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Chapter 9: Models for Proportions: Binomial GLMs

  • Peter K. Dunn
  • Gordon K. Smyth
Chapter
Part of the Springer Texts in Statistics book series (STS)

Abstract

Chapters  5 8 develop the theory of glms in general. This chapter focuses on one specific glm: the binomial glm. The binomial glm is the most commonly used of all glms. It is used to model proportions, where the proportions are obtained as the number of ‘positive’ cases out of a total number of independent cases. We first compile important information about the binomial distribution (Sect. 9.2), then discuss the common link functions used for binomial glms (Sect. 9.3), and the threshold interpretation of the link function (Sect. 9.4). We then discuss model interpretation in terms of odds (Sect. 9.5), and how binomial glms can be used to estimate the median effective dose ed50 (Sect. 9.6).

References

  1. [1]
    Chatterjee, S., Handcock, M.S., Simonoff, J.S.: A Casebook for a First Course in Statistics and Data Analysis. John Wiley and Sons, New York (1995)zbMATHGoogle Scholar
  2. [2]
    Collett, D.: Modelling Binary Data. Chapman and Hall, London (1991)CrossRefGoogle Scholar
  3. [3]
    Crowder, M.J.: Beta-binomial anova for proportions. Applied Statistics 27(1), 34–37 (1978)MathSciNetCrossRefGoogle Scholar
  4. [4]
    Dala, S.R., Fowlkes, E.B., Hoadley, B.: Risk analysis of the space shuttle: pre-Challenger prediction of failure. Journal of the American Statistical Association 84(408), 945–957 (1989)Google Scholar
  5. [5]
    Dunn, P.K., Smyth, G.K.: Randomized quantile residuals. Journal of Computational and Graphical Statistics 5(3), 236–244 (1996)Google Scholar
  6. [6]
    Hand, D.J., Daly, F., Lunn, A.D., McConway, K.Y., Ostrowski, E.: A Handbook of Small Data Sets. Chapman and Hall, London (1996)zbMATHGoogle Scholar
  7. [7]
    Hauck Jr., W.W., Donner, A.: Wald’s test as applied to hypotheses in logit analysis. Journal of the American Statistical Association 72, 851–853 (1977)MathSciNetzbMATHGoogle Scholar
  8. [8]
    Hewlett, P.S., Plackett, T.J.: Statistical aspects of the independent joint action of poisons, particularly insecticides. II Examination of data for agreement with hypothesis. Annals of Applied Biology 37, 527–552 (1950)Google Scholar
  9. [9]
    Hirji, K.F., Mehta, C.R., Patel, N.R.: Computing distributions for exact logistic regression. Journal of the American Statistical Association 82(400), 1110–1117 (1987)MathSciNetCrossRefGoogle Scholar
  10. [10]
    Hu, Y., Smyth, G.K.: ELDA: Extreme limiting dilution analysis for comparing depleted and enriched populations in stem cell and other assays. Journal of Immunological Methods 347, 70–78 (2009)CrossRefGoogle Scholar
  11. [11]
    Irwin, J.O., Cheeseman, E.A.: On the maximum-likelihood method of determining dosage-response curves and approximations to the median-effective dose, in cases of a quantal response. Supplement to the Journal of the Royal Statistical Society 6(2), 174–185 (1939)CrossRefGoogle Scholar
  12. [12]
    Kolassa, J.E., Tanner, M.A.: Small-sample confidence regions in exponential families. Biometrics 55(4), 1291–1294 (1999)CrossRefGoogle Scholar
  13. [13]
    Krzanowski, W.J.: An Introduction to Statistical Modelling. Arnold, London (1998)zbMATHGoogle Scholar
  14. [14]
    Lavie, P., Herer, P., Hoffstein, V.: Obstructive sleep apnoea syndrome as a risk factor for hypertension: Population study. British Medical Journal 320(7233), 479–482 (2000)CrossRefGoogle Scholar
  15. [15]
    Liu, R.X., Kaplan, H.B.: Role stress and aggression among young adults: The moderating influences of gender and adolescent aggression. Social Psychology Quarterly 67(1), 88–102 (2004)CrossRefGoogle Scholar
  16. [16]
    Lumley, T., Kronmal, R., Ma, S.: Relative risk regression in medical research: Models, contrasts, estimators, and algorithms. uw Biostatistics Working Paper Series 293, University of Washington (2006)Google Scholar
  17. [17]
    Maron, M.: Threshold effect of eucalypt density on an aggressive avian competitor. Biological Conservation 136, 100–107 (2007)CrossRefGoogle Scholar
  18. [18]
    Mather, K.: The analysis of extinction time data in bioassay. Biometrics 5(2), 127–143 (1949)CrossRefGoogle Scholar
  19. [19]
    Myers, R.H., Montgomery, D.C., Vining, G.G.: Generalized Linear Models with Applications in Engineering and the Sciences. Wiley Series in Probability and Statistics. Wiley, Chichester (2002)Google Scholar
  20. [20]
    Nelson, W.: Applied Life Data Analysis. John Wiley and Sons, New York (1982)CrossRefGoogle Scholar
  21. [21]
    Shackleton, M., Vaillant, F., Simpson, K.J., Sting, J., Smyth, G.K., Asselin-Labat, M.L., Wu, L., Lindeman, G.J., Visvader, J.E.: Generation of a functional mammary gland from a single stem cell. Nature 439, 84–88 (2006)CrossRefGoogle Scholar
  22. [22]
    Singer, J.D., Willett, J.B.: Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. Oxford University Press, New York (2003)CrossRefGoogle Scholar
  23. [23]
    Venables, W.N., Ripley, B.D.: Modern Applied Statistics with S, fourth edn. Springer-Verlag, New York (2002). URL http://www.stats.ox.ac.uk/pub/MASS4 CrossRefGoogle Scholar
  24. [24]
    Williams, D.A.: Tests for differences between several small proportions. Applied Statistics 37(3), 421–434 (1988)MathSciNetCrossRefGoogle Scholar
  25. [25]
    Xie, G., Roiko, A., Stratton, H., Lemckert, C., Dunn, P., Mengersen, K.: Guidelines for use of the approximate beta-Poisson dose-response models. Risk Analysis 37, 1388–1402 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Peter K. Dunn
    • 1
  • Gordon K. Smyth
    • 2
  1. 1.Faculty of Science, Health, Education and EngineeringSchool of Health of Sport Science, University of the Sunshine CoastQueenslandAustralia
  2. 2.Bioinformatics DivisionWalter and Eliza Hall Institute of Medical ResearchParkvilleAustralia

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