Abstract
Chapters 2] and 3 considered linear regression models. These models assume constant variance, which demonstrably is not true for all data, as shown in Chap. 4. Generalized linear models (glms) assume the responses come from a distribution that belongs to a more general family of distributions, and also permit more general systematic components. We first review the two components of a glm (Sect. 5.2) then discuss in greater detail the family of distributions upon which the random component is based (Sect. 5.3), including writing the probability functions in the useful dispersion model form (Sect. 5.4). The systematic component of the glm is then considered in greater detail (Sect. 5.5). Having discussed the two components of the glm, glms are then formally defined (Sect. 5.6), and the important concept of the deviance function is introduced (Sect. 5.7). Finally, using a glm is compared to using a regression model after transforming the response (Sect. 5.8).
Models are useful distillations of reality. Although wrong by definition, they are the wind that blows away the fog and cuts through the untamed masses of data to let us see answers to our questions.
Keller [4, p. 97]
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References
Data Desk: Data and story library (dasl) (2017). URL http://dasl.datadesk.com
Johnson, B., Courtney, D.M.: Tower building. Child Development 2(2), 161–162 (1931)
Jørgensen, B.: The Theory of Dispersion Models. Monographs on Statistics and Applied Probability. Chapman and Hall, London (1997)
Keller, D.K.: The Tao of Statistics: A Path to Understanding (with no Math). Sage Publications, Thousand Oaks, CA (2006)
Kokonendji, C.C., Khoudar, M.: On strict arcsine distribution. Communications in Statistics—Theory and Methods 33(5), 993–1006 (2004)
Maron, M.: Threshold effect of eucalypt density on an aggressive avian competitor. Biological Conservation 136, 100–107 (2007)
Nelder, J.A., Pregibon, D.: An extended quasi-likelihood function. Biometrika 74(2), 221–232 (1987)
Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S., Boatwright, P.: A useful distribution for fitting discrete data: Revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society: Series C 54(1), 27–142 (2005)
Singer, J.D., Willett, J.B.: Improving the teaching of applied statistics: Putting the data back into data analysis. The American Statistician 44(3), 223–230 (1990)
Smyth, G.K.: Australasian data and story library (Ozdasl) (2011). URL http://www.statsci.org/data
Smyth, G.K., Verbyla, A.P.: Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695–709 (1999)
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
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Dunn, P.K., Smyth, G.K. (2018). Chapter 5: Generalized Linear Models: Structure. In: Generalized Linear Models With Examples in R. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0118-7_5
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DOI: https://doi.org/10.1007/978-1-4419-0118-7_5
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