How to address SE = 0 for estimable parameter at boundary?

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How to address SE = 0 for estimable parameter at boundary?

Postby mherse » Sun Dec 18, 2016 2:59 pm

Hi all,

I am using RDOccupEG (robust-design/multi-season occupancy) in RMark to estimate within-season site occupancy dynamics of a rare and highly mobile grassland bird. Our field crew conducted >10,000 repeat-visit surveys at >1,000 sites during two breeding seasons. We have very sparse data for this particular species (approximately 130 detections at 100 sites across the entire study). The RD approach has worked quite well, but I am running into a problem that I'm not sure how to address...

One of my individual covariates is a simple categorical factor for seasonal Fire (Burned versus Not Burned). Prescribed prairie fires occur during late winter/early spring here. This species requires tall, dense vegetation for nesting and WILL NOT use burned sites during the early breeding season, but will use burned sites later in the season in areas that are not grazed by cattle, where vegetation grows rather quickly. I have been able to capture this pattern: based on multimodel selection, Fire is an important predictor of initial site occupancy (Psi), but not an important predictor of within-season site colonization (Gamma). The effects on Psi are apparent when I plot the data -- Psi increases linearly with % Unburned Grassland Cover as one might expect, whereas Psi is a flat line at 0.0 across the entire range of % Burned Grassland Cover (doesn't matter how much grass there is; if it's burned, Psi = 0.0). So, that's cool.

However, this estimate of Psi = 0.0 is at the parameter boundary, so standard errors for beta estimates are also 0.0. My beta estimates for Psi in the top model are as follows:
Intercept (Burned): beta-hat = -19.95, SE = 0.0 (95% CL = -19.95, -19.95)
Fire (Unburned): beta-hat = 14.47, SE = 0.0 (95% CL = 14.47, 14.47)
% Grassland: beta-hat = 1.28, SE = 0.36 (95% CL = 0.58, 1.98)
% Conservation Reserve Program: beta-hat = 0.57, SE = 0.12 (95% CL = 0.33, 0.81)
% Trees: beta-hat = -0.90, SE = 0.50 (95% CL = -1.88, 0.09)

All estimates are on the logit scale and based on z-transformed covariates. This model provides the single-best fit among my candidate set for Psi, and a plot of the data looks good (again, flat line at 0.0 for burned grass, and a positive effect of unburned grass with fairly tight confidence intervals).

My question: Is the standard error equal to 0.0 for the intercept and Fire a problem? As I understand it, if the parameter is at a parameter but is estimable, then it's okay to ignore the SE = 0. It makes sense -- birds never use burned sites early in the year. But unfortunately this also leads to SE = 0 for the unburned parameter as well? Is it appropriate to retain these effects and present the beta estimates as they are, or would you suggest removing this as a candidate effect in the analysis, and simply provide information on detections at burned versus unburned sites throughout the season as descriptive statistics (i.e., "we encountered zero birds at burned sites during the early season, but found x birds at x burned sites later in the year"...)?

I'd like to retain this effect, but don't want to raise any red flags when I submit these results for publication. I'd appreciate any opinions about this - thanks!!
Best,
Mark
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Re: How to address SE = 0 for estimable parameter at boundar

Postby cooch » Sun Dec 18, 2016 4:41 pm

mherse wrote:I'd like to retain this effect, but don't want to raise any red flags when I submit these results for publication. I'd appreciate any opinions about this - thanks!!


Taken literally, this implies your concern is more over having a 'sigfnificant effect' that (probably a priori) you think is 'correct' (because it makes a compelling story). If so, I will submit that your focus is misplaced -- rather, your interest should be on getting things right, and not on getting things published (since, as you should know, just because its published, doesn't mean its correct).
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Re: How to address SE = 0 for estimable parameter at boundar

Postby darryl » Sun Dec 18, 2016 4:44 pm

A SE of 0 makes sense on the probability scale if the probability is estimated as 0, but not when the SE is on the logit-scale. To me it sounds like something isn't being estimated properly. Check for any warnings in the MARK output file. The variance covariance matrix might not be getting estimated properly, or you've overparameterised the model
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Re: How to address SE = 0 for estimable parameter at boundar

Postby mherse » Sun Dec 18, 2016 5:00 pm

cooch,

Thanks for your message. You are correct that we had an a priori reason to believe that fire is a driving factor of initial site occupancy. Our candidate set of competing models and effects are based on a priori hypotheses. If it wasn't clear in my post, I would absolutely like to get things right in this analysis (or I wouldn't have posted this thread). I'm concerned about whether the standard error being equal to zero in this particular case means that I should not interpret the effect as being real and discard it, or proceed with the top-fit model. I would be grateful for any feedback on how best to address this issue. Thanks again.

Best,
Mark
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Re: How to address SE = 0 for estimable parameter at boundar

Postby mherse » Sun Dec 18, 2016 5:17 pm

Darryl,
Thanks for your input. I will take a closer look at the variance-covariance matrix. Could a boundary estimate give rise to non-positive variances in the VC matrix? I recall seeing this warning at least one time in the past,
"Improper V-C matrix for beta estimates. Some variances non-positive"
but from what I understood, using the 'drop=TRUE' in covariate.predictions() would take care of removing non-positive variances during calculation of real parameter estimates.
Thanks again,
Mark
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Re: How to address SE = 0 for estimable parameter at boundar

Postby darryl » Sun Dec 18, 2016 5:45 pm

Could be that, or it could be that you're trying to do too much with a sparse data set, highly correlated covariates, unestimable parameters, etc. There's a whole host of reasons. The issue may not even be specific to those parameters, but it could be in another part of the model. The first thing is that you need to confirm that there is a problem there before trying to diagnose exactly what the cause is, but having SE of exactly 0 for the beta parameters would be a big red-flag to me that there is a problem with the variance-covariance matrix. Typically, from what I've seen, when you have beta parameters being large +ve or -ve values because they're going to the boundaries of 1 or 0, the SE for that beta parameter also tends to be larger (ie in the 10's or even 100's). It can also be useful to look at the results for similar models to see how consistent estimates are; that can help identify where the problem might be.
Cheers
Darryl
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Re: How to address SE = 0 for estimable parameter at boundar

Postby mherse » Mon Dec 19, 2016 12:29 am

Overall, estimates for similar models without Fire as a predictor are fairly consistent, with beta-hats between -3 and 3 (logit scale), and low standard errors (0< SE <1). It's only when I include Fire as a predictor of Psi that beta-hat is large (+/- 10's) and SE equal to exactly 0. I suspect this is the parameter where the problem lies. Collinearity between individual covariates is fairly low (r < 0.5). I think you could be right that there just isn't enough data to estimate these parameters, and I am over-fitting models for Psi. Perhaps it would be best to simply present our detections of birds in burned versus unburned sites as descriptive results and not try to test for these effects in models.
Thanks,
Mark
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Re: How to address SE = 0 for estimable parameter at boundar

Postby jlaake » Mon Dec 19, 2016 12:05 pm

Mark-

I've been on leave and not following this closely but try swapping the intercept level to be unburned. You can use relevel in R. Right now you have burned as the intercept level which has 0 occupancy so your intercept parameter is at a boundary and then unburned value is relative to it. The value for unburned is 14.47-19.95=-5.48 which is still very small occupancy (0.00415) but you should end up with a more reasonable estimate of std error for unburned if they are switched. You never want to specify your model where the intercept is at a boundary. Alternatively you could not use an intercept and estimate the burned and unburned levels separately.

--jeff
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Re: How to address SE = 0 for estimable parameter at boundar

Postby mherse » Thu Dec 22, 2016 1:16 pm

Jeff,
Thanks for chiming in. I had not thought of that -- I will give it a try. And you are correct, overall occupancy is very low even among unburned sites (< 0.01). We see occupancy climb to ~0.05 among large undisturbed grasslands, but it's definitely a rare bird.
Thanks again,
Mark
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