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Agenda

Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies Feirer V. , Hirn U., Friedl H., Bauer W. Institute for Paper, Pulp and Fiber Technology & Institute for Statistics Graz University of Technology. Agenda. Motivation Generalized Linear Models

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Agenda

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  1. Two Distribution Familiesfor Modelling Over- and UnderdispersedBinomial FrequenciesFeirer V., Hirn U., Friedl H., Bauer W.Institute for Paper, Pulp and Fiber Technology& Institute for StatisticsGraz University of Technology

  2. Agenda • Motivation • Generalized Linear Models • Multiplicative Binomial Distribution • Double Binomial Distribution • Application of the Two Distributions • Summary

  3. Motivation • consider the problem of successful ink transfer on paper (No. of datapoints in sample: roughly 9106 sample size: 3  6 mm²) • explain occurrence of unprinted regions …part of a larger, industry-funded project at the IPZ.

  4. Predictor Variables Topography Formation …the way fibres are arranged

  5. Response true colour image

  6. Basics generalized linear models

  7. Distribution of the Response response …part of the Exponential Family here with the probability for successful ink transmission model for

  8. the Generalized Linear Model* model for linear predictor is linked to the mean by • advances over a linear model: • distribution of the relative frequencies • … member of the Exponential Family • mean lies between 0 and 1 * Nelder & Wedderburn (1972). Generalized Linear Models. Journal of the Royal Statistical Society, 135, 370-384

  9. Model Deviance …a test for goodness-of-fit Deviance = -2 × ( maximized log-likelihood of considered model – maximized log-likelihood of saturated model ) under certain regularity conditions, if Underdispersion Variance of data smaller than assumed by the model if Overdispersion Variance of data larger than assumed by the model

  10. Deviances of the Printability Datasets …values from 11 different data sets distinct deviations from a binomial variance! many few unprinted areas

  11. A Generalization of the Binomial Distribution Multiplicative binomial distribution

  12. Definition • introduced by Altham* as „multiplicative generalization of the binomial distribution“ considers litters of rabbits animals within one litter are treated with the same dosis of a certain drug n… litter size y… number of surviving animals • outcomes from animals from within one litter are not mutually independent Altham introduces an interaction parameter ω *Altham (1978). Two Generalizations of the Binomial Distribution. Journal of the Royal Statistical Society, 27, 162-197

  13. Properties • Member of the 2-parameter Exponential Family • For ω=1, it corresponds to the Binomial Distribution • For n=1, it reduces to the Bernoulli distribution

  14. Comparison With Classic Binomial pdf n = 36  = 0.8 ω=1 gives the classic binomial distribution

  15. Comparison of the Variances n = 36 ω=1 gives the classic binomial distribution

  16. Integration into GLM Context log-likelihood function of distribution log-linear link logit-link  ω > 0  0 <  < 1

  17. A Second Generalization of the Binomial Distribution Double binomial Distribution

  18. Definition introduced by Efron* as part of the Double Exponential Family second parameter  allows variation of variance: variance is smaller than binomial if 0<<1 and larger than binomial if >1 =1 gives the classic binomial distribution *Efron (1986). Double Exponential Families and their Use in Generalized Linear Regression. Journal of the American Statistical Association, 81, 709-721

  19. Comparison With Classic Binomial pdf n = 36  = 0.8 =1 gives the classic binomial distribution

  20. Comparison of the Variances n = 36 =1 gives the classic binomial distribution

  21. Integration into GLM Context member of the 2-parameter exponential family log-likelihood function of distribution log-linear link logit-link   > 0  0 <  < 1

  22. The Printability Dataset An application

  23. Response and Explanatory Variables ~ explained by… + formation topography occurrrence of unprinted areas…

  24. Comparison of Three Models

  25. Comparison of the Means

  26. Comparison of the Means

  27. Comparison of the Means The second parameter influences the mean, too.

  28. Comparison of the Standard Deviations

  29. Comparison of the Standard Deviations

  30. Comparison of the Variances binomial Std. Dev. at n=36: cannot be larger than 3 empirical Std. Deviations: up to 11 Multiplicative and Double Binomial Standard Deviations fit much better to empirical results

  31. Summary Two generalizations of the binomial distribution might compensate over- or underdispersion in the case of classic binomial distribution. Multiplicative Binomial Distribution (Altham, 1978) second parameter ω in GLM context: model  with the logistic link and ω with the log-linear link function

  32. Summary 2 Double Binomial Distribution (Efron, 1986) second parameter  in GLM context: model  with the logistic link and  with the log-linear link function

  33. Thank You for Your Attention

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