Figure 1.
Directed, acyclic graph (DAG) of the (areal) hierarchical model for distance data.
Individual nodes indicate a parameter or vector of parameters, and arrows represent conditional dependence. Notation is defined in Table 1.
Table 1.
Parameter and data definitions.
Figure 2.
True and estimated population size for simulated data.
True abundance is indicated in red, with posterior means and estimated 95% credible intervals for abundance indicated by circles and brackets, respectively. Panel (A) gives results for the simulation with linearly increasing abundance, while panel (B) gives results for the simulation with a quadratic relationship between abundance and a habitat covariate.
Figure 3.
Representation of golf tee population.
Each symbol represents a different group of golf tees, with dark symbols representing yellow tees and gray symbols representing green tees. Groups that were observed by at least one observer are indicated by solid symbols, while open symbols indicate groups that were never observed. Squares represent tee groups that were exposed above surrounding grass, while triangles represent unexposed groups. Group sizes are indicated by the proportional size of each symbol, with the smallest symbols representing groups of 1 animal, and the largest symbols representing a group of 8 individuals. Transect lines are represented by solid black lines, with dotted lines giving survey area boundaries and demarcating the areas surveyed by each transect. The red line serves as the strata boundary (points north comprise the northern stratum).
Figure 4.
Empirical and posterior predictive distributions of golf tee group size.
Bar plots representing the probability mass for group size in the golf tee experiment. Empirical distributions correspond to the actual distribution of group size used in the experiment, while posterior distributions represent estimated posterior predictive distributions obtained after analyzing data with our hierarchical model.
Figure 5.
True values and estimated posterior distributions for simulated data.
Kernel density estimates of marginal posterior distributions are indicated in black, with true values used to simulate data indicated by red, vertical lines. Parameters indexed by “Cov” give covariate parameters, “Det” give detection parameters, “Hab” give habitat parameters, and “N” gives abundance. The first panel (“cor”) gives an estimate of the observer dependence parameter. Species specific parameters are indexed by “sp1” (for species one) or “sp2” (species two).
Figure 6.
True and estimated detection functions for simulated data.
Detection functions for each species are based on mean group sizes for each species (4 and 2, respectively), and are made for observer 2 (who had an intermediate detective ability).
Figure 7.
Posterior distributions for the abundance (number of groups) of golf tees of different types.
Kernel density estimates of posterior distributions are in black, while true values are represented by red vertical lines, and estimates from a conventional mark-recapture distance sampling analysis (see Laake and Borchers [7]) are presented in blue.
Figure 8.
Implied detection probability and observer dependence for covariates maximizing dependence.
The top panel gives detection probability curves for the set of covariates that maximize observer dependence (observer = 2, group size = 1, exposure = 0, species = “green”). “Individual” specifies detection probability for observer 2 only; “Conditional” gives the probability of detection for observer 2 given that the group was detected by observer 1; “Duplicate” gives the probability of detection by both observers; “Pooled” gives the probability of detection by at least one observer. The bottom panel represents dependence, as summarized by the parameter (see [9]) for the same set of covariates.
Table 2.
Modeling choices and justification.