Hierarchical View Of The Population

Consider a survey wherein a particular sampling protocol is applied at various locations judged to be representative of the entire population. In this survey the sample includes a set of spatially-indexed counts observed by sampling a corresponding set of spatially distinct local populations. Conceptually, the population being sampled may be viewed as a metapopulation (Hanski and Simberloff, 1997) of relatively isolated and spatially distinct local populations connected demographically by occasional dispersal of animals. Strict adherence to the metapopulation view is not essential, but it does provide a natural hierarchy for building statistical models of animal abundance and occurrence. For example, at the bottom of this hierarchy, counts are observed at individual sample locations (Figure 8.1). The particular sampling protocol determines, in large part, an appropriate model of the counts. For example, multinomial models are induced by capture-recapture sampling or by removal sampling; however, the precise form of the multinomial cell probabilities depends on which of these sampling protocols is used (see Section 8.2). In both cases the model of counts observed at a particular sample location depends on the unknown abundance of animals at that location and on the unknown probability of detection (per individual) at that location. Therefore, these unknowns (abundance and detection) are intermediate-level parameters in the hierarchy and are defined locally (i.e., for an individual sample location). The top of the hierarchy is reserved for parameters that apply across the entire population of animals. These metapopulation-level parameters specify the extent to which abundance and detection differs among local populations. Such differences may include systematic sources of variation (e.g., effects of habitat) based on measurable covariates and stochastic sources of variation for surveys where location-specific covariates are unavailable.

The hierarchy illustrated in Figure 8.1 is an example of a 'mixed model', a term widely used to describe statistical models that include both 'fixed effects' (metapopulation parameters that are fixed among sample units) and 'random effects' (local parameters that vary among sample units). Scientific interest may be focused on either set of parameters, depending on the context of the study. The hierarchical framework provides a formal connection between these two sets of parameters and allows both to be estimated. This chapter includes a variety of

Figure 8.1. A hierarchy induced by sampling a metapopulation of animals at n spatially distinct locations.

Figure 8.1. A hierarchy induced by sampling a metapopulation of animals at n spatially distinct locations.

examples chosen to illustrate the flexibility of this framework and the pros and cons of alternative modes of inference (classical vs. Bayesian).

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