I'm trying to determine whether there is spatial autocorrelation in a set of data. I am using Moran's I statistic to test for spatial autocorrelation.

The data was point data (instances of an event at specific addresses) that has been aggregated into postal areas. For privacy reasons I cannot get access to the original address data, so have to use the aggregated data.

How do I test for spatial autocorrelation in the underlying data if the data is skewed?

I am asking either:

Are 'events' likely to be near other events or is their distribution random? Or, are postal areas that have many 'events' likely to be nearby to other postal areas that have many 'events'? The issue that I have is that the data is skewed toward zero (Right skewed? Sorry, I don't know what the correct name for this kind of distribution is). In other words, most postal areas have zero 'events' and with decreasing numbers of postal areas the further you get away from zero, with some extreme outliers having hundreds of events. Every example I have read about that uses Moran's has normally distributed data. For example, mean income of census tracts, which is never zero but is normally distributed around some value.

Does this matter in my dataset?

So, will it bias the result?

And if so, how can I correct for it, or is there an alternative test for spatial autocorrelation in this kind of dataset?

What I have done so far:

plotted the data on a Choropleth map. This seems to indicate clustering, with most postal areas that have a non-zero value being clustering an a few regions. Additionally, the fact that there are extreme outliers in the hundreds implies that there is some level of clustering in the underlying data. I have tried to run a Moran's I test using both R and ArcGIS (using both inverse distance and contiguity edges spatial conceptualization). Results give a low p-value, but also a low index value (around 0.1).

EDIT: The data I am using is Intellectual Property filings. As Samuel pointed out, the fact that population is not uniform across all postal areas this will likely bias the analysis. The reason this is interesting is because for this particular dataset the clusters don't appear to be in the large city centers, which is where the largest populations are. Is there a way to correct for population density? Does Moran's work for a rate? So if I corrected my data so it shows filings per capita?


1 Answer 1


Moran's I allows you to test for spatial autocorrelation on data that is expected to be uniform over the study area.

In your case, since the number of addresses is not uniform over the postal areas, I would expect that Moran's I detects autocorrelation no matter what you do. If you corrected the area count for the number of addresses in each postal area, you would test for autocorrelation in the event per address rate, which it is relatively safe to assume is uniform (but this depends on your problem) and this would give a more solid answer.

EDIT: I'll just post the formula for Moran's I and then talk you through it. Moran's I is given by $$I=\frac{n}{\sum_{i=1}^{n}(y_i-\bar{y})^2}\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(y_i - \bar{y})(y_j-\bar{y})}{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}}$$ where $n$ is the number of postal zones on your case. The part that is interesting is the top part of the second fraction, $$\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(y_i - \bar{y})(y_j-\bar{y})$$ which is the covariance between the measurement in zone, $i$ and$j$ given by $y_i$ and $y_j$ weighted by whatever weight $w_{ij}$ you choose between said zones.

First off, Moran's I it does not depend on $y_i, y_j$ being an integer, it will work on rates. Second, if you want to test the impact of various covariates in a rigorous way, you will need to set up a spatial regression model to account for how this happens. Your case, filing rate per capita, seems to be analogous to the standard Scottish lip-cancer case, perhaps you can find what you need in this link.

  • $\begingroup$ Thanks. I edited the question to explain the data I am using. Does Moran's work for a rate (in this case filings per capita)? Also, the data would still be skewed. Since most postal areas have zero filings, the rate for those areas would also be zero. $\endgroup$ Feb 18, 2021 at 21:41

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