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Institute for Paper, Pulp and Fiber Technology 1 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 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

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Institute for Paper, Pulp and Fiber Technology 2 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Agenda Motivation Generalized Linear Models Multiplicative Binomial Distribution Double Binomial Distribution Application of the Two Distributions Summary

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Institute for Paper, Pulp and Fiber Technology 3 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Motivation consider the problem of successful ink transfer on paper explain occurrence of unprinted regions …part of a larger, industry-funded project at the IPZ. (No. of datapoints in sample: roughly 9 10 6 sample size: 3 6 mm²)

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Institute for Paper, Pulp and Fiber Technology 4 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Predictor Variables TopographyFormation…the way fibres are arranged

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Institute for Paper, Pulp and Fiber Technology 5 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Response true colour image

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Institute for Paper, Pulp and Fiber Technology 6 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 GENERALIZED LINEAR MODELS Basics

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Institute for Paper, Pulp and Fiber Technology 7 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Distribution of the Response response model for here …part of the Exponential Family withthe probability for successful ink transmission

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Institute for Paper, Pulp and Fiber Technology 8 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 the Generalized Linear Model* model for is linked to the mean by * Nelder & Wedderburn (1972). Generalized Linear Models. Journal of the Royal Statistical Society, 135, 370-384 linear predictor advances over a linear model: distribution of the relative frequencies … member of the Exponential Family mean lies between 0 and 1

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Institute for Paper, Pulp and Fiber Technology 9 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Model Deviance Deviance = -2 × ( maximized log-likelihood of considered model – maximized log-likelihood of saturated model ) under certain regularity conditions, …a test for goodness-of-fit ifUnderdispersion Variance of data smaller than assumed by the model ifOverdispersion Variance of data larger than assumed by the model

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Institute for Paper, Pulp and Fiber Technology 10 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Deviances of the Printability Datasets distinct deviations from a binomial variance! few many unprinted areas …values from 11 different data sets

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Institute for Paper, Pulp and Fiber Technology 11 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 MULTIPLICATIVE BINOMIAL DISTRIBUTION A Generalization of the Binomial Distribution

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Institute for Paper, Pulp and Fiber Technology 12 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Definition *Altham (1978). Two Generalizations of the Binomial Distribution. Journal of the Royal Statistical Society, 27, 162-197 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 introduced by Altham* as „multiplicative generalization of the binomial distribution“

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Institute for Paper, Pulp and Fiber Technology 13 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 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

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Institute for Paper, Pulp and Fiber Technology 14 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison With Classic Binomial pdf n = 36 = 0.8 ω=1 gives the classic binomial distribution

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Institute for Paper, Pulp and Fiber Technology 15 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Variances n = 36 ω=1 gives the classic binomial distribution

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Institute for Paper, Pulp and Fiber Technology 16 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Integration into GLM Context log-likelihood function of distribution logit-link 0 < < 1 ω > 0 log-linear link

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Institute for Paper, Pulp and Fiber Technology 17 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 DOUBLE BINOMIAL DISTRIBUTION A Second Generalization of the Binomial Distribution

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Institute for Paper, Pulp and Fiber Technology 18 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Definition *Efron (1986). Double Exponential Families and their Use in Generalized Linear Regression. Journal of the American Statistical Association, 81, 709-721 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

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Institute for Paper, Pulp and Fiber Technology 19 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison With Classic Binomial pdf n = 36 = 0.8 =1 gives the classic binomial distribution

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Institute for Paper, Pulp and Fiber Technology 20 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Variances n = 36 =1 gives the classic binomial distribution

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Institute for Paper, Pulp and Fiber Technology 21 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Integration into GLM Context member of the 2-parameter exponential family log-likelihood function of distribution 0 < < 1 > 0 logit-link log-linear link

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Institute for Paper, Pulp and Fiber Technology 22 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 AN APPLICATION The Printability Dataset

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Institute for Paper, Pulp and Fiber Technology 23 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Response and Explanatory Variables occurrrence of unprinted areas… ~ explained by… topography + formation

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Institute for Paper, Pulp and Fiber Technology 24 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of Three Models Distributionclassic binomial multiplicative binomial double binomial 17071845211632 DoF24832482 661458364117 DoF24812480 AIC662058454125

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Institute for Paper, Pulp and Fiber Technology 25 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Means

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Institute for Paper, Pulp and Fiber Technology 26 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Means

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Institute for Paper, Pulp and Fiber Technology 27 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Means The second parameter influences the mean, too.

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Institute for Paper, Pulp and Fiber Technology 28 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Standard Deviations

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Institute for Paper, Pulp and Fiber Technology 29 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Comparison of the Standard Deviations

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Institute for Paper, Pulp and Fiber Technology 30 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 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

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Institute for Paper, Pulp and Fiber Technology 31 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 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

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Institute for Paper, Pulp and Fiber Technology 32 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Summary 2 Double Binomial Distribution (Efron, 1986) second parameter in GLM context:model with the logistic link and with the log-linear link function

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Institute for Paper, Pulp and Fiber Technology 33 Verena Feirer Österreichische Statistiktage Two Distribution Families for Modelling Over- and Underdispersed Binomial Frequencies September 7 th, 2011 Thank You for Your Attention

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