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L20handout

# L20handout - PUBH 7430 Lecture 20 J Wolfson Division of...

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PUBH 7430 Lecture 20 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health November 17, 2011 1 / 25

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Generalized linear mixed models 2 / 25
Linear mixed models So far, we have dealt with the linear mixed model (LMM) Y ij = x 0 ij β + z 0 i b i + ij b i MVN (0 , Σ) , ij N (0 , σ 2 ) Assumptions Given b i , Y ij is Normally distributed... ... with mean μ ij = x 0 ij β + z 0 i b i ... and variance σ 2 (i.e. V ( μ ij ) = 1) 3 / 25

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Linear mixed models Linear mixed model most appropriate for data which are Continuous Approximately normally distributed Have constant variance (no mean-variance relationship) But, as we saw when introducing GLMs, these assumptions may be violated for a variety of data generating processes (eg. binary data, count data, etc.) 4 / 25
Extending the linear mixed model In the linear mixed model, we assume that the mean E ( Y | X ) μ has a Normal distribution. Two main approaches to extend the linear mixed effects model to non-normal outcomes: 1 Generalized Linear Mixed Models: Assume that g ( μ ) has a Normal distribution for some link function g 2 ”Mixture” models: Assume that μ arises from some other specified distribution. Beyond scope of this course. 5 / 25

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Generalized linear mixed models First approach: Assume that g ( μ ) has a Normal distribution Assumptions μ ij is linked to the linear predictor via g ( μ ij ) = x 0 ij β + z 0 ij b i b i arise from a Normal distribution Conditional on b i , Y ij has a distribution from the exponential family, and the Y ij are independent The resulting model is called the Generalized Linear Mixed Model (GLMM) 6 / 25
GLMM - example Gum data Recall that we had oral MS (bacteria) measurements at baseline, 1 week, 4 weeks, and 12 weeks Some measurements were missing We may be interested in predictors of the probability of having a missing measurement Binary data: 1 = missing, 0 = not missing 7 / 25

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GLMM - example
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