NjAj PoAj

log Aj = ß0 + ßi elevj + elev2 + ß3 forestj + ßA length + bj bila - N(0, a2), where a parameterizes the extent of this random variation. A classical statistician might refer to the parameter bi as a log-normal random effect.

Figure 8.2. Comparison of route-level abundances of willow tit territories estimated from point counts (ordinate) or presence/absence data (abscissa). A 45 degree line is superimposed.

To fit this expanded model by the method of maximum likelihood, we would need to integrate both N and bi from the model to obtain a likelihood function based on the metapopulation-level parameters a, fl and a. This approach was used in Chapter 4 to estimate the parameters of an abundance-based occupancy model (see Panel 4.1). Instead, here we opt to conduct a Bayesian analysis of the territory counts. This requires us to complete the model by specifying a set of mutually independent priors for the metapopulation-level parameters; thus we assume fl - N(0,102I), a0 — N(0,1.62), (a1,a2,a3) — N(0,102I), and a — U(0,100), where I denotes an identity matrix of appropriate size. The prior assumed for a0 approximates a U(0,1) prior for expit(a0). The Web Supplement contains R and WinBUGS code needed to fit this model to the data.

Table 8.3 contains summary statistics (posterior means and credible intervals) obtained by fitting this model to the willow tit territory counts. For comparison with the likelihood analysis, we also fit the model without assuming extra-Poisson variation in route-level abundances. The posterior means associated with this simpler model are consistent with the MLEs reported in Table 8.2. However, the parameter estimates obtained by fitting the overdispersed model suggest that the abundance covariates account for some, but not all, of the heterogeneity among routes. For example, the 95 percent credible interval for the overdispersion parameter a clearly exceeds zero, and the posterior of a has negligible probability density near zero.

Table 8.3. Bayesian analysis of willow tit territory counts. Variation in route-level territory abundances was modeled with and without extra-Poisson variation as parameterized by the overdispersion parameter a.

Poisson model Overdispersed model

Table 8.3. Bayesian analysis of willow tit territory counts. Variation in route-level territory abundances was modeled with and without extra-Poisson variation as parameterized by the overdispersion parameter a.

Poisson model Overdispersed model

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