We describe here a model extension developed by Kery and Royle (2008b) for the analysis of data collected in the national Swiss breeding bird survey during 2001. In this survey each of 254 quadrats (=sites) was visited on 2 or 3 separate occasions during the breeding season by a collection of volunteer observers. Each volunteer attempted to cover as much of a quadrat as possible, while traversing an irregular route. Both route length and sampling duration were recorded for each visit, as these were thought to be useful in constructing covariates of detection. Another important covariate of detection was the sampling date because the entire survey required about 3 months (15 April-15 July) to complete. Because the survey included quadrats located throughout Switzerland, covariates which were thought to most influence occurrence of species were forest cover and elevation. An additional covariate, route length, was included to adjust for the increased chance of finding a species along a longer route.
We must extend the notation developed in the previous section to include the possibility that detection differs not only among species, but also among sites and sampling occasions. Therefore, let yjjk denote a binary observation that indicates whether the ith species was detected (yjjk = 1) or not (yjjk = 0) during the jth visit to site k. Using similar notation for detection probability pjjk, we assume the following model of the observations:
yjjk\Pijk, Zjk ~ Bern(pjjfczjfc), where logit(pjjk) = OjO + ajidatejk + a,j2date2k + a,j3 effort^.
The covariate effort is the ratio of time required to sample a route divided by the length of the route. Both linear and quadratic terms are included for the covariate date in an attempt to model differences in activity patterns of species during the breeding season. That is to say, some species may have a peak in activity sometime during the breeding season while others may begin low and steadily increase (or vice versa).
We specify the effects of covariates on species occurrence probabilities similarly, except that the occurrence of each species is assumed to be fixed during all visits to a site. Therefore, our model of species occurrence is logit(ftik) = bio + biielevk + bi2elevk + 6i3forestk + bi4lengthk and wi denotes a latent indicator of whether the ith species is a member of the N species exposed to sampling. This formulation allows occurrence probabilities of each species to differ with elevation (elev), percent forest cover (forest) and route length (length). For numerical reasons, all covariates except route length are standardized to have zero mean and unit variance.
Thus far, we have specified only the contributions of observable sources of heterogeneity in species occurrence and detectability. A component for heterogeneity among species must be added to the model to estimate patterns of occurrence of species that are members of the community but are not observed in the sample. For this modeling component we assume a normal distribution (on the logit scale) as was previously done where bi = (bi0, bi1, bi2, bi3, bu) denotes a vector of logit-scale occurrence parameters for the ith species and ai denotes a vector of logit-scale detection parameters. Heterogeneity among species in occurrence and detection is specified by the symmetric, 9 x 9 matrix S. The diagonal elements of S correspond to the variances of ai and bi, and the off-diagonal elements of S are all zero except for c16 = <761, which allows the 'intercept' parameters ai0 and bi0 to be correlated. This assumption is identical to that described in the previous section where ui (=bi0) and vi (=ai0) were modeled as correlated parameters.
The remaining modeling assumptions are basically identical to those made previously. For the latent variable wi, which indicates membership in the community of N species, we assume wi ~ Bern(^). A set of independent prior distributions is assumed, each of which is intended to be non-informative (see Kery and Royle (2008b) for details).
zifc|ftifc, wi ~ Bern(ftifcwi), where
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