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5_GzLM and Poisson

# 5_GzLM and Poisson - Outline Advanced Topics in Forest...

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Advanced Topics in Forest Biometrics – FOR6934 Poisson Regression Outline Generalized linear models Poisson regression model Poisson regression procedures Poisson regression with categorical variables Poisson regression with continuous variables Generalized linear models (GzLMs) versus General linear models (GLMs) GzLMs are statistical models that combine elements of linear and nonlinear models GzLMs apply if the response variable is in the exponential family (e.g., Bernoulli, Binomial, Poisson, Normal, etc.) The Normal (Gaussian) linear regression (GLM) and ANOVA models are special cases of GzLMs If you know the probability distribution of your response variable (e.g., count data ~ Poisson), it is often more appropriate to use a GzLM GzLM components Systematic component Linear combination of covariates Additive, systematic part of model Link function Transformation of the response variable Maps the response on a scale where covariate effects are additive Ensures range restrictions Random component Distribution of the response chosen from the exponential family The Random component A function is a member of the exponential family if it can be written as: Binomial, Poisson, and Normal are members of this family For all members, the mean is b ΄ ( θ ) For all members except Normal, the mean is a function of the variance [ ] + = ) , ( ) ( 1 exp ) ( ψ θ θ ψ y c b y y f Examples of the Random component Binomial Negative Binomial Poisson Gamma ,... 1 , 0 ), / exp( ) ( 1 ) ( ,... 1 , 0 , ! ) ( ,... 1 , 0 , ) 1 ( 1 ) ( ,..., 1 , 0 , ) 1 ( ) ( 1 Γ = = = = + = = = y y y y f y y e y p y y y k y p n y y n y p y y k y n y β β α λ π π π π α α λ

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The Poisson and Negative Binomial Distributions 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14 16 p(x;l=1) p(x;l=3) p(x;l=5) p(x;l=8) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 2 4 6 8 10 12 14 16 p(x;l=1) p(x;l=3) p(x;l=5) p(x;l=8) Poisson Negative Binomial The Systematic component A linear combination of covariates and/or fixed effects parameters The linear predictor of the i th observation is: The linear predictor in a GzLM is equal to some transformation g (•) of the mean: ki k i i i x x x β β β β β + + + + = ... 2 2 1 1 0 x β i i Y E g x = ]) [ ( The Link function The transformation g (•) of the mean g (•) transforms the mean onto a scale where the covariate effects are additive Æ the link function is a linearizing transformation, and the GzLM is intrinsically linear For example, if the mean function is: Then, the linearizing transformation is ln( µ i ) g (•) also may serve another purpose: to confine predictions to an appropriate range ) exp( ] [ 1 0 x Y E i i β β µ + = = Link functions for count data Characteristics of count data: Whole numbers only Bounded below at zero, but no upper bound ( y Æ ) Æ Link function should map from [0, ] to [- , ]: Log link:
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5_GzLM and Poisson - Outline Advanced Topics in Forest...

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