Jean Richardson writes:

Do you know what might lead to a large negative cross-correlation (-0.95) between deviance and one of the model parameters?

Here’s the (brief) background:

I [Richardson] have written a Bayesian hierarchical site occupancy model for presence of disease on individual amphibians. The response variable is therefore binary (disease present/absent) and the probability of disease being present in an individual (psi) depends on various covariates (species of amphibian, location sampled, etc.) paramaterized using a logit link function. Replicates are individuals sampled (tested for presence of disease) together. The possibility of imperfect detection is included as p = (prob. disease detected given disease is present).

Posterior distributions were estimated using WinBUGS via R2WinBUGS.
Simulated data from the model fit the real data very well and posterior distribution densities seem robust to any changes in the model (different priors, etc.) All autocorrelations and cross-correlations are near zero, except for the correlation between deviance and p which is ~ -0.95 consistently under different model fits. Removing p from the model completely does not seem to change posterior density distributions greatly, but does of course increase the mean deviance value greatly.

I feel certain this large correlation with deviance is not a good thing, but I’m stymied as to what the problem may be.

My reply: I’d expect this to happen if you have a bounded parameter and the maximum likelihood estimate is on the boundary. If the mle is in well in the interior of parameter space, I’d expect the min deviance to be at that mle. That is, I’d expect deviance to be highly correlated with (theta – theta.hat)^2.

P.S. I was right! Richardson reported:

The parameter is indeed near its maximum value of 1 (the prob. of detecting disease on a diseased individual is about 0.94 and as a probability the parameter is of course bounded at 1).