# Statistic for measuring (local) autocorrelation of ordinal data

I have a spatial data set with a bunch of variables in nine-level ordinal scale.

I want to calculate (local) spatial autocorrelation for each of these ordinal scale variables.

I've been using `spdep`-package for R. Join count -statistic seems to be the way to go for categorical data, but I'm unsure whether it works with multi-level and ordinal data.

What is the proper way to measure local autocorrelation or clustering of ordinal data?

In terms of a local autocorrelation (nonstationarity) statistics, there really is not one. Join-counts is adequate for hypothesis testing of global clustering in binary process, albeit very scale dependent, but not for multinomial data. I am not even sure what the underlying hypothesis test would be here, especially with ordinal data. One has to ask, how would you structure the ordered nature of ordinal data in quantifying and testing for spatial process.

Other than testing each level in a separate model, I guess for ordinal process, theoretically you could numerically code the data, following the ordered variable (eg., low=1, med=2, high=3) and then use a LISA statistic. This would at least reveal patterns of low/high, high/low, etc... juxtapositions but, I do not see this as entirely valid. Although, this could be an interesting exploratory step. Just remember that Moran's-I type statistics are about the central tendency (using the mean), which could prove problematic when applied to a small number of integer values.

I would recommend clarifying what you are actually trying to test for and come up with a stated hypothesis, or at least an intent. You are somewhat limited with nominal data to begin with. If your intent here is simply clustering, one option would be to investigate ScanStatistics (Kulldorff 1997). Specifically, Jung et al., (2010) quantify spatial process of multinominal and (Jung et al., 2007) ordinal data using ScanStatistics.

You can implement scan statistics in R using the rsatscan package (on CRAN) or using the standalone freeware SatScan.

References

Jung I, Kulldorff M, Richard OJ. (2010) A spatial scan statistic for multinomial data. Statistics in medicine, 29(18):1910-1918. doi:10.1002/sim.3951

Jung I, Kulldorff M, Klassen A. (2007) A spatial scan statistic for ordinal data. Statistics in Medicine, 26:1594-1607

Kulldorff M. (1997) A spatial scan statistic. Communications in Statistics: Theory and Methods, 26:1481-1496

• Thank you for the detailed answer! For now, my goal is to do an exploratory analysis and find out whether there are clustering going on. Rsatscan looks quite promising. I will try that. Commented Apr 8, 2020 at 12:53
• As an addition to this, could it be worthwhile exploring entropy based approaches such as ELSA? It is a non-parametric method of assessing spatial association for categorical data, which seems like it could be useful. Naimi, B., Hamm, N. A., Groen, T. A., Skidmore, A. K., Toxopeus, A. G., & Alibakhshi, S. (2019). ELSA: Entropy-based local indicator of spatial association. Spatial statistics, 29, 66-88 Commented May 21, 2020 at 4:33