Later in this chapter we describe how estimates of Z may be used to calculate community-level characteristics of scientific interest, such as site-specific species richness.
In the model summarized in Eqs. (12.1.1) and (12.1.2), the probabilities of occurrence —j and detection pj are assumed to be species-specific. Natural communities are composed of species whose abundances vary widely, so it stands to reason that the average occurrence will vary among species. Similarly, differences in abundance, appearance (e.g., size, coloration), or behavior (e.g., time of activity, habitat preferences) of species will almost certainly induce heterogeneity in their detectabilities.
We could treat the species-specific probabilities of occurrence and detection as fixed parameters and compute estimates of —j and pj by analyzing each row of Y separately. However, there are several reasons for not adopting this approach. First, the estimates may be unstable or lack precision, particularly for rare species that may be present in low abundance at occupied sites and difficult to detect. Second, this approach does not yield a parsimonious solution because the number of parameters to be estimated increases with the size of the community. That is, in a community of n species there are 2n parameters to estimate. Another reason to avoid this kind of analysis - actually the most compelling reason - is because this approach does not provide a mechanism for estimating species richness of communities in which N is unknown, as we demonstrate in Section 12.2.
These reasons provide ample motivation for extending the species-specific models of yik and zik given in Eqs. (12.1.1) and (12.1.2). In particular, we require modeling assumptions that characterize the heterogeneity in probabilities of occurrence and detection among species. Thus, let uj = logit(—¿) and vj = logit(pj) denote a logit-scale parameterization of these probabilities. We assume that heterogeneity (among species) in these parameters may be specified using a bivariate normal distribution where S denotes a 2 x 2 symmetric matrix whose diagonal elements, aU and aV, specify levels of the variation in uj and vj, respectively, and whose off-diagonal elements, auv, equal the covariance between uj and vj. The parameters ft and a specify the mean logit-scale probabilities of occurrence and detection, respectively, among all species in the community. We anticipate that estimates of auv will almost surely be positive because probabilities of occurrence and detection are both expected to increase as the abundance of a species increases. For example, if we let Njj denote the abundance of individuals of species i at site j and assume Njj ~ Po(Ajj-), then the probability of occurrence is expected to increase with mean abundance Ajj as follows: — jj = Pr(Njj > 0) = 1 — exp(-Ajj). Similarly, if we let qjj denote the probability of detecting each of the Njj individuals present at site j, and assume that such detections are independent, then the probability of detecting these individuals increases with Njj as follows: pjj = 1 — (1 — qjj)Nij.
This relationship is identical to the underlying assumption of the abundance-based occupancy models described in Chapter 4. Our expectation of positive estimates of auv is therefore based on simple, yet entirely reasonable, arguments.
This model-based characterization of variation in species occurrence and detection probabilities (in Eq. (12.1.3)) is certainly not unique. Other distributional assumptions are possible and, in fact, may exert a strong influence on estimates of species occurrence, species detection, and species richness (Dorazio and Royle, 2005a; Dorazio et al., 2006). Our view is that this modeling assumption, as with any other, should be based on information relevant to the inference problem and can always be revised as needed in the context of the problem.
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