mulation for single-catch traps
As explained in Section 2.1.3, a variety of different detectors or traps are used in CR or SCR studies and their different characteristics determine the specification of the detection process component of the SCR model (Efford et al., 2009a). Capture in either a multi or single-catch trap precludes capture in any other trap during that occasion. The competition between traps for individuals leads to a competing risks formulation for multi-catch traps but single-catch traps have the additional complex- ity that, once they are full, they are effectively unable to catch any other individuals. A suitable capture model for single-catch traps therefore needs to account for a sec- ond kind of competing risk, that of competition amongst individuals for traps (Efford et al., 2009a).
FORMULATION FOR SINGLE-CATCH TRAPS
The nature of single-catch traps induces a dependence in captures between in- dividuals and compared to multi-catch traps it is considerably more complicated to construct a suitable likelihood (Borchers, 2012, 2016). To date no DT likeli- hood function for single-catch traps exists (Efford et al., 2009a; Royle et al., 2013c; Borchers, 2016; Borchers and Marques, 2016) and consequently the DT multi-catch trap estimator is typically used for the analysis of single-catch trap data.
For situations in which actual capture times are recorded, the CT setup en- ables the construction of an appropriate single-catch trap likelihood with only small modifications to the general CT likelihood (Equation 2.16). Traditionally, data from live-trapping studies do not contain actual capture times however devices that record times when a trap is triggered are available and have been used by Cowan and For- rester (2012) to study the activity patterns of possums. This dataset on possums is described in Section 1.5 and analysed later in Chapter 4.
Following Barker et al. (2014), if we were dealing with proximity detectors it would be straight forward to handle latent times of capture by integrating times out of the likelihood as shown in Section 2.2.4. However the fact that single-catch traps induce a dependence between individuals complicates matters and means that a high dimensional integral would need to be solved.
The CT single-catch trap likelihood is presented below in Section 2.3.1. Simu- lations are used in both Chapters 3 and 4 to explore the performance of this new estimator and compare it with that of the DT multi-catch estimator, in the first case with a constant hazard and in the second with a time-varying hazard function.
2.3.1
A CT likelihood for single-catch traps with observed
times
Assuming that the actual times of capture in single-catch traps are available, the pro- cess generating detections can be modelled as a competing hazards survival process (Borchers and Efford, 2008) in which “death” corresponds to being caught and all individuals become “alive” again after release. There are K traps and once caught an animal remains in the detector until it is released. Traps are regularly checked and the period of time preceding a detector check is an occasion, i.e. an interval of time rather than an instantaneous point in time. If release times are the same for all traps then this leads to a natural definition of occasions (for DT SCR models) and the survey durationT can be divided intoJ occasions. Each individual is exposed to trap-specific hazards that we assume are at any time independent of the individual’s capture history up to that time (though the model is easily extended to estimate different hazard levels before and after first capture as per model Mb).
As before, the likelihood for φ and θ is the joint distribution of the number of individuals captured n, and the density of the outcomes “ωik events, at times
tik1 < . . . < tikωikr”, for all i and k. The appropriate likelihood is the same expres-
sion as that given for proximity detectors (Equation 2.16) with a slight change to the term representing the density of capture times, which is fk(tik | si;θ) = Sk(T |
si;θ)
Q
rhk(tikr |si;θ). The survival function termSk(T |si;θ) involves integrating
the detection hazard at the kth trap over the survey duration to produce the cumu- lative hazard, which can be thought of as accumulating the exposure to capture at that trap over time.
The likelihood for single-catch traps needs to account for the consequences of a trap catching and holding an individual. The first consequence is that the trapped individual cannot be caught at any other trap until it is released, i.e. the individual’s exposure to detection by all other traps is zero for the remaining period of capture. The second consequence is that the trap in which the individual is held cannot catch any other individuals until the time of release, i.e. exposure to that trap for all other individuals is zero. The DT multi-catch estimator accounts for the first consequence but not the second.
The proximity detector survival function therefore needs to be modified as follows to be suitable for data from single-catch traps:
1. when a particular individual is caught in a particular trap, the hazard of that individual being caught anywhere else is zero for the period of time that the individual is held in the trap, i.e. until release.
2. once a trap captures an animal, the hazard to all other uncaught individuals of being caught at that trap must be zero until the trap is reset.
To construct a likelihood with these features, we define an indicator variableak(t)
that is 1 if trap k is unoccupied at timet and zero otherwise (k = 1, . . . , K), and we define another indicator variable vi(t) to be 1 if individual i is not in a trap at time
t, and zero otherwise (i= 1, . . . , n). (These variables are readily calculated from the observed capture and release times of individuals at each trap.) The hazard function for individualifor trapkat timetis then conveniently written asvi(t)ak(t)hk(t,si;θ)
from which it can be seen how the two indicator functions “switch” the hazard of detection on and off so that it is zero when capture is not possible.
The survivor function across all traps for individualito timetis therefore defined to be S·∗(t,si;θ) = exp − Z t 0 vi(u) K X k=1 ak(u)hk(u,si;θ)du ! (2.32)
Further details of how the survival function calculation is implemented can be found in Appendix A.
The likelihood forφ and θ for single-catch trap SCR surveys then becomes:
L(φ,θ |n,t) = e −λ(φ,θ) n! n Y i=1 Z A D(si;φ)S·∗(T,si;θ)× K Y k=1 ωik Y r=1 hk(tikr,si;θ)ds (2.33)
where λ(φ,θ) = RAD(x;φ)p·(x;θ) dx, and the integral is over all possible activ-
ity centre locations that could have led to a detection on the survey. The term
p·(x;θ) is the overall probability of being caught during the survey, which de-
pends on the combined detection hazard h·(t,x;θ) over the duration of the sur-
vey. This in turn depends on ak(t) (k = 1, . . . , K), which depend on random
variables (the times of capture in each trap). Calculating p·(x;θ) requires tak-
ing expectation over these K random variables, and since they are dependent for single-catch traps a K dimensional integral would need to be evaluated – some- thing that is prohibitively computationally expensive. The estimator therefore in- volves maximising the above likelihood equation with λ(φ,θ) replaced by λ(φ,θ) =
R AD(s;φ) exp −RT 0 PK k=1ak(u)h·(u,s;θ)du
ds, which depends on the observed
ak(t) (k = 1, . . . , K). Consequently the proposed estimator may not be an MLE
and may not enjoy the asymptotic properties of MLEs and hence the bias of the estimator and the coverage of a confidence interval estimator based on the observed information is evaluated by simulation in Section 3.2.1.