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!