Spatial Autocorrelation II
Content for Tuesday, November 18, 2025
Throughout this class, we’ve talked about the tendency for things that are closer together to be more similar (i.e., spatial autocorrelation). As a starting assumption, this can be really useful for improving our ability to interpolate missing values and to describe spatial trends in our data. Often times, however, our goal is to identify some effect(s) that explain why an outcome is what it is. Identifying causal connections between independent and dependent variables is often the task of statistical models. Unfortunately, many of our statistical models assume that the residuals (i.e., the difference between the modeled output and the measured output) are independent and normally distributed. When the process we’re interested in is autocorrelated AND we’ve failed to capture the independent variables that explain that spatial trend, we end up with spatial autocorrelation in the residuals which violates the assumptions of traditional statistcal models. Today, we’ll talk about methods for incorporating the potential for spatial autocorrelation into your statistical models explicitly.
Resources
Big Picture
Spatial autoregressive models for statistical inference from ecological data by (Ver Hoef et al. 2018) contains the justification for and derivation of autoregressive models for spatial data. It’s a dense read, but one that’s worth sticking with. I learn something new each time I read this paper.
Improving inferences about private land conservation by accounting for incomplete reporting by (Williamson et al. 2021) uses CAR models with occupancy models to explore how inference changes when including different elements of the data generating process. It’s got a more approachable description of CAR models than that of Ver Hoef and was based on a simulation study that I did during my PhD.
Technical Details
Spatial regression models provides a nice overview of using simultaneous autoregressive models using the
spatialregpackage.Spatial Regression in (Pebesma and Bivand 2023) gives examples of spatial regression using a variety of Bayesian packages. We won’t be using these (we’ll use
brms) pacakages, but the data prep is largely the same and this demonstrates a bit of the underlying logic behind the different models.
Objectives
By the end of today you should be able to:
Explain the importance of explicitly incorporating autocorrelation into statistical models
Describe autoregressive approaches for including spatial correlation in statistical models
Explore ways to define adjacency matrices based on areal and continuous data.