05GEE_p2 - PubH8452 Longitudinal Data Analysis Spring 2011...

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Spring 2011 Marginal Model for Categorical Data Marginal Model for Categorical Data Outline • Marginal model – Examples of marginal model • GEE1 • Augmented GEE – GEE1.5 – GEE2 – Modeling the scale parameter φ – A note on modeling correlation of binary responses – Using marginal odds ratios to model association for binary responses 1 PubH8452 Longitudinal Data Analysis - Spring 2011 Marginal Model for Categorical Data Marginal Model • Marginal models (or population-average model) distinguish from the mixed effects models (or subject- specific models) according to the interpretation of their regression coefficients. The marginal models are used to make inferences about population average. • Marginal models emphasize the dependence of the mean response on the covariates of interest, but not both the random effects and the covariates (aka mixed effects models), nor previous responses (aka transition models). • Marginal models do not require the joint distributional assumptions for the vector of responses, which may be difficult for the discrete data. The avoidance of distributional assumptions leads to a method of estimation of generalized estimating equations (GEE). • For longitudinal data, the marginal model separately model the mean responses and within-subject association among the repeated responses. 2 PubH8452 Longitudinal Data Analysis - Spring 2011 Marginal Model for Categorical Data Notation of the marginal models Let the random variable Y ij denote the responses for the i th individual, measured at time t ij , i = 1 , . . . , m , j = 1 , . . . , n i . The response variables for the i th subject is an n i × 1 vector, Y i = ( Y i 1 , Y i 2 , . . . , Y in i ) . Let X ij denote p × 1 vector of covariates X ij = ( X ij 1 , X ij 2 , . . . , X ijp ) that are associated with each response, Y ij . X ij may include covariates whose values do not change over time, i.e. time-stationary or between-subjects covariates, and covariates whose values change over time, i.e., time-varying or within-subject covariates. The covariates can be grouped into an n i × p matrix: X i = X i 1 X i 2 . . . X in i = X i 11 X i 12 . . . X i 1 p X i 21 X i 22 . . . X i 2 p . . . . . . . . . . . . X in i 1 X in i 2 . . . X in i p , i = 1 , . . . , m 3 PubH8452 Longitudinal Data Analysis - Spring 2011 Marginal Model for Categorical Data • A marginal model has the following components: 1. Mean model : the marginal mean of each response, E( Y ij | X ij ) = μ ij , depends on covariates via a known link function g ( μ ij ) = η ij = X T ij β 2. Correlation model (nuisance): Var( Y ij | X i ) = V ij = φV ( μ ij ) Cor( Y ij , Y ik | X i ) = ρ ijk Cov( Y i | X i ) = V i ( φ, α ) = φ C 1 / 2 i R i C 1 / 2 i where R i is the correlation matrix and C i...
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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_p2 - PubH8452 Longitudinal Data Analysis Spring 2011...

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