CH12-4 - Outline 12.1: Random eects modeling of a clustered...

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Chapter 12. Random Efects: Generalized Linear Mixed Models For Categorical Responses Deyuan Li School of Management, Fudan University Feb. 28, 2011 1/55 Outline 12.1: Random efects modeling o± a clustered categorical data; 12.2: Binary responses: logistic-normal model; 12.3: Examples o± random efects models ±or binary data; 12.4: Random efects models ±or multinomial data; 12.5: Multivariate random efects models ±or binary data. 2/55 In Chapter 11 we ±ocused on modeling the marginal distributions o± clustered responses, treating the joint dependence structure as a nuisance. In this chapter we present an alternative approach using cluster-level terms in the model. These terms take the same value ±or each observation in a cluster but diferent values ±or diferent clusters. They are unobserved and, when treated as varying randomly among clusters, are called random efects .The models have conditional interpretations, re±erred to as subject-specifc when each cluster is a subject. This contrasts with marginal models, which have population-averaged interpretations. Random efects models ±or normal responses are well established. By contrast, only recently have random efects been used much in models ±or categorical data. In this chapter we extend generalized linear models to include random efects. 3/55 12.1 Random efects modeling oF clustered categorical data Parameters that describe a ±actor’s efects in ordinary linear models are called ²xed efects. They apply to all categories o± interest, such as genders, age groupings, or treatments. By contrast, random efects usually apply to a sample. For a study using a sample o± clinics, ±or example, the model treats observations ±rom a given clinic as a cluster, and it has a random efect ±or each clinic. GLMs extend ordinary regression by allowing nonnormal responses and a link ±unction o± the mean. The generalized linear mixed model (GLMM) is a ±urther extension that permits random efects as well as ²xed efects in the linear predictor. 4/55
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12.1.1 Generalized linear mixed model y it : observation t in cluster i , t =1 , ..., T i , x it : a column vector of values of explanatory variables, for Fxed e±ect model parameters β ; u i : the vector of random e±ect values for cluster i (common to all observations in the cluster); z it : a column vector of their explanatory variables (normally, the random e±ect is univariate). Conditional on u i ,let μ it = E ( Y it | u i ). The GLMM is g ( μ it )= x ± it β + z ± it u i (12 . 1) for link function g . The random e±ect vector u i is assumed to have a multivariate normal distribution N (0 , Σ). The covariance matrix Σ depends on unknown variance components and possibly also correlation parameters. 5/55 Denote var( Y it | u i φ it v ( μ it ), where the variance function v ( . ) describes how the (conditional) variance depends on the mean.
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CH12-4 - Outline 12.1: Random eects modeling of a clustered...

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