05GEE - PubH8452 Longitudinal Data Analysis - Fall 2011...

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Fall 2011 Generalized Estimating Equations Generalized Estimating Equations Outline Review of Generalized Linear Models (GLM) Generalized Linear Model Exponential Family Components of GLM MLE for GLM, Iterative Weighted Least Squares Measuring Goodness of Fit - Deviance and Pearsons 2 Types of Residuals Over-Dispersion Quasi-Likelihood Motivation Construction of Quasi-Likelihood Q-L Estimating Equations Optimality Impact of Nuisance Parameters Generalized Estimating Equations (GEE) 1 PubH8452 Longitudinal Data Analysis - Fall 2011 Generalized Estimating Equations Review of Generalized Linear Models (GLM) Consider independent data Y i , i = 1 , . . . , m with the covariates of X i . In GLM, the probability model for Y i has the following specification: Random component : Y i is assumed to follow distribution that belongs to the exponential family. Y i | X i f ( i , ) , where is the dispersion parameter. Systematic component : given covariates X i , the mean of Y i can be expressed in terms of the following linear combination of predictors. i = X T i , Link function : associates the linear combination of predictors with the transformed mean response. i = g ( i ) , where i = E( Y i | X i ). 2 PubH8452 Longitudinal Data Analysis - Fall 2011 Generalized Estimating Equations Exponential Family In the random component of GLM, Y i is assumed to follow a probability distribution that belongs to the exponential family. The density functions of the exponential family of distributions have this general form: f ( y ; , ) = exp y- b ( ) a ( ) + c ( y, ) , (1) where is known as the canonical parameter and is a fixed (known) scale (dispersion) parameter. Note that a ( ) and b ( ) are some specific functions that distinguish one member of the exponential family from another. If is known, this is an exponential family model with only canonical parameter of . The exponential family of distribution include the normal, Bernoulli, and Poisson distributions. 3 PubH8452 Longitudinal Data Analysis - Fall 2011 Generalized Estimating Equations Properties of Exponential Family If Y f ( y ; , ) in (1) then E( Y ) = = b ( ) Var( Y ) = b 00 ( ) a ( ) . < Proof > Proof. The log-likelihood is ( , ) = log f ( y ; , ) = y- b ( ) a ( ) + c ( y, ) . Therefore = y- b ( ) a ( ) 2 2 =- b 00 ( ) a ( ) . 4 PubH8452 Longitudinal Data Analysis - Fall 2011 Generalized Estimating Equations < Proof ( cont. ) > Using the fact that E l = 0 , E 2 2 =- E l 2 , we get E y- b ( ) a ( ) = 0 E( Y ) = b ( ) E l 2 = E ( y- b ( )) 2 a 2 ( ) = Var( Y ) a 2 ( ) , hence Var( Y ) a 2 ( ) = b 00 ( ) a ( ) Var( Y ) = b 00 ( ) a ( ) ....
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This note was uploaded on 11/21/2011 for the course PUBH 8452 taught by Professor Xianghualuo during the Fall '11 term at Minnesota.

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05GEE - PubH8452 Longitudinal Data Analysis - Fall 2011...

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