many ages at each marking and extensive aging through study

[WiPhi]

Only after reading your post do I see I was poorly asking if pi could modeled with group factors (I see now that it can). I think I was also searching for general rules when fitting mixtures when other population structure (groups) is also present. Here, is where my thinking is at:

Because mixtures are addressing some invisible structure in the population, it is not clear whether we should expect greater variation in the mixing of mixtures across groups (effects on pi) or in the detection probabilities within in each mixture (additive and interactive effects on p when mixture is also in the detection model). Just wondering if anyone has offered guidance other than fitting candidate models with emphases on variation in pi, p or both and letting AIC sort it out?

I also wonder how frequently factors that define the underlying population structures with effects on detection probability are completely uncorrelated? When facing unknown structure in the population (mixtures) that create individual heterogeneity in detection probability, it would seem best practice would be to consider that pi is linked to other variables, particularly the factors that are thought to influence p.

[Bryan Hamilton]

I think that's one way to look at it. Then test that question with AIC and model selection. Mixtures are based on individual heterogeneity and that may or may not be best explained by adding groups to the mixtures.

Somewhere in the MARK Book there is quote that "Individual heterogeneity is the bane of capture recapture models". When individual heterogeneity in detection exists, the mixtures do a better job than nothing, but they still struggle and always seemed like a "black box" to me.

One thing I try to keep in mind with CJS models, detection is really a nuisance variable. We want to do our best in modeling it but mostly we're interested in detectibility to get at the other parameters.

[WiPhi]

Thanks Bryan,
Yes, i think deciding whether different patterns of heterogeneity within groups (unique mixtures) could be occurring in your system is important. When we aren't sure what is driving the individual heterogeneity (which is why we fit mixture models in the first place), it would seem logical to assume pi may vary (across groups, space, or time) as much as p may vary within each mixture. Interested to see what others encounter when they compare models that differ in whether group, individual covariates, study covariates (time), or environmental covariates are modeled in pi, p, or both. I will share some initial results when I have them.

Yes, I agree, important to stay focused on the demographics. My demographic model is defined by the study question so I am using my time to fret over modelling detection . If detection can be nailed down (and model assumptions are not violated) then I can focus on the inference into survival with confidence.

[Bryan Hamilton]

I've used individual heterogeneity mixtures in closed capture models but never open. Sounds like another rabbit-hole to explore...

[WiPhi]

Into the rabbit hole...

Here is a follow-up with some results. Just to recap, the primary question I was asking was whether there exists a general result when individual heterogeneity in detection is modelled as a mixing of 2 or more mixtures within groups (such as gender) versus variation in detection probability across groups and mixtures? For example in a CJS model with Pledger mixtures (CJSMixture, mixtures=2), how do models generally compare when a group variable like sex is modelled affecting the mixing parameter pi, versus the detection parameter p?
pi(~sex) Phi(…)p(mixture)
versus
pi(~1) Phi(…)p(mixture+sex)
We might go further and ask about interactions between mixtures and grouping variables.
Here is a comparison of 8 models that do just that using 8 years of bimonthly lion detections on several hundred individuals. The two models above are models 6 and 4, respectively. I also include a CJS model without mixtures but a gender effect on p (model 5 below), to compare to the mixture model with no effects on p other than mixture (model 1). An intercept only model for p is also shown (model 8). Survival is modelled the same in each case as a function of 5 age-classes with a sex interaction.

Code: Select all

model   DeltaAICc   weight   npar   form
3   0.00   0.57   15   pi(~1)Phi(~...)p(~mixture * sex)
8   1.79   0.23   15   pi(~sex)Phi(~...)p(~mixture + sex)
7   2.37   0.18   16   pi(~sex)Phi(~...)p(~mixture * sex)
4   8.20   0.01   14   pi(~1)Phi(~...)p(~mixture + sex)
2   9.96   0.00   13   pi(~1)Phi(~...)p(~mixture)
6   10.49   0.00   14   pi(~sex)Phi(~...)p(~mixture)
5   487.44   0.00   12   Phi(~...)p(~sex)
1   494.37   0.00   11   Phi(~...)p(~1)


Models 6 and 4 are separated by 2.29 AICc units suggest modelling the sex effect is best done as an additive effect on detection (in the presence of a mixture) rather than as an effect on mixture. These models, while structurally very different produced a similar result (checking the survival estimates showed <1% disagreement between models except for one age-sex class where model disagreement in Phi was 5%, this was true across all the top 6 models)
Models 6, 7 & 8 suggest that the pi mixture parameter is not the same in each gender group (i.e. unique proportions of males and females within mixtures 1 and 2) but gender effects on p should also be included.
I guess I was expecting further spread in the models along mixture effects so I would say that group effects on pi might be as important to consider as effects on p.