Running Sum Statistic

The statistic is computed by walking down a gene list L of N genes, ranked by differential expression (e.g., signal-to-noise ratio). At each position j, the running sum is updated:

• Hit: If gene j is in the target set S, the sum increases by a weighted amount proportional to the gene's correlation with the phenotype.
• Miss: If gene j is not in S, the sum decreases by a constant penalty. This creates a random walk-like trajectory that reflects the distribution of S within L.

The standard GSEA implementation uses a weighted running sum where the increment for a hit is proportional to the gene's ranking metric (e.g., absolute correlation). This gives more influence to genes at the extremes of the ranked list.

• Weighted (p=1): Increment = |r_j|^p, where r_j is the correlation. Genes with stronger differential expression contribute more to the enrichment signal.
• Unweighted (p=0): Increment = 1 for all hits. All members of S are treated equally, regardless of their individual effect size. The choice of weighting scheme affects sensitivity to leading-edge subsets.

The Enrichment Score (ES) is the maximum deviation from zero of the running sum statistic. It is calculated as:

• Track the running sum across all N positions.
• Identify the point where the sum reaches its peak positive or peak negative value.
• This peak is the ES, reflecting the degree of overrepresentation at the top (positive ES) or bottom (negative ES) of the ranked list. The ES is conceptually similar to a weighted Kolmogorov-Smirnov statistic, testing whether the distribution of S differs from a uniform distribution across L.

To assess statistical significance, the observed ES is compared against a null distribution generated by permutation testing:

• Phenotype permutation: Randomly shuffle sample labels, re-rank genes, and recompute the ES thousands of times. This preserves gene-gene correlations.
• Gene set permutation: Randomly select gene sets of the same size as S from the background. This is computationally faster but assumes gene independence. The nominal p-value is the fraction of permuted ES values exceeding the observed ES. Phenotype permutation is preferred for preserving the correlation structure of expression data.

The running sum trajectory directly identifies the leading-edge subset—the core group of genes driving the enrichment signal:

• These are the genes in S that appear before or at the point where the running sum reaches its maximum deviation.
• The leading-edge subset represents the most biologically relevant members of the gene set, often forming a signaling cascade or protein complex.
• This subset can be extracted for downstream analysis, such as identifying therapeutic targets or constructing diagnostic signatures.

The raw ES is sensitive to gene set size and dataset correlation structure, making direct comparison across gene sets invalid. The Normalized Enrichment Score (NES) corrects for this:

• For each gene set, the observed ES is divided by the mean of all positive (or negative) permuted ES values for sets of that size.
• This normalization accounts for the fact that larger gene sets tend to have smaller maximum deviations.
• NES values enable comparative ranking of enrichment results across multiple gene sets, with |NES| > 1 typically indicating significant enrichment.

The maximum deviation from zero achieved by the running sum statistic during a GSEA walk. It quantifies the degree to which a gene set is overrepresented at the extremes (top or bottom) of a ranked list of genes.

• Calculated as the peak of the cumulative sum
• Positive ES: enrichment at the top of the ranked list
• Negative ES: enrichment at the bottom
• Magnitude reflects the strength of the correlation

The core group of genes within an enriched gene set that drives the enrichment signal. These genes appear clustered at the extreme end of the ranked list before the running sum reaches its maximum deviation.

• Represents the most biologically relevant members
• Used to refine gene sets and identify key drivers
• Can be extracted for downstream network analysis
• Often visualized in Enrichment Maps as overlapping nodes

A resampling strategy used to estimate the statistical significance of the Enrichment Score. Sample labels are randomly shuffled to generate a null distribution of ES values against which the observed ES is compared.

• Preserves the gene-gene correlation structure within the dataset
• Contrasts with gene permutation, which assumes independence
• Computationally intensive but statistically robust
• Essential for calculating nominal p-values and FDR

An Enrichment Score that has been corrected for gene set size and dataset-specific correlation structure. Normalization enables direct comparison of enrichment results across multiple gene sets of varying sizes.

• Accounts for the fact that larger sets tend to have higher ES
• Calculated by dividing ES by the mean of all permuted ES values
• NES > 1.5 typically considered significant
• The primary metric for ranking enriched pathways

A curated or computationally derived library of gene sets stored in GMT file format. Each set represents a shared biological feature, pathway, or regulatory motif used as input for the running sum calculation.

• MSigDB Hallmark: 50 refined biological states
• GO: Biological Process, Molecular Function, Cellular Component
• KEGG: Manually curated metabolic and signaling pathways
• Reactome: Detailed molecular interaction networks

The expected proportion of false positives among all gene sets declared significant. In GSEA, FDR is estimated by comparing the observed NES distribution against the null distribution from phenotype permutations.

• Controls for multiple hypothesis testing across thousands of gene sets
• FDR < 0.25 is the standard GSEA threshold
• More permissive than Bonferroni correction
• Balances sensitivity and specificity in exploratory analyses