Huggins Abundance - Issues with parameter & abundance ests.

questions concerning analysis/theory using program MARK

Huggins Abundance - Issues with parameter & abundance ests.

Postby Abundance_est » Tue Jan 04, 2022 8:22 pm

Hi all,

I'm running a Huggins' closed population abundance analysis on a group of small mammals that were trapped in grids across different habitat types for six consecutive days over two study years. I've run several model combinations, including those outlined in Chapter 14.5 (building models) of the PhiDot book (e.g. {p(.), c(.)}, {p(t)=c(t)} and some of my own (e.g. {p(year), c(year)}). While the models seem to run fine, I have some issues with some of the parameter estimates (particularly of p), and in turn, the derived estimates of abundance.
For example, in the null model {p(.), c(.)}, I got the following estimates (apologies for the formatting, the tables are being shifted in the chat box):
Real Function Parameters of {p(.) = c(.)}
Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:p 0.0400285 0.0550418 0.0025103 0.4085886
2:c 0.6293144 0.0269576 0.5751218 0.6804355

Population Estimates of {p(.), c(.)}
Group N-hat Standard Error Lower Upper
--------- -------------- -------------- -------------- --------------
1 167.58624 222.92166 38.897073 1276.3802
2 195.90894 254.17548 49.982875 1463.6048
3 280.43509 376.84141 58.973851 2138.1585
4 68.049263 95.256703 13.920502 545.54657
5 506.66404 681.81412 102.51236 3853.1983
6 506.09595 669.97491 113.04240 3811.7457
7 137.36420 183.03355 31.828788 1048.3009
8 24.982190 42.180519 3.3406788 246.71736


I know from experience that p cannot be as low as 0.04 - especially with a c value so high. I get even more unreliable values when I use year as a covariate for p and c - {p(year), c(year)}).
Real Function Parameters of {p(year) = c(year)}
Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:p 0.2146716E-04 0.2243204E-03 0.2734184E-13 0.9999407
2:p 0.0949953 0.0724152 0.0197422 0.3536194
3:c 0.5629576 0.0426907 0.4782855 0.6441130
4:c 0.6770823 0.0342902 0.6065907 0.7403498

Population Estimates of {p(year), c(year)}
Group N-hat Standard Error Lower Upper
--------- -------------- -------------- -------------- --------------
1 297361.22 3108224.9 4261.0938 20856038.
2 342395.25 3578311.7 4912.7145 24010710.
3 505432.23 5282559.2 7235.2131 35446167.
4 122670.17 1283099.1 1754.7491 8608938.8
5 222.30880 161.00106 84.852513 858.91905
6 226.47401 157.24962 94.798621 856.39922
7 62.059940 44.065286 26.615829 243.31532
8 10.526836 12.832855 2.2973532 70.958283


I understand that I may get unpredictable estimates when p is time dependent, particularly if no constraint is imposed on p6. However, even in models where p is constant (e.g. {p(.), c(.)}), I am getting very unexpected parameter values. The only ‘reliable’ estimates of parameters and abundance come when p is fixed (e.g. {p(t) = c(t)}, {p(.) = c(.)}, or even {p(t), c(.)} when p6=c.
For example, in {p(t)=c(t)}
Real Function Parameters of {p(t) = c(t)}
Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:p 0.2274658 0.0330256 0.1692381 0.2985282
2:p 0.2856330 0.0362118 0.2201956 0.3615020
3:p 0.3863399 0.0421800 0.3075850 0.4715275
4:p 0.6023720 0.0459486 0.5098386 0.6881226
5:p 0.5560669 0.0451839 0.4666650 0.6419798
6:p 0.6949536 0.0427324 0.6054664 0.7717937

Population Estimates of {p(t) = c(t)}
95% Confidence Interval
Group N-hat Standard Error Lower Upper
--------- -------------- -------------- -------------- --------------
1 30.577812 5.8504369 24.554381 50.804960
2 34.200612 2.1417546 31.971475 41.544706
3 39.671624 5.3800951 33.198988 56.121670
4 11.451802 4.1318313 7.9496508 27.869296
5 68.444487 7.8429649 57.708167 89.945351
6 75.052945 6.2828268 66.862836 93.017133
7 23.427984 4.2456994 19.401267 39.025983
8 1.6601037 1.0544554 1.0726624 6.9967275


However, the models without p fixed to some value of c all performed considerably better by AICc, even though I know abundance estimates from these are unreliable.
Model__________________ AICc __ Del. AICc__AICc Wt.__Model Likel. __# Par__Deviance
{p(year), c(year)}________960.558__0.000 __ 0.674____ 1.000________ 4 __ 952.510
{p(.), c(.)}______________962.022__1.464 __0.324 ____ 0.481________ 2 __ 958.007
{p(t), c(.)} p6=c_________972.547__11.989__ 0.002 ____ 0.003________ 6 __ 960.446
{p(t, year), c(year)} p6=c}_975.482__14.924__ 0.000 ____ 0.001________ 12 __ 951.101
{p(t, year) = c(t, year)}___981.501__20.943__ 0.000 ____ 0.000________ 12 __ 957.120
{p(t) = c(t)}_____________990.659__30.101__ 0.000 ____ 0.000________ 6 __978.557
{p(year) = c(year)}_______1081.029__120.471__ 0.000____ 0.000_______ 2 __ 1077.015
{p(.) = c(.)}_____________1085.306__124.748__ 0.00____ 0.000________ 1 __ 1083.301

How can I resolve the issue with unreasonable parameter estimates leading to unreasonable derived estimates – or, how would I justify choosing one of the lower-performing models with p fixed, such as {p(t) = c(t)}? I've read through the 'book' and searched the internet to no avail. Thanks!
Abundance_est
 
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Re: Huggins Abundance - Issues with parameter & abundance es

Postby gwhite » Tue Jan 04, 2022 8:39 pm

You are running a model Mb, and the numbers of new captures have to decline across time. Show us the data summary from a program capture run of Mb, and I think you'll see a very slight decline, but not one consistent with "removals" from the untrapped population. I suspect you are not trapping a closed population.

Gary
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Re: Huggins Abundance - Issues with parameter & abundance es

Postby Abundance_est » Wed Jan 05, 2022 11:22 pm

Hi Gary,

Thank you so much for the quick and helpful response! If I understand correctly, this may be a problem because in our study system, the number of new captures did not decline over time – in fact, in some areas we caught more individuals on the last couple days than the first! As such, I’d imagine our estimate of p should actually increase with t (as does c). Would this really cause issues with the analysis? Given the small home range and high nest fidelity of our species, plus the short survey period (6 days), I believe it fits the criteria for a closed population. Also, I realize I may have been unclear in my initial explanation – we were not sampling the same grid for two different years; rather, the study simply took place over two years in entirely different areas.

Upon further investigation, I wonder if I caused this problem with my use of the ‘dot’ notation. For some individuals, there were a couple of days in the six-day monitoring period that we were ‘unable’ to search for an individual due to some factor resulting from our trapping methodology – such as mortality, grid disturbance by a bear, or all traps within the effective ‘home range’ of a certain individual being disturbed or occupied by another individual (leaving us unable to trap (search) for them for either initial capture or recapture). In the encounter history, I put a . for these instances, much like you would for an occupancy model, but am now realizing that this may have messed with the ability to derive p effectively (but somehow not c) as it violates an assumption of the model. Do you have a suggestion or solution for how I should handle such instances where I would be unable to search for an individual? I worry that replacing the . with 1s or 0s may bias the final estimates. I may consider the Jolly-Seber model, but am unsure of its efficacy to estimate abundance compared to a closed-population model like Huggins’.

Thank you!
Abundance_est
 
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Re: Huggins Abundance - Issues with parameter & abundance es

Postby murray.efford » Thu Jan 06, 2022 12:14 am

As Gary said, a data summary would help. Dare I ask why you are using a non-spatial model for spatial (trapping grid) data?

Murray
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