Re-parametrization in multivariate models (MCMC)

[NilsGYork]

Hello

I'm struggling to estimate a multivariate (4 outcomes) multilevel (2 levels) model on approx 100,000 observations using MCMC. Correlation between outcomes is allowed through a MVN error distribution at first and second level. The ESS on the constant terms in each of the four models is very low and trajectory plot suggests very bad mixing (ESS=50 with 5000 iterations). The other coefficients are generally fine (ESS>1000 with 5000 iterations). My ultimate interest will be in estimates of the second level unobserved effects (here: hospital effects), but I am concerned that the low ESS on the intercept will cause me problems.

To address this problem, I've started to look into re-parametrization techniques. However, it seems that many of the approaches implemented in MLwiN don't work in a multivariate framework. Can you confirm this? Is there any alternative approach that I should consider?

I've explored this issue by running univariate models (Gaussian) (see pdf file attached). As expected, hierarchical centering solved my problem of high auto-correlation, whereas SMVN greatly reduced the time it took to estimate the model. Orthogonal re-parametrization didn't do much. After that I moved to a bivariate model (both Gaussian) but none of the options seems to work. SMVN resulted in a cryptic error message, hierarchical centering wasn't allowed. Orthogonal was once again very similar to simple Gibbs(?) sampling.

I am using MLwiN 2.29 through x64\mlnscript.exe. I'm also using runmlwin from within Stata 12, but I don't think that's the problem here.

Any comments or suggestions would be highly appreciated.
Thanks
Nils

// Edit: Orthogonal re-parametrization seems to work in the bivariate setting. I've edited my response accordingly, sorry.

[billb]

Hi Nils,
Lots of questions so here are some pointers:
1) Orthogonal reparameterization is in theory only of use when MLwiN has to use Metropolis Hastings i.e. when we don't have normal responses and it improves mixing as it gets around the correlations between the various fixed effects. If MLwiN can do Gibbs sampling i.e. with normal models then it will do block updating of all fixed effects which already gets around this problem.
2) SMVN is very limited in how it is implemented as you note so only for univariate normal responses
3) hierarchical centering should work for MV response normal models as well as univariate - I have just tried it on my machine with the GCSEMV dataset and just intercepts and it works well - one thing to realise is that you need to centre at level 3 and not 2 (which if you go through the interface is the only option) as in a MV response model the responses are level 1 perhaps from runmlwin this wasn't obvious. Another caveat as detailed in my chapter is that hierarchical centering is not quite the standard algorithm as implemented in MLwiN - I'd suggest reading the chapter of the MCMC manual for an explanation.

Hope this helps,

Bill.

[NilsGYork]

Hi Bill,

thank you for your very helpful response. Yes, I think I got tricked by the feedback that I received from runmlwin, saying that hierarchical centering is not an option here. I'll try centering at level 3 and will report back.

Thanks again
Nils