Modeling Colonization and Extinction Events Separately

[Bird Counter]

I give a fair amount of background information here so please bear with me.

I originally used single season models to describe occupancy patterns (within a single habitat type) in each of 3 post-disturbance years. This is an excellent application for multi-season models but I had problems making the models work and abandoned them. Single season models Ψ(cov),p(_) have worked just fine. Nevertheless, there are some interesting differences in occupancy patterns among years that could (should?) be explored with dynamic models. I can (should?) expect reviewers to ask about this issue.

I have a modest dataset: 60 sites surveyed up to 13 times over 3 years (63% were surveyed three years, 26% two years, and 11% in one year). Detectability was not the main issue as the overall probability of detecting the species >83% in any year. My question of interest is did the 3 covariates of interest influence λ/ε and were the effects different each year, so I am modeling cov:yr.

The main gist of the problem is data sparseness of colonization/local extinction events in one or both years. I tended to see more colonization events with few or no extinctions in the 2nd post-disturbance year, and more extinction events with few colonizations in the 3rd post-disturbance year. Obviously, this is not a modeling problem and is in itself an important piece of information, but there are enough λ/ε events to evaluate in at least one year. Note: I am NOT interested in taking a model selection approach, so fitting a single model that makes biological sense and works is all I want, ideally this: Ψyr1(covs), λ(covs), ε(covs), p(yr).

I have tried fixing gamma2 and/or eps1, and supplying initial values. But given this is caused by a data sparseness problem, there isn’t a lot I can do. However, I can make the following models work (all of the covs are the same).

Ψyr1(covs), λ(.), ε(covs:yr), p(yr)

Ψyr1(covs), λ(yr), ε(covs:yr), p(yr)

Ψyr1(covs), λ(covs:yr), ε(.), p(yr)

Ψyr1(covs), λ(covs:yr), ε(yr), p(yr)

This tells me that I do not have enough λ/ε events to model the effects of the covs on both simultaneously, which makes sense.

So now my question. Does it make biological sense to use this model,
Ψyr1(covs), λ(covs), ε(.), p(yr), to explain λ and this model, Ψyr1(covs), λ(covs), ε(.), p(yr), to explain ε separately? I am modeling both at the same time, obviously, but I am saying with these models is that ε is influenced by covs, assuming constant λ probability, and visa versa.

It doesn’t seem right to me because I should (I think) be accounting for the effect of the covariates on both simultaneously to answer my question of interest above. I am double-dipping into the dataset to explore ε and λ separately.

By the way, one of these models does explain the difference in occupancy patterns between the second and third post-disturbance years.

[jhines]

You can use any model to create 'stories' about how parameters are affected by covariates, but poorly fitting models will yield poor (ie. biased) estimates. Generally, you can go to a more general model without introducing bias, but standard errors will be larger. This is the benefit of model fitting procedures. They attempt to balance the reduction in standard errors versus the bias introduced by pooling parameters.

I assume you mis-typed your question: Does it make biological sense to use this model,
Ψyr1(covs), λ(covs), ε(.), p(yr), to explain λ and this model, Ψyr1(covs), λ(covs), ε(.), p(yr), to explain ε separately?

as you've listed the same model twice. If you meant to ask about having gamma constant in one model to explain epsilon, and a model with epsilon constant to explain gamma, then I would not advise it. Constraining gamma to be constant will have an effect on the estimate of the effect on epsilon, and vice-versa. I would let use model selection and let AIC choose the 'best' model. If the top model does not include the covariates for gamma or epsilon, you can still use the estimates from the more general model with the covariates to say something about the 'direction' (positive or negative effect of the covariate), although not being significant.

[Bird Counter]

Hi,

First of all, I am sorry I have not been able to get back to the forum to reply to this until now.

Second, thank you much for your reply, Jim. I have been struggling with this issue for awhile. My thinking was along the lines of what you said, that the constraining one parameter will affect the estimate of the other and produce biased results. I cannot fit the most general model Ψyr1(covs), λ(covs), ε(covs), p(yr) because I have too few observations of gamma and episilon to model them simultaneously. Instead, I need to set these up as alternative hypotheses within a model selection framework (which I have done a lot of with single season models). In this case, one hypothesis is that the probability of colonization is constant across sites, but that the probability of the species becoming locally extinct at a site is influenced by covs.

I have two follow up questions. First, the model Ψyr1(covs), λ(covs), ε(covs), p(yr) is not workable as I noted. I remove it from the candidate model set because its model wt is .9999 making the entire process not useful. When presenting the results with models like this, I have included them in the table but noted that they were not-estimable so people understand that this model was included but did not produce estimates. Does that make sense?

Second, I am again and always thinking about effective sample size. Within this dataset, I have 60 sites surveyed 3 times in year 1 and 3-5 times in years 2 and 3. I asked a question about this for single season models in the general design forum, but it doesn't get much traffic (http://www.phidot.org/forum/viewtopic.php?f=34&t=1513). Multi-season models have a lot of parameters (8 - 18 for my model set). The models are estimable with the exception of the global model.

Thanks much again.

Misty

Ps. Yes, that was a typo in my original question, thanks.