## Multi-season implicit dynamics eps=1-gam parameterization

questions concerning analysis/theory using program PRESENCE

### Multi-season implicit dynamics eps=1-gam parameterization

Hello,

I am looking into modeling two years of data with a multi-season implicit dynamics model using the paramaterization option "seasonal occupancy (eps=1-gam) and detection". I have two questions:

1) Is it possible to obtain seasonal estimates of psi from this model, as it looks like the model only outputs psi1. I found a paper (Groff et al. 2017) that modeled implicit dynamics multi-season data in PRESENCE and obtained yearly estimates of psi, but am not sure how to get these estimates from the program.

2) Can this model handle missing data in year 1? I have data from some sites that were collected both in year 1 and year 2, but additional sites were added in year 2 that were not surveyed in year 1. I read in the MacKenzie et al. 2006 textbook that missing observations can be accommodated by explicit dynamics models, and am wondering if this is also true of implicit dynamics models.

Thank you,

Melissa
Melissa

Posts: 3
Joined: Tue Nov 21, 2017 2:22 pm

### Re: Multi-season implicit dynamics eps=1-gam parameterizatio

The equation for psi(i+1) is:

psi(i+1)= psi(i)*(1-eps(i)) + (1-psi(i))*gam(i)

You've chosen the model where eps(i) = 1 - gam(i), so...

psi(i+1) = psi(i)*(gam(i)) + (1-psi(i))*gam(i)
psi(i+1) = psi(i)*gam(i) + gam(i) - psi(i)*gam(i) = gam(i)

So, psi(2)=gam(1), psi(3)=gam(2),... look for the gam estimates to get occupancy estimates after the 1st year.

Regarding #2, yes, PRESENCE will account for missing data in the multi-season models. Estimates of psi(1), gam(1) and eps(1) will only be based on data collected in both years. (You can even have sites with missing data in both years which won't affect any of the estimates! This is useful for projecting occupancy estimates to unsurveyed sites for maps.)
jhines

Posts: 430
Joined: Fri May 16, 2003 9:24 am
Location: Laurel, MD, USA

### Re: Multi-season implicit dynamics eps=1-gam parameterizatio

Hi Jim,

Thank you for the clarification and quick response! Upon looking at the output from this model parameterization, I have two additional questions regarding estimates of gamma and psi:

1) The output includes a real parameter estimate of gam1, which per the previous equation I am interpreting as equal to psi(2). Why is this estimate computed from covariates at only the 1st site and 1st survey? My apologies if this has been addressed elsewhere. I also have derived gam(1) estimates for each site (values ranging from 1 to 28) and am wondering if such high derived gamma estimates indicate a problem with the model?

2) The output also includes estimates of psi1 for each site, including sites that were surveyed only in year 2. For these sites only surveyed in year 2, are the psi1 estimates actually estimates of psi in year 2, as that was the first year that data was collected for the site?

Thank you again,

Melissa
Melissa

Posts: 3
Joined: Tue Nov 21, 2017 2:22 pm

### Re: Multi-season implicit dynamics eps=1-gam parameterizatio

2) The probability of occupancy applies to all sites in the study area, regardless of whether they are surveyed or not. So, the model assumes that the probability of occupancy of the unsurveyed sites (which you don't have any information about occupancy) is equal to the probability of occupancy of the surveyed sites (which you do have info about). That is why estimates are printed for unsurveyed sites.

1) Yes, gam(1) is equal to psi(2) for the model. To save paper, PRESENCE doesn't print individual site estimates for parameters which don't have covariates since all sites will have the same estimate.

You mentioned that the gam(1) estimates range from 1 to 28. The parameters, psi, gam, eps, and p are probabilities which should be between zero and 1. In PRESENCE, the mechanism which forces this is the logit link function. Internally, parameters are allowed to range between minus to plus infinity, and the real parameters are computed using the logit link function. In the output, the first set of estimates are those unconstrained "beta" parameters, which can range from minus to plus infinity. The real parameters (probability parameters) appear after the "beta" parameters. Without covariates, you're probably not interested in the "beta" parameters.

A nuance of the multi-season models is that colonization can only be estimated from sites which are unoccupied. So, if occupancy is high, there aren't many unoccupied sites to provide data in the estimation of gamma and it will have a very high variance and std. error. Similarly, extinction is computed from occupied sites, so if occupancy is low, there aren't many occupied sites to provide data to estimate eps.
jhines

Posts: 430
Joined: Fri May 16, 2003 9:24 am
Location: Laurel, MD, USA