Advertisement

Journal of Statistical Theory and Practice

, Volume 7, Issue 2, pp 401–420 | Cite as

Connecting Logistic Probability Models With Basic Dynamic Processes

  • Jeremy A. LauerEmail author
  • Sander Greenland
Article
  • 7 Downloads

Abstract

The logistic distribution is a popular probability model yet it is usually not motivated with reference to the processes under study. While its popularity can be attributed to its simplicity, it can also be derived from basic contextual considerations. Although it has been shown that logistic growth is the limiting form of a class of Markov processes on a lattice, the connections of these microscopic models with statistics are less widely appreciated. We review some history of logistic distributions in classical models of infection, and describe how the apparent density dependence emerges as a consequence of a particular lattice embedding. We then review how logistic growth arises from microscopic random behavior. We also derive a “square-logistic” model from basic considerations. Finally, we describe how the underlying discrete model relates to ordinary logistic regression modeling of time dependence.

Keywords

Logistic Logit Contact process Percolation Hydrodynamic limit Interacting particle system Lattice Diffusion Hyperbolic differential equation Hyperbolic conservation law Infection Transmission 

AMS 2000 Subject Classifications

Primary 60J70 Secondary 35L40 60K35 62-09 62E10 62J05 62J12 62M05 92D25 92D30 92D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldrich, J. H., and F. D. Nelson. 1999. Linear probability, logit, and probit-models. Beverly Hills, CA, Sage.Google Scholar
  2. Armitage, P., G. Berry, and J. N. S. Matthews. 2002. Modelling categorical data. In Statistical methods in medical research, 4th ed., 485–502. Oxford, UK, Blackwell.Google Scholar
  3. Banks, R. B. 1994. Growth and diffusion phenomena: Mathematical frameworks and applications. Berlin, Germany, Springer.zbMATHGoogle Scholar
  4. Bressan, A. 2000. Hyperbolic systems of conservation laws: The one-dimensional Cauchy problem. Oxford, UK, Oxford University Press.zbMATHGoogle Scholar
  5. Cantrell, R. S., and C. Cosner. 2003. Spatial ecology via reaction-diffusion equations. Chichester, UK, John Wiley and Sons Ltd.zbMATHGoogle Scholar
  6. Cardy, J. 1996. Scaling and renormalization in statistical physics. Cambridge, UK, Cambridge University Press.zbMATHGoogle Scholar
  7. Cox, D. R., and E. J. Snell. 1989. Analysis of binary data, 2nd ed. Monographs on Statistics and Applied Probability, 32. London, UK, Chapman and Hall/CRC.zbMATHGoogle Scholar
  8. Cox, C. 1996. Nonlinear quasi-likelihood models: Applications to continuous proportions. Comput. Stat. Data Analy., 21, 449–461.zbMATHGoogle Scholar
  9. Cox, D. R., and D. Oakes. 1984. Analysis of survival data. New York, NY, Chapman and Hall.Google Scholar
  10. Cramer, J. S. 2004. The early origins of the logit model. Stud. History Philos. Sci. Part C 35(4), 613–626.Google Scholar
  11. Daley, D. J., and J. Gani. 1999. Epidemic modelling: An introduction. Cambridge, UK, Cambridge University Press.zbMATHGoogle Scholar
  12. De Masi, A., P. A. Ferrari, and J. L. Lebowitz. 1985. Rigorous derivation of reaction-diffusion equations with fluctuations. Phys. Rev. Lett., 55 1947–1949.MathSciNetGoogle Scholar
  13. Dickman, R. 1997. Critical phenomena at absorbing states. In Nonequilibrium statistical mechanics in one dimension, ed. V. Privman, 51–70. Cambridge, UK, Cambridge University Press.Google Scholar
  14. Durrett, R. 1984. Oriented percolation in two dimensions. Ann. Probability, 12(4), 999–1040.MathSciNetzbMATHGoogle Scholar
  15. Durrett, R., and C. Neuhauser. 1994. Particle systems and reaction diffusion equations. Ann. Probability, 22, 289–333.MathSciNetzbMATHGoogle Scholar
  16. Enright, J. T. 1976. Climate and population regulation: the biogeographer’s dilemma. Oecologia, 24, 295–310.Google Scholar
  17. Fisher, R. A., 1937. The wave advance of advantageous genes. Ann. Eugenics, 7, 355–369.zbMATHGoogle Scholar
  18. Fletcher, R. I. 1974. The quadric law of damped exponential growth, Biometrics, 30, 111–124.MathSciNetzbMATHGoogle Scholar
  19. Gardiner, C. W. 2004. Handbook of stochastic methods for physics, chemistry and the natural sciences, 3rd ed. Berlin, Germany, Springer.zbMATHGoogle Scholar
  20. Good, I. J., 1983. Good thinking: The foundations of probability and its applications. Minneapolis, MN, University of Minnesota Press.zbMATHGoogle Scholar
  21. Greenland, S. 2003. Generalized conjugate priors for Bayesian analysis of risk and survival regressions. Biometrics, 59, 92–99.MathSciNetzbMATHGoogle Scholar
  22. Greenland, S. 2007. Prior data for non-normal priors. Sta. Med., 26, 3578–3590.MathSciNetGoogle Scholar
  23. Greenland, S. 2008a. Introduction to regression models. In Modern epidemiology, 3rd ed., ed. K. J. Rothman, S. Greenland, and T. L. Lash, 345–380. Philadephia, PA, Lippincott Williams and Wilkins.Google Scholar
  24. Greenland, S. 2008b. Variable selection and shrinkage in the control of multiple confounders. Am. J. Epidemiol., 167, 523–529.Google Scholar
  25. Greenland, S., and T. L. Lash. 2008. Bias analysis. In Modern epidemiology, ed. K. J. Rothman, S. Greenland, and T. L. Lash, 3rd ed., 345–380. Philadephia, PA, Lippincott Williams and Wilkins.Google Scholar
  26. Grimmet, G. 1999. Percolation, 2nd ed. (A series of comprehensive studies in mathematics, Volume 321). Berlin, Germany, Springer.Google Scholar
  27. Harris, T. E. 1974. Contact interactions on a lattice. Ann. Probability, 2(6), 969–988.MathSciNetzbMATHGoogle Scholar
  28. Hosmer, D., and S. Lemeshow. 2000. Applied logistic regression, 2nd ed. New York, NY, John Wiley and Sons.zbMATHGoogle Scholar
  29. Jones, M. C. 2004. Families of distributions arising from distributions of order statistics. Test, 13, 1–43.MathSciNetzbMATHGoogle Scholar
  30. Kendall, D. G. 1956. Deterministic and stochastic epidemics in closed populations. Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, 4, 149–165.MathSciNetzbMATHGoogle Scholar
  31. Kermack, W. O., and A. G. McKendrick. 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A, 115, 700–721.zbMATHGoogle Scholar
  32. Kipnis, C., and C. Landim. 1999. Scaling limits of interacting particle systems. Berlin, Germany, Springer.zbMATHGoogle Scholar
  33. Lauer, J. A., A. P. Betrán, C. G. Victora, M. de Onís, and A. J. D. Barros. 2004. Breastfeeding patterns and exposure to suboptimal breastfeeding among children in developing countries: review and analysis of nationally representative surveys. BMC Med., 2, 26.Google Scholar
  34. LeFloch, P. G. 2002. Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves. Basel, Switzerland, Birkhauser.zbMATHGoogle Scholar
  35. Liggett, T. M. 2005. Interacting particle systems. Berlin, Germany, Springer.zbMATHGoogle Scholar
  36. Lloyd, P. J. 1967. American, German and British antecedents to Pearl and Reed’s logistic curve. Population Stud., 21, 99–108.Google Scholar
  37. McCullagh, P., and J. A. Nelder. 1989. Generalized linear models, 2nd ed. 1–19. Cambridge, UK, Chapman and Hall.zbMATHGoogle Scholar
  38. Mueller, C., and R. Tribe. 1994. A phase transition for a stochastic PDE related to the contact process. Probability Theory Related Fields, 100, 131–156.MathSciNetzbMATHGoogle Scholar
  39. Mueller, C., and R. Tribe. 1995. Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probability Theory Related Fields, 102, 519–545.MathSciNetzbMATHGoogle Scholar
  40. Rudin, W. 1976. Principles of mathematical analysis, 3rd ed. New York, NY, McGraw-Hill.zbMATHGoogle Scholar
  41. Spitzer, F. 1970. Interaction of Markov processes. Adv. Math., 5, 246–290.MathSciNetzbMATHGoogle Scholar
  42. Verhulst, P.-F. 1838. Notice sur la loi que la population suit dans son accroissement. Corresp. Mathé. Phys., publiée par A. Quetelet, 10, 113–120.Google Scholar
  43. Wedderburn, R. W. M. 1974. Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439–447.MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2013

Authors and Affiliations

  1. 1.Department of Health Systems FinancingWorld Health OrganizationGeneva 27Switzerland
  2. 2.Department of Epidemiology and Department of StatisticsUniversity of California Los AngelesLos AngelesUSA

Personalised recommendations