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by **Kenup17** » Thu Apr 28, 2016 10:08 am

Hello,

I'm running RDPNE analysis with a small number of marked individuals (14), and while real estimates of U (number of Unmarked Individuals) seem reasonable,
derived estimates of N are much higher than expected (the difference between U and N is greater than the maximum number of marks in the population). This only
occurs when modelling alpha as a function of any individual covariates.

Nevertheless, estimates of $\alpha_j$ and $\lambda_j$ (intercept of resighting rate and mean resighting rate) also seem reasonable.

Clearly there is something wrong going on on the deriving of $\hat{n}_j$ , the exact number of marked individuals on the population.
The Good Book says it is derived as $\frac{n^*_j}{1-\exp(-\lambda_j)}$ (18.4.1, p688), but my by-hand calculation doesn't match the results.
When using individual covariates the deriving of $\hat{n}_j$ is done differently?

Maybe the low number of individuals marked, and low number of individuals in some groups is able to bias $\hat{n}_j$ ?

Any insight on this would be very appreciated!

Cheers,

Caio Kenup

***

Encounter histories and estimates below
- I kept only one individual covariate, but all the others (sex, age etc) present the same problem)
- U is modelled as 'U = ~ time' and all other parameters are constant

Encounter History:

Code: Select all: ch captive anathema 030102-00103050504070402080104030602060411070402040505030403 0 bjork ..............................+005-0-0-0-0-0-0-0-0-0-0-0-0-0 0 grey ..............................05030101-0-0-0-0-0-0-0-0-0-0-0 0 kanayeva ..................................................0902020106 0 lili ..........+00201-001-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0 0 lolita ..............................+0-0-0-0-0-0-0-0-0-0-0-0-0-0-0 0 luke ..............................04-0060605-0-0-0-0-0-0-0-0-0-0 0 luna ..........020203030402-0-00707020301010202050503040702080910 0 maia ..............................+0-0-002-0-0-0-0-0-0-0-0-0-0-0 0 malu ....................07110407060207020504-00101-0-00502010101 0 negra 0510060407050805-006050901-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0 1 sol ........................................+0-001-0-0-0-0-0-0-0 0 wallace ..............................010302-0020401-0-001.......... 1 x 0203050805060403-006060405080808050505-00906030305010202-005 1

Comparison of estimates

alpha= ~1 + time

alpha - real (last 5 intervals)

Code: Select all: estimate se lcl ucl 5.3332387 1.0825030 3.5971441 7.9072271 2.5750835 0.7280453 1.4953436 4.4344692 3.1649404 0.8125586 1.9288759 5.1931012 3.4553933 1.0134059 1.9678061 6.0675403 4.9412111 1.0358542 3.2908123 7.4193132

alpha - beta (last 5 intervals)

Code: Select all: estimate se lcl ucl 0.4817914 0.3833180 -0.2695118 1.2330947 -0.2462853 0.4310450 -1.0911336 0.5985630 -0.0400330 0.4143167 -0.8520938 0.7720278 0.0477690 0.4373981 -0.8095312 0.9050693 0.4054432 0.3868360 -0.3527553 1.1636417

Lambda (last 5 intervals)

Code: Select all: estimate se lcl ucl 5.780045 1.0899755 4.0069378 8.337768 2.598571 0.7336351 1.5101127 4.471569 3.769199 0.8380829 2.4505262 5.797475 4.213964 1.0420259 2.6140737 6.793036 5.582182 1.0452474 3.8796027 8.031946

Unmarked Individuals

Code: Select all: estimate se lcl ucl 17.5527260 3.7767529 11.5680380 26.6335730 29.8155940 9.1015770 16.6098120 53.5207520 24.5348360 6.0475842 15.2423900 39.4923740 20.5210930 5.5553673 12.1849990 34.5601410 17.4592410 3.7498138 11.5149520 26.4721120

N (last 5 intervals)

Code: Select all: estimate se lcl ucl 22.582193 3.802787 16.271369 31.34066 35.243927 9.404081 21.078965 58.92767 29.769366 6.220069 19.852314 44.64039 24.661776 5.684443 15.789039 38.52060 22.501699 3.785882 16.218003 31.22002

alpha = ~1 + captive

alpha - real

Code: Select all: estimate se lcl ucl 3.7645482 0.2194680 3.3583824 4.2198361

alpha - beta

Code: Select all: estimate se lcl ucl alpha:(Intercept) 1.2810913 0.0677920 1.1482191 1.4139635 alpha:captive 0.2078373 0.1016335 0.0086357 0.4070389

Lambda (last 5 intervals)

Code: Select all: estimate se lcl ucl 4.293518 0.2059123 3.908529 4.716429 3.902010 0.2041100 3.522025 4.322990 4.489273 0.2068137 4.101876 4.913256 4.461688 0.2400205 4.015506 4.957447 4.489273 0.2068137 4.101876 4.913256

U (last 5 intervals)

Code: Select all: estimate se lcl ucl 23.6103530 2.8586746 18.6386420 29.9082290 19.8254520 2.7494567 15.1263920 25.9842880 20.5773480 2.5664912 16.1298950 26.2510850 19.3622570 2.5249805 15.0113120 24.9743000 21.6910890 2.6451310 17.0947230 27.5233100

N (last 5 intervals)

Code: Select all: estimate se lcl ucl 49.448587 2.933112 44.025782 55.53934 45.663685 2.820848 40.461088 51.53525 46.415581 2.637441 41.527389 51.87916 35.958628 2.582978 31.241881 41.38749 47.529323 2.717223 42.494990 53.16007

by **bmcclintock** » Thu Apr 28, 2016 1:39 pm

Hi Caio,

When using individual covariates, $\hat{n}_j$ is indeed calculated differently. It is calculated as $n^*_j / p^*_j$

, where

$p^*_j = 1/n^*_j \left[ \sum_{s=1}^{n^*_j} \left(1-\int_{-\infty}^\infty \exp( - \exp ( \sigma_{sj} z_{sj} + \alpha_{sj} ) ) \phi(z_{sj})dz_{sj} \right)\right]$

and $$\phi(z_{sj})$ is the standard normal density. In terms of the betas reported by MARK, $$\alpha_{sj}$ above is on the same log scale, but I believe MARK reports $\log (\sigma_{sj})$ , so use $\exp (\sigma_{sj})$ if calculating by hand from the betas.

With small sample sizes and/or considerable individual heterogeneity in resighting probability, it's certainly possible to get $\hat{n}_j > n$ . MARK does not know the maximum number of marks that could be in the population, but is doing it's best to estimate $n$

based on the encounter histories, and, in this case, the individual covariate on $\alpha$ .

It's possible that a subset of your marked individuals has pretty low sighting rates (I'm looking at you "Lili", "Lolita", "Maia", and "Sol"). RDPNE assumes that the marked sample is representative of the population as a whole with respect to sighting rates (including any covariates). If neither of these seem to be the issue, this could of course just be an artifact of a relatively small marked sample size.

Cheers,
Brett

by **Kenup17** » Thu Apr 28, 2016 2:57 pm

Thanks Brett, that was really clarifying!

Regards,

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RDPNE overestimating number of marked individuals

RDPNE overestimating number of marked individuals

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