Generalized Multilevel Models for Binary and Count Outcomes I PDF

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Generalized Multilevel Models for Binary and Count Outcomes Lecture Notes

 Generalized Multilevel Models  Relatively new methods for modeling non-normal outcomes in longitudinal or cluster sample studies (mid- to late-1990s)

 Combines ideas of generalized linear models with random-effects modeling

 More highly complex alternative to GEE  Using random-effects to account for within-subject variation, rather than specifying covariance matrix  Generalized Multilevel Models  Similar to MLMs  Subset of fixed-effects assumed to vary randomly across clusters  Random-effects represent heterogeneity due to unmeasured factors  Mean of outcome is conditional on random-effects and is linearly related to fixed-effects (covariates) via a link function  Continuously distributed outcome related via ‘identity’ link  Assumes highest-order clusters are independent  Generalized Multilevel Models  Differ from MLMs

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Generalized Multilevel Models for Binary and Count Outcomes Lecture Notes  Distribution of random-effects unlikely normal  Assumption of normally distributed residuals untenable  Binomial: Can only assume 1 of 2 values (0,1)  Predicted values of outcome are restricted within certain bounds, depending on distribution of outcome  E.g., Probabilities of a binary outcome are between 0 and 1 and predicted values of a Poisson model are non-negative integers  Variance of random-effects depends on predicted values of outcome and are not homogeneous  Generalized Multilevel Models  Three components…  Conditional distribution of Yij (on random-effects) belongs to a member of the exponential family of distributions (e.g., Binomial, Poisson)  Conditional mean of Yij is assumed to depend on fixed- and randomeffects and is linearly related to fixed-effects via link function

 Structural linear model consisting of covariates  Random-effects are assumed independent of covariates  Generalized Multilevel Models  Now distribution of random-effects represents unobserved latent variable(s)

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Generalized Multilevel Models for Binary and Count Outcomes Lecture Notes  Simple assumptions about distribution are made (i.e., normality)

 However, for models with non-normal outcomes, interpretation of regression coefficients can be highly sensitive to difficult-to-verify assumptions about the distribution of random-effects, particularly dependence of latent variable distribution on covariates  Generalized Multilevel Models  For computational convenience and simplicity, random-effects are typically expressed in standardized form  Random-effects variance term becomes part of generalized MLM equation (not case in standard MLMs)  Regression coefficients and variance are on same scale…logit  Thus, coefficients from generalized MLMs are amplified by amount of variance in random-effects, those in standard MLMs and GEEs are not  Generalized Multilevel Models  β represent influence of covariates on a specific subject’s mean response, holding other covariates, as well as that individual’s random-effects, fixed or constant

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Generalized Multilevel Models for Binary and Count Outcomes Lecture Notes  Thus, prediction of outcome must include random-effect related to cluster to which participant belongs, in addition to linear combination of predictors  Interpretation of parameter estimates is now subject-or unit-specific (I prefer cluster-specific), as opposed to population-averaged, as in marginal (GEE) models

 Fixed-effect coefficients represent effects for the ‘average subject’, rather than averaged across subjects  Generalized Multilevel Models  Interpretation of time-invariant or between-subjects effects is difficult as there is no subject-specific change occurring (models are subjectspecific)  Change might be inferred from individuals who posses same randomeffects, but differ by 1-unit in terms of fixed-effects

 Generalized Multilevel Models  Why was this not the case with standard MLMs?  When an identity link function is used (outcome is continuously and normally distributed), regression coefficients can be interpreted as either subject-specific or population-averaged

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Generalized Multilevel Models for Binary and Count Outcomes Lecture Notes  When averaged over distribution of random-effects, population means flow from a linear model with regression coefficients β  Mathematically, this relationship does not hold for non-linear links used in presence of discrete outcomes  Generalized Multilevel Models  Why was this not the case with GEEs?  GEEs models are only marginal/population-averaged  No possibility for subject-specific interpretations  Estimates in GEEs represent true averages of individual effects (regression lines)

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