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

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Unformatted text preview: GENERALIZED LINEAR MODELS I Consider the normal theory Gauss-Markov linear model y = X + , N ( , 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 ( i , 2 ) , where i = X i 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 ( i ) , where i = exp ( X i ) 1 + exp ( X i ) for all i = 1 , ..., n and y 1 , ..., y n independent. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 1 / 16 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 i . 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 I The distribution is in the exponential scale family I i.e., y i has density (or p.m.f.) exp ( ) T ( y i )- b ( ) a ( ) + c ( y i , ) (1) where () , T () , a () , b () , and c () are known functions and is a vector of unknown parameters depending on X and is either a known or unknown parameter. I Exponential family / exponential class is (1) without the a ( ) I a ( ) includes overdispersed distributions in the family c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 2 / 16 I For example, the pdf for a normal distribution can be written as: exp- 1 2 2 y 2 i + 2 2 y i- 2 2 2- 1 2 log ( 2 2 ) I from which: I ( ) = ( 2 ,- 1 2 2 ) I and T ( y i ) = ( y i , y 2 i ) I This family includes many common distributions: Distribution ( ) T ( y i ) a ( ) Normal ( 2 ,- 1 2 2 ) ( y i , y 2 i ) 1 Bernoulii log 1- y i 1 Poisson log y i 1 Overdisp. Poisson log y i Gamma (- 1 , ( k- 1 ) ) ( y i , log y i ) 1 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 26 3 / 16 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...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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

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