Joint hierarchical generalized linear models with multivariate Gaussian random effects

Likelihood based inference for correlated data involves the evaluation of a marginal likelihood integrating out random effects. In general this integral does not have a closed form. Moreover, its numerical evaluation might create difficulties especially when the dimension of random effects is high. H-likelihood inference has been proposed where the explicit evaluation of the integral is avoided. The approach also allows extensions handling e.g. (1) complex design experiments, (2) REML type of inference beyond the class of a linear model and (3) overdispersion modeling. The h-likelihood approach to multivariate generalized linear mixed models is extended. The h-likelihood computational algorithms is blended with a Newton-Raphson procedure for the estimation of the correlation parameters. This allows that components of the joint model are interlinked via correlated Gaussian random effects. Further, correlated random effects are allowed within each component. This approach can serve as a basis for further developments of joint double hierarchical generalized linear models with correlated random effects. The methods are illustrated with a rheumatoid arthritis study dataset, where the correlation between latent trajectories of three endpoints is evaluated.

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