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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 Pearson’s χ 2 – Types of Residuals – OverDispersion • QuasiLikelihood – Motivation – Construction of QuasiLikelihood – QL 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 ( θ i , φ ) , where φ 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. η i = X T i β , • Link function : associates the linear combination of predictors with the transformed mean response. η i = g ( μ i ) , where μ 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 ; θ, φ ) = exp ‰ yθ b ( θ ) a ( φ ) + c ( y, φ ) , (1) where θ is known as the canonical parameter and φ 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 φ is known, this is an exponential family model with only canonical parameter of θ . 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 ; θ, φ ) in (1) then E( Y ) = μ = b ( θ ) Var( Y ) = b 00 ( θ ) a ( φ ) . < Proof > Proof. The loglikelihood is ‘ ( θ, φ ) = log f ( y ; θ, φ ) = yθ b ( θ ) a ( φ ) + c ( y, φ ) . Therefore ∂‘ ∂θ = y b ( θ ) a ( φ ) ∂ 2 ‘ ∂θ 2 = b 00 ( θ ) a ( φ ) . 4 PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Estimating Equations < Proof ( cont. ) > Using the fact that E ∂l ∂θ ¶ = 0 , E ∂ 2 ‘ ∂θ 2 ¶ = E ∂l ∂θ ¶ 2 , we get E y b ( θ ) a ( φ ) ¶ = 0 ⇒ E( Y ) = b ( θ ) E ∂l ∂θ ¶ 2 = E ‰ ( y b ( θ )) 2 a 2 ( φ ) = Var( Y ) a 2 ( φ ) , hence Var( Y ) a 2 ( φ ) = b 00 ( θ ) a ( φ ) ⇒ Var( Y ) = b 00 ( θ ) a ( φ ) ....
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 Fall '11
 XianghuaLuo
 Regression Analysis, Variance, Maximum likelihood, Estimation theory, Longitudinal Data Analysis

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