Abundance estimation with sightability data: a Bayesian data augmentation approach

Authors


Correspondence author. E-mail: jfieberg@umn.edu

Summary

  1. Steinhorst & Samuel (Biometrics 1989; 45, 415–425) showed how logistic regression models, fit to detection data collected from radiocollared animals, can be used to estimate and adjust for visibility bias in wildlife population surveys. Population abundance is estimated using a modified Horvitz–Thompson (mHT) estimator in which counts of observed animal groups are divided by their estimated inclusion probabilities (determined by plot-level sampling probabilities and detection probabilities estimated from radiocollared individuals). The sampling distribution of the mHT estimator is typically right-skewed, and statistical inference relies on asymptotic theory that may not be appropriate with small samples.
  2. We develop an alternative, Bayesian model-based approach which we apply to data collected from moose (Alces alces) in Minnesota. We model detection probabilities as a function of visual obstruction, informed by data from 124 sightability trials involving radiocollared moose. These sightability data, along with counts of moose from a stratified random sample of aerial plots, are used to estimate moose abundance in 2006 and 2007 and the log rate of change between the 2 years.
  3. Unlike traditional design-based estimators, model-based estimators require assumptions regarding stratum-specific distributions of the detection covariates, the number of animal groups per plot and the number of animals per animal group. We demonstrate numerical and graphical methods for assessing the validity of these assumptions and compare two different models for the distribution of the number of animal groups per plot, a beta-binomial model and a logistic-t model.
  4. Estimates of the log rate of change (95% CI) between 2006 and 2007 were −0·21 (−0·53, 0·12), −0·24 (−0·64, 0·16), and −0·25 (−0·64, 0·15) for the beta-binomial model, logistic-t model and mHT estimator, respectively. Plots of posterior-predictive distributions and goodness-of-fit measures both suggest the beta-binomial model provides a better fit to the data.
  5. The Bayesian framework offers many inferential advantages, including the ability to incorporate prior information and perform exact inference with small samples. More importantly, the model-based approach provides additional flexibility when designing and analysing multi-year surveys (e.g. rotational sampling designs could be used to focus sampling effort in important areas, and random effects could be used to share information across years).

Introduction

Aerial surveys are frequently used to monitor wildlife populations over large spatial scales, and several methods are available for adjusting counts of observed animals when detection rates are <1 (see Lancia et al. 2005, for a review). A popular approach, pioneered by Steinhorst & Samuel (1989), uses auxiliary ‘test trial’ or ‘sightability trial’ data collected from radiocollared animals to model the detection process as a function of individual-level covariates (e.g. group size, measures of visual obstruction); see Fieberg (2012) for a list of example applications. Abundance is estimated from ‘operational’ surveys using a modified Horvitz–Thompson (mHT) estimator, in which animal counts are divided by their estimated detection probabilities from fitted logistic regression models applied to the auxiliary data, along with plot-level sampling probabilities. The variance of the estimator can be expressed as the sum of three variance components that are due to: (i) random sampling (of aerial plots); (ii) random detection (and missed detection) of independent groups of individuals; and (iii) estimation of parameters related to the detection process (Steinhorst & Samuel 1989).

Inference using the mHT estimator is ‘design-based’ – that is, confidence intervals are formed by considering variation across repeated samples (which never occur), under the assumption that the estimator follows an approximate Gaussian distribution when the sample size is large. Yet, operational surveys typically involve small numbers of sample units, and as a result, the sampling distribution of the mHT estimator is typically right-skewed. In addition, the variance estimator developed by Steinhorst & Samuel (1989) includes several terms that are derived under an asymptotic normality assumption for the logistic regression parameter estimators. Fieberg & Giudice (2008) showed that for small numbers of sightability trials (e.g. ≤100), these approximate terms tend to underestimate the variance. In simulation studies, normal-based confidence intervals have often resulted in less than nominal coverage (Cogan & Diefenbach 1998), whereas intervals based on a log-normal assumption often perform better (Wong 1996). Lastly, as is often the case with Horvitz–Thompson estimators, the mHT is inefficient when inclusion probabilities are small (either due to small probabilities of sampling the aerial unit or low detection probabilities; Little 2009).

Model-based estimators for survey data, often formulated within a Bayesian framework, have been suggested as an alternative to traditional design-based estimators (Ericson 1969; Valliant, Dorfman & Royall 2000; Little 2009). Popularity of these methods has increased with advances in computing power and the availability of Markov Chain Monte Carlo (MCMC) samplers capable of approximating posterior distributions. Inferences based on Bayesian models that use vague, or noninformative, priors are often quite similar to inferences made using design-based methods (Ericson 1969). Models should, however, incorporate important survey design features to ensure estimators are consistent (e.g. covariates and random effects may be used to account for stratification and clustering, respectively). Model-based estimators provide a natural approach to modelling survey non-response and can also lead to efficiency gains compared with traditional design-based estimators (Little 2009). Little (2009) lists several other advantages of Bayesian (model-based) inference for survey data, including the potential for better inference in small samples and the ability to incorporate prior information. These advantages, in part, motivated us to develop a Bayesian alternative to the mHT estimator.

Non-detection of individuals in wildlife surveys is similar to non-response in traditional sample surveys (Steinhorst & Samuel 1989); however, an important difference is that a sampling frame (i.e. a list of all individuals in the population) is lacking in the former case. As a result, the dimension of the problem (i.e. size of the population) is itself an unknown parameter that must be estimated. Similar problems arise with other popular abundance estimators, including capture–recapture methods that include individual covariates (Royle 2009) and models of species richness that account for detection rates that are <1 (Dorazio et al. 2006). Maximizing the complete data likelihood, formulated to capture both the sampling and detection processes, requires numerical integration techniques that are particularly challenging when the number of covariates in the detection model is large (Royle 2009). Alternatively, Dorazio et al. (2006) and Royle, Dorazio & Link (2007) developed a Bayesian data augmentation scheme for abundance estimation. We show how a similar approach can be applied to sightability data and discuss the advantages afforded by the Bayesian approach.

Data

We will refer to sightability surveys (or sightability trials) when discussing aerial surveys used to estimate detection parameters and operational surveys when describing formal surveys used to estimate abundance.

Operational surveys

Since 2005, the Minnesota Department of Natural Resources has conducted an (annual) aerial survey in northeastern Minnesota in an effort to monitor moose numbers and determine the calf/cow and bull/cow ratios. These data are used to determine population trends and set the harvest quota for subsequent hunting seasons. Surveys are conducted in the winter (usually, early January) once mean snow depth is >20 cm. Rectangular (5 × 2·67 mi.) plots are sampled using a stratified random sampling design from an aerial sampling frame (Fig. 1). Plots are flown using a Bell OH-58A helicopter (Bell Helicopter Textron, Fort Worth, TX, USA) using east–west transects spaced 0·5 km apart. Surveys are conducted by two teams, each consisting of a pilot and two experienced observers (one seated behind the pilot). Observers work together to jointly detect groups of moose and to count individuals within each group once a group of moose is detected. In addition, observers record various covariate data associated with the detection process (see 'Sightability surveys'). For the purposes of developing the approach, we consider data from operational surveys conducted in 2006 and 2007 (Table 1).

Figure 1.

A map of the operational survey sampling frame consisting of rectangular (5 × 2·67 mi.) plots in northeastern Minnesota.

Table 1. Number of plots per stratum (number of sampled plots per stratum) for operational surveys of Minnesota moose conducted in 2006–2007
YearStratum 1Stratum 2Stratum 3
2006238 (18)180 (14)35 (5)
2007248 (16)170 (18)35 (6)

Sightability surveys

We provide a brief overview of the data used to model the detection process, but refer the reader to Giudice, Fieberg & Lenarz (2012) for more details. A total of 124 sightability trials were conducted concurrent with operational surveys of moose in northeastern Minnesota, taking advantage of individuals that were radiocollared during 2002–2007 as part of a survival study ( Lenarz et al. 2009, 2010). Test plots were delineated so as to provide areas likely to contain radiocollared moose, and these plots were flown using the same procedures as in operational surveys. When a moose (marked or unmarked) was sighted, the helicopter left the transect and circled the moose to determine group size, classify individuals according to sex and age (calf or adult) and identify any marked animals. A suite of potential sightability covariates was also recorded for each observed moose group when the group contained at least one radiocollared individual. If the helicopter crew failed to detect a radiocollared moose during a sightability survey, they attempted to locate the target animal using telemetry immediately after the test plot was surveyed. If the radiocollared moose was still within the boundaries of the test plot, the crew collected the same suite of covariate information.

Detection probabilities varied with the amount of visual obstruction present (Giudice, Fieberg & Lenarz 2012). To capture this effect, we estimated the amount of screening cover within four animal lengths (c. 10-m radius circle) of the first animal seen, measured from the location and angle of initial sighting. In the case of missed individuals, visual obstruction was measured from an oblique angle while circling the missed animal or group. Additional covariates had negligible effects on detection rates (Giudice, Fieberg & Lenarz 2012), so we only consider visual obstruction in our example application.

Notation

For the sightability surveys, let:

  • R = the number of sightability trials conducted during the test trial period (in our case, R = 124).
  • math formula = a random variable, equal to 1 when the lth group containing a radiocollared moose is detected and 0 otherwise (l = 1, …, R).
  • math formula = visual obstruction associated with the moose group in the lth sightability trial.

For the observational surveys, let:

  • N = the number of spatial sampling units (hereafter ‘plots’) in the population sampling frame.
  • n = the number of plots in the sample.
  • H = the number of strata in the population sampling frame.
  • math formula = the number of spatial sampling units in stratum h.
  • math formula = the number of sampled plots in stratum h.
  • math formula = the number of animal groups in the ith plot located within the hth stratum.
  • math formula = the number of animal groups observed or sighted in the ith sampled plot located within the hth stratum.
  • math formula = the number of animals in the jth group located in the ith plot located within the hth stratum.
  • math formula = random variable, equal to 1 if the jth animal group in the ith plot located within the hth stratum is observed and 0 otherwise.
  • math formula = visual obstruction associated with the jth animal group in the ith plot located within the hth stratum.
  • math formula = the probability that the jth animal group located in the ith sampled plot located within the hth stratum is observed (given the plot was sampled).
  • math formula ≈ math formula, an ‘inflation factor’ associated with the jth observed animal group in the ith sampled plot located within the hth stratum (see 'Modified Horvitz–Thompson estimator' section for details).
  • math formula = the number of animals in the ith plot located within the hth stratum.:
  • display math
  • ϒ = total population size:
  • display math

Note that we use a ‘math formula’ to distinguish data collected during the sightability trials math formula from those collected during the operational surveys math formula. In addition, the former data have a single subscript (identifying the sightability trial), whereas the latter data have subscripts indexing stratum (h), plot (i) and animal group (j). Lastly, we note that we will use a subscript to index year when comparing the estimates of ϒ from 2006 to those from 2007.

Modified Horvitz–Thompson estimator

We estimated abundance in 2006–2007 using the ‘SightabilityModel’ package in program r (R Development Core Team 2011; Fieberg 2012). This package implements the mHT estimator suggested by Steinhorst & Samuel (1989) via the following steps. First, a logistic regression model is fitted to the sightability data (math formula, math formula. The fitted model is then used to estimate ‘inflation factors’, math formula = inverse probability of detection, for each independently sighted group during the operational survey (eqn 1):

display math(eqn 1)

where math formula is a 2 × 1 vector containing the estimated logistic regression parameters and math formula is the estimated variance–covariance matrix of math formula. Using large-sample maximum likelihood theory and the result that math formula, Steinhorst & Samuel (1989) showed that this estimator is consistent for math formula. The total number of animals in the study area, ϒ, is then estimated using:

display math(eqn 2)

where math formula is the probability that the ith plot located within the hth stratum is selected.

Steinhorst & Samuel (1989) proposed a variance estimator for math formula, but it was later shown to be biased (Thompson & Seber 1994; Wong 1996). Here, we consider a consistent estimator derived by Wong (1996), applicable to any sampling design (stratified or not) with selection probability math formula for sampling unit i (for convenience, we drop the subscript h and allow the plot index, i, to run from 1 to n in the following equations):

display math(eqn 3)

where math formula and math formula is the probability that both the ith and math formulath plots are selected. We used the following expression (as suggested by Steinhorst & Samuel (1989) and Wong (1996)) for math formula:

display math(eqn 4)

This expression, like (eqn 1), is motivated by the asymptotic normality of math formula.

We constructed confidence intervals for the mHT estimator under the assumption that (math formula) is log-normally distributed, where T is the total number of animals seen (Wong 1996; Fieberg 2012). Specifically,

display math(eqn 5)
display math(eqn 6)

where LCL and UCL are the lower and upper confidence limits, respectively, math formula, and math formula.

Model-based estimator

Detection model

Conditional on covariates math formula, we assume the detection indicators from the sightability trials, math formula, are independent Bernoulli random variables:

display math(eqn 7)

We specified vague N(0, 10) priors for math formula and math formula.

Population and sampling model

To develop a model-based estimator, we must also specify distributions describing:

  1. The number of moose groups per plot, math formula
  2. The number of moose per moose group, math formula.
  3. The visual obstruction associated with each moose group, math formula. These values, along with the detection model (eqn 7), determine each group's probability of detection during the operational survey.

Strata in the operational survey were defined in terms of expected moose density (Fieberg & Lenarz 2012). To be faithful to the survey design, we allowed each of these distributions to vary by stratum.

In developing a Bayesian model-based estimator, we are faced with a further challenge. The number of animal groups per plot determines the dimension of math formula and math formula (i.e. standard implementations of MCMC sampling would require the dimensions of math formula and math formula to change with each MC iteration). In similar settings, Dorazio et al. (2006) and Royle, Dorazio & Link (2007) demonstrated how data augmentation can be used to create a zero-inflated version of the complete data likelihood in which the dimension of the problem is fixed. The method works by augmenting the observed data with an additional set of B unobserved records (all with detection indicators, z, set equal to 0). These additional records can be unobserved because they do not belong to the study population or because they are not detected. A set of indicator variables, q (equal to 1 for all observed data and missing for all augmented data), are modelled as q∼ Bernoulli (ψ), where ψ, a zero-inflation parameter, determines the probability that each augmented observation is part of the study population. Estimation of ψ provides a means to estimate (or impute) unobserved q's (for the augmented data records). The size of the study population, ϒ, can then be estimated as a derived parameter: math formula. Royle, Dorazio & Link (2007) showed that this approach to inference is equivalent to placing a discrete [0, B] uniform prior on the size of the population (ϒ).

To adapt the approach to the sightability model framework requires augmenting the vector of observed groups in each sampled plot with math formula unobserved groups. An important step in the data augmentation process is to ensure that the augmented population size, math formula, exceeds the true population size, math formula (Royle, Dorazio & Link 2007). To determine reasonable values of math formula, we estimated the distribution of math formula in each stratum as follows:

  1. We used the glm function in Program r (R Development Core Team 2011) to fit the logistic regression model (eqn 7) to the sightability data.
  2. We used the fitted model and data from operational surveys conducted in 2006–2007 to estimate the number of moose groups in each sampled plot, math formula:
    display math(eqn 8)
    where math formula was determined using (eqn 1).

Maximum values of math formula were = (24, 25 and 40) in the (low, medium and high) density strata, respectively. We choose to use constant values of math formula within each stratum, setting math formula = (40, 60 and 100) for plots in (low, medium and high) density stratum, respectively, and inspected the posterior distribution of the math formula to make sure most of the probability mass was far from this upper bound.

In contrast to applications of data augmentation where interest lies in estimating a single abundance parameter, we sought to capture plot-to-plot variability in the number of moose groups, math formula, in a way that would allow us to predict variable numbers of moose groups in plots that were not sampled. Elsewhere, Langtimm et al. (2011) modified the data augmentation approach to allow for habitat-specific zero-inflation parameters when modelling abundance of manatees (Trichechus manatus latirostris) in spatially referenced sample units. A similar approach could have applied to the moose survey data (i.e. we could have considered stratum-specific ψ's). This approach would be equivalent to assuming a prior distribution for math formula Binomial (math formula, math formula). To allow for addition plot-to-plot variability (i.e. overdispersion relative to the binomial distribution), we used hierarchical models to specify plot-level ψ's.

We considered two different sets of parametric assumptions for the plot-level zero-inflation parameters, math formula. In the first case, we assumed:

display math(eqn 9)

We explored a few different hyperpriors for math formula and math formula. Using vague hyperpriors for these parameters often led to bimodal priors for math formula, with much of the probability mass associated with values close to either 0 or 1. Ultimately, we followed the general approach of Dorazio, Gotelli & Ellison (2011) and assumed that math formula followed a t-distribution with σ = 1·566267 and d.f. = 7·763170. We specified γ(20,20) hyperpriors for precision parameters = math formula. We refer to this model as the ‘logistic-t’ model.

In the second case, we assumed:

display math(eqn 10)

We specified diffuse gamma (1,0·01) hyperpriors for math formula and math formula. We refer to this model as the ‘beta-binomial’ model.

For both models (logistic-t and beta-binomial), we defined indicator variables, math formula, equal to 1 for groups that were observed and NA (i.e. missing) for the augmented groups. These indicator variables were modelled as:

display math(eqn 11)

Both specifications (logistic-t and beta-binomial) resulted in near uniform prior-predictive distributions for the number of animal groups per plot, math formula (Fig. S1, in Appendix S1).

Similar to Royle (2009), we assumed the distribution of group sizes followed a shifted Poisson:

display math(eqn 12)

We used vague N (0, 10) hyperpriors for each math formula. We assumed the visual obstruction measurements, math formula, followed a beta distribution, with vague gamma hyperprior distributions assigned to the beta parameters (math formula):

display math(eqn 13)

We also considered an alternative parameterization of the beta distribution, with two different sets of hyper priors, as part of a sensitivity analysis (Appendix S2). Lastly, we assumed the same detection model (eqn 7) applied to both the sightability and operational surveys, although detection in the latter case was also dependent on the group belonging to the study population (i.e. math formula):

display math(eqn 14)

We combine the individual model components to provide the complete data likelihood in Appendix S3.

Implementation and population estimation

We implemented the approach using open-source software, Program r (R Development Core Team 2011) and jags (Plummer 2003), with the r package r2jags used to communicate between the two software platforms (Su & Yajima 2012). We considered operational survey data from 2006 and 2007, with the goal of estimating the log rate of change between the 2 years. Because the two population estimates rely on the same detection model (and sightability data), they will be positively correlated (Fieberg & Giudice 2008; Fieberg 2012). This correlation must be accounted for when characterizing uncertainties associated with estimates of change between years. We show how the Bayesian framework can easily accommodate this correlation by fitting a single model (simultaneously) to both data sets. To mimic the way the mHT estimator is typically applied to multi-year surveys, we assumed that all parameters were independent across years except for the sightability parameters math formula and math formula. However, we return to this point in the discussion, where we allude to alternatives and discuss the additional flexibility offered by the Bayesian, model-based framework.

We assessed convergence by running three independent chains and then inspecting the Gelman–Rubin statistic (Brooks & Gelman 1998). This statistic compares ‘between chain’ and ‘within chain’ variation, with values close to 1 suggesting convergence. We also visually inspected the full trajectory of the Markov chain simulations to see if the independent chains had ‘settled down’ to a similar range of values. After 20 000 iterations, all Gelman–Rubin statistics were  < 1·05, posterior distributions for model parameters were unimodal, and these distributions were similar for each of three chains.

We then generated an additional 20 000 iterates from each of three chains and applied a 50% thinning rate, resulting in 30 000 samples from the posterior distribution of math formula and math formula (logistic-t model) or math formula (beta-binomial model) for both survey years. In addition, we generated 30 000 samples from the posterior-predictive distributions of all unobserved math formula and math formula, corresponding to the augmented groups (we use a superscript, K, to track these iterates, below). These values allowed us to generate samples from the posterior-predictive distribution of the number of moose in each of the sampled plots, math formula, via:

display math(eqn 15)

We then generated 30 000 values from the predictive posterior distribution of math formula in each of the unsampled plots (i.e. for math formula, using the following steps:

  • 1.We generated a plot-specific data augmentation parameter, math formula, as follows:
  • logistic-t model We generated a normal random variable with mean = math formula and standard deviation math formula and then applied the inverse logit transformation (see (eqn 9)).
  • beta-binomial model We generated a random variable from a math formula distribution.
  • 2.We determined the number of moose groups in each unsampled plot, math formula, by generating a binomial random variable with math formula trials and probability of success equal to math formula.
  • 3.We determined the number of moose in each unsampled plot, math formula, by generating a Poisson random variable with mean =  math formula and adding math formula (taking advantage of the fact that the sum of Poisson random variables is again Poisson).

We determined the total population size by summing estimates for each of the sampled and unsampled plots:

display math(eqn 16)

Lastly, we generated 30 000 samples from the posterior distribution for the log rate of change from 2006 to 2007 by taking the difference in log population sizes in the 2 years, math formula.

Estimates were computed using the median of the sampled values from each parameter's posterior or posterior-predictive distribution, and 95% credible intervals were formed from the 2·5 and 97·5 percentiles of these distributions. Monte Carlo standard errors were calculated for median and credible interval estimates using overlapping batch means with batch size equal to math formula (Flegal & Jones 2011).

Goodness-of-fit

Unlike the design-based mHT estimator, the Bayesian model-based estimator requires several distributional assumptions. We used a variety of plots to explore the validity of these assumptions. First, we compared stratum-specific prior- and posterior-predictive distributions of the number of observed animal groups per plot (math formula) to kernel density estimates constructed using the observed data (Fig. 2). Similarly, we compared observed numbers of animals per animal group (math formula) with stratum-specific prior- and posterior-predictive distributions (Fig. 3). The observed distribution of math formula in the operational surveys will be biased towards small values because detection probabilities will be lower for groups in high cover. Recognizing this problem, we considered two different diagnostic plots to determine whether the assumed beta distributions provided an adequate fit to the visual obstruction data. First, we compared prior- and posterior-predictive distributions of math formula with a weighted kernel density estimator (Fieberg 2007) in which the observed math formula were weighted by math formula to correct for detection biases (Fig. S2 of Appendix S1). Second, we constructed posterior-predictive distributions of visual obstruction measurements for detected groups only (i.e. math formula), which we compared with an unweighted kernel density estimator applied to the math formula from the operational surveys (Fig. 4).

We also explored fit of the models to the group size math formula and observed visual obstruction math formula data by calculating Bayesian P-values for each year × strata combination using math formula goodness-of-fit (GOF) measures (Gelman et al. 2004, pp. 167–174). We made use of the 30 000 samples from the posterior distribution of math formula and from the posterior distributions of math formula for evaluating model fit to math formula and math formula, respectively. For each set of sampled parameter values, we generated a new data set, math formula or math formula In addition, we determined (math formula, math formula) and (math formula, math formula). In the case of math formula, closed form expressions for the mean and variance do not exist, so we used the ‘integrate’ function in program r (R Development Core Team 2011) to approximate these values. We then calculated:

display math(eqn 17)
display math(eqn 18)

We calculated similar math formula discrepancy statistics for the simulated and observed visual obstruction measurements. Bayesian P-values were then calculated as follows:

display math(eqn 19)

Lastly, we compared logistic-t and beta-binomial models using a posterior-predictive loss (PPL) criterion formulated using the number of independently sighted moose groups in each plot, math formula:

display math(eqn 20)

where math formula and math formula represent posterior means and variances, respectively. The first term provides a measure of model fit, whereas the second term serves as a penalty for model complexity (Gelfand & Ghosh 1998). We provide more details regarding the steps required to calculate PPL in Appendix S4.

Figure 2.

Prior-predictive, posterior-predictive and empirical stratum-specific distributions of the number of observed animal groups per plot (math formula) in low (a, d), medium (b, e) and high (c, f) density strata, in 2006 (top row) and 2007 (bottom row).

Figure 3.

Prior-predictive, posterior-predictive and empirical stratum-specific distributions of the number of animals per group (math formula) in low (a, d), medium (b, e) and high (c, f) density strata, in 2006 (top row) and 2007 (bottom row). The prior distribution for math formula was specified in the same manner for both the beta-binomial and logistic-t models. Posterior-predictive distributions were also nearly identical for the two models, so we only show the posterior-predictive distribution for the beta-binomial model.

Figure 4.

Prior-predictive, posterior-predictive and empirical stratum-specific distributions of visual obstruction measurements associated with each observed moose group (math formula) in low (a, d), medium (b, e) and high (c, f) density strata, in 2006 (top row) and 2007 (bottom row). The empirical estimate was formed using an unweighted kernel density estimator applied to groups detected during the operational surveys.

Results

Estimates of abundance in 2006 and 2007 were 7895 and 6420 for the beta-binomial model, 9874 and 7747 for the logistic-t model, and 8840 and 6917 for the mHT estimator (Table 2). Estimates of the log rate of change (95% CI) between 2006 and 2007 were −0·21 (−0·53, 0·12), −0·24 (−0·64, 0·16), and −0·25 (−0·64, 0·15) for the beta-binomial model, logistic-t model and mHT estimator, respectively. Because each of these intervals contains 0, we cannot rule out the possibility that the population was stable or even increased between the two surveys. Nonetheless, credible intervals for the abundance estimates and log rate of change were shorter for the beta-binomial model than the mHT estimator. Assuming the credible intervals have good frequentist coverage rates (i.e. 95% intervals include the true abundance 95% of the time), these results illustrate the potential advantage of parametric, model-based inferences from survey data (Little 2009).

Table 2. Estimates of abundance in (2006 and 2007) and log rate of change between the 2 years, along with goodness-of-fit statistics for logistic-t and beta-binomial models
ModelEstimates (95% CI)Posterior-predictive lossa
math formula math formula Rate of change G P PPL
  1. a

    math formula. The first term (G) provides a measure of model fit, whereas the second (P) serves as a penalty for model complexity (Gelfand & Ghosh 1998).

  2. b

    Traditional sightability estimator developed by Steinhorst & Samuel (1989)

Logistic-t9874 (7017, 13 821)7747 (5631, 10 716)−0·24 (−0·64, 0·16)231 665164 211395 876
Beta-binomial7895 (6030, 10 420)6420 (4886, 8597)−0·21 (−0·53, 0·12)206 84160 952267 792
mHTb8840 (6438, 12 578)6917 (5092, 9740)−0·25 (−0·64, 0·15)

The beta-binomial model had lower values of both math formula and math formula than the logistic-t model (Table 2), suggesting that it provided a better fit to the data and resulted in less variable predictions. The superior performance of the beta-binomial model was also evident in plots comparing posterior-predictive distributions of the number of observed animal groups per plot, math formula, to the observed data (Fig. 2). Whereas the beta-binomial model fit the data well, the logistic-t model resulted in right-skewed posterior-predictive distributions in the high density stratum that were not well supported by the data (Fig. 2c,f). This behaviour likely explains the relatively large estimates of ϒ produced by the logistic-t model. Similarly, the logistic-t model exhibited greater sensitivity to the amount of data augmentation (i.e. math formula) in early test runs.

All other parameters and assumptions were consistent across the two models (beta-binomial and logistic-t; see Appendix S5 for tables summarizing the posterior distributions for all model parameters). As a result, posterior-predictive distributions of the number of animals per animal group (math formula) and visual obstruction measurements (math formula) were similar for the beta-binomial and logistic-t models (Figs S2–S4 in Appendix S1). Visually, the shifted Poisson model appeared to provide a reasonable fit to the distribution of math formula (Fig. 3), although Bayesian goodness-of-fit tests rejected the null hypothesis (that the data come from a shifted Poisson distributed) for the medium stratum in 2006 (P = 0·03) and for the high stratum in 2007 (P = 0·005); the latter estimate was largely influenced by an outlying group size with nine animals. All other P-values were >0·15.

Posterior-predictive distributions for math formula were often similar to the prior-predictive distributions (Fig. S2b–d in Appendix S1), which suggests the data may not be very informative with respect to the distribution of detection covariates; alternatively, the visual obstruction measurements may be distributed fairly uniformly over the [0, 1] interval. [Note that the kernel density estimates of math formula in Figs S2 and S4 of Appendix S1 may be subject to a (negative) boundary bias near 0 and 1 (Wand & Jones 1994).] Posterior-predictive distributions of visual obstruction for observed groups (math formula) better fit the data, and these plots suggest the model was capable of greater ‘learning’ (Fig. 4). Further, Bayesian P-values for the distribution of math formula were all >0·45. In addition, we obtained similar point estimates and credible intervals with two alternative prior specifications for the distribution of math formula, including a more informative prior with values of math formula centred around 0·5 (Appendix S3). Thus, we conclude that the assumed beta distribution provides a reasonable fit to the data, and the results appear to be robust to the assumed prior distribution.

Discussion

Survey statisticians have traditionally favored design-based inferential procedures because of the robustness afforded from the lack of parametric assumptions. With design-based inference, sample units are selected according to a random sampling design, and estimators of population quantities (or targets), along with estimates of their variance, are constructed by considering all possible samples that could be obtained from the design (referred to as the randomization distribution Dorazio 1999; Little 2004; Särndal 2010). The main advantage of design-based inference is that assumptions about the distribution of population-level responses are not required for valid inference (although confidence intervals typical rely on large-sample asymptotic theory).

Nonetheless, applications of survey sampling rarely conform to the idealistic scenarios necessary for performing exact, design-based inference (Little 2004; Särndal 2010). As a result, models are often used to correct survey data for non-response or errors associated with the sampling frame (i.e. the list of all survey units from which the sample is drawn). Further, models are often considered when constructing design-based estimators (Särndal, Swensson & Wretman 2003). Thus, the line between ‘design-based’ and ‘model-based’ inference is not always black and white. The mHT estimator originally proposed by Steinhorst & Samuel (1989) also contains several model-based features, including the use of a logistic regression model to adjust counts for detection rates <1. Our proposed estimator goes a step further, formally adopting a model-based form of inference with parametric assumptions for several population-level variables (x, y and M).

The model-based framework offers many potential inferential advantages compared with design-based mHT estimator, especially when we consider the best-fit beta-binomial model. Estimates produced by the beta-binomial model were more precise than the mHT estimates. This improved precision likely reflects the inefficiency of the mHT estimator when inclusion probabilities are small, but also the potential gains that can follow from the use of parametric assumptions. This strength of the model-based approach is also its main drawback; that is, the Bayesian framework may be less robust than the mHT estimator (as evidenced by the logistic-t model). At a minimum, careful thought is required in choosing appropriate model forms. Posterior loss criterion may be used to choose among competing models (Gelfand & Ghosh 1998), but posterior-predictive plots and goodness-of-fit measures should also be used to assess model fit.

We did not find any evidence that detection rates were influenced by group size in our sightability data (Giudice, Fieberg & Lenarz 2012). Our group sizes were small and also had comparable means (1·45, 0·93, 1·36, 0·95, 1·28 and 1·12) and variances (1·48, 1·19, 1·97, 1·08, 1·55 and 1·86) for all year × strata combinations. Yet, larger groups are common in other wildlife populations, and as a result, group size often influences detection rates. Further, for many applications, group sizes may be overdispersed relative to a Poisson distribution. In such cases, one could use a negative binomial distribution or a Poisson-normal model to account for large group sizes (Breslow 1984). Even though our group sizes were not highly overdispersed, formal goodness-of-fit tests rejected the shifted Poisson distributed in two of six cases. Yet, there was no general pattern suggestive of a consistent bias (Fig. 3). Further, it is not clear whether the lack of fit is of any real consequence (goodness-of-fit tests often have high power to detect small deviations from the null distribution). Nonetheless, additional simulation work would be helpful to assess bias, precision and model robustness relative to the mHT esitmator under a range of data generating scenarios, including situations involving larger and more variable group sizes and cases where detection probabilities are influenced by group size.

In addition to improved efficiency, the Bayesian approach offers the ability to perform exact inference in small samples and to incorporate prior information to improve precision. Several telemetry studies have been recently initiated in an attempt to understand why moose have been declining in Minnesota, USA (Lenarz et al. 2010). Information on habitat use from these studies could conceivably inform the distribution of visual obstruction measurements in each stratum. These data could also be used to inform parameters related to population density (e.g. λ) or to improve the stratification scheme (Habib, Moore & Merrill 2012); the latter would improve the precision of both the mHT and the model-based estimators.

Having developed a framework for model-based inference, we see many potential opportunities for extensions. Arguably, the most attractive feature of the model-based approach is the flexibility it affords when analysing multi-year survey data. Rotational or panel designs that repeatedly sample the same survey units are often used to increase the precision associated with estimates of population trends. Although design-based theory is available for analysing data from panel designs (Kasprzyk et al. 1989), the estimators are complex, and the theory has not been extended to the sightability modelling case. By contrast, random effects could offer a simple approach to analysing these data within a model-based framework. Further, a hierarchical model for year-specific parameters (e.g. math formula) could be used to ‘borrow strength’ across years. Another exciting alternative would be to explore spatially explicit models of abundance parameters rather than the exchangeable (within-stratum) models considered here (see Schmidt et al. 2012 for a similar approach applied to data collected using distance sampling). Lastly, we suspect other complications, including measurement error associated with estimates of group size (Walsh et al. 2009), might be best addressed using a model-based approach. In conclusion, we suspect the mHT estimator will remain popular due to its robustness, but the model-based estimator will allow for more future growth and extensions.

Acknowledgements

The Minnesota Department of Natural Resources, the Fond du Lac Band of Lake Superior Chippewa and the 1854 Treaty Authority provided funding and field support for the sightability and operational moose surveys. The United States Geological Survey, Northern Prairie Wildlife Research Center provided in-kind support. The United States Fish and Wildlife Service's Tribal Wildlife Grants Program provided additional funding. We thank K. Carlisle, A. Edwards, D. Litchfield, M. Nelson, T. Rusch, B. Sampson, M. Schrage, and MNDNR pilots A. Buchert, J. Heineman, M. Trenholm and B. Maas for aerial telemetry and survey contributions. Comments and suggestions from B. Dorazio, O. Gimenez and M. Samuel improved the manuscript.

Data Accessibility

Data from this paper have been deposited in the Dryad repository (Fieberg et al. 2013).

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