In Section 8.2.1 we noted that a multinomial observation model is induced by a variety of different sampling protocols (e.g., mark-recapture, removal sampling, and double-observer sampling) and that the counts that arise by applying a protocol at a particular site depend on the same multinomial index parameter (say, Ni for the ¿th site). This latent parameter is, in effect, the local abundance of animals at that site. When the multinomial observation model is combined with a model of heterogeneity in local abundances (i.e., heterogeneity among sample locations), we obtain a mixture of multinomials. For example, assuming Ni|A ~ Po(A) produces a multinomial-Poisson mixture model.
As with the binomial mixture models, multinomial mixture models have been developed and used to solve a variety of inference problems. For example, Royle et al. (2004) used counts observed in spatially replicated distance samples to estimate spatial variation in local abundance. Similarly, hierarchical models of multinomial counts have been used to improve site-specific estimates of abundance in removal surveys (Dorazio et al., 2005, 2008) and to produce maps of the spatial distribution of abundance and occurrence from spatially-indexed, capture-recapture data (Royle et al., 2007b). The following subsections include two examples, one involving double-observer counts of manatees detected in an aerial survey and another involving removal counts of stream salamanders. In each example, we develop multinomial mixture models and fit them using classical (likelihood-based) and Bayesian methods.
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