When the average hides the story: Moran's I and its local map

Video lesson · 01:45 Separate “is there clustering?” from “where is it?”

Vera begins with whether nearby values look alike, then separates the questions answered by global Moran's I and local LISA. The video connects the global statistic to the local map; the article handles significance, multiple testing, and interpretive limits.

The video distinguishes a global clustering test from local pattern location; this article remains the reference for the formulas, inference conditions, multiple-testing correction, and causal limits.

Start with the global question: are nearby values alike? Global Moran’s I answers it with one number — essentially the correlation between each unit’s (mean-centred) value $z_i$ and the average of its neighbours, $Wz_i$:

$$I = \frac{n}{S_0}\frac{\sum_i\sum_j w_{ij}z_i z_j}{\sum_i z_i^2},\qquad S_0 = \sum_i\sum_j w_{ij}$$

With a row-standardised $W$ this simplifies to $I = \frac{z^{\mathsf T}Wz}{z^{\mathsf T}z}$. Positive $I$ means like clusters with like (hot and cold zones); negative $I$ means a checkerboard of highs beside lows.

Two things trip people up on the inference:

• Under the null of spatial randomness, $\operatorname{E}[I] = -1/(n-1)$ — slightly negative, not zero (it tends to 0 only as $n$ grows). Compare against that, not against 0.
• Significance can come from a closed-form (normal / randomisation) variance, or from permutation: shuffle the values across locations many times and build an empirical null distribution. Permutation is safer when normality is doubtful.

But a global I has a blind spot: it is an average. One number can hide a map where clustering is fierce in one district and absent in another — or where two opposite patterns cancel out. That is what the local version is for.

LISA (Local Indicator of Spatial Association; Anselin, 1995) decomposes the global statistic to one value per unit:

$$I_i = z_i\sum_j w_{ij}z_j$$

and classifies each location by the sign of its own value against its neighbours’ average:

type	unit	neighbours	reading
HH	high	high	hot spot
LL	low	low	cold spot
HL	high	low	a high outlier in a low area
LH	low	high	a low outlier in a high area

HH/LL are clusters (positive local association); HL/LH are spatial outliers (negative). The local pieces sum back to the global $I$ — it is an exact decomposition.

The trap that ruins LISA maps: one hypothesis test per location means thousands of simultaneous tests, and uncorrected pseudo p-values scatter false “significant” clusters everywhere. Apply a multiple-comparison correction (FDR / Benjamini–Hochberg, or the conditional adjustments Anselin discusses) before you believe a coloured map. And, as always, the whole picture depends on $W$ — change the neighbour definition and the hot spots can move.

One last caution worth stating: if you compute Moran’s I on a variable that has already been spatially smoothed — for instance, coefficients estimated with overlapping kernels — part of the autocorrelation you measure is a mechanical artefact of the smoothing, not evidence of a real spatial process. Know what produced your variable before you celebrate its clustering.