Table 1.
Simulated scenarios (marked with asterisk * those also tested by WLD).
Figure 1.
Log-likelihood surface displaying the maximum-likelihood estimate (P1) and a saddle point at the boundary (P2).
This example corresponds to a constant hierarchical occupancy model and a data set where S = 200 sites, K = 2 replicate visits, Sd = 80 sites with detection and dT = 134 detections. P1 is located at { = 0.416,
= 0.806} and P2 at {
= 1,
= 0.335}.
Table 2.
Counts of different estimation results obtained when fitting hierarchical occupancy models to simulated data from Scenario A1.
Figure 2.
Simulation results of fitting hierarchical and naïve occupancy models to 5000 data sets from Scenario A1 with 55 sites.
The first three columns correspond to the hierarchical model: in column 1 estimates of occupancy probability (‘psi-hat’), in column 2 estimates of the conditional single-survey detection probability
(‘p-hat’) and in column 3 estimates of the unconditional detection probability after K surveys
(‘pdet-hat’). Column 4 presents the estimates for the naïve model that assumes perfect detection. Rows represent increasing number of replicate surveys per site, from K = 1 to K = 5. Where
the naïve model was fitted to data collapsed to a single record per site (1 if species detected at least once, 0 otherwise). In this particular scenario (also presented by [14]) the imprecision in the hierarchical model is large compared to the bias in the naïve model. The true occupancy was 0.4, and the true detection probability increased with the value of the x-variable. In each figure a solid line represents true values. For reference, in columns 3 and 4 a dashed line represents the true occupancy probability.
Figure 3.
Simulation results of fitting hierarchical and naïve occupancy models to 5000 data sets from Scenario A2 with 55 sites.
For details in figure arrangement see Figure 2; here the true occupancy was 0.8. In this example the hierarchical model clearly outperforms the naïve model, which is greatly biased. A comparison of the estimates for K = 2 illustrates with those in Figure 2 illustrates how the naïve model can produce the same estimates for very different scenarios.
Table 3.
Mean square error (MSE) for the occupancy estimator in the hierarchical/naïve models, and their ratio, obtained from simulations of (a) Scenario A1 and (b) Scenario A2 (see Table 1 for details).
Figure 4.
Beta distributions used to generate detectability in the “abundance” scenarios for the different covariate categories.
Lines correspond to (solid),
(dashed) and
(dotted). Panel (a) displays the probability density functions (pdf) for the distributions used by WLD (Scenario B1) and panel (b) for the distributions used in our Scenarios B2 and B3. The distribution that WLD used for
has considerable mass for detectability very close to zero:
. Panels (c-d) display the pdf of the probability of detecting the species in at least one of K surveys (
) at sites
(from darker to lighter, lines correspond to
).
Table 4.
Mean square error (MSE) for the occupancy estimator in the hierarchical/naïve models, and their ratio, obtained from simulations of three “abundance” scenarios: (a) Scenario B1 from [14], (b) Scenario B2 and (c) Scenario B3.
Figure 5.
Simulation results of fitting hierarchical and naïve occupancy models to 5000 data sets from Scenario B2 with 165 sites.
For details in figure arrangement see Figure 2. This example shows that, even if detectability is heterogeneous, the hierarchical model has smaller bias and that this bias is reduced with the sample size.
Figure 6.
Asymptotic bias of the naïve and hierarchical occupancy estimators as a function of heterogeneity in detectability.
In the data-generating model, occupancy is constant and detectability at each site is drawn from a single distribution . In the fitted model both occupancy and detectability are assumed constant across sites (i.e. heterogeneity is not modelled). Heterogeneity is expressed in the x-axis as the coefficient of variation of the distribution (CV). Black thick lines represent the hierarchical model and red thin lines the naïve model (solid lines for K = 2 and dashed lines for K = 5; horizontal grey lines correspond to a naïve model where K = 1). In extreme heterogeneity conditions (high CV such that detectability switches between 0 and 1) both models lead to the same bias. For more realistic scenarios, where heterogeneity is still substantial, the hierarchical model has lower asymptotic bias. The hierarchical model is asymptotically unbiased in the absence of heterogeneity (i.e. CV = 0). Plots in the lower row (A–C) illustrate the heterogeneity in detectability represented by three different CVs when mean detectability is 0.33. Note that the relative asymptotic bias is independent of occupancy probability.