Calculating overall occupancy

[jbauder]

I am developing a long-term monitoring protocol for eastern indigo snakes using occupancy as our state variable. We conduct our surveys each winter over a five month time period and our goal is to compare a single, overall occupancy rate for each winter across time. Based on previous posts and MacKenzie et al.'s occupancy book, I have found two possible methods for calculating a single, overall estimate for a single season. I cannot determine which method is appropriate for my application and would like clarification as to the consequences of using either method. I have single best model I am making inferences from (AICc weight = 0.7417) containing three continuous site covariates and constant detectability (p(.)), with 34 sites and four repeat surveys. My naive occupancy is 0.2941 and detection probability is 0.4038.
The first method I have seen is to take the logit transformation of the beta estimate for the intercept of psi, which in my case would be (exp(-0.858294))/(1+(exp(-0.858294)) = 0.2977. The second method is to average the individual site estimates of psi, which comes to 0.3642. If my goal is a single estimate of occupancy that incorporates the information provided by the site covariates and detection rate that I can compare over time (along with SE and confidence intervals), which of these values are valid? I understand that the delta method can be used to generate SE's for the former method but could you clarify how they would be derived for the second method?

Thank you very much for your help

Javan Bauder

[jhines]

Your first method only compares the estimates of psi for sites where all covariates are zero. It doesn't take into account other sites where covariates are nonzero. I think what you want to do is compare the distribution of psi's for one year versus the distribution of psi's in another year. The first thing that comes to mind is to use program CONTRAST (www.mbr-pwrc.usgs.gov/software) to compare the two groups of estimates. I'm not sure if the assumption of normality of the estimates holds though.

My suggestion would be to use model selection to test for differences in occupancy in years. Use a multi-season model with one of the year-specific psi parameterizations. Then run a model with yearly occupancy as a function of year and your covariates, and a model with yearly occupancy as a function of only the covariates (constant with regard to year). If the first model has the lowest AIC (by at least 2), then occupancy differs between years. If the 2nd model has lowest AIC, then occupancy is (relatively) constant between years. If AIC's are close, then there might be a difference in occupancy, but not a big difference.

Jim

[jbauder]

Thank you very much for your prompt response. I was planning to use the multi-year occupancy models once we had a few years of data. For now, I was looking for a single value to report to the agency funding the project in our annual progress report.
I have another very basic question. I am trying to run a single-season multi-method model to test for differences in detection rates between indigo snake shed skins and the snakes themselves but I cannot figure out how to set up the data input form for the multi-method model. Could you please explain how the form should be set up?

Thanks
Javan

[jhines]

The data format for the multi-method model is the same as a multi-season model. Instead of having k surveys per season, you have k methods per survey. So, if you have 3 methods, and 2 surveys, there will be 6 columns in the detection data. Column 1 represents the status for method 1 in survey 1, column 2 for method 2, and column 3 for method 3. Columns 4-6 represent the status for the 3 methods for survey 2. Just be sure to enter 3 surveys per season it the appropriate box near the top of the form.

Jim

[darryl]

Hi Javan,
Getting back to your original question, what exactly do you regard as a 'site', how did you select them from your region of interest, how many other 'sites' are there within your region of interest that you could have sampled, but didn't, and finally what are the covariates you've used and do you have the respective values for places you haven't surveyed?

Cheers
Darryl

[jbauder]

Hi Darryl
I defined my sites as sandhills (ie, polygons) within a sandhills GIS layer for our study area, which is a biologically meaningful unit since sandhills are distinct from surrounding habitat and during the winter indigo snakes concentrate on these sandhills to overwinter and there is little or no movement among sandhills. We randomly selected 34 sandhills out of 77, stratifying our sampling by subdrainage (five strata) and ownership (public or private). For the time being (mostly due to small sample sizes) I am not interested in generating separate occupancy estimates for each stratum; their purpose was just to ensure uniform spatial coverage of our entire study area. The covariates in our best selected model were the number of tortoise burrows on the sandhill, the size of the sandhill, and the density of tortoise burrows on the sandhill. All the beta estimates were positive, which makes sense from the biology of the indigo snake. However, we do not have these covariates for the other 43 sandhills, since the number of tortoise burrows is something we can only collect by surveying each sandhill. Were you thinking of using the regression equation from this best model to generate predicted occupancy estimates for the other 43 sandhills and then take the average predicted occupancy for all 77 sandhills to get an overall occupancy estimate for this first season?
Thank you very much for your response

Javan

[darryl]

I was going to suggest something along those lines, except that for those sites you've actually surveyed you use the conditional estimate of occupancy (conditional on the observed history, it's in the book) rather than just the estimated probability of occupancy. However, because you don't have the covariate values for your other sites, then you'll just have to use a psi(.) model (or you could include stratum effects, but sample sizes will be low) rather than something fancier. This should be reasonable if you really have a random sample. You should also look in the book for the section on finite samples as that is the situation you're in. It will be to your benefit as by taking the extra steps you're able to get a smaller SE. Note, that even if you don't include covariates on psi, you still want to check them out for p as they may be required to account for heterogeneity.

Good luck.

Cheers
Darryl

[mrose]

Greetings. I'm using Presence to model single species occupancy (with habitat covariates) for birds found in one of two categories of ice damaged forest, high and low). I would like to calculate mean occupancy (and SE) estimates for these 2 categories. Based on the following quote (and subsequent post by Darryl), it almost sounds like it would be okay to have Presence generate model averaged estimates for each site, and then take an average of the site estimates that fall into my two ice damage categories. I seem to remember reading somewhere else that this is not acceptable. Am I interpreting this thread incorrectly?

Hi Darryl
Were you thinking of using the regression equation from this best model to generate predicted occupancy estimates for the other 43 sandhills and then take the average predicted occupancy for all 77 sandhills to get an overall occupancy estimate for this first season?

Cheers,
Maureen

[darryl]

It's probably reasonable provided that the values of the other habitat variables you've collected are representative of what's out in the real world. If you've preferentially selected sites in a certain habitat, that's could lead to misleading results. However, the question I posed previously in this thread still need to be considered.
Darryl

[mrose]

My sites (i.e., point count surveys) were selected prior to the ice damage, so there should not be a bias in terms of seeking out specific habitats. I have all covariates for all points sampled, so I don't share Javan's problem of missing covariates. I do have a follow up question that gets back to Javan's original post... can you recommend a way to calculate SE's for average occupancies if I take the route mentioned in my previous post?