Linkage Disequilibrium (LD) Score Regression

Estimates SNP heritability (h²) using only GWAS summary statistics and an LD reference panel, without requiring individual-level genotype data. The method regresses chi-square test statistics against LD Scores—the sum of squared correlations between a variant and all others in a window. Under a polygenic model, variants in high-LD regions tag more causal effects, producing elevated test statistics. The slope of this regression provides an estimate of heritability attributable to common SNPs, while the intercept captures confounding due to population stratification and cryptic relatedness.

The regression intercept provides a formal test for residual confounding in GWAS. An intercept significantly greater than 1.0 indicates inflation of test statistics beyond what polygenicity predicts, signaling bias from:

• Population stratification: systematic allele frequency differences across subpopulations
• Cryptic relatedness: unmodeled kinship among study participants
• Uncorrected case-control imbalance Unlike genomic control (λ), which applies a uniform correction and can over-correct polygenic traits, LD Score Regression distinguishes true polygenic signal from spurious inflation, enabling researchers to diagnose data quality issues without discarding genuine associations.

Bivariate LD Score Regression extends the framework to estimate the genetic correlation (r_g) between two complex traits using only their respective GWAS summary statistics. The method regresses the product of Z-scores from two studies against LD Scores. A genetic correlation of 1.0 indicates identical genetic architectures; 0.0 indicates no shared genetic basis. This enables systematic cross-trait analyses without requiring overlapping cohorts, revealing shared biology between, for example, schizophrenia and bipolar disorder or type 2 diabetes and body mass index.

Stratified LD Score Regression partitions heritability across functional genomic annotations to identify enrichment in specific categories. By computing separate LD Scores for each annotation—such as coding regions, conserved elements, enhancers, or cell-type-specific regulatory marks—the method quantifies the proportion of heritability attributable to each category relative to its size in the genome. Significant enrichment in a functional category suggests that causal variants concentrate in those regions, providing biological insight into disease architecture and prioritizing annotations for fine-mapping.

LD Score Regression corrects for winner's curse and LD-related attenuation bias in heritability estimates. Naive estimators that sum squared effect sizes from marginal GWAS summary statistics produce downwardly biased heritability estimates because:

• Marginal effect sizes capture only the component of a causal variant's effect not tagged by correlated neighbors
• Winner's curse inflates discovery effect sizes while leaving non-significant variants at zero By modeling the expected relationship between LD and test statistics, the regression framework recovers the total additive genetic variance explained by all common SNPs, providing unbiased estimates even when causal variants are imperfectly tagged.

Accurate LD Score computation requires a well-matched reference panel reflecting the ancestry of the GWAS sample. Key considerations include:

• Sample size: panels should contain at least several thousand individuals for stable LD estimates
• Ancestry matching: using a European reference panel for a European GWAS minimizes bias; cross-ancestry applications require careful calibration
• Variant coverage: the panel must include the variants present in the summary statistics Common reference panels include the 1000 Genomes Project and UK Biobank. Mismatched panels can introduce systematic errors in both heritability and genetic correlation estimates.

The proportion of phenotypic variance in a population attributable to the additive effects of all measured single nucleotide polymorphisms. LD Score Regression estimates this by regressing GWAS test statistics against LD Scores, leveraging the fact that a variant in high LD tags more causal variation and thus has an inflated χ² statistic under polygenicity.

• Key distinction: Differs from broad-sense heritability (H²) by excluding dominance and epistatic effects
• Intercept interpretation: An intercept > 1 indicates residual confounding from population stratification or cryptic relatedness
• Attenuation ratio: (Intercept - 1) / (Mean χ² - 1) quantifies the proportion of inflation attributable to confounding vs. true polygenic signal

A measure of the shared genetic architecture between two complex traits, quantifying the extent to which the same causal variants influence both phenotypes. LD Score Regression estimates genetic correlation using bivariate LD Score regression, which regresses the product of Z-scores from two GWAS against LD Scores.

• Range: -1 (perfect negative correlation) to +1 (perfect positive correlation)
• Distinction from phenotypic correlation: rg isolates the genetic component, unaffected by environmental confounders
• Applications: Identifying pleiotropic loci, predicting cross-trait effects of interventions, and understanding disease comorbidity

A measure of the total amount of linkage disequilibrium a particular genetic variant has with all other variants in a genomic window. Calculated as the sum of squared correlations (r²) between the variant and all neighboring variants within a specified distance.

• Computation: LD Score = Σ r²ᵢⱼ for all variants j within a window (typically 1 centimorgan)
• Reference panels: Pre-computed from large cohorts like 1000 Genomes Project for each ancestry
• Interpretation: High LD Score variants tag more of the genome, capturing more causal signal under polygenic architectures

The foundational method that regresses GWAS χ² test statistics on LD Scores to estimate SNP heritability and quantify confounding. Under a polygenic model, the expected χ² statistic for variant j is proportional to its LD Score.

Regression equation: E[χ²ⱼ] = 1 + N·(h²g/M)·ℓⱼ + a

Where:

• N = sample size

• M = number of variants

• ℓⱼ = LD Score for variant j

• a = confounding bias term

• Key output: The slope provides an estimate of h²g, while the intercept detects population stratification

An extension that estimates the genetic correlation between two traits using only GWAS summary statistics. Instead of regressing χ² statistics, it regresses the cross-product of Z-scores from two independent GWAS against LD Scores.

Regression equation: E[Z₁ⱼ·Z₂ⱼ] = (√N₁·√N₂·rg·h²g_cross / M)·ℓⱼ + ρ·(Ns_overlap / √N₁·√N₂)

• Sample overlap correction: The intercept term accounts for phenotypic correlation due to shared samples
• Advantage over individual-level methods: Computationally efficient and privacy-preserving, requiring only publicly available summary statistics
• Standard error: Derived from a block jackknife across the genome to account for correlated LD Score estimates

An advanced method that partitions heritability across functional genomic annotations to identify enrichment in specific categories. Extends univariate LD Score Regression by modeling multiple annotation-specific LD Scores simultaneously.

Key annotations analyzed:

• Coding regions and conserved elements

• Cell-type-specific regulatory elements (e.g., DNase I hypersensitivity sites)

• Histone modification marks (H3K4me3, H3K27ac)

• Evolutionary conserved regions

• Enrichment metric: Proportion of h²g in annotation divided by proportion of SNPs in annotation

• Application: Identifying biologically relevant tissues and cell types for complex diseases, informing functional follow-up studies