Results
Aggregation and cohesion influenced the numbers of individuals detected,
total detections, and traps with detections. The mean number of
individuals detected and total detections changed little with
aggregation and cohesion: when p0 = 0.05,
approximately one-third of the population was consistently detected (min
– max: 43 – 47, of N = 140) a total of 65 – 71 times. However, the
variance in these measures increased as aggregation and cohesion
increased (Figure 4, Appendix 2 Table 1). For example, the standard
deviation (SD) in individuals detected increased three-fold from SD = 5
to SD = 18 when changing from the independence (aggregation = 1,
cohesion= 0) to maximal non-independence (aggregation = 10, cohesion =
1); SD in total detections similarly increased 3.5-fold from SD = 9 to
SD = 32. In contrast, the number of traps with detections decreased with
increasing aggregation and cohesion, from a maximum of 43 ± 5 (mean ± 1
SD) under the independence scenario to a mean of only 7 ± 3 traps under
maximal non-independence when p0 = 0.05. The
patterns of fewer traps with detections and greater variance in unique
individuals and total detections with increasing non-independence also
occurred with p0 = 0.20. (Appendix 2 Table 1,
Appendix 2 Figure 1).
Aggregation and cohesion
increased over-dispersion in the number of unique individuals detected
per sampling location (Figure 5). The overdispersion factor \(\hat{c}\)ranged from a minimum of 1.8 ± 0.2 to a maximum of 10.4 ± 0.8 when
p0 = 0.05. Generally, \(\hat{c}\) < 4 when
cohesion was 0 and at low to intermediate values of cohesion and
aggregation. Over-dispersion further increased with p0 =
0.20 (Appendix 2 Figure 2).
Abundance
SCR estimates of abundance (\(\hat{N}\)) were consistently the least
biased across all grouping scenarios compared to SC and SPIM (Figure 6,
Appendix 2 Figure 3). Mean bias was negligible
(|RB|<6%) under low cohesion (0, 0.3) and
increased to a maximum of RB = 16 – 22% when animals exhibited maximal
aggregation (10) and high cohesion (0.67, 1) with p0 =
0.05 (Appendix 2 Table 2). Estimates were relatively precise with mean
CV ≈≤ 20% but suffered when both aggregation and cohesion were high,
whereby the variance in CV increased to 23% (Appendix 2 Table 3).
Coverage was nominal under the independence scenario (97%) and averaged
80 ± 19% across all grouping scenarios, but decreased to a minimum of
38% at maximal aggregation and cohesion when p0 = 0.05
(Figure 7, Appendix 2 Table 4). After correcting the variance for
overdispersion, coverage averaged 97 ± 5% and decreased to 82% only
under maximal aggregation and cohesion (Appendix 2 Figure 4). Patterns
with p0 = 0.20 were similar but with less bias, more
precision, and greater coverage (Appendix 2 Tables 2, 3, 4 and Appendix
2 Figure 4).
In comparison, SC consistently
produced biased \(\hat{N}\), as expected (Figure 6). Mean bias was
negative, underestimating N with inflated precision when there was any
aggregation or cohesion (mean RB ranged -34
± 20% to -88 ± 6%; Appendix 2
Table 2). The amount of bias (|RB|) increased with
both aggregation and cohesion, although bias was less affected by
increasing cohesion at greater levels of aggregations (i.e., 10).
Precision increased with detection probability and with aggregation
under intermediate cohesion (0.3, 0.67), with the greatest mean
precision (CV = 29 ± 12%) under maximal non-independence when
p0 = 0.20 (Appendix 2 Table 3). Coverage with SC
averaged 29 ± 32% across all grouping scenarios and detection
probabilities (Figure 7). Coverage reached nominal levels only under the
independence scenario, and decreased quickly with aggregation and
cohesion to a minimum of 0 (Appendix 2 Table 4). Even after correction
with \(\hat{c}\), mean coverage improved to 44 ± 33% (Appendix 2 Figure
4).
SPIM estimation of N performed differently depending on the
levels of aggregation and cohesion, and number of partial identity
covariates. Estimates were generally overestimated, with the exception
of high aggregation (10) and low cohesion (0, 0.3) in which medianN was underestimated (Figure 6). Otherwise, as with SC, bias
increased with aggregation and cohesion, reaching a maximum of RB = 185
± 96% under maximal non-independence with only the collar covariate
when p0 = 0.20 (Appendix 2 Table 2). Precision also
increased with aggregation and cohesion; mean CV with only the collar
covariate when p0 = 0.05 decreased from CV = 59 ± 13%
under the independence scenario by 30% to 29 ± 12% under maximal
non-independence (Appendix 2 Table 3). Having more partial identity
covariates, or more specifically a greater probability of identity,
helped to moderate bias and inflated precision from aggregation and
cohesion; estimates became less biased and more stable in precision with
more partial identity covariates, such that use of all 4 partial ID
covariates when p0 = 0.20 consistently resulting in
negligible mean bias (RB: 2 - 6%) and stable mean precision of CV = 9 -
10% across all combinations of aggregation and cohesion (Appendix 2
Tables 2, 3). Average coverage across all grouping scenarios was 62 ±
23% (Figure 7), with a minimum of 45% under maximal non-independence
even when all 4 spatial identity covariates. Coverage only reached
nominal or near-nominal levels when cohesion was low and probability of
identity was greater than 54% (i.e., at least 2 partial identity
covariates if collar was not one of the 2 covariates; Appendix 2 Table
4). Coverage improved when variance was corrected with \(\hat{c}\),
averaging 89 ± 11% across all scenarios (Appendix 2 Figure 4).
Sigma
SCR estimates of sigma (\(\hat{\sigma}\)) exhibited similar but less
pronounced patterns of bias, precision, and coverage compared to\(\hat{N}\) (Figures 6, 7). SCR \(\hat{\sigma}\) was generally unbiased
(mean |RB|≤ 8%) and precise (mean CV ≤ 15%) across
all levels of aggregation and cohesion (Appendix 2 Tables 5, 6),
although the variance in bias and precision increased with aggregation
and cohesion. Coverage was nominal under the independence scenario and
less than 90% only when cohesion was high (0.67, 1) (Figure 7, Appendix
2 Table 7). After correcting variance for overdispersion, coverage was
an average 96 ± 7% and only decreased below 90% (to 82%) with maximal
non-independence when p0 = 0.05 (Appendix 2 Figure 4).
For SC, \(\hat{\sigma}\) was less biased (range of mean
|RB|: 0 ± 5% - 70 ± 4%) and more precise (CV
< 20%) compared to \(\hat{N}\), but followed similar patterns
of more bias and inflated precision with increasing cohesion and
aggregation (Figure 6, Appendix 2 Tables 5, 6). Coverage of SC\(\hat{\sigma}\) was nominal or near-nominal when cohesion = 0 and
decreased dramatically when cohesion and aggregation increased (Figure
7, Appendix 2 Table 7). Patterns in coverage did not improve with
variance correction and reached an average of 62 ± 40% (Appendix 2
Figure 4).
Finally, SPIM \(\hat{\sigma}\) had low bias (|RB|:
1-12%) and accordingly acceptable precision (CV ≤ 20%) across all
levels of aggregation, cohesion, and detection probabilities (Figure 6,
Appendix 2 Tables 5, 6). As expected, estimates exhibited greater bias
(or variance in bias), more precision, and less coverage with increasing
cohesion and aggregation. However, as with SPIM \(\hat{N}\), more
partial identity covariates helped to moderate these effects. Notably,
at high levels of cohesion (0.67, 1), SPIM \(\hat{\sigma}\) had greater
coverage than SCR \(\hat{\sigma}\) (Figure 7). Once corrected for
overdispersion, coverage for SPIM \(\hat{\sigma}\) improved from an
average of 83 ± 12% across grouping scenarios and detection
probabilities to a mean of 93 ± 7% (Figure 7, Appendix 2 Figure 2,
Table 7).