Model averaging pi's and p's from closed capture mix models

[jlaufenb]

This is a complex question, so I will try to make this as concise as possible.

Here's the scenario:

-Full Closed Capt w/ Het data type w/ 2 groups (M&F)
-10 capture occasions resulting in full DM of 80 rows
-12 candidate models
-some w/ 2 mixtures, some w/o and properly constrained
-some 2-mixture models w/ group specific mixtures (additive)
-data set supports mixtures w/ high probability of coming from mixture w/ low p
-I want to report model-averaged estimates of p for each group and measure of heterogeneity


Problem:
-sometimes, for individual mixture models w/ similar structure, MARK outputs pi's representing different mixtures with corresponding reversals in p's outputed (as does RMark)
-Example: for 2 models with same mixture structure; model 1 outputs pi=.75, pA=.1, pB=.4 and model 2 outputs pi=.25, pA=.4 and pB=.1.

My concern is that MARK model averages corrresponding rows from each model according to the full DM. In the above example, this would result in model weights being applied to parameter estimates that essentially represent different mixtures.

I planned on using the formulas from Carothers (1973, Biometrics 29) that Boulanger et al. (2006, Ursus 17) used to calculate mean capture probabilities based on 2 mixture distributions and CV's as indices of heterogeneity. I just need clarification on how to ensure that the model-averaged estimates I use are right.

Thanks
Jared

[ganghis]

Hi Jared,

Have you tried setting initial values for all models close to those estimated for one of the modes of the two point mixture? This is a bimodal likelihood surface so you want to make sure the 'hill-climbing' algorithm knows which hill to climb. My guess is if you set initial values for pi and p to correspond to one of those solutions that you will get similar solutions for each model and model averaging will make more sense.

Cheers, Paul

[cooch]

Hi Jared,

Have you tried setting initial values for all models close to those estimated for one of the modes of the two point mixture? This is a bimodal likelihood surface so you want to make sure the 'hill-climbing' algorithm knows which hill to climb. My guess is if you set initial values for pi and p to correspond to one of those solutions that you will get similar solutions for each model and model averaging will make more sense.

Cheers, Paul

Paul makes an excellent suggestion, but I want to add a `cautionary' note that in many (perhaps most?) situations, the pi is not interpretable 'biologically'. The finite mixture is an approximation, and its easy to come up with radically different underlying continuous distributions for encounter probability that would yield the same fixed points in the finite mixture model. Which, of course, is nonsense, since the true underlying distributions are different. Not to mention the fairly strong assumption (at least in some cases) that Pi is fixed over the sampling interval.

I'm not saying don't pursue this - I'm simply pointing out that quite often, there is little -> no biological information in Pi (do you really think encounter probability is split into discrete classes of individuals?). The Pledger models (and related things) focus on using mixtures as an approximation - they can improve precision of estimates of N, but beyond that. See recent work by Dorazio and Royle on continuous mixture models for a pretty decent discussion of all this.

[murray.efford]

Jared
I think Evan means 'improve accuracy' rather than 'improve precision' (finite mixture models are almost guaranteed to blow your SE out of the water, and continuous mixtures don't solve that problem). You risk getting mired in the ritual of N estimation (ritual sensu Guthery 2008 JWM 72:1872-1875). Better to use a natural model for the heterogeneity, in the shape of spatially explicit capture-recapture, but I know you know about that...
Murray

[cooch]

Jared
I think Evan means 'improve accuracy' rather than 'improve precision' (finite mixture models are almost guaranteed to blow your SE out of the water, and continuous mixtures don't solve that problem).

Indeed. Sorry, my mistake (was rushing to finish the note in time to get to a class).

You risk getting mired in the ritual of N estimation (ritual sensu Guthery 2008 JWM 72:1872-1875). Better to use a natural model for the heterogeneity, in the shape of spatially explicit capture-recapture, but I know you know about that...
Murray

If heterogeneity can be plausibly modeled based on some ultrastructural relationship to a covariate (or several) in space/time, then absolutely. But, in lieu of such insight (the key word being plausible - a strong mechanistic expectation that some set of covariates strongly links to heterogeneity), mixture models have their place. I would agree (I think we're agreeing) that they're often used uncritically, but their performance as an omnibus tool to 'do a better' job of estimating N (which is the objective, by and large) is pretty decent - my point about continuous mixtures is that they're worth considering. Quite frankly, most people don't use them simply because they're not sitting as a 'click here' option in MARK (or equivalent).

[murray.efford]

Responding to Evan -

I think we're very much on the same wavelength regarding mixture models. (In fact, the beta binomial for N is "sitting as a 'click here' option" (actually a drop down) for nonspatial analysis in Density, where you can also see plots of the unpleasant likelihoods that result (MLE tab) and compare with 2-part finite mixture...).

"If heterogeneity can be plausibly modeled based on some ultrastructural relationship to a covariate (or several) in space/time, then absolutely"

I take that as an absolute endorsement of SECR .

Murray