Negative value for derived gamma estimate

[JWatson]

I am running a robust design occupancy model that estimates epsilon and derives gamma. In all models the derived estimates for gamma are very small (e.g., 0.03 to 0.07) but when SEs are applied dip into the negative range (e.g., -0.15 to 0.20). Is this a function of gamma being a derived estimate and the estimation process or is there something wrong with my models? The very small probability of colonization makes sense ecologically.

[cooch]

I am running a robust design occupancy model that estimates epsilon and derives gamma. In all models the derived estimates for gamma are very small (e.g., 0.03 to 0.07 but when SEs are applied dip into the negative range (e.g., -0.15 to 0.20). Is this a function of gamma being a derived estimate and the estimation process or is there something wrong with my models? The very small probability of colonization makes sense ecologically.

In other words - what to do if the standard CI bounds 0, such that 'negative values' are included. Which is the 'logical problem' you seem to be describing.

First step at solution - switch to profile likelihoods. There are others, but this is the generally recommended starting point.

[cooch]

In re-reading, I've realized I don't fully understand what you mean when you say 'SE's were applied'. I presumed (perhaps incorrectly) that you're constructing a CI using the estimated SE. Perhaps not.

And it isn't clear which of the many approaches you're using to estimate gamma and epsilon. MARK has more than a few - some where epsilon and or gamma are derived parameters, some where one or both are explicit parameters in the likelihgood.

[JWatson]

To clarify, yes, 95% CIs are computed from the estimated SEs.

I am estimating occupancy (ground squirrel study) and most interested in estimating probability of local extinction and colonization. We used a removal design for data collection so want to keep p constant in models. We also collected a few covariates that we include in models so have had to work hard to minimize over-parameterization and get models to converge.

I am running the "Robust Design Occupancy Estimation with psi, epsilon so gamma is derived". The profile likelihood CI option does not allow for derived estimates, at least these parameters are not included in the drop down options for estimated parameters.

I looked at the two options that directly estimate both epsilon and gamma (psi[1] and heterogeneity) but it is not clear how the former model can be designed to model covariate effects by survey.

[cooch]

First, real parm +/- 1.96SE is a silly way to constuct CI for parameters back-transformed from logit (typically) to to the real probability scale, since you will fairly oftnen end up with CI that make no sense. Thats why MARK doesn't do it. It calculates CI on the transformed scale, and then back-transforms everything from there. There are ots of discussions of this in Chapter 6, in various places. [You can also use MCMC to get there from here in MARK -- see the MCMC appendix E in the MARK book.]

If the parameter is dynamic, and in the likelihood, no reason you can't model it as a function of *temporal* (environmental) covariates. You simply handle it all in the design matrix. See the muti-season occpancy example in Chapter 22, section 22.6.1. The DM at the bottom of p. 30 shows epsilon and gamma both modeled as a function of an elevation covariate, for example.

[cooch]

In other words, no obvious reason you can't use the data type with epsilon and gamma in the likelihood.

[jhines]

My 2 cents:

"I looked at the two options that directly estimate both epsilon and gamma (psi[1] and heterogeneity) but it is not clear how the former model can be designed to model covariate effects by survey."

With the psi(1),gamma,epsilon parameterization, you have 1 initial occupancy parameter and K-1 colonization (gamma) and extinction (epsilon) parameters (assuming K seasons). The colonization and extinction parameters apply to the intervals between seasons, so you need to decide if colonization between seasons i and i+1 is affected by your covariate collected in season i or season i+1. Since there is only 1 colonization and extinction estimate for each interval between seasons, you cannot use a "survey" covariate for them. You could create an average value of the covariate during each season and use that, if you wish.

If your covariate is something like "habitat quality"(HQ), then you might use HQ in in season 1 for both psi(1) and gamma(1) (and possibly epsilon(1)). Or, you might use it for psi(1), then HQ in season 2 for gamma/epsilon in the first interval (gamma(1),epsilon(1)). In any case, you'll need K-1 covariate values for the gamma's (and/or epsilon's). Since you've already run the seasonal psi and gamma model, you must have covariates for each season. You'll just have to decide how to assign them to psi(1) and the K-1 gamma's and epsilons.

The most common reason folks use the seasonal psi and gamma model is that they only want to estimate the effect of some covariate on the seasonal psi's. You cannot do that if psi(2)-psi(K) are derived estimates. I think it is more informative to make your hypothesis about what causes the change in occupancy over seasons and model the gamma's and epsilon's with whatever covariates are needed than to only model seasonal occupancy as a function of covariates. Another benefit of the psi(1),gamma,epsilon parameterization is that converges in cases where the seasonal psi,gamma parameterization might not. This is due to the way the epsilon parameter is computed using the seasonal psi's and gamma's. Also, it should solve the problem you're having with the negative lower confidence limits.

[cooch]

My 2 cents:

"I looked at the two options that directly estimate both epsilon and gamma (psi[1] and heterogeneity) but it is not clear how the former model can be designed to model covariate effects by survey."

Of course, Jim is right. I misinterpreted 'survey sample'. In most of what I do, the survey sample is the annual sample from some common population. For the occupancy crowd, survey sample means something else.

My apologies. I guess I'm curious what survey specific covariates will tell you about parameters that are mean field estimates over all sample unit?