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).