# Is there no such thing as a multivariate generalized linear mixed model?

Fong, Y., Rue, H., & Wakefield, J. (2010). Bayesian inference for generalized linear mixed models. Biostatistics, 11(3), 397-412.

and was surprised to see that the Generalized Linear Mixed Model (GLMM) is defined such that the outcome variable is a scalar (see Section 2).

Two reasons for my surprise include:

• The construction defines the mean of the exponential family distribution as a function of covariates and regression weights, and the exponential family includes multivariate distributions (e.g. multivariate normal).
• I expected that multivariate normals could be handled as a special case.

Is there no such thing as a multivariate generalized linear mixed model? If not, why not?

• multivariate exponentially families are very complicated; for instance, they are not uniquely determined by their mean-(co)variance relationship. Commented Mar 7, 2021 at 3:07
• Multivariate probit? en.m.wikipedia.org/wiki/Multivariate_probit_model
– Dave
Commented Mar 7, 2021 at 4:17

Yes, there is such a thing as a Multivariate (multi-response) Generalized Linear Mixed Model (MGLMM)

Many popular software packages for fitting GLMMs are unable to handle multiple responses, especially those that work utilise the frequentist paradigm. However if you adopt a Bayesian approach then there are a number of options, such as BUGS, JAGS, Stan and the R package MCMCglmm. The latter even has a good vignette: "MCMC Methods for Multi-response Generalized Linear Mixed Models: The MCMCglmm R Package":
https://cran.r-project.org/web/packages/MCMCglmm/vignettes/Overview.pdf

There are also a number of relevant journal papers:

Bailey, T.C. and Hewson, P.J., 2004. Simultaneous modelling of multiple traffic safety performance indicators by using a multivariate generalized linear mixed model. Journal of the Royal Statistical Society: Series A (Statistics in Society), 167(3), pp.501-517.