Local Inverse Simpson's Index (LISI)

LISI is derived from the Inverse Simpson's Index, an ecological diversity metric. For a given cell, it computes the inverse of the sum of squared proportions of each category (e.g., batch label) among its k-nearest neighbors. A score of N indicates that the local neighborhood is as diverse as perfectly mixing N different categories. The formula is: 1 / Σ p_i², where p_i is the proportion of neighbors belonging to category i.

LISI is deployed as two complementary metrics to diagnose integration quality:

• iLISI (integration LISI): Measures the effective number of batches in a cell's neighborhood. A high iLISI score indicates successful batch mixing. The ideal value approaches the total number of batches in the experiment.
• cLISI (cell-type LISI): Measures the effective number of cell types in a cell's neighborhood. A high cLISI score indicates that distinct cell types remain separated. The ideal value is 1.0, representing a pure, unmixed cell-type neighborhood.

The metric operates on a k-nearest neighbor (k-NN) graph constructed in a shared, integrated embedding space (e.g., PCA or Harmony output). The choice of k (number of neighbors) is a critical parameter:

• Small k: Highly sensitive to local mixing, penalizing even small batch-specific clusters.
• Large k: Evaluates mixing over a broader region, providing a more global assessment. A typical default is k = 90, but this should be tuned based on dataset size and expected cluster granularity.

LISI provides a per-cell score distribution, enabling fine-grained diagnostics:

• Uniformly high iLISI: Indicates successful batch integration across the entire manifold.
• Bimodal iLISI distribution: Reveals that some cell types integrated well while others retained strong batch effects, a common failure mode masked by global metrics.
• High cLISI for a known cell type: Signals overcorrection, where the algorithm blended a distinct biological population with a dissimilar one, destroying true biological signal.

LISI complements other batch correction metrics by focusing on effective diversity rather than statistical acceptance rates or silhouette widths:

• kBET: Performs a chi-squared test for local batch label distribution. It is a binary accept/reject test, while LISI provides a continuous, interpretable diversity score.
• Average Silhouette Width (ASW): Measures cluster cohesion and separation. ASW requires pre-defined cluster labels, whereas LISI operates directly on batch and cell-type annotations without assuming discrete clusters. LISI is less sensitive to neighborhood size than kBET and more directly interpretable as a mixing coefficient.

LISI is implemented in the lisi R package and is natively integrated into the Seurat and Harmony single-cell analysis workflows. The function compute_lisi() takes a reduced-dimension embedding matrix (e.g., PCA coordinates) and a metadata vector of categorical labels. It returns a per-cell score vector. For large datasets (>100k cells), approximate nearest neighbor methods are recommended to maintain computational feasibility, as the naive k-NN construction scales quadratically.

A quantitative metric that evaluates the degree of batch mixing by comparing the local batch label distribution in a k-nearest neighbor graph to the global batch distribution. For each cell's neighborhood, a chi-squared test determines if the batch composition deviates from the overall dataset. A perfect mix yields an acceptance rate near 1.0, while a low acceptance rate indicates residual batch structure. Unlike LISI, which provides a continuous diversity score, kBET offers a binary accept/reject decision per neighborhood, making it a stringent test for local batch effects.

A metric that measures the cohesion of cell-type clusters against the separation of batch labels to evaluate integration quality. The silhouette score ranges from -1 to 1, where:

• Cell-type ASW: High values indicate tight, well-separated biological clusters
• Batch ASW: Values near 0 indicate batches are well-mixed and indistinguishable

A successful integration shows high cell-type ASW and low batch ASW. The batch silhouette score is often computed as 1 - |batch ASW| to create a 0-1 scale where 1 represents perfect mixing.

An information-theoretic metric that quantifies the randomness of batch labels within a defined local neighborhood of cells. High entropy indicates a well-mixed, successfully integrated dataset where no single batch dominates any local region. This metric is closely related to LISI but uses Shannon entropy rather than the inverse Simpson index. While LISI reports the effective number of batches, entropy reports the uncertainty in batch identity, with maximum entropy achieved when all batches are equally represented in a neighborhood.

A kernel-based statistical test used as a loss function in deep learning to measure the distance between two probability distributions. In batch correction, MMD quantifies the divergence between the latent feature distributions of different batches. A low MMD value indicates that the distributions have been successfully aligned. Unlike LISI, which operates on discrete neighborhood compositions, MMD compares the continuous distributions of the full embedding space, making it a complementary global assessment of distributional alignment.

The process of evaluating whether a batch correction method has removed true biological variation alongside technical noise. Key indicators of overcorrection include:

• Loss of known cell-type clusters that were present before correction
• Reduced variance explained by biological covariates
• Homogenization of distinct biological states

LISI's cLISI variant (cell-type LISI) directly measures this by quantifying the effective number of cell types in each neighborhood. A low cLISI after integration suggests overcorrection has blended distinct biological populations.

A critical experimental design flaw where the batch variable is perfectly correlated with the biological condition of interest. For example, if all control samples are processed on Monday and all treatment samples on Tuesday, it becomes statistically impossible to separate technical artifacts from true biological signal. In such cases, even perfect LISI scores (iLISI near the total number of batches) are meaningless because the batch effect and biological effect are indistinguishable. Proper experimental design with balanced batch allocation is essential for valid integration assessment.