Dear Allstat,

I'm trying to identify an appropriate, calculable, Information criterion
for model selection in the following circumstances. I've fitted a Markov
disease progression model to a set of censored (several types) individual
level
data for patients with HIV. The transition rates are modeled on the log-rate
scale and the baseline parameter allows for the possibility of tracking
(i.e. where persons who's Cd4% has declined more quickly in the past are
more likely to do so in the future). For each person, there is an estimated
probability that his or her baseline rate is that of a fast tracker vs. a
slow tracker.

This means I have fitted a mixture model where the likelihood function for
each observation includes multiple parameters that are nonlinearly
correlated (due to censoring). I have a couple of questions relating to this
problem:

Firstly, Aside from the problem of mixing: - Because the data are censored,
the likelihood function for each observation includes multiple non-linearly
correlated parameters. This means, I think, that even if the mixing
distribution is properly accounted for using the methods described in Celeux
et al 2006, the DIC will still not be calculable. This is because when
estimating the Deviance calculated at the posterior mean/median/mode (D.hat)
the results are incorrect because of the non-linear correlations. Is this
correct? If so does the same problem arise when estimating the BIC or the
AIC? I have looked at the slideshow from IceBUGS and didn't quite understand
how the first term in the BIC and AIC should be calculated for a model that
is not hierarchical (D.Hat,D.Bar?). Also, is it reasonable to use the pV as
an estimate of the parameters in the AIC or BIC.

Secondly: Is a mixture model a type of hierarchical model? and are we
expecting in this example for the effective number of parameters to be very
high (1 per person for the mixture model + a few other covariates). Does the
pV (or pD(2) in Gelman et al 2004) provide a useful estimate in a mixture
model? I calculated it for the eyes Normal mixture model example and found
that pV = 58. I was unsure if this is a robust estimate of the effective
number of parameters for this model and entirely unsurprising, or if I have
inappropriately used the statistic?

Thirdly, I have found that all of these different measures are fairly
disparately reported and I have not found a single source that explains
everything fully. Does anyone know of a textbook that does this?

As far as I can make out the only calculable statistics would be the
posterior mean deviance + either the pV, or the total number of parameters
in the linear predictors for the log-rates excluding the individual level
mixing probabilities. Possibly adjusted to account for sample size as in the
BIC.

Thank you very much for any help you can offer

Best Wishes

Malcolm Price
PHD Student
University of Bristol - Dept. Social Medicine
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