Multi-Season Parameterizations

[Miguel Licona]

Howdy,

We've been trying to model psi for 3 seasons with alternate parameterizations of the multi-season model. Our models have a number of covariates. We get a message that says Numerical Convergance Not Reached and our SEs are astronomical or 0. When we use the original parameterization for the multiseason method this does not happen. But this is not useful because we only want to estimate psi for each season and the original parameterization will not do this. Is there any way around this or should we model each season separately?

Thank you for any insight.

Miguel

[gwhite]

Miguel:
Do you realize that the estimates of psi are provided as derived parameters?
Gary

[jhines]

Miguel,

As you've found out, the other parameterizations of multi-season occupancy can be difficult to converge. This is due to the fact that you have one parameter that is computed from the others, but certain values for the estimated parameters can yield impossible values for the computed parameter. With the default parameterization this can't happen. The computed (derived) parameter is:

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

As long as the estimated parameters, psi(t),eps(t), and gam(t) are between zero and one, the derived parameter (psi(t+1)) will be valid (between zero and one).

In the alternate parameterization, the derived parameter is:

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

Obviously, if psi(t)=1, gam(t) is undefined due to division by zero, and if psi(t) is very near 1, gam(t) gets near infinity (way bigger than 1)!

A similar situation occurs if eps(t) is derived instead of gam(t).

So, if you only need the seasonal psi's, use the default model and get psi's as derived parameters.

If you need to model psi's as a function of covariates, I'd suggest trying to find a good guess for the beta parameters to use as starting values and start with simple models initially (only one covariate). If you suspect a positive relationship between your covariate and psi(t), try 1.0 as a starting value for that beta.

Jim

[darryl]

Note that you don't have to think about gamma and epsilon (actually, 1-epsilon) as vital rates, they can also be interpreted as probabilities of occupancy, similar to psi. You're just saying that the probability of a unit being occupied depends on whether the species was absent or present at the unit in the previous year.

Darryl

[Miguel Licona]

Thank you all for your replies.

Jim: In response to your answer, what are my options if not all of the estimated parameters are between 0 and 1? Also, how are the SEs calculated for the derived Psi's?

Miguel

[jhines]

Miguel,

The logit link function forces all of the estimated parameters to be between zero and one, so, no problem there. With the alternate parameterizations, the derived parameter can sometimes be outside this interval. Your options would be 1) just use the default parameterization and model colonization and extinction with the covariate(s), as Darryl mentioned, or 2) Try to get the alternate parameterization to converge on reasonable estimates (where the derived parameters are between zero and one) as I mentioned earlier.

The 2nd choice may take some time, as the alternate paramterization can be sensitive to starting values. You might have to try quite a few combinations of starting values to find a set that converge.

Jim

[jCeradini]

Hi Folks.
My question fits nicely into this thread, so I did not think it was necessary to open a new post.

I'm interested in modeling psi as a function of covariates, so I am running multi-season models with the alternate parameterization where epsilon is a derived parameter (RMark, RDOccupPG).

My beta estimates are converging. However, some of my derived epsilon estimates are > 1 for the lcl and ucl. I understand that is a function of how the derived parameter is estimated in this parameterization. The reals are converging, which is what I'm interested in ecologically, and the derived epsilon estimates are sometimes illogical, which I'm not planning to use/interpret, so can I simply ignore the derived estimates and use the reals to answer my questions? Is there any reason, in general, to question the validity of the beta estimates given the issues with the derived estimates (I understand this is hard to answer without seeing my data)?

Thanks!
Joe

P.S., UCL for all derived epsilon estimates are < 1.7

[darryl]

This is really a MARK/RMark question so Evan may want to bump it across somehow.

Even though you're not interested in the epsilon values, that some of them (or even the CI limits) are outside of 0-1 is a violation of the biological assumptions of the model. Therefore personally, I'd be very hesitant to use the results. This is one of the issues with these reparameterized models as implemented in MARK; unlike PRESENCE, there are no contraints to ensure the derived parameter is between 0-1, which biologically it should be. The upside is that the estimation is more numerically stable, particularly with covariates. The downside is that you can get nonsensical results, particularly with covariates.

Focusing on seasonal occupancy is a naturally thing to do, but I think the original parameterization is much better as it looks more closely at the underlying dynamics and from that you can easily derive seasonal occuapncy estimates if you want them.

Cheers
Darryl

[jCeradini]

Thanks for the response, Darryl!

I understand you cannot speak to the specifics of RMark, but I have one follow-up question if you get a chance. I'm trying to better understand why you would be hesitant to use the results. If the derived estimates are outside of 0-1 then the biological assumptions of the model are violated for the derived parameter or for the whole model, or hard to say? The derived estimates do not influence the betas (since they're derived from the betas), correct?

Thanks again!
Joe

For the sake of showing this in the post again:

jhines wrote:

In the alternate parameterization, the derived parameter is:
gam(t)=[psi(t+1)-psi(t)*(1-eps(t))] / (1-psi(t))

Obviously, if psi(t)=1, gam(t) is undefined due to division by zero, and if psi(t) is very near 1, gam(t) gets near infinity (way bigger than 1)!

[darryl]

Whole model. If you're getting nonsensical dervied estimates that can only happen if the combination of estimated parameters from which the values are derived are not biologically reasonable. For example, if psi[1]=0.5, psi[2]=0.4 and gamma[1]=0.9 (all estimated directly from betas, say), that would imply epsilon[1] =1.1. We can't have that so therefore that combination of the 3 parameter values aren't biologically reasonable in the first place. As max(epsilon[1])=1, psi[2] must either be higher, or gamma[1] must be lower.
Darryl