Discussion

Unmarked density estimation is a challenging but worthy endeavor due to the many species and populations that are unmarked and require information about population status for informed conservation. However, when individuals are both unmarked and non-independent, our simulations showed that two primary SCR-based unmarked approaches, SC and SPIM, perform poorly. Population abundance (N ), and the spatial scale parameter (σ ) to a lesser extent, were often estimated with increasing bias and imprecision as aggregation and cohesion increased. The reference SCR model remained the least biased and most precise despite non-independence (Bischof et al., 2020; López-Bao et al., 2018), while aggregation and cohesion affected SC and SPIM differently. In particular, we identified no conditions of non-independence under which SC could reliably estimate parameters. In contrast, SPIM was more robust to violating the independence assumption and could estimate N andσ under some conditions with low cohesion. Unmarked density models should be applied selectively and carefully to populations exhibiting non-independence among individuals.
Our simulations strongly suggest that SC models should not be used when individuals aggregate or cohere, and are likely only appropriate for independent species or periods when individuals are independent. Abundance was severely underestimated with misleading precision at all levels of aggregation and cohesion, consistent with our expectations but opposite to the overestimation found in previous work with independent populations (Chandler & Royle, 2013; Ruprecht et al., 2021). Perhaps fortunately, the CV never reached 20% or lower to afford confident application of the biased estimates for informing conservation or management action (Morin et al., 2022). Estimates of σ were slightly better but still biased, rendering SC unfit for inferences about movement and space use when individuals exhibit grouping behaviors. These issues could not be resolved by applying \(\hat{c}\) as a variance inflation factor, as it failed to recover sufficient coverage for parameter estimates. Indeed, \(\hat{c}\) was ineffective because the relative variance (variance for a given grouping scenario divided by the variance for the independence scenario) far exceeded \(\hat{c}\), in contrast to the SCR simulations of Bischof et al. (2020). Furthermore, overdispersion when \(\hat{c}\) > 4 is ideally addressed with additional model structure (Anderson et al., 1994; Fletcher, 2012). Realistically, other measures of overdispersion would be more appropriate for SC models because empirical or non-simulated SC data do not have the counts of unique individuals used for the formulation of\(\hat{c}\) in these simulations. We also recognize the potential to consider other distributions to model the SC data, such as the negative binomial, to allow for greater variance (Lindén & Mäntyniemi, 2011) rather than the standard Poisson distribution used here.
While SPIM estimation also suffered with increasing non-independence, SPIM may be acceptable when cohesion is minimal provided there are sufficient partial identity covariates. Applying SPIM under larger values of cohesion resulted in overestimated populations, which Sun et al. (2022) also found with empirical and simulated SPIM modeling under independence. Indeed, the variance in RB and CV (across the iterations for any single grouping scenario) were large even with high probability of identity (i.e., 99% from all 4 partial identity covariates) and p0 = 0.20, despite small mean RB and CV values across grouping scenarios. The underestimated population observed with low cohesion and high aggregation was consistent with our expectations, while the otherwise mostly positive bias observed with SPIM was consistent with Augustine et al. (2019) – who also reported positive bias under independence due to high local densities that increased identity uncertainty and ultimately assigned detections to more individuals than actually detected. The change in pattern of bias with increasing cohesion, from under- to over-estimation, with few partial identity covariates is unexplained and motivates further examination. Broadly, the increase in bias with increasing non-independence is consistent with increasing uncertainty in individual identification due to overlapping space use.
The decreased bias and greater precision from using more SPIM partial identity covariates highlight the importance of maximizing identity resolution by increasing the numbers of partial identity covariates and their categorical values. More combinations of partial identity covariates could have been considered because probabilities of identity ranging 53 - 90% were not assessed. However, increasing the number partial identity covariates is not a panacea because in practice the number of partial identity covariates for a species or population will be limited (motivating the use of SPIM in the first place), and estimates were still biased under high aggregation and cohesion even when identities were completely known (SCR). Additional investigation of SPIM is nonetheless warranted to further explore the ecological and sampling conditions in which SPIM models could still be reliably used. Future simulations should explore more probabilities of identity, values of aggregation and cohesion, and different spacings of sampling locations. Density estimation for unmarked populations expected to have low levels of aggregation and cohesion may benefit from sampling spaced closer than generally recommended under independence in order to increase spatial variation among group members (Augustine et al., 2019; Chandler & Royle, 2013).
In addition to SCR-based unmarked approaches, other approaches instead rely on information about movement speed (or its inverse: staying time within a viewshed). Two such unmarked models, the random encounter model (Rowcliffe et al., 2008) and random encounter and staying time (Nakashima et al., 2018), have demonstrated potential to be robust to non-independence but also poor model fit that would hamper interpretation of spatial or temporal patterns (Hayashi & Iijima, 2022). Two other non-SCR approaches, space to event (Moeller et al., 2018) and time in front of camera models (Becker et al., 2022), have shown either bias or a lack of concordance with other approaches when non-independence is added (Fisher et al., 2021; Hayashi & Iijima, 2022). Hayashi and Iijima (2022) also showed that bias in the non-SCR approaches is sensitive to the choice of statistical distribution for modeling count data, including Poisson, zero-inflated Poisson, exponential, and negative binomial. Comparing SCR-based and non-SCR-based approaches under identical simulating conditions would be informative for helping wildlife researchers and managers choose the appropriate sampling and modeling approaches for obtaining unbiased population estimates. Indeed, we recommend researchers run a small simulation study using expected population conditions and the anticipated sampling designs to confirm that robust density estimation is feasible before implementing in the field.
Continued efforts to identify optimal sampling designs and estimation approaches are essential for robust wildlife populations monitoring. If SCR-based unmarked approaches are to be tenable for non-independent populations, models must have procedures to account for – and ideally attempt to incorporate – cohesion and aggregation. In addition to using statistical distributions that account for overdispersion, some efforts to estimate density of group-living species have corrected group densities with average group size (e.g., Mattioli et al., 2018). However, including a mechanism for how aggregation and cohesion impact the detection process and lead to overdispersion would further advance the understanding of the focal species’ ecology. The clustered SCR model of Emmett et al. (2021) begins this work and presents some potential for modification to fit unmarked populations for ecological inference, but caution is necessary. The current formulation requires knowledge about group membership for detected individuals, which is difficult to obtain when identities are unknown unless groups are being monitored intensively, in which case detections with unknown identities are usually the exception and can be dealt with in ways previously described. Group membership might instead be estimated, informed by spatial patterns and supported with informative priors about range of movement and group size. A model that uses the spatial pattern of detections and any available partial identities to estimate the distribution of individuals and groups when identities and group membership remain unknown would be of great appeal and value. However, attempting to develop a model that weaves a tapestry of gold from relatively little information risks neglecting careful sampling design for statistical machismo (Gimenez et al., 2014). As a result, we encourage the continued development of unmarked models but recommend that applications of current SCR-based approaches to unmarked population estimation recognize that they are generally unsuitable for most cases of non-independence.