Moran's I

Moran's I is defined as a weighted cross-product of deviations from the mean, normalized by the variance. The formula is:

I = (N / W) * (Σᵢ Σⱼ wᵢⱼ (xᵢ - x̄)(xⱼ - x̄)) / (Σᵢ (xᵢ - x̄)²)

• N: Number of spatial units (spots or cells)
• W: Sum of all spatial weights
• wᵢⱼ: Spatial weight between locations i and j
• xᵢ: Gene expression value at location i
• x̄: Mean expression across all locations

The statistic produces a value typically ranging from -1 (perfect dispersion) to +1 (perfect clustering), with 0 indicating random spatial distribution.

The output of Moran's I provides immediate insight into tissue organization:

• I > 0 (Positive Autocorrelation): Nearby locations have similar expression levels. This indicates spatial clustering—common for cell-type-specific marker genes confined to anatomical regions.
• I ≈ 0 (No Autocorrelation): Expression is randomly distributed across the tissue. Typical for housekeeping genes with uniform expression.
• I < 0 (Negative Autocorrelation): Nearby locations have dissimilar expression. This indicates spatial dispersion or a chessboard-like pattern, often seen in genes with mutually exclusive expression domains.

The magnitude indicates the strength of the pattern, while the sign indicates the direction.

The spatial weights matrix (W) defines what 'nearby' means and is the most critical user-defined parameter:

• Distance-based weights: wᵢⱼ = 1 if distance ≤ threshold, else 0. Requires choosing a biologically meaningful radius.
• K-Nearest Neighbors (KNN): wᵢⱼ = 1 if j is one of the k nearest neighbors of i. Adapts to local spot density.
• Inverse distance weights: wᵢⱼ = 1 / dᵢⱼᵖ. Nearby points have stronger influence; distant points have weaker influence.
• Binary contiguity: wᵢⱼ = 1 if spots share a border. Common for grid-based spatial transcriptomics platforms like Visium.

The choice of weights matrix directly impacts the Moran's I value and must be reported for reproducibility.

A raw Moran's I value alone is insufficient; its statistical significance must be assessed against a null hypothesis of spatial randomness:

• Permutation test: Randomly shuffle gene expression values across spatial locations and recalculate Moran's I. Repeat 999+ times to generate a null distribution.
• Z-score: z = (I_observed - E[I]) / √Var[I]. Measures how many standard deviations the observed value is from the expected value under randomness.
• P-value: The proportion of permuted I values as extreme or more extreme than the observed I.
• Multiple testing correction: When testing thousands of genes, apply Benjamini-Hochberg or Bonferroni correction to control false discovery rate.

Only genes with an adjusted p-value < 0.05 are considered spatially variable genes (SVGs).

Moran's I exists in two complementary forms for spatial transcriptomics analysis:

• Global Moran's I: A single summary statistic for the entire tissue. Answers: 'Is this gene spatially autocorrelated anywhere in the tissue?' Best for initial screening of spatially variable genes.
• Local Indicators of Spatial Association (LISA): Decomposes the global statistic into a value for each spatial location. Answers: 'Where exactly are the clusters and outliers?' Identifies:
  • High-High clusters: Hot spots of high expression
  • Low-Low clusters: Cold spots of low expression
  • High-Low outliers: A high-value spot surrounded by low values
  • Low-High outliers: A low-value spot surrounded by high values

LISA is essential for identifying anatomical boundaries and transition zones.

Moran's I has important limitations that must be considered in spatial transcriptomics:

• Stationarity assumption: Assumes the spatial process is constant across the tissue. Violated in complex tissues with multiple distinct regions.
• Sensitivity to scale: Results depend heavily on the chosen distance threshold or number of neighbors. Always test multiple scales.
• Global summary masking: A global Moran's I near zero can occur when a gene has both clustered and dispersed regions that cancel each other out.
• Spot-based data: For Visium data (55μm spots), each spot contains multiple cells. Moran's I measures spot-level, not single-cell, autocorrelation.
• Not a causal measure: High Moran's I does not imply that spatial proximity causes the expression pattern—only that a pattern exists.

Complement with Ripley's K function or SPARK for multi-scale or model-based alternatives.

The foundational concept measured by Moran's I. It quantifies the degree to which a variable's values at nearby locations are more similar than expected by random chance. In spatial transcriptomics, positive spatial autocorrelation indicates that a gene's expression is clustered in specific tissue regions, while negative autocorrelation suggests a dispersed, checkerboard-like pattern. This principle underpins the identification of spatially variable genes (SVGs) and the definition of tissue microenvironments.

A complementary spatial autocorrelation statistic that is more sensitive to local differences between neighboring observations than Moran's I. While Moran's I focuses on covariance from the global mean, Geary's C uses squared differences between adjacent values, making it particularly effective at detecting sharp boundaries or edges between tissue domains. A value of C < 1 indicates positive autocorrelation, C = 1 suggests randomness, and C > 1 indicates negative autocorrelation. Often used alongside Moran's I for a more complete spatial characterization.

A decomposition of global Moran's I into location-specific statistics that identify hot spots, cold spots, and spatial outliers within a tissue. LISA maps visualize where statistically significant clustering of high or low gene expression occurs, enabling researchers to pinpoint the exact anatomical regions driving global autocorrelation patterns. This is essential for discovering spatially variable genes that define specific histological structures or pathological lesions.

The data structure that defines spatial weights for Moran's I calculation. It represents each cell or spot as a node, with edges connecting k-nearest neighbors or locations within a distance threshold. The choice of graph construction—such as Delaunay triangulation for single-cell data or grid-based adjacency for Visium spots—directly impacts the autocorrelation statistic. Proper graph definition is critical for accurately modeling the spatial relationships in heterogeneous tissues.

A non-parametric statistical method for assessing the significance of an observed Moran's I value. It works by randomly shuffling gene expression values across spatial locations to generate a null distribution of autocorrelation statistics under the assumption of complete spatial randomness. The observed I is then compared against this distribution to compute an empirical p-value. This approach avoids parametric assumptions about the data's underlying distribution and is standard practice in spatial transcriptomics packages like Squidpy and Giotto.

A multi-scale spatial point pattern analysis tool that evaluates clustering or dispersion across a range of distances, rather than a single neighborhood scale like Moran's I. It counts the number of neighboring points within concentric circles of increasing radius around each point. In spatial transcriptomics, it is used to determine whether specific cell types are randomly distributed, clustered, or regularly spaced at different length scales, revealing hierarchical tissue organization.