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26GLM(1) - GENERALIZED LINEAR MODELS Consider the normal...

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GENERALIZED LINEAR MODELS I Consider the normal theory Gauss-Markov linear model y = X β + , N ( 0 , σ 2 I ) . I Does not have to be written as function + error I Could specify distribution and model(s) for its parameters I i.e., y i N ( μ i , σ 2 ) , where μ i = X 0 i β for all i = 1 , ..., n and y 1 , ..., y n independent. I This is one example of a generalized linear model. I Here is another example of a GLM: y i Bernoulli ( π i ) , where π i = exp ( X 0 i β ) 1 + exp ( X 0 i β ) for all i = 1 , ..., n and y 1 , ..., y n independent. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 1 / 16
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I In each example, all responses are independent and each response is a draw from one type of distribution whose parameters may depend on explanatory variables through a known function of a linear predictor X 0 i β . I The normal and Bernoulii models (and many others) are special cases of a generalized linear model. I These are models where: I The parameters are specified functions of X β I The distribution is in the exponential scale family I i.e., y i has density (or p.m.f.) exp η ( θ ) T ( y i ) - b ( θ ) a ( φ ) + c ( y i , φ ) (1) where η () , T () , a () , b () , and c () are known functions and θ is a vector of unknown parameters depending on X β and φ is either a known or unknown parameter. I Exponential family / exponential class is (1) without the a ( φ ) I a ( φ ) includes “overdispersed” distributions in the family c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 2 / 16
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I For example, the pdf for a normal distribution can be written as: exp - 1 2 σ 2 y 2 i + μ 2 σ 2 y i - μ 2 2 σ 2 - 1 2 log ( 2 πσ 2 ) I from which: I η ( θ ) = ( μ σ 2 , - 1 2 σ 2 ) I and T ( y i ) = ( y i , y 2 i ) I This family includes many common distributions: Distribution η ( θ ) 0 T ( y i ) 0 a ( φ ) Normal ( μ σ 2 , - 1 2 σ 2 ) ( y i , y 2 i ) 1 Bernoulii log π 1 - π y i 1 Poisson log λ y i 1 Overdisp. Poisson log λ y i φ Gamma ( - 1 θ , ( k - 1 ) ) ( y i , log y i ) 1 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 3 / 16
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I A lot of stat theory results follow immediately from the exponential form I e.g. T ( y i ) is the vector of sufficient statistics I A lot of theory is much nicer when distribution parameterized in terms of η ( θ ) instead of θ I e.g. use log π 1 - π as the parameter of a Bernoulli distribution
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