*Statistical Modelling* **1** (2001), 177–193

##
A Multivariate Generalized Linear Mixed Model for Joint Modeling of
Clustered Outcomes in the Exponential Family

R. Gueorguieva

Associate Research Scientist, Division of Biostatistics

Department of Epidemiology and Public Health

Yale University School of Medicine, 60 College Street

P.O. Box 208034 New Haven, CT 06520, U.S.A.

e-mail: ralitza.gueorguieva@yale.edu

### Abstract:

Clustered and
repeated measures data are very common in biomedical applications, for example
when one or more variables are measured on each patient at a number of hospital
visits, or when a number of questions are asked at a series of interviews.The
Generalized Linear Mixed Model (GLMM) can be used for fully-parametric
subject-specific inference for clustered or repeated measures responses in the
exponential family. In the current paper a multivariate generalization is
proposed to deal with situations when multiple outcome variables in the
exponential family are present. Separate GLMM's are assumed for each response
variable and then the responses are combined in a single model by imposing a
joint multivariate normal distribution for the variable-specific random
effects. This allows maximum-likelihood estimation approaches such as
Gauss-Hermitian quadrature and Monte Carlo EM algorithm to be extended from the
univariate to the multivariate case. Two data sets are used for illustration.
The outcome variables are assumed to be conditionally independent given the
random effects which is a restrictive assumption in some cases. Score tests for
checking this assumption are proposed and alternative models are considered for
one of the data examples.
#### Keywords:

Conditional independence; Gaussian quadrature; Monte Carlo EM algorithm;
Multivariate response; Random effects.
#### Downloads:

Data and software in
zipped
archive
Software uses
SAS and Ox, an object-oriented matrix language with a comprehensive
mathematical and statistical function library. For the latter refer to
http://www.nuff.ox.ac.uk/Users/Doornik
or http://www.timberlake.co.uk.

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