07GLMModels - PubH8452 Longitudinal Data Analysis Fall 2011...

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Modeling Strategies - Marginal Models, Random Effects Models, and Transition Models Outline Summary of Modeling Strategies Modeling Mean Responses Modeling Covariances Interpretation Continuous Response Binary Response Count Data 1 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Summary of Modeling Strategies Modeling Mean Responses Marginal models: The marginal expectation of the response, E( Y ij ) = ij , depends on the covariates, x ij , through a link function g , E( Y ij ) = ij = g- 1 ( x T ij ) . Random effects (conditional) models: Conditional on subject specific, unobserved random variables b i , E( Y ij | b i ) = * ij = g- 1 ( x T ij * + z T ij b i ) . 2 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Transition models: Let H ij = ( Y i 1 , . . . , Y ij- 1 ) denote the history of Y ij , E( Y ij |H ij ) = ** ij = g- 1 ( x T ij ** + s X r =1 f r ( H ij , ) ) . 3 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Modeling Covariances Marginal models: The marginal variance of the response depends on the marginal mean, Var( Y ij ) = V ( ij ) , where V is a known function and the scale parameter may also depend on some covariates. The correlation between Y ij and Y ik is a function of the marginal mean: Cor( Y ij , Y ik ) = ( ij , ik , ) , where is a known function and the correlation parameters may depend on covariates. 4 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Random effects (conditional) models: Typically, conditioning on the random effects, b i , the responses Y i 1 , . . . , Y in i are independent with an exponential family distribution. The random effects b i have mean and variance D . Typically the random effects are assumed to be multivariate Gaussian. Correlation among observations from the same person arises from their sharing unobservable variables, i.e., random effects. 5 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Transition models: Typically, Y ij |H ij is assumed to have an exponential family distribution with variances Var( Y ij |H ij ) = V ( ** ij ) . Correlation among Y i 1 , . . . , Y in i exists because the past response values explicitly influence the present observation. 6 PubH8452 Longitudinal Data Analysis - Fall 2011 Modeling Strategies Interpretation Marginal model: is the mean (expected) difference in the response for the two populations (indi- viduals) with the identical covariate values but differ in X by one unit....
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07GLMModels - PubH8452 Longitudinal Data Analysis Fall 2011...

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