When should multivariate GLM(M)s be used? Suppose I have a set of $k$ dependent variables which are all correlated with each other and known a priori that they are dependent. I also have a set of $p$ independent variables (predictors) which could potentially be correlated (or potentially repeated measures). This could include both independent variables of interest which are modeled as fixed effects, and nuisance covariates which are modeled as random effects.
Under what circumstances is it better to use $k$ independent generalized linear mixed models vs. one multivariate GLMM? Does it depend on the assumptions we make about the distributions of the dependent variables?

One multivariate GLMM will capture correlation in your dependent variables.
$k$ independent GLMM will not.
