Removal models: time varying p and closure

[abrereton]

I am using removal models to estimate abundance and capture probabilities of squirrels. I have used the MARK book, electrofishing literature and threads on this forum but still can’t resolve two key questions:

1) Can you allow time varying p in removal models?
White (1982) says there is no way to deal with time varying p (in CAPTURE), and that removal models are based on the premise that capture probability is constant over time, disregarding effect of individual heterogeneity. I haven’t found any papers using removal models where p varies with time. However threads on this forum, and the section in the MARK book on removals suggest p can be varied with time. I have found that p(time) models return estimates of N equal to the number of squirrels in my capture histories, even if I constrain the final p to equal the penultimate p. This also happens if I fit a trend in p. So although it is easy in MARK to allow p to vary with time, does this mean that this is actually a valid thing to do? Similarly I have seen suggestions to constrain p to effort, although in theory removal models assume constant effort.

2) Is there a way to assess closure in removal models?
I know ‘close test’ exists for closed captures models but haven’t found examples of it being used with removals: obviously as it is designed to look for permanent and temporary emigration as well as immigration I would expect it to always fail for removal models as a removal is permanent! But would the test statistics related to immigration still be valid? Otherwise the only option seems to be to compare estimates of p or N from different parts of the capture history e.g. days 1-5 vs days 1-10 to see whether estimates change with time, and how these compare to the total number caught over the full trapping session (up to 15 days). Alternatively, using model Mbh, p(time) or p(Time) in MARK, I have seen some evidence for increasing capture probability – animals becoming used to traps, or evidence of non-closure?

Thanks, Amelia.

[abreton]

Can you allow time varying p in removal models?

Yes, but note that for the removal model estimator to give a valid result (valid estimate) the number of animals caught on each occasion must decline through time. See sidebar on page 14-16 in the Gentle Intro to Mark, http://www.phidot.org/software/mark/doc ... chap14.pdf.

I assume you fixed the c's in your model to zero?

Regarding your second question, removal data are very limited because animals are removed on initial capture ... this doesn't leave a whole lot of information to 'model' -- in particular, no recaptures of course. Let's see if we can figure out why you're getting an N-hat estimate = m t+1 and then move on to your second question.

[abrereton]

C is fixed to zero.

I played around with some encounter histories simulated with constant p, then fit models with Time or p varying every day except the last day - mean estimates of p for each occasion always increase over time, with increasing standard error - hence the N estimate equal to the number caught, as models estimate that all individuals were caught by the end of trapping as p is very high. This makes me suspect that I shouldn't use time variation!

I am interested in using time variation because my first day of trapping has very low numbers of captures as the traps were set only for the afternoon, whereas the second day has high captures as traps are set for a full day, and there is a decline in captures from day three onwards. So I would like to allow the first day to have a different capture probability to the other days.

Any suggestions?
Thanks, Amelia.

[murray.efford]

If time variation in catch is due to known variation in effort (duration of sampling) it can be modeled by scaling the 'daily' detection probability to a constant unit of time (half-day?) without estimating additional parameters (e.g. Efford Borchers & Mowat MEE 2013). I'm not sure how this should be done in MARK - maybe by providing a suitable covariate and fixing the parameter for the corresponding beta coefficient to 1.0. Suggestions from MARK afficionados?
Murray

[abrereton]

Hi, I had thought of trying this but it's a bit tricky - I am pooling data from several sites, some sites have traps set and checked on the afternoon of the first day (so effort could be fixed to e.g. 0.5, compared to the rest of the days effort of 1), and some sites are set on the first day but not checked until the second day, so no captures recorded on the first day (so effort should be fixed to 0 for day one). Unfortunately the way each site was trapped has not been recorded, so all I could do is pick a figure for effort between 0-0.5 for all sites.

[abreton]

my first day of trapping has very low numbers of captures as the traps were set only for the afternoon, whereas the second day has high captures ... numbers decline from there

The literature on removal models, including the section in the MARK book that I made reference to yesterday, always states that numbers of captures must decline from the first to last occasion for estimates from a removal model to be valid. To my knowledge, there is no way of getting around this issue, numbers must decline. In your study, the fewest captures come on the first occasion rather than the last. Unless you can find a way to get past the requirement that numbers decline, then one option might be to drop the first occasion since numbers decline from the second occasion onward. Of course, you'd be estimating population size at occasion 2, not occasion 1, if you did this. To estimate the pop size on occasion 1 you could add captures from occasion 1 to the occasion 2 population estimate ... easy ... estimating a SE for this occasion 1 pop estimate would be more difficult.

I propose that you fit a time-varying removal model with all of the adjustments (e.g., c=0) that you've outlined above ... but this time drop the first occasion. I suspect the pop estimate will not equal m t+1 when you do this ... if so you know what's causing the problem.

[murray.efford]

Dropping the first occasion and adding them back in later does seem a good way to go.

My intuition is that the condition for successful removal estimation is not quite as restrictive as implied above. If the cause of low numbers in early samples is known and modelled, good estimates can still emerge. My evidence is a bit indirect, using secr code below. I have generated some screwy data and fitted a spatial removal model by fixing the spatial scale parameter to its true value (this doesn't work for all datasets). (If you had the effort information by trap, and a good idea of the spatial scale, this might be a solution).

I would be fairly confident that this translates to nonspatial models.

Murray

Code: Select all

library(secr)
tg <- make.grid()  ## default 36-trap grid
usage (tg) <- matrix(1, nrow=36, ncol=8)  ## effort
## zero effort for half traps on first 4 occasions
usage (tg)[1:18,1:4] <- 0
## simulate some captures
CH <- sim.capthist(tg, popn=list(D=20, buffer=100))
## convert to removal data (there must be an easier way)
onlyfirst <- function(x) {
   s <- length(x)
   i <- match(1,cumsum(x)>0) + 1
   if (i<s) x[i:s] <- 0
   x
}
CH[,] <- -t(apply(CH,1,onlyfirst))
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
Usage range by occasion
    1 2 3 4 5 6 7 8
min 0 0 0 0 1 1 1 1
max 1 1 1 1 1 1 1 1
Counts by occasion
                   1  2  3  4  5  6  7  8 Total
n                 16 14 17 15 33 31 31 36   193
u                 16  6  4  7 15  4  1  3    56
f                 11  8  8 17  4  4  2  2    56
M(t+1)            16 22 26 33 48 52 53 56    56
losses             0  0  0  0  0  0  0  0     0
detections        16 14 17 15 33 31 31 36   193
detectors visited 11 10  9 10 24 22 20 23   129
detectors used    18 18 18 18 36 36 36 36   216

fit <- secr.fit (CH, fixed=list(sigma=25), verify = F, start=c(log(20),logit(0.2)))
predict(fit)
    link   estimate SE.estimate         lcl        ucl
D    log 16.6768853   2.6725336 12.20591952 22.7855429   ## cf true density 20/ha
g0 logit  0.1809619   0.0607604  0.09002512  0.3304037


[murray.efford]

There's an obvious flaw in my example. Will have to rethink. Sorry about that.
Murray

[murray.efford]

The flaw: I copied and pasted the wrong summary table (no 'losses' in this one, so not removal data). Nevertheless, the method does work with removal data.
Murray

[abreton]

My intuition is that the condition for successful removal estimation is not quite as restrictive as implied above

Despite not being aware of a solution when only removal data are available my intuition is perfectly aligned with Murray's. One would think that if they modeled the p's to account for low caps on occasion 1 (in the present study) followed by a decline after occ 2 then the model could sort out an estimate of N. Nonetheless, for the removal model option available in MARK my understanding (certainly not conclusive) is that the model will fail to converge on an estimate of N when caps do not decline through time ... and instead default to m t+1 (a message to the MARK user that something has gone wrong).

I'm hoping a solution consistent with Murray's intuition might be posted ... so I can apply it to my own work! Thanks for the valuable discussion.

andre