PDF
Similar content being viewed by others
Explore related subjects
Find the latest articles, discoveries, and news in related topics.Avoid common mistakes on your manuscript.
References
Andrews DF, Herzberg AM (1985) Data. Springer, New York
Ascher H, Feingold H (1981) Repairable systems reliability. Marcel Dekker, New York
Berman M (1981) Inhomogeneous and modulated gamma processes. Biometrika 68: 143–152
Berman M, Turner TR (1992) Approximate point process likelihoods with GLIM. Appl Stat 41: 31–38
Chan JSK, Kuk AYC, Bell J (1997) A Likelihood approach to analysing longitudinal bivariate binary data. Biom J 39(4): 409–421
Chan JSK, Kuk AYC, Bell J, McGilchrist C (1998) The analysis of methadone clinic data using marginal and conditional logistic models with mixture or random effects. Aust N Z J Stat 40: 1–10
Chan JSK, Lam Y, Leung DYP (2004) Statistical inference for geometric processes with gamma distributions. Comput Stat Data Anal 47: 565–581
Chan JSK, Yu PLH, Lam Y, Ho APK (2006) Modeling SARS data using threshold geometric process. Stat Med 25: 1826–1839
Chan JSK, Lam CPY, Yu PLH, Choy STB (2010a) Bayesian conditional autoregressive geometric process model for range data. submitted to journal
Chan JSK, Wan WY, Lee CK, Lin CK, Yu PLH (2010b) Predicting drop-out and committed first time blood donors: the use of Poisson GP model. submitted to journal
Cox DR, Lewis PA (1966) The statistical analysis of series of events. Mathuen, London
Feller W (1949) Fluctuation theory of recurrent events. Trans Am Math Soc 67: 98–119
Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov chain Monte Carlo in practice. Chapman and Hall, UK
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97–109
Lakey MJ, Rigdon SE (1992) The modulated power law process. In: Proceedings of the 45th annual quality congress, pp 559–563
Lam Y (1988a) A note on the optimal replacement problem. Adv Appl Probab 20: 479–482
Lam Y (1988b) Geometric process and replacement problem. Acta Math Appl Sin 4: 366–377
Lam Y (1992a) Optimal geometric process replacement model. Acta Math Appl Sin 8: 73–81
Lam Y (1992b) Nonparametric inference for geometric processes. Commun Stat Theory Methods 21: 2083–2105
Lam Y (2007) The geometric process and its application. World Scientific Publishing Co. Pte. Ltd, Singapore
Lam Y, Chan JSK (1998) Statistical inference for geometric processes with lognormal distribution. Comput Stat Data Anal 27: 99–112
Lam Y, Zhang YL (1996) Analysis of a two-component series system with a geometric process model. Nav Res Logistics 43: 491–502
Lam Y, Zhang YL, Zheng YH (2002) A geometric process equivalent model for a multistate degenerative system. Eur J Oper Res 142: 21–29
Lam Y, Zhu LX, Chan JSK, Liu Q (2004) Analysis of data from a series of events by a geometric process model. Acta Math Appl Sin, English Series 20: 263–282
Lawless JF, Thiagarajah K (1996) A point-process model incorporating renewals and time trends, with application to repairable systems. Technometric 38: 131–138
Lindqvist BH, Elvebakk GE, Heggland K (2003) The trend-renewal process for statistical analysis of repairable systems. Technometric 45: 31–44
Maguire BA, Pearson ES, Wynn AHA (1952) The time intervals between industrial accidents. Biometrika 39: 168–180
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH (1953) Equations of state calculations by fast computing machines. J Chem Phys 21: 087–1091
Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Series B 55: 3–23
Spiegelhalter D, Thomas A, Best NG, Lunn D (2004) Bayesian inference using Gibbs sampling for Window version (WinBUGS) Version 1.4.1. The University of Cambridge. http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml
Walker SG, Gutiérrez-Peña E (1999) Robustifying Bayesian procedures. Bayesian Stat 6: 685–710
Wan WY, Chan JSK (2009) A new approach for handling longitudinal count data with zero inflation and overdispersion: Poisson geometric process model. Biom J 51: 556–570
Wan WY, Chan JSK (2010) Bayesian analysis of robust Poisson geometric process model using heavy-tailed distributions. submitted to journal
Acknowledgments
We gratefully acknowledge Dr. James Bell of the Drug and Alcohol Unit, Prince of Wales Hospital, Sydney, Australia, for kindly providing the data set on Methadone Maintenance Treatment Programme analyzed in this paper.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chan, J.S.K., Leung, D.Y.P. Binary geometric process model for the modeling of longitudinal binary data with trend. Comput Stat 25, 505–536 (2010). https://doi.org/10.1007/s00180-010-0190-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-010-0190-8