Modeling correlated random effects


I have a question about modeling correlated random effects. The model is:

I converted it to non-centered parametrization:
where image is 10 dimensional vector of parameters and f is system of DOEs.
However, stan gave me two modes for imagewhich are visible for some i. Pairs plot for vectorimageshows that scales of random effects are correlated. Therefore, my guess is that by introducing covariance matriximagewould help to remove those correlations and hopefully multiple modes. The problem is in PK domain and i is a patient while t is time when a sample of blood is drawn (imageis plasma concentration for patient i at time t). Each patient has 10 PK parameters and there are two groups of patients. Because of physical considerations (different transporters) some parameters in the second group should be fixed to 0. Can anybody suggest how to define covariance matrix imagein such case. Actually I was planning to use Cholesky decomposition to speed up stan simulation.

Thanks for any advice,

This is not easy to do unless you can assert that Sigma is block-diagonal, i.e. a dense correlation matrix among group one and a different dense correlation matrix among group two but zero correlation between groups.

Otherwise, if you can arrange things so that the zeros occur in the bottom right of the correlation matrix, then if the non-zeros are sufficiently close to zero, the entire correlation matrix may satisfy the positive-definiteness condition. But then you are basically looking at building a Cholesky factor yourself and solving a bunch of equations.

If there is really one one correlation that has to be zero, you can go uniform on the entire correlation matrix plus a very strong prior on the one that must be (near) zero.