**6. Acknowledgments**

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In our hierarchical model of replicated, presence-absence surveys, the parameter of primary ecological interest is the community's incidence matrix. This matrix is only partially observable because a species may be present at a sample location but not observed in the surveys. We use a binomial sampling model to specify the probability of detection (or capture) of each species and thereby to account for detection errors in the observed data. In this way estimates of the community's incidence matrix are automatically adjusted for the imperfect

In our approach, measures of biodiversity are estimated indirectly as functions of the estimated incidence matrix of the community. Thus, species richness and measures of alpha or beta diversity depend on a set of model-based estimates of species- and site-specific occurrences. This approach differs considerably with classes of statistical models wherein species richness is treated as a single random variable – usually a discrete random variable – that represents the aggregate contribution of all species in the community. This "top-down" view of a community may yield incorrect inferences if heterogeneity in detectability exists among species or if the effects of environmental covariates on occurrence differ among species,

The inferential benefits of using hierarchical models to estimate measures of biodiversity are not free. As described earlier, the price to be paid for the ability to estimate probabilities of species occurrence and species detection is replication of presence-absence surveys within sample locations. In our opinion the improved understanding acquired in modeling the community at the level of individual species and the versatility attained by having accurate estimates of a community's incidence matrix far outweigh the cost of additional sampling. That said, there are other, perhaps less obvious, costs associated with these hierarchical models. Specifically, estimates of species richness and other community-level parameters may be sensitive to the underlying assumptions of these models, and these assumptions can be difficult to test using standard goodness-of-fit procedures. For example, the choice of distributions for modeling heterogeneity among species or sites may exert some influence on estimates of species richness. We assumed a bivariate normal distribution for the distribution of logit-scale, mean probabilities of occurrence and detection, but other distributions – even multimodal distributions – also might be useful. In single-species models of replicated, presence-absence surveys, estimates of occurrence are sensitive to the distribution used to specify heterogeneity in detection probabilities among sample sites (Dorazio 2007, Royle 2006); therefore, similar sensitivity can be expected in multispecies models, though this aspect

Another assumption of our model that is difficult to test is absence of false-positive errors in detection. In other words, if a species is detected (or captured), we assume that its identify is known with certainty. However, in surveys of avian or amphibian communities where species are detected by their vocalizations, misidentifications of species can and do occur (McClintock et al. 2010a,b, Simons et al. 2007). These misidentifications are even more common in circumstances where surveys are conducted by volunteers whose identification skills are highly variable (Genet & Sargent 2003). If ignored, false-positive errors in detection induce a positive bias in estimates of species occurrence because species are incorrectly "detected" at sites where they are absent. While it is possible to construct statistical models of presence-absence data that include parameters for both false-positive and false-negative detection errors (Royle & Link 2006), these models are prone to identifiability problems. To

detectability of each species.

as illustrated in our analysis of the ant data.

of model adequacy has not been rigorously explored.

Collection of the original ant dataset was supported by NSF grants 98-05722 and 98-08504 to AME and NJG, respectively, and by contract MAHERSW99-17 from the Massachusetts Natural Heritage and Endangered Species Program to AME. Additional support for AME's and NJG's research on the distribution of ants in response to climatic change is provided by the U.S. Department of Energy through award DE-FG02-08ER64510. The statistical modeling and analysis was conducted as a part of the Binary Matrices Working Group at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from The University of Tennessee, Knoxville.

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