where S is an M x M variance-covariance matrix. Typically S is described by some covariance function having parameter(s) 0. For example, the exponential covariance function is

where ||sj — Sj|| denotes the distance between locations si and Sj. This basic formulation - a multivariate normal model with a parametric covariance matrix - underlies the method usually referred to as 'kriging'. Diggle et al. (1998) adapted this formulation in the development of spatial models for counts within a generalized linear modeling framework. The main deficiency with this approach is that it can become computationally prohibitive as the number of spatial samples increases owing to the dimensionality of S.

An efficient class of models that has seen widespread use are the conditional autoregressive (CAR) models. Like the auto-logistic models introduced in Section 9.4, the CAR models are ideally suited for situations in which the spatial process is inherently discrete, such as in analyses involving states, counties or other geopolitical units. The CAR model relates ai to a at other locations in the conditional mean, according to:

E[aj|a_j] = Y^Cij aj, j where 7 is a correlation parameter, Cj are weights reflecting spatial association between units and a_i denotes the vector of random effects except the ith. As described here, the permissible range of 7 is constrained by the eigenvalues of the matrix of weights C = {Cij}. For highly irregular lattices, this can limit the strength of the correlation significantly. A more flexible version of this model is the intrinsic CAR model (Besag et al., 1991), which has the form where Ni is the neighborhood of cell i, which is the collection of grid-cell identities of neighboring cells, and n is the number of neighbors of grid cell i, i.e., n = dim(Ni). In this analysis we used a first order 'queen's neighborhood', so the 8 cells surrounding a cell are its neighbors. Banerjee et al. (2004) provide an accessible treatment of CAR models.

To apply the intrinsic CAR model to the yellow warbler data from the MAPS program, we created a 2-degree grid (2 deg. longitude by 2 deg. latitude) over North America, and assigned each MAPS station to a grid cell. The CJS model for the warbler data was extended to include a 'degree-block effect,' which was assumed to be a spatially-correlated random effect. The degree-block effects were assigned a CAR prior. As such, the basic model is similar in structure to the American redstart nest survival example (e.g., Panel 11.3), having random spatial effects with (potentially) multiple observations being associated with each random spatial effect. Essentially, this construction produces a relatively fine-scale stratification scheme (relative to BCRs), but not so fine so as to produce many empty cells (i.e., without data).

We consider a model with year-specific detection probabilities and a spatially varying survival probability, governed by a CAR prior distribution on the logit-transformation of In particular, let be the survival probability for a bird located within degree block k. Then, the model for survival probability for a MAPS station i within degree-block k is the additive model where the vector of random effects a is given the intrinsic CAR prior described in the previous section.

The WinBUGS description of this model for the warbler data is shown in Panel 11.6. Specification of the CAR model requires 3 data objects. First weights which is defined (for the intrinsic CAR model) to be a vector of 1s. The length of weights is sumNumNeigh (also data) which is equal to the number of adjacent pairs of grid cells. The data object adj, describes the adjacency structure of the logit(^ifc) = + ak, grid, and num, is a vector having elements equal to the number of neighbors of each grid cell. These can be produced in WinBUGS using the adjacency tool under the maps menu. We provide an example of their calculation using R functions on the Web Supplement. Finally, the data object gridid is a vector having length equal to the number of individual birds in the data set. Each element of gridid indicates the grid-cell the individual is associated with which.

The model was fitted in WinBUGS and the resulting map of survival probability is shown in Figure 11.2. Each point on the map represents a prediction made at the center of a 2-degree grid cell. The shading of each point is related to the magnitude of the estimated survival probability, whereas the size of each point is inversely proportional to the posterior standard deviation. Thus, more precise predictions are indicated with larger circles and vice versa. This map of predicted survival probability shows strong gradients in survival probability over North America. Survival is very low in the east and southwest coast, and relatively higher in the interior of North America. We note that areas having higher MAPS station densities yield more precise predictions of local survival probability, as a result of the relatively greater contribution of data (i.e., relative to the spatial model).

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