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Unformatted text preview: PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Linear Mixed Models Generalized Linear Mixed Models Outline Subjectspecific Models Conditional Inference Hierarchical Generalized Linear Model BetaBinomial Model PoissonGamma Model Generalized Linear Mixed Model 1 PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Linear Mixed Models SubjectSpecific Models Assumptions Given the subjectspecific effects b i (a qvector), the responses Y ij ( i = 1 , . . . , m, j = 1 , . . . , n i ) are independent and follow a distribution from the exponential family Y ij  b i f ( y ij  b i , ) . Let E( Y ij  b i ) = ij then g ( ij ) = ij = X T ij + Z T ij b i , where ij is the linear predictor and g is the link function. X ij and Z ij are p and qvector of covariates, with Z often being a subset of X . 2 PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Linear Mixed Models Three Ways to Handle Subjectspecific Parameters Treated as fixed unknown parameters. Neyman and Scott (1948) showed that the ML estimates may be inconsistent due to the fact that the number of unknown parameters increases with the sample size. Conditional likelihood approach appropriate when only interested in regression coefficients that do not vary across subjects; subjectspecific effects b 1 , b 2 , . . . , b m are treated as nuisance parameters ; estimate using the conditional likelihood given the sufficient statistics for b i . Full likelihoodbased approach appropriate when subjectspecific coefficients are of interest or conditioning discards too much information. treat b i as unobserved random variables and integrate them out to get the marginal likelihood of 3 PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Linear Mixed Models the parameters . The random effects b i are independent and identically distributed with mean and variance D ( ). Its distribution G is completely specified with parameters . That is, G does not depend on any covariates. Estimation of inference for is obtained from ML estimation, based on the marginal density for Y i . Examples include linear mixed model, hierarchical generalized linear model (betabinomial model, poissongamma model) and generalized linear mixed model. 4 PubH8452 Longitudinal Data Analysis  Fall 2011 Generalized Linear Mixed Models Conditional Inference Sufficiency Suppose a random vector Y has density indexed by parameter , and s = s ( y ) is a statistic. s is said to be sufficient for if f ( y ; ) g ( s ; ) h ( y  s ) . The inference for can be based on the marginal density of s and no information is lost. The conditional density h ( y  s ) is useful for model checking but not in inference for ....
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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.
 Fall '11
 XianghuaLuo

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