# Difference between spatially autocorrelated logit methods

I am seeking advice on different methods to account for spatial autocorrelation in logit models. I've seen a lot of different models attempt to address all of the issues with spatial logit models (here is a good summary, though it doesn't include the ngspatial autologistic model discussed below). I am aware of 5 models available in R. My data is areal, not points, so some of the models aren't applicable (spatstat::slrm). I am having trouble figuring out which is most appropriate in my case. I have found three separate methods, implemented in R. These are the packages and the best I can understand as their differences.

The basic spatial model is:

$Y_i=\rho WY_j+X_i\beta+\epsilon_i$

Where $X_i$ is a vector of covariates for unit i, $\beta$ their coefficients, $\epsilon$ the error term, W a contiguity matrix of weights for neighboring units (W=1 if j is a neighbor, 0 otherwise), $Y_j$ being the value of Y for the neighboring unit. When applied to dichotomous variables, the idea is similar, except $Y_i$ is the propensity to have the outcome=1 relative to 0. Similarly, $Y_j$ is the propensity of the neighbors. The biggest challenge with logistic modeling of spatial data seems to be the intractability of calculating integrals over all of the random effects in maximum likelihood estimation, which in a linear model isn't necessary.

• splogit in the McSpatial package. Implements the spatial logit model from Klier-McMillen (2008). This uses a GMM estimator, where an instrument is used for WY, based on the predicted value of regressing WY on a set of instruments, Z. The GMM estimator minimizes $(y-p)'Z(Z'Z)^{-1}Z'(y-p)$, and is done in two stages: first estimate the standard logit model, calculate gradiant terms, and regress those gradiant terms on the instruments Z. The coefficients from the second model become the estimates for $\beta$ and $\rho$.
• autologistic in the ngspatial package. Known as a Centered Autologistic Model, authored by John Hughes, based on a model by Caragea and Kaiser (2009) (see here). Uses maximum pseudolikelihood estimation or MCMC for Bayesian inference.
• geoRglm - uses Bayesian/MCMC approach
• spGLM in the spBayes package. Also uses Bayesian/MCMC approach.

I am a novice in spatial statistics - I understand the principles, but this is my first time modeling spatial data. Some of these models take a long time to converge, which is part of the challenge I'm having in working them out. I was able to run my model with ngsptial and splogit, and the results were very different which made me nervous to dive in without really understanding the differences between the various approaches.

My scenario: I have data from a foundation that funds organizations addressing poverty, education gaps, and health needs. I want to identify whether the presence of a grantee in a county correlates with the level of economic, educational, and health needs in the county. I have county level data for all counties in the continental US, for a number of indicators of need, and a 0/1 indicator for absence/presence of a grantee (which is the dependent variable). Moran's I shows significant spatial correlation, and it makes theoretical sense: grantees tend to cluster together spatially for a number of reasons. I want to account for this correlation in the modeling.