Improving inference for aerial surveys of bears: The importance of assumptions and the cost of unnecessary complexity

Abstract Obtaining useful estimates of wildlife abundance or density requires thoughtful attention to potential sources of bias and precision, and it is widely understood that addressing incomplete detection is critical to appropriate inference. When the underlying assumptions of sampling approaches are violated, both increased bias and reduced precision of the population estimator may result. Bear (Ursus spp.) populations can be difficult to sample and are often monitored using mark‐recapture distance sampling (MRDS) methods, although obtaining adequate sample sizes can be cost prohibitive. With the goal of improving inference, we examined the underlying methodological assumptions and estimator efficiency of three datasets collected under an MRDS protocol designed specifically for bears. We analyzed these data using MRDS, conventional distance sampling (CDS), and open‐distance sampling approaches to evaluate the apparent bias‐precision tradeoff relative to the assumptions inherent under each approach. We also evaluated the incorporation of informative priors on detection parameters within a Bayesian context. We found that the CDS estimator had low apparent bias and was more efficient than the more complex MRDS estimator. When combined with informative priors on the detection process, precision was increased by >50% compared to the MRDS approach with little apparent bias. In addition, open‐distance sampling models revealed a serious violation of the assumption that all bears were available to be sampled. Inference is directly related to the underlying assumptions of the survey design and the analytical tools employed. We show that for aerial surveys of bears, avoidance of unnecessary model complexity, use of prior information, and the application of open population models can be used to greatly improve estimator performance and simplify field protocols. Although we focused on distance sampling‐based aerial surveys for bears, the general concepts we addressed apply to a variety of wildlife survey contexts.


| INTRODUCTION
Obtaining useful estimates of wildlife abundance or density requires thoughtful attention to potential sources of bias and precision affecting the population estimator (Williams, Nichols, & Conroy, 2002).
For instance, properly accounting for bias associated with incomplete detection of individuals has been consistently difficult to address when surveying wildlife populations. This difficulty has led to the development and implementation of a wide range of survey tools such as distance sampling, mark-recapture, occupancy, and double observer approaches (e.g., Buckland et al., 2001;MacKenzie et al., 2002;Nichols et al., 2000;Rosenstock, Anderson, Giesen, Leukering, & Carter, 2002;Williams et al., 2002). Each of these techniques addresses specific sources of incomplete detection, although the detection process consists of four individual detection components: p s (the probability that the home range of an animal overlaps the sampled space), p p (the probability that the animal is present within the survey area during the survey period), p a (the probability that the animal is available to be detected given that it is present within the sampled area), and p d (the probability that the animal is detected given that it is present and available; see Nichols, Thomas, & Conn, 2009;Schmidt, McIntyre, & MacCluskie, 2013). Many commonly implemented field sampling techniques focus on correcting for p d < 1.0, and although p s is typically addressed through design (Nichols et al., 2009), p a and p p are less often estimated explicitly or dealt with through design considerations. Instead, inference is often based on the assumption that all individuals are present and available to be detected, although temporary emigration (Kendall, Nichols, & Hines, 1997) or incomplete availability (Schwarz & Arnason, 1996;Wilson, Schmidt, Thompson, & Phillips, 2014) can cause bias relative to the population of interest.
Avoidance of these sources of bias is important if results are to be interpretable, and hence useful for addressing research or management questions.
Situations where p p and p a approach 1.0 certainly exist, although this requires careful assessment within each survey context. For example, if the species of interest has a small home range and exhibits slow movement rates relative to the survey period and study area size, assuming p p ≈ 1.0 might be reasonable. Conversely, highly mobile species (e.g., passerine birds) may move to areas outside of a particular sample unit on any particular visit . Similarly, highly visible species in open habitats may meet the assumption of p a ≈ 1.0 (e.g., Dall's sheep; Schmidt, Rattenbury, Lawler, & MacCluskie, 2012), whereas in other cases, behavior or individual characteristics could be expected to cause p a ≈ 0 for some individuals during part of the survey period (e.g., diving cetaceans; Laake, Calambokidis, Osmek, & Rugh, 1997). When it is unreasonable to assume that all individuals are present and available, p p and p a must be addressed directly through careful attention to design and/or analysis; otherwise inference is limited to some unknown proportion of the population, despite correcting for p d < 1.0. Although partially corrected population indices can be useful in some contexts (e.g., Johnson, 2008), inference to the entire study population is generally of primary interest to scientists and managers. When assumptions related to p p and p a cannot be met through design modifications (e.g., use of radiotelemetry), analytical solutions become necessary to achieve estimates that are interpretable and support comparisons across time or space.
In addition to addressing bias in the overall detection process, consideration must be given to important tradeoffs among bias, precision, and survey costs (both logistical and monetary) within the context of an estimator's total error (i.e., total error = √ bias 2 + varience; Reynolds, 2012). Designs employing mark-recapture methods are often preferred in wildlife work, in part because p d , p p , and p a can be estimated directly, thereby minimizing bias. However, the costs of capturing and marking sufficient numbers of animals may not always be practical, particularly when inference over large areas is required (Pollock et al., 2002). The expense of the marking efforts can limit feasible sample sizes, reducing estimator precision and increasing total error relative to an approach that does not attempt to discern the different sources of detection bias. In many contexts, methods based on detections of unmarked individuals (e.g., occupancy surveys; MacKenzie et al., 2006;distance sampling;Buckland et al., 2001Buckland et al., , 2004 are simpler and cheaper to implement than other approaches, leading to larger sample sizes, fewer parameters to be estimated, and potentially smaller total error. For example, conventional distance sampling (CDS) only requires a single visit and relies on the assumption that p a = 1.0 (although see Amundson, Royle, & Handel, 2014), whereas multiple visit approaches (e.g., occupancy surveys) can be used to incorporate p a < 1 directly. A primary consideration when assessing the appropriateness of any particular survey approach is the relative balance it provides between bias and precision for feasible sample sizes and, specifically, the magnitude of bias it may be susceptible to while meeting precision objectives.
Although realized sample sizes are influenced by factors such as design requirements, cost, and animal density, model fitting in a Bayesian analytical framework provides another option for increasing estimator efficiency (precision) after data have been collected.
However, this approach requires large sample sizes that can be cost prohibitive, limiting utility for long-term monitoring of bear populations (e.g., Reynolds et al., 2011).
Here, we explore the application of alternative analytical tools in the interest of cost reduction and improved inference for bear datasets collected under the MRDS approach.
We used three brown bear datasets to examine methodological and analytical assumptions, detect and reduce apparent bias, increase precision, and simplify field methods. Our specific objectives were to (1) explore the tradeoff between apparent bias and precision between the MRDS method and a simpler CDS framework for aerial bear survey data, (2) investigate the potential for improved inference through the use of informed priors on the detection process, and (3) explicitly test the assumption p a = 1.0 through application of open-distance sampling models.

| Study area
The study area consisted of portions of Lake Clark National Park and

Preserve [LACL] and Katmai National Park and Preserve [KATM] in
southwestern Alaska, USA ( Figure 1; Table 1). The maritime climate is characterized by cool wet summers. Winter months tend to be drier on average, but deep snow accumulates on the higher, glaciated peaks. Habitat consists of glaciated mountains, sparsely vegetated alpine plateaus, alder slopes, and forested bottomlands. Brown bears also use coastal sedge meadows and beaches. Bear densities in these coastal areas are approximately 10-fold higher than in interior areas (Miller et al., 1997).  Table 1, Figure 1). We considered these datasets to be well-suited for our work because they were collected using a MRDS approach specifically developed for bears (Becker & Quang, 2009) and included sufficient sample sizes for analysis. The KATM survey was conducted over two consecutive spring periods to meet sample size objectives; population closure between years was assumed for the purposes of analysis. Transects were surveyed for brown bears from tandem-seat, fixed-wing aircraft flown at approximately 100 m above the ground. The pilot and observer independently searched for bears on the uphill side (or a random side in flat areas) of the aircraft.

| Data collection
Detected groups were not immediately announced, allowing the other observer an opportunity to detect the group independently. To maintain observer independence, a curtain was installed in the aircraft to ensure the detection of a bear group by one observer did not alert the other observer to its presence. After a detected group had passed from the observable space, the detection was announced and the detection history (i.e., pilot only, observer only, both pilot and observer) was recorded. The location and size of the group were recorded, and the distance of the group from the line was later calculated using GIS.

Further details on survey methodology can be found in Becker and
Quang (2009)

| Data analysis
In order to assess the underlying assumptions of the sampling design and analytical tools for data of this type, we used four basic approaches to analyze these three datasets: (1) MRDS following the basic approach of Becker and Christ (2015), (2)

| Becker-Christ ("B-C") approach
Histograms of the observed distances collected under the MRDS bear survey protocol are typically "humped," increasing to an apex at some distance from the line and then decreasing thereafter (Becker & Quang, 2009). This distribution violates the typical assumption of CDS that numbers of detections decrease monotonically with increasing distance from the transect (Buckland et al., 2001). Initially such datasets were analyzed using a gamma-shaped detection function (Becker & Quang, 2009;Walsh et al., 2010), which can accommodate humped detection data. However, this formulation was found to violate the assumption of full independence, producing negative bias in the resulting estimators (Benson, 2010). Recently, the gamma-function was replaced with a two-piece normal distribution assuming point independence at the apex of the detection curve (i.e., Borchers, Laake, Southwell, & Paxton, 2006), thereby correcting for the negative bias in the original model formulation (Becker & Christ, 2015). Hereafter we refer to this analytical approach as the "B-C approach." Because the B-C approach and its earlier variants have routinely been applied to datasets similar to ours, we used estimates based on it as a baseline for comparison with the other approaches in the context of apparent bias and precision.
We obtained the two piece normal source code for the B-C approach from the original publication (Becker & Christ, 2015) and used it to estimate the number of bears in LACL, KATM, and KATM.
PR. Effective search distance (ESD: how far out the observers were searching at the time the bear was detected) is a covariate thought to be necessary to account for heterogeneity that would affect the mark-recapture part of the MRDS approach (Becker & Christ, 2015).
We found that both ESD and equal search-distance categories were highly correlated with distance; therefore, we did not include ESD in our analysis. We modified the input code to fit the intercept model T A B L E 1 Details of survey effort for each of the three example brown bear datasets collected in southwestern Alaska, USA for each dataset (i.e., no covariates were included on the detection process). Doing so simplified direct comparisons among the different analytical frameworks.

| CDS approach
The B-C approach generally requires much larger sample sizes (i.e.,  (Becker & Christ, 2015;Becker & Quang, 2009;Walsh et al., 2010), suggesting that joint p d at the apex may approach 1.0 in many cases. To test these hypotheses, we created a dataset appropriate for a CDS analysis by left-truncating each dataset at the apex of the composite detection curve estimated using the B-C approach above, which left a monotonically declining subset of distances from that point ( Figure 2). We also discarded the capture history information associated with each detection, which is the equivalent of the pilot and observer working together as a team to detect bear groups. We then fit each reduced dataset in a CDS framework (Buckland et al., 2001), assuming a half-normal detection function and the basic Bayesian hierarchical model described by Schmidt et al. (2012) (see Supporting Information). We used the Gelman-Rubin diagnostic (Brooks & Gelman, 1998) and a visual inspection of the chains to assess convergence. For simplicity, we did not include covariates on the detection process, but instead relied on the "pooling robustness" characteristics of the CDS approach to unmodeled heterogeneity in detection (Buckland et al., 2001;Marques, Thomas, Fancy, & Buckland, 2007).
Group size was assumed to be Poisson distributed, and we used minimally informative priors on all model parameters. Using this basic model structure, we then estimated the total number of bears within each study area. We compare these results to those based on the B-C approach with respect to apparent bias and precision of the resulting estimators.

| Bayesian approach: Informed priors
To investigate the influence of sharing detection information among surveys, we used an informed prior on the scale parameter, σ, of the detection function to refit the less data-rich LACL and KATM.PR datasets (Table 1)

| Open-distance model
We also suspected violation of the assumption that all bears were available to be sampled (Becker & Quang, 2009). Variation in factors such as viewing angle, coloration, or cover type may render some proportion of individuals "unavailable" for sampling during a particular Open-distance models use variation among repeated surveys to estimate p a and produce estimates of the entire population exposed to sampling over the survey period. Because the study areas were large relative to bear movement rates, p p was presumed to be approximately 1.0. If closure was not maintained at the level of the study area (i.e., p p < 1.0), then estimates of p a would reflect the combination of incomplete p p and p a .
In order to test the assumption of p a ≈ 1.0, we selected a subset of the KATM data from 6 days when sampled transects were spatially distributed throughout the study area (Figure 1). This reduced dataset consisted of 168 detections on 323 transects, providing a set of six representative samples of the population in the study area for analysis. We fit a version of the temporary emigration open-distance model described by Kery and Royle (2016) to these data within the framework of Schmidt et al. (2012). The resulting estimator of population size represents the superpopulation (i.e., Schwarz & Arnason, 1996) of bears exposed to sampling during the six sampling days, whereas estimators for the KATM data based on both the basic CDS and the B-C approaches represent the population of bears available to be sampled throughout the duration of the survey (21 days total).

| RESULTS
The B-C approach resulted in the least precise estimators of bear abundance for each of the three areas ( Figure 3). Precision was especially low (i.e., CV = 22%) for the KATM.PR dataset where <50% of the recommended number of detections had been recorded. Estimators using the CDS approach were approximately 30-40% more precise (based on reduction in CVs) and exhibited little apparent bias (Figures 3 and   4). In comparing the point estimates between the two methods, we found little evidence that discarding the mark-recapture information in the CDS approach caused appreciable or consistent apparent bias relative to the B-C results (i.e., −7%, −9%, +2% for the datasets from LACL, KATM.PR, and KATM, respectively; Figure 4). Error bars around

| DISCUSSION
This work was motivated by a desire to develop logistically feasible and cost-efficient survey and analytical methodologies that would yield reasonably unbiased and precise estimators of abundance of brown bears in Alaska. The MRDS protocol is commonly employed when assessing bear populations in large landscapes; however, obtaining the recommended number of bear group detections for model fitting in the B-C analytical framework is often cost prohibitive. This led us to reevaluate model complexity in the context of the assumptions inherent in the B-C approach. We found that this well-established and widely implemented approach for estimating bear abundance was suboptimal in the context of apparent bias and estimator efficiency. Our findings are especially important given the potential for long-term reductions in cost and effort under simplified protocols (see " Step 10" in Reynolds, Knutson, Newman, Silverman, & Thompson, 2016). Although the B-C approach formally accounted for the entirety of the complex detection process inherent in the bear survey data, the additional model complexity did not improve inference relative to much simpler approaches requiring estimation of many fewer pa-  (Nichols et al., 2009;. In the context of long-term monitoring of bear populations, these sources of bias can have important consequences for confounding trend estimators (Kery, Royle, & Schmid, 2005;Pollock et al., 2002). Violations of the assumption of p a = 1.0 is known to cause large negative bias in a variety of settings (e.g., Bailey, MacKenzie, & Nichols, 2014;Efford & Dawson, 2012;Kendall & White, 2009;Wilson et al., 2014). In the case of occupancy models, such bias alters survey inference to some form of "use" (Mordecai, Mattsson, Tzilkowski, & Cooper, 2011;Schmidt, Flamme, & Walker, 2014), the utility of which must be carefully considered relative to survey objectives. Careful attention to the role of each component of the detection process is critical to proper inference and is an important consideration when designing bear surveys.
Field data on bear populations are generally expensive to collect, and sample size requirements play a large role in assessing whether a given survey approach is practical in the context of study objectives (e.g., Reynolds et al., 2011). Our work suggested that critical assess- Sample size demands are directly related to estimator precision, although the use of informed priors in a Bayesian framework may be advantageous (Gelman et al., 2004;King et al., 2010;Link & Barker, 2010). Despite the possible benefits of sharing information among surveys, the use of informed priors remains uncommon in wildlife surveys, possibly due to concerns that objectivity may be compromised (Morris, Vesk, McCarthy, Bunyavejchewin, & Baker, 2015). We found that sharing information on a common detection process dramatically increased the precision of our estimates of bear abundance with little effect on apparent bias. The use of informed priors on detection parameters has been found to have similar advantages in other distance sampling applications (i.e., Schmidt & Rattenbury, 2013) and would likely be useful for bear surveys in particular because the basic field methodology has been standardized and can be repeated in both time and space. Unless survey conditions are expected to be highly variable, we argue that the use of prior information on the detection process should be a default approach to increase the precision of estimators bear abundance collected using distance sampling protocols.
Assessing assumptions regarding each component of the detection process is often difficult without ancillary data. Recent developments of open population models for unmarked individuals (i.e., Dail & Madsen, 2011) provide a suite of tools that may be used to assess some of these assumptions directly as has been done in a markrecapture framework (e.g., Kendall et al., 1997;Schwarz & Arnason, 1996). We used open-distance models to assess p a and found that a large proportion of bears were likely unavailable during the average survey day. We expect other similar applications (e.g., dynamic occupancy models) might prove equally useful in assessing assumptions for other survey types. When wildlife surveys are repeated over time (e.g., annually) for monitoring purposes, open population models may also be used to extract information on population dynamics (Dail & Madsen, 2011;Schmidt et al., 2015;Zipkin et al., 2014), further increasing their value for science and management. We expect this class of models will be useful in increasing the amount of available information on bear populations, as well as a variety of other species.