Hierarchical spatial modeling

I have a couple of questions after reading the paper by Mitzi Morris. Perhaps someone here with background in spatial modeling can advise.

  1. The authors modeled Poisson distributed count data within census tracts, and mention that other distributions could be easily modeled as well. Are there approaches within the spatial-hierarchical framework which model dis-aggregated data, rather than single number summaries (count, mean, etc.)? For example, if you had n individual observations within each region, and were interested in estimating each region’s mean while accounting for spatial correlation between neighboring regions. A generic example here might be home prices.

  2. Can anyone offer insight on how one might approach the added complexity of data points defined by both origins and destinations (edges in a graph)? Could the problem still fit in the spatial-hierarchical framework if “neighbors” were defined such that two edges are neighbors if both their origins and destinations are neighbors, respectively?

Apologies for 2) being vague " how would you… " type question. It seems like it could be related to or an extension of this work, and I’m just trying to figure out how to properly conceptualize it.

Tagging @mitzimorris and @anon75146577.

Yes. You can use this just like you’d use a random effect in a multilevel model.
Eg if each region j had i = 1,\ldots, n_j observations, then a model like
y_{ij} \mid \mu, \beta, u, \sigma \sim N(\mu + x_{ij}^T\beta + u_j, \sigma^2)
u \mid \tau \sim ICAR(\tau)
would make sense.

If you’ve got case data that have locations attached to them, it’s possible that a point process model (like a log-Gaussian Cox process) model might be appropriate. This model ends up with a more complex neighbourhood structure (and more complex precision matrix) than the ICAR in the paper, but it’s possible to fit in Stan (using a 2 dimensional random walk prior on a lattice).

I don’t understand this question, sorry.

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I guess the question is whether one can replace the usual incidence matrix by a weighted matrix representing actual proximity in space. One example is two regions A and B that are only a few kilometres apart, but are separated by a mountain range and a region C that is farther away but connected to A, say, by good roads. Then d(A,B) > d(A,C), for instance. One could come up with a binarisation of this matrix of distances and use that as a neighbourhood structure, but I don’t know much about this.

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I guess the question is whether one can replace the usual incidence matrix by a weighted matrix representing actual proximity in space.

no can do with the ICAR model. CAR model, yes - there’s a case study on CAR models too -
but what you need to understand is that computing phi in the CAR model is cubic on the number of regions in your neighborhood map. so doable only for small numbers of regions.

here’s a non-paywalled version of the paper: http://www.stat.columbia.edu/~gelman/research/published/bym_article_SSTEproof.pdf
the math is laid out in section “Conditional Autoregressive Models” - with that in hand, take a look at Max Joseph’s case study:



Yes but it’s difficult to ensure that the resulting matrix is symmetric positive semi definite. I don’t have a good reference of something that does it successfully.

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Thanks @anon75146577, the hierarchical modeling extension makes sense.