closed captures N-hat = Mt+1

[brookieblaster]

Hi All

I am working with a data set from a project in which fish in a closed population were marked and released on the first capture occasion, then all fish captured in subsequent capture occasions were removed from the population. Input files look something like:

10000 12
11000 -7
10100 -3
etc.

Oddly, even in models where the final p is constrained (say, M(N, p(.), c(.))) the parameter estimates have suspiciously small standard error values and N-hat almost always equals the actual number of individual fish captured (i.e., Mt+1). The only model for which this is not true is M(N, p(t) = c(t)). Although this model gives reasonable parameter estimates it does not have the lowest AIC value. Also, it doesn't seem to matter if heterogeneity is allowed for or not. I thought maybe the negative frequency values were the problem, but it seems like MARK is well equipped to handle them (in fact the bible says that's how to handle them). Perhaps it is not so with respect to closed captures models? Has anyone seen anything similar or have ideas about what the issue could be? Thanks for any help...

Drew

[ganghis]

You're trying to estimate more parameters than the data can support - getting N-hat=Mt+1 is what often happens in such cases. I think what you want to do is to estimate p for the first period, and c thereafter. So create a time-specific model for p and set p_2 through p_5 = 0. You'll also need to set p_1 = c. I don't have time to verify whether you can get away with having some of the c's be time specific or whether you'll have to make all the detection parameters be constant - probably best to start with them all constant to see if you can get these working.

[murray.efford]

Like Drew I'm mystified why his original analysis shouldn't work. It does happen that N-hat collapses to (near) Mt+1 for some datasets. I don't see that there are inherently too many parameters in his model, although that of course can depend on the data. For my own amusement I simulated some (spatial) data in R and had no problems fitting models to capture histories censored after the second capture (code and results follow)
Murray

Code: Select all

library(secr)

# simulate some data
tempgrid <- make.grid()   # 6 x 6 trap grid
temppop <- sim.popn (tempgrid, D = 20, buffer = 100)
CH <- sim.capthist(tempgrid, pop = temppop, detectpar = list(g0 = 0.2, sigma = 25))
## use binary capture histories for recoding
binCH <- CH
binCH[] <- CH>0

# remove on second capture (negative trapsite)
cum <- t(apply(binCH, 1, cumsum))
CH[cum > 2] <- 0
CH[cum == 2] <- -CH[cum == 2]
summary(CH)

# MLE
fit <- secr.fit(CH, buffer = 100, trace = F)
fit.b <- secr.fit(CH, model = g0~b, buffer = 100, trace = F)
fit.t <- secr.fit(CH, model = g0~t, buffer = 100, trace = F)

nrow(temppop)   # true population
region.N(fit)
region.N(fit.b)
region.N(fit.t)
AIC(fit, fit.b, fit.t)

# view a subset
> head(CH)
Session =  1
   1   2 3 4   5
1  7 -31 0 0   0
2 22 -10 0 0   0
3 19 -25 0 0   0
4 18 -18 0 0   0
5 26   0 0 0 -29
6 15 -33 0 0   0

> summary(CH)
Object class      capthist
Detector type     multi
Detector number   36
Average spacing   20 m
x-range           0 100 m
y-range           0 100 m
Counts by occasion
                   1  2  3  4  5 Total
n                 25 29 19 13  7    93
u                 25 15  6  4  5    55
f                 17 38  0  0  0    55
M(t+1)            25 40 46 50 55    55
losses             0 14 13  9  2    38
detections        25 29 19 13  7    93
detectors visited 18 26 18 13  6    81
detectors used    36 36 36 36 36   180

> nrow(temppop)   # true population
[1] 172

> region.N(fit)
    estimate SE.estimate      lcl     ucl  n
E.N 172.6916    29.73226 123.5347 241.409 55
R.N 172.6917    26.67050 130.9036 237.486 55

> region.N(fit.b)
    estimate SE.estimate      lcl      ucl  n
E.N 167.6546    29.57570 118.9612 236.2793 55
R.N 167.6547    26.59074 126.3758 232.8066 55

> region.N(fit.t)
    estimate SE.estimate      lcl      ucl  n
E.N 170.3527    30.77818 119.8927 242.0500 55
R.N 170.3527    27.87371 127.3195 238.9926 55

> AIC(fit, fit.b, fit.t)
                 model   detectfn npar    logLik     AIC    AICc dAICc  AICwt
fit   D~1 g0~1 sigma~1 halfnormal    3 -266.1497 538.299 538.770 0.000 0.7143
fit.b D~1 g0~b sigma~1 halfnormal    4 -265.9258 539.852 540.652 1.882 0.2788
fit.t D~1 g0~t sigma~1 halfnormal    7 -265.8296 545.659 548.042 9.272 0.0069

## all models may be fitted; null model preferred to learned response or time model


[cooch]

Hi All

I am working with a data set from a project in which fish in a closed population were marked and released on the first capture occasion, then all fish captured in subsequent capture occasions were removed from the population.

In other words, a removal model. A straightforward way to build removal models in MARK is to build an {N, p(t), c(.)} or {N, p(.), c(.)} model. Then simply fix c = 0, and run the model. Note - a removal model requires that the number of captures decrease with occasion. All you need is your encounter histories as they are, but without using the minus to indicate removal.

Not that if you use this approach, you don't have to worry about constraining final p, since c is fixed to 0.

Give it a try.

[murray.efford]

Grrr. My speed read failed again - I see all fish were removed from the second occasion, not from second capture as I read. Ignore my comment if indeed all occasions after first were removal occasions.
Murray

[ganghis]

I'd stick with my original recommendations as opposed to what Evan suggests. My reading is that this differs from a traditional removal model in that there is an initial marking event and then only marked animals are removed in subsequent periods. It's not clear to me what this buys you over a traditional removal experiment...certainly the first p is not separable estimable from the c's, but I'm uncertain whether you could get away with something other than a p_1=c_2=c_3=... model. Moral of story: check out parameter ID stuff before conducting field experiment.

[murray.efford]

My (second) reading was that the fish removed on occasions after the first could include individuals not marked on the first occasion, although no such capture histories were shown. In that case, it still seems that several models are potentially identifiable. I can't say how to fit them in MARK (globally constraining c=0 doesn't work for the fish released on occasion 1, which may be recaptured (c>0) until next caught).
Murray

[brookieblaster]

Hi All,

Thanks for the replies! I wish I could say it's all been worked out, but, well, I can't. I see there is some confusion about the data itself. It is the case that fish were marked and released on the first occasion, then ALL fish (that is to say, n(j)) captured on subsequent occasions were removed. The following are actual input data:
/*Site A*/ /*Site B*/
1000 2; 1000 17;
0100 -6; 0100 -16;
0010 -1; 0010 -7;
0001 -1; 0001 -3;
1100 -11; 1100 -4;
1010 -1; 1010 -1;
1001 -1; 1001 -0;

I see that I could run removal models, but I don't understand why mark-recapture models (e.g., N, p(.), c(.)) fail. For instance the model Mb, taking notation from White et al. (1982), is it not the case that m., M., and Mt+1 remain calculable even when individuals are removed upon capture, and thus Nb should remain estimable?

Now to make things more (or less!) interesting: I have found that the Mb will produce reasonable estimates of p, c, and N for some data and not others. When I run Mb on Site B data (above), I get reasonable estimates for p, c, and N. However, when I run this model on Site A data, though I get reasonable p and c estimates, N = Mt+1 and a std error = 0.0000. It is apparent from the data that p and c are quite high at Site A, and thus Mt+1 may be quite close to N. However, SE = 0 makes me think something is not right in the analysis. Any more thoughts that might come to anyone will be genuinely appreciated. Thank you!

Drew

[murray.efford]

I guess my (second) reading was correct regarding the data collection...

I have confirmed to my own satisfaction using simulated data and other software that the mixed CR/removal models are in general estimable. Of course some datasets may fail, and this has always been an issue with removal-type models.

Presumably you use Mb because you have reason to expect a learned response (i.e. over and above the removal issue)?

For the record, I cannot get M0 or Mb estimates from either of the datasets you show (discarding Area B 1001 -0;). Sadly, I think your data are just too few. (Note also rather low recapture probability of fish tagged in Area B on first occasion).

Murray