Streamlining some multi-year analysis

[bacollier]

All,
I have been fiddling with a data set collected under a robust-design framework, 4 years, 6 sampling occasions per year. My interest is in using the robust-design approach, but I am not interested in estimating gamma/epsilon (for biological, not statistical reasons). I have both site specific covariates and survey-dependent covariates and I have ran the data through a standard RD design where I can estimate gamma/epsilon.

I am wondering (and looking for a reference/example), is there a simple way to keep the structure and ease of implementation of an RD design for multi-season study when gamma and epsilon are not of interest, or am I going to have to brute force this analysis and break the dataset into 4 years and run each year separately as I am interested in annual occupancy estimates and associate detection both within and between years? Should I be looking at fixing the parameters for gamma/epsilon?

Note, I have been working in MARK via RMark.

Bret
Tx A&M

[darryl]

Hi Bret,
You have a few options.

1. Just derive estimates of occupancy from your estimates of gamma/epsilon using the recursive equation psi[t+1]=psi[t]*(1-epsilon[t])+(1-psi[t])*gamma[t]. PRESENCE now reports this at the bottom of the multi-season output.

2. Use a reparameterised version of the model (available in both MARK and PRESENCE) to estimate occupancy directly. Only point here is that there are constraints on the possible values of psi otherwise you can end up with values for gamma or epsilon (whichever is now the derived parameter) that are outside of 0-1. In PRESENCE this constraint is enforced which can make things a bit numerically unstable (in terms of getting convergence), particularly if you've got a few covariates. In MARK this constraint isn't enforced (at least not when I last checked) so you could get an answer that gave you biologically unreasonable estimates. Therefore you need to check the derived parameter values for all sites (if you have covariates).

3. Use the 'random changes in occupancy' reparameterisation that's available in PRESENCE (not sure about in MARK; model occurs through setting the constraint epsilon=1-gamma). This is equivalent to fitting a series of single-season models without rearranging your data.

Cheers
Darryl

[bacollier]

Thanks Darryl,

All,
I finally got back to this and I have a more complicated (I think) follow up question. I used what Darryl describes in No. 3 (random changes in occupancy) to model the robust design data. Consider the situation where the occupancy model I am interested in has Psi as a function of Patch Area (patch size in hectares) and eps/gam is a function of Patch Proximity (how close the patch is to the nearest other patch in the landscape). Note: I am not including detection as it is not relevant to this question.

So, from a model description standpoint I have

Psi(PatchArea), eps=1-gam(PatchProximity);

both PatchArea and Patch Proximity are continuous variables.

I can easily estimate Psi[t] from the beta’s provided in the output across the range of the covariate. Normally, this would be fine as we are often interested in predicting occupancy as a simple function of some covariate. But, I would also like to use one of the recursive equations to evaluate occupancy in Psi[t+1] and Psi[t+2], etc. In most examples, Psi, eps, and gam are shown to be either constant or time specific (the book and some of the online help stuff from Vermont Coop Unit), which make estimation of Psi[t+1],etc. fairly easy using the following equation:

Psi[t+1]=Psi[t]*(1-eps[t])+(1-Psi[t])*gam[t]

because constant or time-specific estimates of Psi/eps/gam allow for straight plug and chug into the above formula.

However, in my situation, to estimate occupancy in future years, I ‘think’ that I would have to either 1) model Psi[t+1] using the average of the covariate values for Psi[t] and eps/gam for each year (year specific covariates since patches enter and exit the dataset) or 2) would I need to simulate all possible combinations of Psi/gam/eps relationships and plot the predicted Psi[t+1] to show the predicted impacts of changes in the covariates?

Am I right, or did I miss something?

Bret

[darryl]

Under the 'random changes' parameterisation, epsilon[t]=(1-gamma[t])=(1-psi[t+1]). So you don't need to use the recursive equation. In fact with the model you're fitting you're effectively saying that only first year occupancy is a function of patch area, and in later years occupancy is a function of patch proximity. How biologically reasonable is that?

In what sense are you trying to interpret psi? The probability of a patch being occupied (which may be a function of certain patch characteristics), or what fraction of patches are occupied in a give year? The 2 are subtly (and importantly) different.

Just because you have covariates doesn't stop you from using the recursive equation, you just have to apply it to each patch individually (which should be trivial if you're a regular R user). See the house finch example in chapter 7 of the book where distance was included in the models as a covariate.

Cheers
Darryl

[bacollier]

Darryl,
Ok, that makes more sense. I understand your statement on patch area and patch proximity, sometimes you just stare at some things way too long (guess I should work faster).

My interest on interpreting Psi is in the probability of a patch being occupied |patch characteristics. So, I will need to estimate Psi for each of the 3 years for each patch and just develop a set of plots showing Psi given PatchSize.

Thanks,
Bret

[darryl]

My interest on interpreting Psi is in the probability of a patch being occupied |patch characteristics. So, I will need to estimate Psi for each of the 3 years for each patch and just develop a set of plots showing Psi given PatchSize.

I can tell you now that it will be a flat relationship is years 2 and 3 if you stick with your current model set....