Estimates of caribou herd size using post-calving surveys in the Northwest Territories and Nunavut , Canada : A meta-analysis

Post-calving surveys to estimate herd size of barren-ground caribou (Rangifer tarandus groenlandicus, R. t. granti, and R. t. caribou) have been used for caribou herds in Alaska, Yukon, Northwest Territories, Nunavut, and Québec/Labrador. The main field procedure uses relocation of collared caribou to locate aggregated groups of hundreds or thousands of caribou during times of high insect harassment that usually occur in July. These groups are then photographed to obtain a count of the caribou in the aggregated groups. Often some caribou are missed, and the count of caribou may be a negatively biased estimate of total herd size, unless a high proportion of the herd is found and photographed. To address this, some previous studies have used the Lincoln-Petersen estimator, which estimates the proportion of the herd counted based on the percentage of available collared caribou found during the survey. However, this estimator assumes equal probabilities of all groups of caribou being found, regardless of group size and the numbers of collared caribou in the group. These assumptions may not be valid, as larger groups are more likely to be found than smaller groups, particularly if there are several collared caribou present. This may lead to estimates that are biased low, along with an estimate of variance that may also be biased low. A two phase estimator developed by Rivest et al., in 1998 became available in R statistical software in 2012. We analyzed 20 data sets from post-calving surveys in the NWT and NU carried out between 2000 and 2015 using the Rivest estimator to explore working characteristics of this estimator. We compared the Rivest estimates with Lincoln-Petersen estimates and total counts on each survey. We considered factors that influence precision of the Rivest estimator with a focus on sampling factors such as the proportion of collars found, the number of collars available, and natural factors such as the degree of aggregation of caribou in each survey (as indexed by the negative binomial dispersion parameter). In general, the Rivest estimator displayed acceptable precision when high proportions of caribou groups with collars were detected and counted, collar numbers were sufficient, and aggregation was adequate. Notable exceptions occurred in years of lower aggregation which resulted in many small groups with 0 or few collared caribou, and in these cases herd estimates had large variances and low precision. Estimates from the Rivest estimator, Lincoln-Petersen estimator, and total counts converged when sampling effort was high, collar numbers relative to herd size were high, and caribou were well aggregated in a limited number of groups. In other cases, estimates of the Rivest estimator were generally higher than Lincoln-Petersen estimates, presumably due to negative bias with the Lincoln-Petersen estimator. We provide a set of working recommendations to optimize field sampling to ensure reliable estimates of herd size using post-calving methods.


Introduction
Post-calving surveys have been used to estimate population size of migratory caribou (Rangifer tarandus groenlandicus, R. t. granti, and R. t. caribou) herds in Alaska, Yukon, Northwest Territories, Nunavut and Québec/Labrador with the first survey in Alaska in 1961 (Davis et al., 1979;Valkenburg et al., 1985;Russell et al., 1996;Patterson et al., 2004;Harper, 2013;Adamczewski et al., 2017).The main field procedure for this method is the use of collared caribou to locate aggregated groups of hundreds or thousands of caribou during times of high insect harassment that usually occur in July.The main objective of field procedures is to photograph the aggregations of caribou found and thereby obtain a near-count of total herd size.
The main challenge with this technique is that it is often difficult to locate all the aggregated groups and it is likely that some groups are missed; the total count of all photographed groups is thus an underestimate of total herd size by an unknown amount.From a statistical perspective this total count is problematic in that it is negatively biased as an estimate of herd size and has no associated estimate of variance.For trend monitoring, the total count becomes an index and trend estimates will only be valid if it can be assumed that the total amount missed is the same each year (Anderson, 2001), or if the estimation of herd size reliably accounts for the proportion of the herd that was likely missed in each survey.
An estimator for post-calving surveys of herd size that has been applied in the NWT and elsewhere is an adaption of the Lincoln-Petersen mark-recapture estimator (Lincoln, 1930) to collar data where the proportion of available radio collared caribou that are detected during the survey estimates the proportion of the herd that is found by survey flying (White & Garrott, 1990;Russell et al., 1996).Fundamental assumptions of the Lincoln-Petersen estima-tor are that all collared caribou will have equal probability of detection, and that each collared caribou will be a random representation of all caribou so that the recapture rate of the collared caribou will reflect the true proportion of the population sampled.In the context of postcalving surveys, this assumption can be problematic given that the number of collared caribou is a very small proportion of total herd size and often the number of radio collared caribou in large groups is larger than in small groups.In addition, the survey is generally built around flying to the collared caribou, thus groups with multiple collars have a high likelihood of being found while smaller groups with one or no collars are more likely to be missed.Therefore, detection probabilities of caribou groups and collared caribou may not be equal and the varying size of groups and varying numbers of collars will mean that some groups have higher detection rates than others (Patterson et al., 2004).As a result, estimates from the Lincoln-Petersen estimator may be negatively biased, and associated estimates of variance (confidence intervals) may also be negatively biased.This leads to a biased but apparently precise estimate, which can be misleading if used for management purposes.Some ad-hoc methods have been proposed to account for bias issues with the Lincoln-Petersen estimator (Russell et al., 1996), however, these are subjective and often result in the loss of data from smaller group sizes (Rivest et al., 1998).
As an alternative to the Lincoln-Petersen estimator, Rivest et al. (1998) proposed a twophase estimator of population size from postcalving surveys that circumvents many of the issues with the Lincoln-Petersen estimator.The main distinction of the Rivest estimator is that it more appropriately defines caribou groups rather than collared caribou as the sample unit for estimates and treats photographed groups of caribou with collars as a sample of all the groups in the herd.Using this approach allows for various models of how collared caribou represent aggregated groups to be proposed and allows for a more robust estimation of population size that better accounts for the effect of varying group sizes and varying numbers of collared caribou on estimate precision.Until recently, the Rivest model was not applied to post calving data sets with the exception of Patterson et al. (2004), who conducted a limited analysis of the Bluenose-East 2010 post calving data set.In 2012, the estimator became available as the caribou package (Crepeau et al., 2012) in R statistical software (R Development Core Team, 2009) allowing fitting of a full suite of Rivest models.This estimator has been adopted in Alaska (Harper, 2013) and in the Northwest Territories (Adamczewski et al., 2017, this paper).
The main objective of this paper is to assess trends across 20 post-calving survey data sets from the Bluenose-East (BE), Bluenose-West (BW), Cape Bathurst (CB), and Tuktoyaktuk Peninsula (TP) Herds carried out in the NWT and NU between 2000 and 2015 (Figure 1).We compared the general performance of the Rivest estimator across the range of post-calving data sets and compared Rivest estimates with Lincoln-Petersen estimates and total counts.We note that the data sets analyzed ranged from relatively large herd sizes (BE herd that sometimes exceeded 100,000 caribou) with resulting limited collar coverage (i.e.relatively low number of collared caribou relative to overall herd size) to relatively small herds (CB and TP herds with 4,000 or fewer caribou), and higher collar coverage and sampling effort.Comparison of these data sets provided a useful way to determine sampling thresholds and guidelines that will help ensure reliable post-calving estimates of herd size.

Field methods
The general method of post-calving surveys is aerial survey of groups of caribou that are aggregated due to insect harassment.This may occur as early as late June or later in July, but occurs most often in the first half of July.Radio collared caribou are used to locate groups and often the majority of groups contain radio collared caribou.Survey flying usually begins near July 1, and continues until either the survey is completed or it is clear in late July that the post-calving period has ended.
Surveys occur within a narrow window of time to minimize mixing of groups and possible double counting of caribou.For smaller herds, a single aircraft has generally been sufficient to photograph all groups in one day or occasionally over 2-3 days.For the BE herd in 2010 that exceeded 100,000, two aircraft were used to find all the collars and cover the full summer range in a day (Adamczewski et al., 2017).In some surveys, an initial systematic reconnaissance survey has been used across the known summer range of the herd and guided by collared caribou locations (e.g.Adamczewski et al., 2017); however, the movement rates of caribou in the insect season can be high (30-40km/day) and the caribou can have a very clumped distribution that changes frequently, so the distribution defined by such a reconnaissance survey is only useful for a day or two.More commonly, the survey crew center their flying around the last known set of collar locations and focus on locating a high proportion of these collars.In earlier years (before 2007), some of the collars used were satellite collars and some were VHF collars that could only be found if the aircraft was within a few miles.Since 2007, collars used have been almost entirely satellite or GPSsatellite, with daily locations during the survey period.Under these conditions, the day's flying is focused on the last set of collar locations, although the exact locations when photos are taken still depend on homing in on the VHF signal.Additional groups with no collars are generally found incidentally in the vicinity of groups with collars or in flying to and from recent collar locations.Groups with and without collars are located and photographed to allow accurate counts of caribou within each group (Russell et al., 1996;Rivest et al., 1998).
The chief weakness of this method is that it is weather-dependent; the survey is most likely to succeed during warm dry weather with limited winds.If caribou do not aggregate sufficiently, or if part of the herd does not aggregate, then photography is not possible and the survey fails.This occurred, for example, with the BE herd in 2001, 2009 and 2012, and in the past has failed in multiple years for the Porcupine herd (http://www.pcmb.ca/herd).In the field it is usually readily apparent if a caribou group is sufficiently aggregated for photography, with well-defined edges; many groups will be contained within a single photo or they may be spread over a series of overlapping photos.Caribou groups that are more dispersed where the edges of the group are difficult to define are not suitable for photography.Multiple photo passes are made over each group.Photos are converted to GPS map files and caribou are counted on-screen by placing waypoints on each caribou 1 year old or older (Adamczewski et al., 2017).Young calves born in June are not usually counted as they may be hidden behind adults, particularly in groups that are tightly ag-Rangifer, 38, (1) 2018 gregated.All photos are counted independently by at least two observers.In our experience, counts from two obervers are usually very similar (e.g.totals of 915 caribou vs. 918 caribou for a single photo) and the difference in counts by two obervers is usually well below 1% (Adamczewski et al., 2017).
Appendix 1 provides details on each survey.

Rivest estimator
The Rivest estimator considers the sampling of post-calving aggregations as a two phase sampling process.The first phase involves the distribution of collared caribou within the postcalving groups encountered during the survey.For this estimator it is assumed that n caribou are collared and that these caribou randomly distribute themselves into m groups during the post-calving period when the survey occurs.In general, the probability of a group containing at least one collared caribou P̂≥ 1collar increases with group size.The second phase of sampling involves the actual aerial search for groups.For this phase, various models are proposed as to how groups with collared caribou are detected P̂g roup .Three models are considered: 1.The homogeneity model.This model assumes that caribou groups (with collared caribou in the groups) are missed as a completely random event that is independent of the number of collared caribou in the group or other factors.Therefore, each group will have the same probability P̂g roup of be-ing detected by the aerial survey.The Lincoln-Petersen estimator essentially assumes a homogeneity model of detection of groups.
2. The independence model.This model assumes that each collared caribou in the group has the same independent probability of being detected and therefore the overall probability of detecting a group P̂g roup increases as a function of the number of collared caribou in the group.
3. Threshold model.This model assumes that all groups with more than a threshold level of collared caribou (symbolized by B) have a detection probability of 1.For example, it might be that once more than 3 collared caribou occur in a group the group will always be detected whereas groups with 1 or 2 collars are not always detected.For this model, all groups with 3 or more collared caribou get a detection probability of 1 and detection probability P̂g roup is estimated for groups with 1 or 2 collars.
Each of these models can potentially describe detection probability variation in the data set.As part of the estimation procedure a log-likelihood score is produced and the model with the highest log-likelihood is considered to best fit the data.Threshold models are run across the range of observed sizes of collars in groups.
The estimate of herd size (symbolized T̂) is then basically the summation of each group size divided by the probabilities of the group being observed and having at least one collared animal included (which is estimated by the product of P̂g roup and P̂≥ 1collar . and NU (e.g.Nagy & Johnson, 2006) and elsewhere (e.g.Russell et al., 1996) to obtain estimates of herd size.The Lincoln-Petersen estimate of herd size was calculated using the total count of caribou observed during the survey (C), the number of collared caribou available (M), and the number of collared caribou that were observed in groups (R); (Russell et al., 1996;Patterson et al., 2004).Herd size is then estimated as: It is through an iterative likelihood-based optimization procedure that each of these parameters is estimated to produce estimates of herd size.Given that collared caribou are used to estimate detectability of groups, the Rivest estimator does not use data for groups of caribou photographed with no collars.Intuitively, if caribou are aggregated into larger groups and therefore likely contain at least 1 collar then P̂g roup and P̂≥ 1collar will be close to 1 and the resulting estimate will be close to the total count of caribou observed.
An assumption of this method is that the collared caribou are randomly distributed within the separate caribou groups that are photographed.It is possible to test this assumption using a test for over-dispersion of the Poisson probability distribution.Over-dispersion applies to a case when non-independence of collared caribou produces a distribution of collared caribou relative to group size that is different from the distribution if the caribou were randomly distributed.If over-dispersion occurs, then estimates of population size and variance from the Rivest estimator will both be negatively biased (Rivest et al., 1998).
All Rivest estimator calculations were conducted using the R-package (R Development Core Team, 2009) entitled "caribou" (Crepeau et al., 2012).Confidence limits were based upon multiplication of the standard error of the estimate times 1.96.The lower limit of the confidence limit was constrained to be equal or greater than the minimum number of caribou counted during the survey.

Lincoln-Petersen estimator
The Lincoln-Petersen method has been used in several post-calving surveys in the NWT Some authors have suggested that only counts of groups with collars (C in the LP equation) should be used with the Lincoln-Petersen estimator (Russell et al., 1996;Patterson et al., 2004) whereas other studies have included counts of groups observed without collars (Nagy & Johnson, 2006) under the assumption that groups without collars were often in close proximity to collared groups and therefore constituted part of the population represented by collared caribou.We calculated the estimate using both methods to assess the sensitivity of estimates to this assumption.
If all the available collared caribou are found in photographed groups, then the M and R terms in the Lincoln-Petersen herd size equation cancel each other and the Lincoln-Petersen estimate equals the count of caribou observed in all groups.The M-R term in the variance estimate becomes 0 leading to an estimate of 0 variance.In this case it is assumed that a census of the herd has occurred with all individuals counted.

Analysis of factors affecting estimates
As with any statistical estimator, the performance of the Rivest estimator will depend on sample size, sampling effort, and how animals are distributed relative to sampling efforts.We initially explored factors that can be controlled by the survey crews such as the sample size of collars available, and on the proportion of the collars that are located (which is proportional to overall survey effort) and their effect on estimate precision.
The reliability of post-calving herd estimates is also based on how strongly the caribou aggregate during the survey.This factor cannot be controlled in terms of study design but often it can determine the relative success of a survey.To index aggregation, we estimated the mean group size and the negative binomial dispersion parameter (θ) (Anscombe, 1948;White & Bennetts, 1996;Krebs, 1998) from the distribution of observed group sizes for each survey.If the groups are well aggregated, then θ should be small (<0.5).Larger values of θ indicate a more random (Poisson) distribution of groups which is not desirable for estimates, since it becomes less likely that a substantive portion of groups will be found and photographed.We used values of θ as a covariate to help explain differences in levels of precision from the Rivest estimator.The MASS package (Venables & Ripley, 2013) in program R was used to estimate θ using a maximum likelihood estimator (theta.ml (counts, mean) command where counts is a vector (list) of group counts and mean is the mean group count).We used all groups observed to estimate θ, including groups that had no collared caribou.
We used graphical methods and multiple regression analyses (Zar, 1996) to determine optimal working properties of the Rivest estimator as defined by estimated precision of estimates.We defined adequate precision as an estimate with a coefficient of variation of less than 20%, which is generally deemed suitable for manage-ment studies (Pollock et al., 1990), however, we suggest managers assess precision needed for estimates dependent on management objectives.A regression analysis was used to determine ranges of sample sizes and sampling effort, as indicated by the proportion of collared caribou that would be needed to achieve adequate estimate precision at different levels of aggregation (as estimated by the negative binomial θ).The fit of the regression model was assessed using r 2 (coefficient of determination) as well as parameter significance (Zar, 1996).

Summary of data sets
The herds sampled using post-calving methods varied in size from the relatively large BE herd of over 100,000 caribou to the TP herd that was less than 4,000 caribou at the time the surveys occurred (Table 1).The number of collars used in surveys ranged from 24 to 63 per herd.Levels of aggregation varied from well aggregated (θ=0.2) to much less aggregated (θ=0.9).There were 20 surveys in total between 2000 and 2015.Specifics of each survey are given in Appendix 1.

General comments on performance of the Rivest estimator
Estimates were derived from the Rivest model estimates with the highest likelihood score (Table 2).Precision of the Rivest estimator was adequate (CV<20%) in 14 of the 20 data sets used in the comparison.Threshold models with various cut-points in terms of collar group size had the highest likelihoods in 15 of the 20 data sets.This indicates that the general pattern for group detection was for groups with lower numbers of collars to have detection rates that were less than 1 but detection rates became 1 once a critical sample size of collars was achieved per group.The actual number of collars per group needed for detection to be 1 was dependent on the total group size as well as the number of collars per group.A homogeneity model was selected in one study, and the independence model was not selected in any of the data sets.For the TP herd (2006 and 2015) and CB Herd (2012), all estimators had similar likelihood scores and estimates, presumably due to a large proportion of the herd being counted during the survey.
The assumption of randomness of collars across caribou groups was violated in 5 of 20 surveys.One example of this was the BW 2009 survey where the distribution of collars and groups was irregular with one group of approximately 2,500 caribou having 12 collared caribou and a group of similar size only containing 1 collared caribou.In this case, it was likely that there may have been an aggregation of collared caribou that was not easily related to group sizes observed.As a result, the hypothesis that collared groups were distributed randomly was rejected and it is possible that estimates from this survey were negatively biased.

Effect of proportion of collars located during the survey
One of the most immediate factors that might influence performance of the Rivest estimator is the relative proportion of available collars located and photographed during the survey.This proportion is a rough indicator of the ability of the aerial survey to find the majority of the caribou herd and should have some bearing on the ultimate precision of the estimate.Results from the comparison of data sets indicated that acceptable levels of precision occurred when 80% or more of the collars were located with 3 studies showing lower levels of precision (CV>20%) when less than 80% were located (Figure 2).

Effect of sample size of collars relative to herd size on the Rivest estimator
A related question in terms of sample size is the effect of the number of collared caribou available during the survey relative to the size of the caribou herd.Intuitively, larger caribou herds should require a larger number of collared caribou to adequately sample the herd, which can be indexed by the number of collared caribou relative to estimated herd size.Smaller herds such as the CB and TP herds had higher numbers of collared caribou per caribou in the herd, which is one potential reason why precision of estimates for these herds was more uniformly acceptable.In contrast, surveys of the BW and BE herds had higher numbers of caribou per collar and more variance in precision of estimates.For these herds, the degree of aggregation of animals played a larger role in determining estimate precision (Figure 3).

The effect of aggregation of groups on the Rivest estimator
In general, stronger aggregation of caribou increased the precision of Rivest estimates (Figure 4).When strong aggregation occurred, larger numbers of collared caribou were likely to be found in a limited number of larger caribou  groups.As a result, these larger groups often had high detection rates and the probability that these groups contained at least one collared caribou approached 1. Subsequently, the estimate of herd size was usually precise and often similar to the total number of caribou counted.This relationship is shown if the negative binomial dispersion parameter (θ) is plotted against Rivest estimator precision.In most studies, θ had to be below 0.5 for adequate precision.Two exceptions to this were the BW surveys which had aggregation indices of less than 0.4 but higher coefficients of variation.For these two surveys a lower (<70%) of collars were located which also reduced estimate precision.The BE 2000 survey had the lowest level of aggregation and the highest coefficient of variation of the surveys compared.

Regression analysis of factors affecting estimate precision
We conducted a regression analysis to determine the factors strongly influencing Rivest estimator precision.These included the proportion of available collars located and pho-tographed (Figure 2), the number of caribou collared relative to herd size (Figure 3) and the degree of aggregation of the herd (Figure 4).Of the covariates considered, the degree of aggregation and the proportion of collars located and photographed were significant predictors

Correspondence of Lincoln-Petersen and Rivest estimates
In 4 of the 20 data sets the Lincoln-Petersen estimate was equal to the number of caribou counted due to all of the collared caribou being observed in photographed groups (Table 4).In this case, the estimated variance was 0. Estimated precision was high for all LP estimates with coefficients of variation of less than 20%.Estimates from groups that contained collars were 5.9% lower (std.dev.=7.0%, min=0%, max=16.4%,n=20) than estimates that used total counts of all caribou.The variance was minimally affected by whether groups with collars were included or excluded, with similar coefficients of variation.We used the estimates that used all caribou groups observed, with or without collars, for comparison with the Rivest estimates, given that it was likely that collared groups and non-collared groups were in close proximity and it was likely that collared groups helped in detecting non-collared groups during the aerial search process.In other words, it was likely that all groups (with or without collars) had a probability of containing collars and of estimate precision (Table 3).The r 2 value for the model was 0.81, indicating that the regression model explained 81% of the variation in the data set.
The regression results are illustrated by a plot of predicted precision as a function of the proportion of collars located across the range of observed levels of aggregation (Figure 5).In this case, a θ value of 0.19 was the highest level of aggregation observed (CB herd, July 9, 2006) and 0.9 was the lowest level of aggregation (BE herd, 2000) with the mean level of aggregation at 0.37 and a lower level of aggregation represented by the 90 th percentile of theta of 0.53.The main conclusion is that if caribou are very aggregated (θ=0.2),acceptable levels of precision (CV<20%) can be achieved with less dependence on the proportion of collared caribou located.At mean levels of aggregation (θ=0.37), at least 70% of the collars need to be located, and up to 90% of the collars need to be located if aggregation is relatively low.If caribou are not substantially aggregated (θ=0.9), then it is not possible to get a precise estimate even if all the collared caribou are located.the majority of collared caribou were located, and most of the caribou were in a few groups.In these cases, the detection probabilities of groups (P̂g roup ) for the Rivest estimator were 1 for all groups or for the majority of groups which had larger numbers of caribou.In these situations, the probability of at least one collared caribou in a group was close to 1 indicating that the herds had effectively been censused (all caribou in the herd were counted) in the survey, as also indicated by similar likelihood scores and estimates for the Rivest models.This mainly occurred in the TP and CB herds which were small and contained in relatively small areas where it was possible to conduct intensive surveys when compared to the larger BW and BE herds.

Performance of the Rivest estimator
From a statistical perspective, the Rivest estimator provides an improvement in post-therefore inclusion of non-collared groups was justifiable.
Lincoln-Petersen estimates were 21.0% lower (std.=14.3%min=2.0%,max=56.7%,n=20) on average than the Rivest estimates (Table 2).The difference was most pronounced for the BW and BE herds where the relative sample size of collars to overall herd size was lower.Confidence limits from the Rivest estimator included the Lincoln-Petersen estimate in 17 of 20 surveys.However, confidence limits of the Lincoln-Petersen estimator did not overlap the Rivest estimator in 12 of 15 surveys.For the remaining 5 surveys there was no variance or confidence interval estimate for the Lincoln-Petersen estimate (Table 4), because all the collared caribou were found.
Rivest and Lincoln-Petersen estimates were most similar when the majority of caribou groups were counted by the Rivest estimator as indexed by the total caribou counted divided by the Rivest herd estimate (Figure 6).This occurred when herds were well aggregated, Figure 5. Predicted levels of precision as a function of roportion of available collars located across highest observed level of aggregation (θ=0.19),mean levels of aggregation (θ=0.37),lower levels of aggregation (90th percentile of θ=0.53) and the lowest level of aggregation (θ=0.9)from regression analyses (Table 3) for caribou herds in Northwest Territories and Nunavut, Canada, 2000-2015.The main constraint of applying the Rivest estimator is having a suitable number of collared caribou to allow the modeling of detection probabilities of groups as indicated by groups with radio collars.The Rivest estimator cannot use data for groups without collars and therefore an imprecise estimate becomes more likely if collar sample size is reduced relative to the size of the herd (Figure 3) or a lower proportion of available collars is located (Figure 2).In addition, if caribou are less aggregated due to lower insect harassment then it will be more likely that multiple smaller groups with no collared caribou will occur unless collar sample size is very high.Therefore, it is essential that suitable numbers of collars are employed, and that sampling is conducted during times of peak aggregation.If aggregation does not occur sufficiently, then it is likely that no estimator or count will provide a reliable estimate of herd size from post-calving surveys (Figure 5).An earlier simulation study based upon the CB, BW and BE herds in 2006 recommended a sample of 30 collars for the relatively small CB herd, 40-60 collars for the BE herd, and 60 collars for the BW herd to allow an 80% probability of detecting at least 90% of the herd (Rettie, 2017).This analysis also identified the importance of group size distribution (many small groups vs. few large groups) as a key factor in the likely success of post-calving surveys The assumption of randomness of collared caribou relative to groups was violated in 5 of 20 studies.Rivest et al (1998) discussed possible methods to confront this issue such as modelling the distribution of collars in groups as a negative binomial as opposed to a Poisson distribution.However, these enhancements have not been incorporated into the caribou program in R. The net result of violation of this assumption is a potential negative bias in estimates.

Performance of the Lincoln-Petersen estimator
The results of this study suggest that estimates from the Lincoln-Petersen model will generally be negatively biased due to heterogeneity of capture probability of collared caribou within groups: larger groups with multiple collars are more likely to be found than smaller groups with one collar or particularly groups with no collars.The true number of caribou in each herd is not known and therefore bias cannot be inferred by comparison of the Rivest and Lincoln-Petersen estimates.However, model selection results from the Rivest estimator demonstrated different levels of detection for groups as a function of the number of collars within each group (Table 2) as indicated by selection of the threshold models in 15 of the 20 data sets.If the assumption of equal probabilities of detection of groups (P̂g roup ) was met, then the homogeneity model would most likely be chosen.The modelling of unequal probability of detection of groups (P̂g roup ) as well as the probability of a group having at least one collar (P̂≥ 1collar ) with the Rivest estimator addresses two sources of heterogeneity bias that are not accounted for with the Lincoln-Petersen estimator.Subsequently, estimates from the Rivest estimator are generally higher than from the Lincoln-Petersen estimator.The Lincoln-Petersen estimator is likely to be negatively biased both in terms of the point estimate but also in terms of the estimate of precision, resulting in a "biased but apparently precise" estimate unless herds are well aggregated and the vast majority of caribou in the herd are enumerated.In this case estimates from the Lincoln-Petersen estimator, Rivest estimator and total count of caribou converge (Figure 6).
In post-calving surveys of the Western Arctic Herd (WAH), which has at times numbered more than 400,000, collar numbers have generally been 90-100, survey coverage has been intensive with multiple survey aircraft and Rivest estimates have shown a high degree of convergence with totals counted on photos (Alaska Department of Fish and Game, 2011; Harper, 2013).These results suggest that the WAH has effectively been censused during multiple postcalving surveys, as apparently occurred in the much smaller CB herd in the NWT in 2006, 2012 and 2015.Results for the small TP herd in the NWT likewise have shown a fairly close correspondence between Rivest estimates and total counts.Under these conditions which include reasonable levels of aggregation and relatively high collar numbers, the Lincoln-Petersen estimates would likely be very similar to the Rivest estimates and total counts.Post-calving surveys of the Teshekpuk herd in Alaska (Harper, 2013) generally used lower collar numbers (35-60) than in the WAH, and in these surveys the Rivest estimates were higher than the total counts by an average of 16.3%, similar to the differences we found.As collar numbers increased over the years for the Teshekpuk herd, the difference between total counts and Rivest estimates grew smaller (Harper, 2013), suggesting that higher collar numbers led to increased group detection probabilities.
A recent report from Québec (Brodeur et al., 2017) suggested that the Lincoln Petersen estimator is suitable for post-calving surveys of the George River and Leaf River herds, given recent advances in collaring technology, high collar numbers and a large survey effort used to locate all caribou groups.While the Lincoln Petersen estimate may be close to herd size in some circumstances, we suggest that the use of the Rivest estimator and associated aggregation index introduced in this manuscript allow a statistical test of whether a near-census of a herd has occurred, in which case the minimum count, the Lincoln Petersen estimate, and Rivest estimate will be similar.This type of information would not be available with exclusive use of the Lincoln-Petersen estimator.We therefore suggest that Rivest estimates be used for all post-calving surveys especially given the relative ease of obtaining estimates using the caribou R package.

Use of the negative binomial aggregation index to assess performance of Rivest estimator
We suggest that the negative binomial index can be used to retrospectively analyze data and potentially determine scenarios when reliable post-calving estimates are not possible.A notable example of a year with a poor Rivest estimate was the BE herd in 2000.In this year, the herd did not aggregate well as exemplified by the aggregation index (θ=0.90)suggesting the poorest aggregation of any data set considered.In addition, a lower proportion of the collars were located which may also have been caused by poor aggregation (Patterson et al., 2004).The majority of the groups for this year only contained 1 to 3 collars with many groups having 0 collars, and the numbers of collars available and found were low (Appendix 1).As a result, the estimate was imprecise (CV=35%) and not reliable.Basically, there was not enough information in the data set to adequately model the relationship between collared caribou and group sizes.Some claims have been made that the Rivest estimator was "biased" when aggregation is poor based upon the Bluenose 2000 results (Patterson et al., 2004).In terms of statistics, bias cannot be inferred from this result given that bias is not really meaningful when precision of an estimate is low (wide confidence limits).Basically, a coefficient of variation much over 20% should raise a "red flag" from the Rivest estimator or any other estimator.It could be argued that the general requirements for the Rivest estimator are not necessarily unique to this estimator alone.The general challenge of the post-calving method is assuring that survey conditions are sufficient for reasonable enumeration of aggregations.In addition, collar sample sizes should be adequate so that most groups can have a reasonable probability of being detected.If they are not, then it is likely that no estimator can provide a reliable population estimate from these surveys.The advantage of the Rivest estimator in this con-text is that it provides a statistical assessment of how survey conditions were met through statistical interpretation of the precision of herd size estimates.

Recommendations
We make the following recommendations for the use of the Rivest estimator and successful post-calving surveys.First, the precision of the Rivest estimator depends partially upon finding a substantial proportion of the collared caribou in the herd.In general, at least 80% of collared caribou should be located and photographed in groups to ensure adequate precision (Figure 5).Under ideal conditions, nearly 100% of the collars will be found and photographed; however, we have occasionally had surveys (e.g.BW herd in 2012) where a limited portion of the herd, with associated collars, did not aggregate sufficiently for photos.If less than 80% of collars are located after substantial effort, then it may be that the herd is not well aggregated, and a lower precision estimate will result (Figure 5).Second, the sample size of collars required depends on the estimated herd size.Estimates with adequate precision were obtained for the Tuktoyaktuk Peninsula and CB herds with sample sizes of 20-30 collars whereas sample sizes of greater than 50 collars were required for the BE and West herds.Aggregation of caribou is a critical factor with results demonstrating the difficulty of obtaining reliable estimates when aggregation is low even with substantial sampling effort.The negative binomial θ term provides a way to compare the degree of aggregation across surveys.We suggest that this coefficient can provide a diagnostic of the influence of aggregation on estimate precision.

Management context: herd status and management for four herds
Management of the CB, BW and BE herds in the NWT and NU is primarily defined in a management plan finalized in 2014 (ACCWM Rangifer, 38, (1) 2018 2014) while a management plan for the TP herd, whose range is solely in one land claim region, remains to be developed.The ACCWM plan uses a color chart with four phases (red, low numbers; green, high numbers; yellow, intermediate and increasing; and orange, intermediate and declining) and numerical thresholds between phases.As an example, the red phase for the BE herd is 20,000 or less, the green phase is above 120,000, and the threshold between green and either yellow or orange is at 60,000.The co-management boards making up the ACCWM hold an annual status meeting where each of the three herds is assessed, using available demographic data and other monitoring, including reports from the communities within each herd's range.Management of harvest and other actions is tied to the color phase that each herd is considered to be in, with the strongest actions for herds in the red phase.Population estimates for each herd are key to defining herd size and trend, although they are not the sole information used in assigning herd status.In 2016-2017, through a number of meetings that included the GNWT and co-management boards in the ACCWM, a transition from LP estimates to Rivest estimates was agreed to as a more robust way of estimating herd size and variance from post-calving surveys in the TP, CB and BW herds.
In 2016 the Wekèezhìı Renewable Resources Board (WRRB) held a hearing to consider management actions for the BE herd, which had by 2015 declined to about 38,600 caribou from more than 100,000 in 2010 and was assessed as being in the orange phase (WRRB 2016).The WRRB determined that harvest in that land claim area should be limited to 750 bulls/year (WRRB 2016).Similar hearings resulted in limitation of BE harvest in NU of 340 caribou/year and 150 caribou in the Sahtú region of the NWT.Population surveys for the BE herd switched in 2010 from a post-calving survey to a calving photo survey (Adamczewski et al., 2017) after multiple failed post-calving surveys.
The BW herd has been roughly stable between 21,000 and 28,000 between 2005 and 2015 (this paper, Rivest), after a large decline 2000-2005 (ACCWM 2014).The Wildlife Management Advisory Council (Northwest Territories) (WMAC(NWT)), Gwich'in Renewable Resources Board (GRRB) and SRRB made recommendation to limit harvest for this herd to a 4% annual rate and 80% bulls (AC-CWM 2014).Harvest limitations for the herd are unlikely to change unless the herd shows clear evidence of recovery.
The CB herd has been roughly stable at about 2500 caribou 2006-2015 (this paper, Rivest) after a large decline 2000-2005 (AC-CWM, 2014).This herd is considered to be in the red phase, thus the WMAC(NWT) and GRRB made recommendations to close harvest of the CB herd by creating no-hunting zones (ACCWM, 2014).The TP herd has declined steadily from 4,188 caribou in 2006, when it was first surveyed, to 1,930 in 2015 (this paper, Rivest), leading to increased discussions about harvest limitation.
The overall trends in the four herds where post-calving surveys have been used, and associated management, have not been altered by the transition from LP estimates to Rivest estimates, although individual survey estimates have increased by varying percentages and an average increase of 21.0%.A new round of population surveys for all four herds in 2018 is planned, and thereafter herd status will be assessed and management may be re-evaluated.We suggest that greater attention be paid in future in management discussions to the quality of Rivest-generated estimates, including the degree of aggregation, the adequacy of collar numbers, the proportion of collars found in photographed groups, and the overall precision of the estimates.The 2012 BW survey had poor aggregation in groups associated with 17 of 55 collared caribou, for which photos were not feasible, and the coefficient of variation was a relatively high 24.4%.These results mean that this population estimate should be used with caution when compared to other surveys of this herd with better aggregation and higher precision, such as the 2009 and 2015 surveys.This section provides listings of field data and summaries of each of the post-calving survey data sets used in the paper.Lower confidence limits were constrained to be equal to the total count of caribou during the survey.A threshold Rivest model with groups of 2 or more collars having detection probabilities of 1 had the highest likelihood.Groups with less than 2 collars had detection probabilities of 0.5.

Bluenose-West (BW) herd 2005
The Bluenose-West 2005 survey was conducted on July 6 (Nagy & Johnson, 2013).Sixty three caribou with collars were available during the survey, of which 54 were detected in photographed groups.Overall, 17,875 caribou were counted of which 16,824 were in groups that contained one or more collared caribou.Tests for randomness of collared caribou across groups suggested that this assumption was met for all the models that were considered in the BW 2005 survey.

2006
The 2006 BW survey was conducted on two sampling occasions; a smaller number of groups were counted on July 4, and then sampling was repeated on July 7 and 8, with a larger number of groups being counted (Nagy & Johnson, 2006).
A Rivest threshold model with group sizes of 6 or greater having a detection probability of 1 and groups of less than 6 having a detection rate of 0.78 had the highest likelihood.The resulting herd estimate was reasonably precise with a CV of 11.4%.The model estimates ranged between 25,370 and 27,863.For the July 4 data set, a Rivest model with detection rates of groups with greater or equal to 9 collared caribou showing detection probabilities of 1 displayed the highest likelihood score.Detection rates were relatively low (.22) for groups with less than 9 collared caribou.Rivest model estimates for the July 4 sampling session varied between 25,000 and 30,000 caribou with good to marginal precision.In comparison, the Lincoln Petersen estimate was 22,827.
For the July 7 and 8 th data set, a threshold model with groups with 6 or more collared caribou showing detection probabilities of 1 and groups with less than 6 collars still showing high detection rates (0.97) was most supported.All models estimated high detection probabilities and in general estimates were very close and in the range of 28,000 caribou.The two estimates basically suggest that the majority of groups were counted on July 7 and 8 compared to July 4 th .Reassuringly, the estimates from the 2 sessions are relatively close despite the differences in the number of groups counted.This result suggests that the Rivest estimator was effectively estimating the fact that groups were missed on July 4 th , but bias cannot be inferred from these results given that the true number was not known.The July 7-8 estimate is preferred due to higher precision and a higher proportion of collars found.(Nagy & Johnson. 2006).The Lincoln-Petersen estimate is based on Nagy & Johnson (2006).

2012
Fifty five collared caribou were available during the 2012 post calving survey (Davison et al., 2016).Of these 38 were detected in photographed groups.Field observations suggested that the herd did not aggregate as well as in other years, which was the main reason that 17 collared caribou were not in photographed groups.A threshold model with groups of 9 or more collars having detection probabilities of 1 and other groups with lower detection rates was most supported.Estimates were generally imprecise with coefficients of variation > 20%.Tests for randomness of collar distribution across groups suggested this assumption was violated and as a result, herd estimates may be negatively biased.

2015
In 2015, 25 groups of caribou were counted of which 22 contained collared caribou (Davison et al., unpublished).Forty nine of 55 available collars were located within photographed groups.The most supported Rivest model was the homogeneity model, however, other model estimates were relatively similar.Tests for randomness of collared caribou distribution across groups suggested a non-random distribution, and therefore it is likely that these estimates are negatively biased.

2006
Three sampling sessions for the CB herd were conducted in July 2006 (July 6, 9 and 13) and 33 collared caribou were monitored (Nagy & Johnson, 2006).The largest number of groups detected was on July 9.  Estimates were run for each survey date.In general, estimates were reasonably similar for each sampling session with the highest level of precision obtained on July 13, which was presumably due to the higher level of aggregation at this time (19 collared caribou in one group of 1,367 caribou).This estimate is the preferred one for the herd in 2006.Rivest model analysis suggested that a threshold model with detection probabilities of 1 for groups with 5 or more collars was most supported.Groups with less than 4 collars had a detection probability of 0.57.This model produced an estimate of 2,925 caribou compared to an estimate of 1,934 caribou from the Lincoln-Petersen estimator.Estimates had marginal precision (CV>20%).The Rivest model estimates were similar for the homogeneity and threshold models with detection probabilities of groups equal to 1 in all cases.In this case, the Rivest models basically estimated that a high proportion of the herd had been found and therefore all models converged on the same estimate of caribou.The Lincoln-Petersen estimate equaled the number of caribou counted with no estimate of standard error.Tests for randomness of collared caribou were similar (Z=-0.375,p=0.646) for all models which suggested the assumption of randomness was not violated.

2015
In 2015, 50 of 51 collared caribou were observed in 9 groups totaling 2203 caribou in the CB herd.In addition 3 groups composed of 13 caribou were observed without collared caribou (Davison et al., unpublished).A threshold model with all groups of 2 or more collars having sighting probabilities of 1 had the highest log-likelihood score.Groups with less than 2 collars had a sighting probability of 0.83.
Estimates from this model were precise and relatively similar to estimates from the other Rivest models.Tests for randomness of collars across groups indicated that this assumption was potentially violated as indicated by p-values of less than 0.05.In this case the Rivest estimates may be slightly negatively biased.

Tuktoyaktuk Peninsula (TP) Herd 2006
On July 9 and 13, 2006, the Tuktoyaktuk Peninsula Herd was sampled with all 27 collared caribou detected in photographed groups (Nagy & Johnson, 2006).On both dates, collared caribou were detected as single individuals so that the collar size equaled the group size.Identical log-likelihood scores and estimates were returned for all models for both survey dates suggesting the herd had been effectively censused during the survey.The estimate for July 13 is preferred as it had higher precision.A model that assumed detection probabilities were one for groups that had 2 or more collars was most supported with detection probabilities of 0.67 for groups that had less than 2 collars.The estimate of herd size from this model was 2,889 (±765) caribou with a CV of the estimate of 13.5%.Interestingly, this estimate was reasonably close to the Lincoln-Petersen estimate.

2012
Twenty three collars were available during the 2012 Tuktoyaktuk Peninsula herd post calving survey.Of these, 22 were observed with the majority in a single group of caribou (Davison et al., 2016).

Figure 1 .
Figure 1.Annual ranges of the caribou herds whose post calving survey data were used in this study.The majority of the herds occurred in Northwest Territories with some overlap of the Bluenose East into Nunavut, Canada.

Figure 2 .
Figure 2. The relationship between proportion of available collars located and estimate precision with herd surveyed, for caribou herds in Northwest Territories and Nunavut, Canada, 2000-2015.Estimates are displayed from the Bluenose-East (BE), Bluenose-West (BW), Cape Bathurst (CB) and Tuktoyaktuk Peninsula (TP) herds.

Figure 3 .
Figure 3.The relationship between the number of caribou collared (left graph) and the number of estimated caribou per collared caribou (T̂)/No. of collared caribou; right graph) with Rivest estimator precision (CV) for caribou herds in the Northwest Territories and Nunavut, Canada, 2000-2015.Estimates are displayed from the Bluenose-East (BE), Bluenose-West (BW), Cape Bathurst (CB) and Tuktoyaktuk Peninsula (TP) herds.

Figure 4 .
Figure 4. Relationship between Rivest estimator precision and aggregation as estimated by the dispersion parameter of the negative binomial distribution (θ) for caribou herds in the Northwest Territories and Nunavut, Canada, 2000-2015.Estimates are displayed from the Bluenose-East (BE), Bluenose-West (BW), Cape Bathurst (CB) and Tuktoyaktuk Peninsula (TP) herds.

Figure 6 .
Figure 6.Comparison of difference between Lincoln Peterson and Rivest estimates ((LP estimate-Rivest estimate)/Rivest estimate) as a function of estimated proportion counted (total caribou counted/Rivest herd size estimate) for post-calving surveys of caribou herds in the Northwest Territories and Nunavut, Canada, 2000-2015.Estimates are displayed from the Bluenose-East (BE), Bluenose-West (BW), Cape Bathurst (CB) and Tuktoyaktuk Peninsula (TP) herds.

Table 1 .
Summary of post-calving data sets for caribou herds in the NorthwestTerritories and Nunavut, Canada,  between 2000 and 2015assessed in this study.Further details on each data set are given in Appendix 1.

Table 2 .
Summary of herd estimates using the Rivest estimator for post-calving surveys used in meta-analysis for caribou herds in the NorthwestTerritories and Nunavut, Canada, 2000-2015.The model used for estimates had the highest likelihood score of models considered.Further details on each analysis are given in Appendix 1. Estimates with the lowest CVs and highest likelihoods were the preferred ones.P-value for test of randomness of collared caribou relative to group size.b TB refers to a Rivest threshold model with the number referring to the number of collars where detection probability = 1.H refers to the Homogeneity model and I refers to the Independence model.c All models returned the same likelihood score and estimates. a

Table 4 .
Lincoln-Petersen estimates of herd size from post-calving surveys using all caribou counted and caribou only in groups that contained collared caribou for caribou herds in NorthwestTerritories and Nunavut, Canada,  2000-2015.

Table 3 .
Regression analysis results for factors affecting the precision of post-calving survey estimates as indicated by the coefficient of variation for caribou herds in the NorthwestTerritories and Nunavut, Canada, 2000-2015.
calving survey estimation methodology over the Lincoln-Petersen estimator.It provides a model-based method to estimate the number of caribou missed in the surveys that properly uses caribou groups and associated collared caribou in the group as the sampling unit.By doing this, more robust estimates of herd size and associated estimate variance are produced if the general assumptions of the post-calving method are met.The level of precision of the Rivest estimator is lower than that of the Lincoln-Petersen estimator.However, this variance estimate most likely reflects the true degree of statistical uncertainty in estimates, and the low variance or 0 variance from Lincoln-Petersen calculations is likely unrealistically low.Despite lower precision, the coefficients of variation for Rivest estimates were still within levels considered acceptable by managers (CV<20%;Pollock et al., 1990)for most of the data sets we analyzed.

Table 1 .
Field data for the Bluenose-East 2000 post calving survey.(2004)intheirTable 1 derived a total count of 84,412 adult caribou for the BE herd in 2012; the reduction from 96,036 was based on estimated overlap with BW collared caribou.The adjusted 84,412 total was used in Fig.15.The Rivest estimate in Fig.15was adjusted downward by the same factor to 245,545; however, all Rivest estimates for this survey had low precision.
2000The BE herd was surveyed in 2000(Patterson et al., 2004)from July 2 to July 6.Of 33 collars that were available, 23 were detected, with 1 to 3 collars per group of caribou observed.Patterson et al.

Table 3 .
(Adamczewski et al., 2017)llar distribution for the Bluenose-East 2000 survey.2010TheBluenose-Eastherdwas primarily surveyed from July 6-12, 2010, at which time caribou groups congregated into 3 geographic areas (Main, Southern and Northern).During this time 47 collared caribou were monitored of which 44 were located within photographed groups.Thirty nine groups were counted on photos which amounted to a total count of 92,481 caribou(Adamczewski et al., 2017).

Table 4 .
Field data for the Bluenose East 2010 post-calving survey.All Rivest herd estimates were imprecise (CV>31%) and ranged between 204,944 and 279,358.Lincoln-Petersen estimates were substantially lower.Patterson et al. (2004)used only groups with collars for Lincoln-Petersen estimates so this estimate is included in addition to an estimate using counts from all groups.Tests for random distribution of collars suggest this assumption was not violated in the BE 2000 survey.

Table 5 .
Adamczewski et al. (2017)d LP estimate for the Bluenose-East 2010 survey 1 .In earlier analyses of this data set, a threshold model with B=5 was found to have the best log-likelihood.Those results were used byAdamczewski et al. (2017).The difference is 1005 caribou for the herd estimate.A threshold model with group sizes of 8 or more caribou having a probability of detection of 1 had the highest log-likelihood score.Estimates were precise and relatively similar between the Rivest models. 1

Table 6 .
Tests for randomness of collared caribou across groups for the Bluenose-East 2010 survey.

Table 7 .
Field data for the Bluenose-West 2005 post calving survey.

Table 8 .
Rivest estimator results and LP estimate for the Bluenose-West 2005 survey.

Table 9 .
Tests for randomness of collared caribou across groups for the Bluenose-West 2005 survey.

Table 10 .
Field data for the Bluenose-West 2006 post calving survey.

Table 11 .
Rivest Estimator results for the Bluenose-West July 4 and July 7-8, 2006 data sets

Table 12 .
(Davison et al., 2014)f collared caribou distribution suggested that collared caribou were randomly distributed within groups for both the July 4 and July 7-8 data sets in the BW 2006 survey.Tests for randomness of collared caribou for the Bluenose-West July 4 and July 7-8, 2006 data sets.2009For the BW 2009 survey, larger groups had more collared caribou with one notable exception where group 42 of 2,515 had only one collared caribou(Davison et al., 2014).There were 54 collared caribou during the survey.Of these, 50 collars were found during the survey in 21 groups with 15,108 caribou counted in all groups that had collars.If groups without collars are considered then 16,595 caribou were counted.

Table 13 .
Bluenose-West 2009 post calving field data.Model selection results from the Rivest estimator suggested that a threshold model with groups with 12 or more collars displaying detection rates of 1 and groups that had less than 12 collars displaying detection probability of 0.90.The homogeneity model assumes that collar mixing in groups is random and that all groups will have the same detection probability (of 0.93).The estimate of herd size from the best threshold model was 21,773 (± 4,884) caribou with a CV of 11.4% for the estimate, and similar to the other estimates.

Table 14 .
Rivest estimator results and LP estimate for the Bluenose-West 2009 data set.Tests for randomness of collar distribution across groups suggested this assumption may have been violated.Therefore, estimates of herd size for the Rivest estimator may be negatively biased for this survey.Regardless, they are higher than the Lincoln-Petersen estimates for the BW 2009 data set.

Table 15 .
Tests for randomness of collared caribou for the Bluenose-West 2009 data set.

Table 16 .
Caribou groups counted for the 2012 Bluenose-West survey.

Table 17 .
Rivest Model estimates and LP estimate for Bluenose-West 2012 survey.

Table 18 .
Tests for randomness of collared caribou for the Bluenose-West 2012 data set.

Table 19 .
Summary of collar and group data for the 2015 Bluenose-West post calving survey.Model selection results indicated that a threshold model with groups of 3 or more having a sighting probability of 1 had the highest likelihood.The estimate from this model (21,535) was reasonably precise with a CV of 12.2%.Other Rivest model estimates were similar.

Table 20 .
Model selection and herd size estimates for the Bluenose-West 2015 post-calving survey.

Table 21 .
Tests for randomness of collar distribution across groups suggested that this assumption was not violated during the 2015 survey with non-significant tests for all models.Tests for randomness of collars across group sizes for the 2015 Bluenose-West survey.

Table 22 .
Field data for the 2005 Cape Bathurst post calving survey.

Table 23 .
Rivest estimator results and LP estimate for Cape Bathurst 2005 survey.

Table 24 .
Tests for randomness of collar distribution for the Cape Bathurst 2005 survey.

Table 25 .
Summary of field data on three dates for the Cape Bathurst herd in 2006.
(Davison et al., 2014)caribou were available during the Cape Bathurst 2009 survey of which 22 were observed in photographed groups(Davison et al., 2014).Overall, 1,534 caribou were counted.Only 111 caribou in 3 groups were seen without collared caribou within the groups.

Table 28 .
Post calving field data for Cape Bathurst 2009 survey.

Table 29 .
Rivest estimator results for the Cape Bathurst 2009 data set.The Lincoln-Petersen estimate is based on Davison et al. (2014).Tests for randomness of collared caribou across groups suggested that this assumption was met with the CB 2009 data set.

Table 30 .
(Davison et al., 2016)f collared caribou for the Cape Bathurst 2009 data set.2012For the 2012 Cape Bathurst post calving survey, 24 collared caribou were available during the survey and all 24 were found in photographed groups, with 2,427 caribou counted in collared and non-collared groups(Davison et al., 2016).

Table 31 .
Field data collected for Cape Bathurst 2012 post calving survey.
A 19 observations of single caribou

Table 32 .
Rivest model estimates and LP estimate for Cape Bathurst 2012 post calving survey.

Table 33 .
Summary of field data for the 2015 Cape Bathurst survey.

Table 34 .
Rivest model estimates and LP estimate for the Cape Bathurst 2015 post calving survey.

Table 35 .
Tests for randomness of collar distribution for the Cape Bathurst 2015 post calving survey.

Table 36 .
Summary of sampling for the Tuktoyaktuk Peninsula Herd on two dates in July 2006.

Table 37 .
Rivest Estimator results and LP estimate for the Tuktoyaktuk Peninsula herd on July 9, 2006.

Table 40 :
Davison et al. (2014)lts for the Tuktoyaktuk Peninsula Herd in 2009.The Lincoln-Petersen estimate is based onDavison et al. (2014).Tests for randomness of collared caribou across groups suggested that this assumption may have been weakly violated which could cause a negative bias in estimates and associated variances.

Table 41 .
Tests for randomness of collared caribou for the Tuktoyaktuk Peninsula herd 2009 data set.

Table 42 .
Field data for the 2012 Tuktoyaktuk Peninsula Herd post calving survey.

Table 46 .
Rivest model estimates and LP estimate for the 2015 Tuktoyaktuk Peninsula herd post calving survey.Tests for randomness of collars indicated that this assumption was not violated during the TP 2015 survey.

Table 47 .
Tests for randomness of collars across groups for the 2015 Tuktoyaktuk Peninsula herd survey.