Multi-Species Occupancy Model (MSOM)
Species occupancy and detection were estimated using Multi-Species Occupancy Models (MSOM; Iknayan et al. , 2014). Similarly to single-species occupancy models (MacKenzie et al. , 2002), MSOM uses repeated surveys (Winkler extractions) to tease apart the real absences of species from detection errors (false zeroes), and therefore estimates species presence even in those patches where they went undetected. By estimating detection and occupancy rates for each species, the model provides unbiased estimates of species presence in each patch (unlike naïve/raw observations of presence; Tingley et al. , 2020). MSOM also combines the detection and occupancy of all species into a single model and, similarly to mixed-effects models, estimates the variability among species (random effects) in their detection, occupancy, and the effect of covariates on these rates. By estimating the overall parameters (fixed effects; hyperparameters) and the variability in model coefficients across species in the community (random effects) in a single model, MSOMs provide (1) more precise estimates of species detection and occupancy of individual species compared to simpler single-species models (rare species borrow strength from common species), (2) high-precision estimates of the number of missing species in each patch and regionally (alpha and gamma richness estimators; Tingley et al. , 2020), and (3) correct measures of similarity in species composition (Jaccard) while taking into account imperfect detection (beta-diversity estimator; see Chao et al. 2005 for the implications of imperfect detection on beta-diversity).
To estimate the number of missing species across the landscape (gamma richness estimator), we used a data augmentation approach by adding 300 species with zero occurrences in the data (Iknayan et al. 2014). These represent species potentially present at the landscape that went completely missing in the study due to detection errors. As the number of missing species estimated by the model was much lower than the 300 additional species included (see Results), the inclusion of more than 300 species did not change the results.
In addition to using the MSOM, we also conducted Mixed-Effects Models using species presences/absences within patches as the response variable. These models used raw data assuming no detection errors. The predictor variables included patch size, landscape area, connectivity, and cattle presence. Species were included as random factors for intercepts and all slopes. Most of the model results were qualitatively similar to the MSOMs, with species consistently showing positive associations with habitat patch area. For a few species that exhibited distinct patterns in the mixed-effects models, we ran single-species occupancy models. However, we did not find significant and distinct results from the MSOM, which indicates that differences among occupancy models (including MSOM) and raw-data models (Mixed-Effects models) arise primarily from changes in detectability among patches rather than true changes in occupancy. Given that the MSOM provides corrected estimates of diversity and occupancy, even for rare and unobserved species, we only present results from this model in the main text (see Fig. S1 for results of mixed-effects models).