50 100 200 300 400

Population size, N

Figure 6.1. Profile likelihood of N for the tiger data. The profile likelihood is minimized at N = 111.69.

Table 6.1. North American Breeding Bird Survey data from a route in Maryland. Frequencies are number of species detected k times for k = 1, 2,... , 50.

# detections frequency (# species)

1-18: 15 7 5 2 5 6 3 2 1 2 4 4 0 1 0 2 3 F" 19-36: 1 11000110101000001

no species was detected at more than 36 stops. The number of species detected 1, 2, . . . , 36 times is given in Table 6.1.

This model can be fit in R using the integrate function, as demonstrated by Panel 6.1. The estimates for these data were V = -2.44, V = 1.49 and ¬ęc) = 11.53 so that N = 82.31. The 95 percent profile likelihood interval for these data is (73.69,104.29).

6.3.1 Spatial Subsamples as Replicates

The classical design for estimating N in closed populations (Chapter 5) is that in which a population is sampled J times. Implicit in the conceptual formulation is that the population occupies a single location or area. Often, as we have seen in the previous examples (voles, lizards, tigers), we may only be able to describe this area approximately (e.g., a 9 ha plot, or a polygon around camera trap locations). However, in the present application of closed population models to estimating species richness, we do not have this situation of replicate samples of a population that is (conceptually) static. Rather, we have used spatial samples as replicates so that, potentially, each sample exposes a different 'population' of species. In the BBS data, each route consists of 50 stops, and we view these stops as replicates of the population of species that is exposed to sampling by the collection of 50 routes. The use of spatial subsamples as replicates is common in this context, having been used by almost all of the applications of which we are aware (Burnham and Overton, 1979; Boulinier et al., 1998; Cam et al., 2002b,c; Dorazio and Royle, 2003).

Given this distinct spatial sampling context, it is natural to question whether this has any relevance to inference or interpretation of N. The spatial sampling should be relevant because there is a component of variation that is now strictly due to spatial sampling (occurrence or not at the level of a subsample) and also a component of variation due to detectability at occupied subsamples.

1 Logit-normal estimates for these data are slightly different than those reported in Dorazio and Royle (2003, Table 3) which we believe are in error.

One view that yields a clear interpretation of model parameters is that in which we can assert that species occur, or not, on all subsamples along the route. That is, if the route is occupied, then all 50 stops are also occupied and vice versa. In this case, variation in p is due to variation in detectability of species alone. A consequence of this view is that N is the number of species present on the 50 particular spatial subsamples (not the region from which those subsamples were randomly chosen). This view is consistent with the closed population sampling view from which the method derives. We (Dorazio and Royle, 2005a) incorrectly stated that the use of this spatial subsampling design implies 0 = 1. It does not. The view that all spatial replicates are occupied (and 0 = 1) is only one possible interpretation of the design. An alternative and prevailing view is that, by neglecting the occurrence process in the model, there is an implicit confounding of a subsampling occurrence probability for species i, say 0j, with detection probability. That is, if occurrence of a species is independent across spatial subsamples then, when spatial subsamples are used as

replicates, this induces a confounding of conditional detection probability, say p( ),

with 0j so that the pj = p( )0i. Here p( ) is the normal 'conditional on capture' detection probability (i.e., that contained in the models of Chapter 5). In this case, the view that N is the size of the community that was exposed to sampling by the spatial subsamples (Boulinier et al., 1998; Cam et al., 2002c) appears to be justified.

It is possible to formulate models, under more general sampling designs, in which one can obtain explicit information about both subsample occurrence probabilities and also conditional-on-occurrence detection probabilities (Dorazio and Royle, 2005a). The design in this case has formal replicate samples at each spatial sample unit. Applications of this design to modeling community structure can be found in Dorazio et al. (2006); Kery et al. (2008); Kery and Royle (2008a,b). These models are the topic of Chapter 12.

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