model selection metrics for MCMC, negative 2*p_D value

[dpeters]

Dear Forum,

I’m fitting CJS model to estimate survival of hatchery-reared juvenile steelhead smolts during their seaward migration through the Snake and Columbia Rivers. There are 8 “occasions” (dams with PIT interrogation sites), 7 groups, and <17,000 individuals in the encounter history for the year in question. When I use the logit link, MARK is having a hard time estimating the number of parameters so I have been fitting models using the sin link.

I am exploring the use of the MCMC estimation capability in MARK because: (a) there are a number of parameters estimates near the [0,1] boundary, and (b) I need to derive survival estimates and CI across multiple dams (e.g., from release to dam 5) that are the product of the estimates between the individual dams. I realize that the Delta method could be used to derive the variance, but there is some advantage to having the posterior distribution for easier interpretation and comparing differences between groups.

During preliminary analyses and review of the MCMC output, I noticed that the penalty term (2*p_D) of the Deviance Information Criterion (DIC) was a negative number, such that the DIC < -2logL. E.g.,
-2log Likelihood for means of beta estimates = 46467.439
Penalty function 2*p_D for DIC = -10012.086
DIC = 36455.353
WAIC = 41530.705

Any ideas why the negative value for 2*p_D? I realize these model selection metrics should be applied with caution; I’m just trying to understand what is happening here.

Another thing I have noticed that may provide some context: for the models examined so far, the DIC values differ depending on whether sin or logit link is passed to MARK during the MCMC estimation. When the logit link is passed, the DIC is nearly equal to WAIC, but when sin link is passed DIC and WAIC are quite different.

Thank you.

[cooch]

Another thing I have noticed that may provide some context: for the models examined so far, the DIC values differ depending on whether sin or logit link is passed to MARK during the MCMC estimation. When the logit link is passed, the DIC is nearly equal to WAIC, but when sin link is passed DIC and WAIC are quite different.

Thank you.

Yes, because the logit link is monotonic, and the sin link isn't. You should use only the logit (or some other monotonic) link function for MCMC in MARK (this is noted in passing in the MCMC appendix - p. 6). The use of the sin for ML fitting is fine (except for certain covariate structures in the DM), but the advantages of the sin link (estimation near the boundary) aren't as critical when using MCMC (meaning, MCMC does reasonably well with describing the posterior for boundary estimates).

So, I'd suggest not comparing apples and oranges: if using MCMC in MARK, stay with the logit link, inless you're really sure the sin link isn't causing major weirdness (which, in my experience, it often does...).