# Choosing value of Moran's I to say existence of spatial correlation?

I am a GIS beginner. Usually we use Moran's I to evaluate spatial autocorrelation. Everyone knows the Moran's I coefficient ranges from -1 (uniform) to 1 (clustering); `I = 0` means no correlation.

If `I = 0.2, 0.3, 0.4` (suppose `I` is statistically significant; i.e. p-value is small, like `p < 0.05`), do we say there exists spatial correlation?

How large I coefficient we claim `strong` spatial correlation empirically?

Or we should proceed to run a spatial regression model, and then observe how significant on lag y or lag error coefficients to make conclusion?

• Well, if you think back to your course work, would this not depend on a p or z value? Without a significance test, the question is somewhat irrelevant. – Jeffrey Evans Jan 24 '18 at 2:49
• @Jeffery I slightly revised the statement. Suppose the p-value is small enough to reject null hypothesis. Thanks. – Hsiang Hung Jan 24 '18 at 21:03

I myself am still learning as much as I can about Moran's I, but I think I help figure out the answer to this question. There is a great video on coursera about spatial correlation:

Based on the Z-score, a statistical test is feasible to check if a given variable is spatially autocorrelated or not. The statistical test can be formulated like this, Null hypothesis, H0, is spatial autocorrelation does not exist. Alternative hypothesis, H1, is spatial autocorrelation exist. The Z-score is the test statistic. And dependent on the value of Z-score, we can either accept H0, null hypothesis, or reject H0. For example, Z-score is bigger than 1.96, then you can say at the confidence level of 95 percent, this variable has a positive spatial autocorrelation. Or if the value of Z-score is a smaller than 1.96, then you can say, at the confidence level of 95 percent, the null hypothesis is accepted, meaning that no spatial autocorrelation exists.

So like Frank mentioned you need to calculate a Z-score. Now to calculate the Z-score you need the mean which for Moran's I `-1/(N -1)` where N is the number of samples. This number serves as a baseline for what your correlation values should be like.

From what I have read about spatial correlation generally most people either choose p-value of .10 or .05 to say that the autocorrelation is statistically significant. In the quote above the professor considers using a p-value of .05 for statistical significance, while in ARGIS's documentation you will find they use a p-value of .10.

Because this is slightly subjective, I have reproduced a more detailed table for Z-scores to P-values to Confidence Intervals for the Z-test.

Here is a brief table for Z-score assuming its just the basic Z-test:

``````+---------------------+------------------+------------------+---------+
| Confidence Interval | Positive Z-Score | Negative Z-Score | Pvalue  |
+---------------------+------------------+------------------+---------+
| 99.9%               |             3.27 |            -3.27 |   0.001 |
| 99.73%              |             3.00 |            -3.00 |   0.020 |
| 99%                 |            2.576 |           -2.576 |   0.010 |
| 98%                 |            2.326 |           -2.326 |   0.020 |
| 95.45%              |             2.00 |           -2.000 |   0.046 |
| 95%                 |             1.96 |            -1.96 |   0.050 |
| 90%                 |            1.645 |           -1.645 |   0.100 |
+---------------------+------------------+------------------+---------+
``````

P.S. I also learned a little myself, as I thought that strength rules for spatial correlation matched the strength rules for correlation (I >.8 being the very strong relationship and .6 < weak relationship ). Though Moran's I is a weighted Pearson correlation, it not true that you can interpret the values similar to regular correlations when you compare. Like Jeffery Evans mentioned, you need to consider the p and z-values to test statistical significance to really interpret the spatial autocorrelation because tails represent a different spatial process (vs. regular correlation). According to Yanguang Chen spatial auto-correlation is only one piece figuring the spatial relationship between two variables, you need to consider the spatial cross-correlation. In fact, the Pearson Correlation between any two spatial variables is the combination of the direct correlation and this spatial-cross correlation.

• Common statistical practice would evaluate statistical significance at a minimum of the 95% confidence or p=0.05 which gives you a +/- critical z-value of 1.96. I would not go with ESRI's recommendation of p=0.10. As far as the "strengths of correlation" one should keep in mind that Moran's-I is really nothing more than a spatially weighted Pearson's correlation. There is some interesting work by Chen (2015) showing the indexes local and global decomposability into an eigenvector. – Jeffrey Evans Apr 19 '18 at 15:50
• Hrmm... Wait... was I then right the scale or "strength of correlation" for spatial was similar to how we determine the strength or magnitude for Pearson correlations? I thought I might have been wrong. Also do you have the link by Chen (2015) or his first name? It might be interesting to add. – mlane Apr 19 '18 at 16:08
• I have these cross-correlation methods available in R via the crossCorrelation function in the spatialEco package. Here is the citation: Chen., Y. (2015) A New Methodology of Spatial Cross-Correlation Analysis. PLoS One 10(5):e0126158. doi:10.1371/journal.pone.0126158 – Jeffrey Evans Apr 19 '18 at 16:14
• I think I'd rather see a variogram than a Moran's I statistic. – Spacedman Apr 19 '18 at 16:14
• @Spacedman, indeed! Although, I really like the Borcard & Legendre "principal coordinates of neighbour matrices" method as well. – Jeffrey Evans Apr 19 '18 at 16:17