Known Mortality of Tagged Fish...

[Fish_Boy]

What would be the best way of treating mortality of tagged fish in calculating an Nt estimate?

I have been thinking that the best route would be simply to delete those fish from the encounter history and thus the Nt estimate.

The project was not planned from the outset to estimate Nt, but it is now a requirement. The data collection was carried out basically at the same time every year for 4 years. So the data are consistent enough to analyse and phi(.)p(t) is the best fitting model.

Prehaps an LDLDLD model may be more appropriate? Any suggestions from the LDLDLD experienced would be greatly appreciated.

[abreton]

This is easily accomplished by using "-" (negative) to identify losses on recapture in the group column(s). For example, I supect you were previously considering one of the closed model types such as 'Huggins' closed capture' parameterization. An example encounter history for this model type from a four occasion/2 group study might look like:

0101 1 0;

To idenitfy that this animal was lost (killed, removed, etc.) on the last recapture (occasion four) all we need to do is the following:

0101 -1 0;

MARK will handle the rest. Note that you need to include a "1" on the occasion when the animal was recaptured and 'lost', i.e., you wouldn't want to replace

0101 -1 0;

with,

0100 -1 0;

if the animal was recaptured and lost on occasion four.

See Chapter 2 and the Chapter on Closed models for more details in the MARK book. - andre

[Fish_Boy]

Andre, I understand the -1 concept from the workshop this June. I have been using a CJS estimate so far.

The workshop got pretty busy at towards the end and there was not a lot time/tutorials regarding the Huggins etc. although we did get the rapid version. I guess I should jump into Chapter 15 in more depth. I keep ending up there anyway. The system I am working on is 'closed' per se or at least bound on both ends by dams. I am certain that the phi(.)p(t) is a function of a spatially focused sample area and some individuals using the system at a larger spatial scale.

I will give the Huggins models a try. I have been curious about the misidentification, tag losses, and other such inevitable realities of field work.

[abreton]

If abundance is the primary parameter of interest, this leaves you with several options but most of these are 'closed' CMR models (including Huggins and others). Recall that the 'closed' assumption is that no marked or unmarked individuals enter (immigrate or are born) or leave (emmigrate or die) the study area from the start to the end of the study. This assumption assures that the population size of interest is not a 'moving target' as animals are being marked and re-encountered. Given your data are from four years, the closure assumption is probably not realistic. However, you're probably aware of a model type know as the Robust Design?

Assuming you sampled repeatedly within each summer, you might be able to use the Robust Design to estimate abundance over the relatively 'closed' period of your study (summer) and survival over the relatively open intervals between summers (fall and winter). You can learn more about the RD (and its variations) in Chapter 16 of 'the book'. Keep in mind that the RD, like multi-state models, is fairly data hungry. Best bet is to do as Gary White likes to suggest, "Pull yourself up by your bootstraps", i.e., insert estimates from simple models as starting values in the more complex models in your set.

[Fish_Boy]

Andre, thanks for the advice. I will run some today and see how they run... see how they run.

I will post success/failure soon.

[Fish_Boy]

Andre,

I have finally got back to getting these data analysed. 6 primary occassions, each with two secondary intervals. Basic results...

First the model of 'choice' is typically constant S, constrained gammas, time dependent c, p, N. This makes sense given the system is physically bound at each end and the fact that acoustic and radio telemetry indicate that lake sturgeon utilize the entire area, often leaving the sampling area for a period, season or even year, but always returning eventually to the sample area.

Second, the population estimates have no SE and they are virtually identical to number of animals caught during each primary occassion. So if there were 50 new fish tagged at secondary occasion 1 and 50 more tagged during secondary occassion 2, plus 5 recaptures from secondary occassion 1 as well as 7 fish reacpatured from an earlier primary occassion the N(t) estimate equals 107 +/- 0.

If I run a CJS model on the annual captures the best model is always constant phi time dependent p. But the population estimates are significantly higher than those using the closed captures Robust Design. We know that the population size is greater than the number of fish captured during each primary occassion. BUt the closed captures robust design defaults to number captured during primary occassion regardless of model.

[Fish_Boy]

Using robust design closed captures with heterogeneity provided results that make biological sense compared to simple closed captures. The best models, however, regarless of study area have constant pi. Would this have any meaning biologically speaking?

[Fish_Boy]

Are there any hard rules about fixing parameters and using models with parameters that cannot be estimated?

Real Function Parameters
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
-------------- -------------- -------------- -------------- --------------------
1:S 0.8529515 0.0512293 0.7225987 0.9281420
2:Gamma'' 0.9219646 0.0813536 0.5629279 0.9908575
3:Gamma'' 0.7807124E-12 0.5734493E-06 -0.1123960E-05 0.1123961E-05
4:Gamma'' 0.4032330 0.1769808 0.1378178 0.7406808
5:pi 0.4500000 0.0000000 0.4500000 0.4500000 Fixed
6:p1 0.1813255 0.0637845 0.0871004 0.3395678
7:p2 0.1306828 0.0444962 0.0652156 0.2446686
8:p3 0.0907363 0.0181173 0.0609347 0.1330482
9:p4 0.1070505 0.0192953 0.0746845 0.1511518
10:p5 0.0711430 0.0151592 0.0465798 0.1072027
11:p6 0.1772546 0.0383159 0.1140506 0.2650076
12:N1 187.50806 62.668412 111.78114 378.43056
13:N2 417.04026 137.02662 241.33044 814.33799
14:N3 443.97089 96.262370 298.33388 685.43663
15:N4 636.09282 119.26346 450.79381 928.09285
16:N5 516.89959 120.53022 335.96631 821.38386
17:N6 522.57389 107.30774 366.61035 801.63131

[Fish_Boy]

Sorry forgot to complete the description of results first... These are a Robust Design closed captures with heterogeneity results. pi was fixed at 0.45, which is ~ the same value for two other RD models on the same species, but from different reproductive populations in the same region. gamma" is not estimable. Is it possible that the gamma" value is not estimable because of variable time intervals between primary occassions?

Any pointers would be greatly appreciated.

[abreton]

Off the top of my head, I can think of three scenarios when it is reasonable to fix a parameter(s): (1) when you know a priori the value of the parameter; (2) when the parameter is not estimable given the model structure; (3) when a a biologically plausible value is available to fix a parameter is normally estimable given the model but, due to data sparsity or some other issue with the data, it cannot be estimated with your dataset.

I recently completed a simple CJS analysis and applied scenario 1: all animals released on occasion 3 were immediately seen on occasion 4 to all survived with probability 1.0. In this case, since I knew the valueof the parameter a priori, it made no sense to ask the model to 'estimate' it for me. In fact, if I relied on the model in this case to estimate the parameter this may have caused convergence problems since this parameter was 'on' the upper boundary (=1).

There are instances when a parameter cannot be estimated by the model no matter how much data are available; in these cases, you are advised to fix the parameter(s) to some value or to set it/them equal to an estimable parameter in the model. Here is an example from the Robust Design from Chapter 16, Section 16.3.3, 2nd paragraph: "To provide identifiability of the parameters for the Markovian emigration model (where an animal
“remembers" that it is off the study area) when parameters are time-specific, Kendall et al. (1997) stated that γ′′ k and γ′k need to be set equal to γ′′ t and γ′t , respectively, for some earlier period. Otherwise these parameters are confounded with St−1. They suggested setting them equal to γ′′ k−1 and γ′k −1, respectively, but it really should depend on what makes the most sense for your situation. This problem goes away if either movement or survival is modeled as constant over time."

In my experience, convergence failure is an extremely common experience when for analysts fitting capture-recapture models. And this may be for the simple reason that many of us are slow to realize that estimability not only declines as a function of decreasing data - it also declines as the number of parameters in a model are increased. People love to dream up complex models but they often fail to realize how much data they'll need to get reasonable or even any estimates at all from the model.

In my view, all estimable parameters in a model should be succesfully estimated given the model and the data otherwise the model should not be included in the model set (not available for inference). The only exception I'd make is when a biologically plausible value is available for an estimate that cannot be found by the numerical search procedure/algorithm deployed by the software. If this is available (e.g., based on a previous study), then it might be reasonable to fix the paramter to this value in the model and retain it in the model set. I should note here that when one or a small fraction of the parameters in a model are not estimated given the data, model, and solution strategy deployed by MARK (and other software) analysts have several options before giving up on the model or fixing parameters - e.g., a different form of the same model might do the trick see CHapter 7, MARK manual); or a different link function when one or more parameters are on a boundary (close to 0 or 1). But in the end, if a parameter(s) are not estimated by the model and a biologically supported value is not available, my view is that the model should not be included in the results/inference.

This latter suggestion is often not taken well by collaborators - especially when the model is necessary to test their hypothesis(es). Unfortunately, what they need to do in this scenario is back-up, design/redesign their study and acquire more (or the right) data. Of course, this advice will not make you popular. These are just a few thoughts, some of which may be helpful...