question about SE and confidence intervals

[gurutzeta]

Hi,

In one of the models I am trying I got a psi estimate of 100% for some sites. Obviously the SE (calculated thru the delta method) for that estimate is zero, since the derivatives of psi are basically psi*(1-psi)*c. However the confidence intervals calculated by PRESENCE (based on the likelihood I believe) give something different (I guess probably also closer to “reality”?). Actually the fact is that for some of these sites I got a very big confidence interval (9.1%-100%)....

Anyway, I guess my question is whether in some cases the SE calculated from the betas-VCmatrix through the delta method should be disregarded or taken with caution. I was thinking for instance that if one wants to calculate the variance for the overall level of occupancy then zeros like those above will pull down the variance a bit artificially.

Sorry if the question is too basic, but I am quite new in this ;-)

Thanks,
Gurutzeta

[dhewitt]

Are you referring to derived estimates of Psi for sites based on covariates? If not, what do you mean by Psi for particular sites?

If you are referring to derived Psi estimates, perhaps you simply have very little data in that "cell" and thus the estimate itself is suspect (thus the huge confidence interval returned by PRESENCE - it can't find a good answer). I don't think this is so much a question of what standard error you should trust when an estimate hits a boundary, but why it hit the boundary.

[darryl]

Also, CI's are back-transformed from logit scale so what tends to happen when you get near the boundary, the SE's for beta parameter estimates get large -> wide confidence interval on logit-scale -> CI of 0-1 on real scale. I don't trust the CI when you have estimates on the boundary. Profile likelihood might be a better way to go, but neither Jim or I have looked into implementing them.

[gurutzeta]

Hi,

Thanks a lot to both for your answers!

Yes, I was referring to the derived estimates of psi for each of the sites (i.e. solving the logit with the betas found in the maximization taking into account the value of the covariates in each of the sites).

I believe it is not completely illogical that I am getting an occupancy of 100% for one of the "types" of sites (i.e. one of the combinations of covariate values). I am modelling the occupancy of a species of lemur living in a marsh and there is an area where the habitat is particularly good and the hunting pressure (one of the main threats for the species) is almost zero. In my model I have as covariates for occupancy "habitat quality" (as 3 dummy variables representing 4 categories) and "nearest village" as a proxy for hunting pressure (again as 3 dummy vars representing 4 different villages). The model shows 100% probability of occupancy for sites of good habitat quality in one of the villages. The fact is that in many of those sites the species was detected when surveyed, so the occupancy seems to be very high. It is true though that for some of the "types" of sites the number of samples (sites) available for the analysis is low, so that could explain the wide confidence interval in those cases I guess.

Darryl, do you mean that the 95% confidence intervals that PRESENCE show for the "individual site estimates of Psi" are not based on the profile-likelihood? I am not sure I understand what you mean when you say that the "CI's are back-transformed from logit scale". I understand that the SE for the individual site estimates of Psi are calculated through the delta method from the SE of the betas, but these SEs are not utilized to calculate the CIs, right? Could you please explain me a bit more how the CIs are calculated by PRESENCE then?

Thanks again!
Gurutzeta

[gurutzeta]

about my last question, I see now what you meant by back-transformed, sorry ... however I am not sure how the calculations are done when there is more than one parameter (i.e. there are covariates) in the expression... any hints about where I could read about this? thanx!

[darryl]

Once you have an estimate and SE on the logit scale for a specific site, then an approximate 95% CI would be +/- 2*SE, you then take these limits and transform them back on to the real scale.

The issue is that as you approach the boundaries, the logit-SE tends to get pretty big but with the delta method the real-SE is near zero like you'd expect. However because the logit-SE is big, when you go +/- 2*SE then you might end up with limits like (-5, 20), which when back transformed would be (0,1) on the real scale.

[gurutzeta]

Ok, thanks, I see... it was not so complicated after all :-)

[darryl]

K.I.S.S.