Materials and Methods
We simulated populations with varying degrees of grouping behavior,
while setting N = 140 and movement scale of σ = 3 over a
rectangular state-space of 60 x 30 = 1800 units2 to
mirror previous inferences and camera-trap sampling of the Algar herd of
boreal woodland caribou in northern Alberta, Canada (part of the East
Side Athabasca River population, Hervieux et al. 2013; for further
details see (Beirne et al., 2021; Sun et al., 2022; Tattersall et al.,
2020). We generated different population scenarios by varying
individual- level aggregation and cohesion, the number of available
partial marks, and detection probability. Aggregation values included 1,
4, and 10 in order to range from single individuals (1) to the largest
group size (10) observed in the Algar herd; cohesion values included 0,
0.3, 0.67, and 1 to range from none (0) to complete (1). This resulted
in 9 grouping scenarios in which individuals could group together and
move, ranging from a baseline independence scenario in which individuals
are solitary and move independently (aggregation = 1; cohesion = 0) to a
maximal non-independence scenario in which large groups of individuals
always move together (aggregation = 10, cohesion=1) (Figure 1). When
aggregation is 1, within-group cohesion is irrelevant.
For partial marks, we created 4 categorical independent marks or
covariates: coat color (e.g., black, white, gray, brown, piebald; each
with 0.20 population-level proportions), sex (equal 1:1 sex ratio),
presence of GPS collar (4% yes / 96% no), and total number of points
on left and right antlers (sequential integers from 0 to 17, each with a
mean proportion of 0.049 and range from 0.004 – 0.180; Sun et al.
2022). Covariate values were randomly assigned to individuals according
to these proportions. We considered 6 combinations in which identity
covariates could be available, thereby spanning a range of expected
probabilities of identity from 8% to 99% (Figure 2). We defined
probability of identity as the probability that two randomly drawn
individuals from the population would have the same set of partial
identity covariate values. Finally, we considered populations to have
either a low baseline detection probability (p0 = 0.05)
or higher (p0 = 0.20). In total, we assessed 108
different population combinations of grouping scenarios, available
identity information, and detection probability.
We simulated 100 datasets per population combination, each time
subjecting individuals to 4 sampling occasions. Sampling occurred with a
5 x 15 grid of 75 sampling locations centered within the state-space to
approximate camera trapping of the Algar caribou herd (Tattersall et al.
2020), with 3-unit spacing and a 9-unit buffer to the state-space edge
(Figure 3). We adapted custom code from Bischof et al. 2020, which uses
the sim.pop function in the ‘secr’ R package (Efford 2022)
(Appendix 1). We summarized data by reporting the mean numbers of unique
individuals detected, total detections, and traps with detections. To
describe the potential consequence of non-independence, we used the
counts of unique animals detected per sampling location to calculate
Fletcher’s \(\hat{c}\), a measure of over-dispersion that describes the
degree to which the observed variance is greater than predicted by
homogeneous density (Fletcher, 2012). The statistic is also a variance
inflation factor as corrected variances may be obtained under low levels
of overdispersion (\(\hat{c}\lessapprox 4)\) multiplying variance by\(\hat{c}\) (Anderson et al. 1994).
Simulated detections were manipulated for density estimation in three
ways. For SCR analysis, unique individual identities were retained and
fit to a frequentist null SCR model using the ‘secr.fit’ function in the
R package ‘secr’. For SC, identity information was removed to derive
trap-and-occasion-specific counts of total detections that were fit to a
Bayesian-formulated null SC model using the ‘nimble’ (de Valpine et al.
2022) package in R. For SPIM analyses, partial identity covariates were
retained and associated with individual detections and then data were
fit using the ‘SPIM’ package (Augustine et al. 2019) in R. For the
Bayesian SC and SPIM, we used a slightly informative prior of
gamma(24,8) on σ (Burgar et al., 2019), resulting in a
mean (± 1 standard deviation, SD) σ = 3 ± 0.6 units. Data
augmentation was set to either M = 400 or M = 600 as
necessary to prevent truncation of the posterior distribution inN as determined with visual inspection. We used an initial
burn-in of 5,000 iterations and a subsequent 45,000 iterations split
into 3 chains each with 15,000 iterations. Model results were assessed
for convergence with the r-hat statistic and removed if r-hat
> 1.1 (Gelman & Rubin, 1992). SC and SPIM were run using
Microsoft Azure virtual machines [Standard D64s v4 (64 vcpus, 256 GiB
memory)] to reduce computation time.
Parameters of interest were abundance (N) and the spatial scale
parameter for individual movement (σ) . We report the mean (for
SCR) and median (for SC and SPIM) estimates of N and σ ,
and similar metrics as used in Bischof et al. (2020) to summarize model
performance across the iterations of each scenario: relative bias,
coefficient of variation, and coverage. Relative bias describes the
extent of over-or under-estimation and is calculated as
\(RB\ =\frac{\hat{\theta}-\ \theta}{\theta}\)
where \(\theta\) is the true value of the parameter of interest (in this
case the value of N or σ used in simulations) and\(\hat{\theta}\) is the point estimate. Coefficient of variation is a
measure of precision and is calculated as
\(CV\ =\frac{SD(\hat{\theta})}{\hat{\theta}}\) .
where SD is the standard deviation. For conservation and management
purposes, CV < 20% is preferred (Morin et al., 2022).
Coverage was calculated as the proportion of simulations that include\(\theta\) within its 95% credible intervals.
\(Coverage=\frac{\sum_{i=1}^{100}{\ I({95CI}_{\text{ilower}}\ \leq\ \text{θ\ }\leq\ {95CI}_{\text{iupper}})}}{100}\)