Moran's I — Global Spatial Autocorrelation Index
Moran's Index of Spatial Autocorrelation · Also known as: Moran's I statistic, global Moran's I, spatial autocorrelation index, Moran index
Moran's I is the standard global statistic for detecting spatial autocorrelation: whether nearby locations tend to share similar values. The index ranges from approximately −1 (perfect dispersion) through 0 (spatial randomness) to +1 (perfect clustering), allowing researchers to test whether a geographic pattern differs from complete spatial randomness with a single, interpretable number.
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When to use it
Use Moran's I as a first diagnostic step whenever you have georeferenced data and want to test whether the spatial pattern is random before proceeding to spatial regression or interpolation. It is appropriate for continuous or count outcomes mapped to areal units or point locations. Do not use it when locations are not geographically meaningful, when the spatial weights matrix cannot be justified theoretically, or when you need to identify where clustering occurs rather than whether it exists overall — for local detection, use Local Moran's I (LISA) instead.
Strengths & limitations
- Provides a single, interpretable global summary of the degree of spatial clustering or dispersion.
- Well-established theoretical framework with known analytical sampling distribution under normality and randomisation assumptions.
- Permutation-based inference is distribution-free and robust to non-normal data.
- Widely implemented in GeoDa, R (spdep), Python (PySAL), and ArcGIS, making replication straightforward.
- Directly indicates whether spatial autocorrelation is present, guiding the choice between OLS and spatial regression models.
- Returns a single global value; spatially heterogeneous clustering (some areas cluster, others do not) is masked.
- Sensitive to the choice of spatial weights matrix — different neighbourhood definitions can yield different conclusions.
- Assumes stationarity: the same autocorrelation process is assumed to operate everywhere in the study region.
- Edge effects can bias results for observations at the boundary of the study area.
Frequently asked
What does a Moran's I value of 0 mean?
A value equal to E[I] ≈ −1/(n−1) (approximately 0 for large samples) indicates that the observed spatial pattern is consistent with complete spatial randomness. Positive values above this baseline indicate clustering; negative values indicate dispersion.
How do I choose the spatial weights matrix?
The weights matrix should reflect the theoretical mechanism of spatial interaction in your study (e.g., shared borders for administrative data, inverse distance for diffusion processes, k-nearest neighbours for point data). The choice must be justified before analysis, not selected to maximise Moran's I.
Should I use the analytical or permutation test?
Permutation testing is generally preferred because it makes no distributional assumption about the underlying variable. It is especially important when data are skewed, count-based, or have a small n. The analytical z-test is acceptable for large samples with approximately normal residuals.
If Moran's I is significant, must I use spatial regression?
A significant Moran's I on the raw variable does not necessarily require spatial regression — what matters is whether the OLS residuals are spatially autocorrelated. Run Moran's I on OLS residuals (and examine Lagrange Multiplier tests) to decide between a spatial lag model, spatial error model, or OLS.
What is the difference between Moran's I and Geary's C?
Both measure global spatial autocorrelation, but Moran's I is based on cross-products of deviations from the mean (making it more sensitive to global trends), while Geary's C uses squared differences between neighbouring pairs (making it more sensitive to local dissimilarities). Moran's I is more widely used and more directly comparable across studies.
Sources
- Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1/2), 17–23. DOI: 10.2307/2332142 ↗
- Cliff, A. D., & Ord, J. K. (1981). Spatial Processes: Models and Applications. Pion. ISBN: 9780850860818
How to cite this page
ScholarGate. (2026, June 3). Moran's Index of Spatial Autocorrelation. ScholarGate. https://scholargate.app/en/spatial-analysis/morans-i
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Geary's CSpatial analysis↔ compare
- Geographically Weighted RegressionSpatial analysis↔ compare
- Local Getis-Ord Gi*Spatial analysis↔ compare
- Local Indicators of Spatial AssociationSpatial analysis↔ compare
- Local Moran's ISpatial analysis↔ compare
- Spatial AutocorrelationSpatial analysis↔ compare