I have calculated Moran's I in R and reject the null hypothesis in both data sets, there is evidence of strong spatial autocorrelation.

However, how do I compare the two?

First set:

Moran I statistic standard deviate = 9.7817, p-value < 2.2e-16

alternative hypothesis: greater

sample estimates:

Moran I statistic: 0.1736815445

Expectation: -0.0009970090

Variance: 0.0003188969

Second set:

Moran I statistic

standard deviate = 18.575, p-value < 2.2e-16

alternative hypothesis: greater

sample estimates:

Moran I statistic: 0.3307119647

Expectation: -0.0009970090

Variance: 0.0003189016

Both indicate a strong spatial autocorrelation, however, how can I precisely define between the two which one is stronger?

Is it simply by looking at which Moran's I value is greater?

Both have the same boundary (shapefile) and are in the same study area, but different time stamps.

1 Answer 1


Yes. If I_A > I_B for two data sets A and B, then there's greater spatial autocorrelation, where spatial autocorrelation is defined by the formula for the Moran I (other measures of spatial autocorrelation exist and may give different results). In short, neighbours of region i are more similar to i in data set A than in B, averaged all over.

Usually the next question asked is "But is it statistically significantly different?". That would require a statistical model for uncertainty in Moran's I, which I think can be computed under the assumption of no autocorrelation, but you've already rejected that for each data set. Moran's I is a statistic, not a random variable.

You could make it a random variable if you've got another hypothesis to test - for example suppose your measurements have a 5% uncertainty in them. That might not be so much that even at the extremes it affects the significance of the Moran's I test, but it might cause a difference in I_A and I_B to be down to the noise in the measurement. You could simulate 1000 noisy versions of A and B, compute 1000 I_A and I_B values and hence 1000 differences, then see where your actual difference ranks among those 1000 - this would be a Monte Carlo test of significance of the difference in spatial autocorrelation under the hypothesis of 5% noise in the measurements.

  • Excellent. Thanks for your thorough response.
    – perrier
    Commented Dec 5, 2019 at 21:54

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