The closure assumption with SCR

HOME Passive detectors can be left out for long periods, providing more data on each animal captured and thus giving better estimates of detection parameters in spatial capture-recapture (SCR) studies. But with long study periods, the risk of animals coming and going or changing activity centres (ACs) increases.

In this post I want to explore what is "closure" in SCR, why it is important, and what happens if it is violated. I'll also explore some ideas for open population models which can be used to mitigate closure issues.

Coming and going

Most capture-recapture studies that aim to estimate abundance or density work with the assumption that marked animals are not entering or leaving the population during the study: no births, deaths, immigration or emigration.

Effect of violation of closure

The plot below shows simulated capture histories for a long survey with camera traps deployed for 480 days (almost 18 months). Time is on the x-axis and each horizontal dotted line is one captured animal; points represent detection events.

The calculation of probability of detection makes the assumption that if an animal is detected at all during the 480 days, then it's present for all 480. This applies to animal #2 in the plot, only detected on Day 8, and we assume that non-detection on the remaining 479 days reflects low probability of detection: 1 success in 480 trials indicates p = 1/480.

But now let's look behind the scenes (these are simulated data) and see when these animals were present in the study area. This is indicated by the grey lines in the plot below:

In reality, animal #2 is only in the study area for 40 days, so our estimate should be 1 success in 40 trials, p = 1/40. This is 12 times higher than our previous estimate.

Four of the 17 animals captured violate the closure assumption, either leaving early (#2, #13), arriving late (#16) or both (#10). Violation of the closure assumption - animals come and go when the calculation assumes they don't - means that estimates of detection probability are too low.

Our main purpose with SCR is to estimate population density. That means estimating the number of animals we did not detect. If probability of detection is very high, there will be only 1 or 2 animals we didn't detect, but if it's low, there will be lots. If our estimate of probability of detection is too low, our estimate of density will be too high.

Violation of the closure assumption results in inflated estimates of density and abundance.

Keep the survey period short

One way to reduce the effect of comings and goings is to keep the survey period short. Let's see what the above data would look like if we discarded all except the first 120 days:

Animal #2 is still leaving early, but now we're erroneously assuming it's there for 120 days, not 480. But obviously we have far less data: only 6 animals and 9 captures instead of 17 animals and 39 captures for the 480 day period. We may have reduced the bias, but now the uncertainty in our estimates will be much larger.

Use open population models

A better alternative seems to be to use models that allow for animals to come and go, ie, open population models.

A strategy that seems to work well is to divide the study period into a series of sessions, for example, 480 days become 8 sessions of 60 days:

We no longer assume that the animal is there for the whole period: prior to the first detection and after the last, the probability of being present is less than 1. Changes are controlled by recruitment and survival parameters.

We are admittedly still assuming the status is the same for the whole 60 day session, but that is a good deal closer to reality than the original closure assumption of presence during the whole 480 days.

Changes in Activity Centre

In SCR, we model detection probability as a function of the distance from a trap to the animals activity centre. We assume that the activity centre does not change over the period of the survey, or at least that it is appropriate to use a single 'average' location.

Violation of this assumption was investigated by Royle, Fuller and Sutherland (2016) using simulations. (They use the term "transience", but I think of a transient animal as one with no home range and no AC; often we are dealing with animals with established and defended territories, but with boundaries that evolve with time. I prefer to talk about "AC drift").

The good news is that estimates of density are remarkably robust to AC drift (see Tables 1-3 in their paper). Even so, with long-term data sets we should try these models and see the difference in the estimates.

Changes in AC are modelled as a random walk, with the AC coordinates for each session drawn from a bivariate normal distribution centred at the AC for the previous session.

Royle et al apply this to the Fort Drum black bear data set used in the SCR book (Royle et al 2014). Hair snares were checked weekly for 8 weeks, and different ACs were estimated for each week. The plots below are for the bear with most movement, having been caught twice in the northern part of the trap array and then again in the south:

(In the plot, the red crosses are traps, yellow dots capture locations, grey dots 1000 draws from the posterior distribution for the AC location with the black dot the centroid, and the blue dots are the centroids for previous weeks.)

For the bears, the drift parameter is small compared with σ and difficult to estimate (tiny effective sample sizes). With other data sets I've worked with, it cannot be distinguished from the usual SCR movement, but is obviously small.

Recommendations

My current thinking is that we should try to fit models with both AC drift and survival/recruitment to data sets covering periods so long that closure is likely to be violated. If it turns out that the drift parameter is small and can't be separated from short-term animal movement or that recruitment = 0 and survival = 1, then we can proceed with the simpler models with confidence.

Updated 26 May 2018 by Mike Meredith