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.