Burden Heritability Regression (BHR)

Summary¶

Burden Heritability Regression (BHR) is a random-effects regression framework developed by Weiner, Nadig et al. (2023) to quantify the total phenotypic variance explained by rare coding variants ($h^2_{\text{burden}}$) and estimate rare-variant genetic correlations ($r_g$) using summary-level exome association statistics.[1] Analogous to Linkage Disequilibrium Score Regression (LDSC) for common variants, BHR regresses gene-level burden test statistics on gene burden scores (reflecting cumulative rare-allele variance), using the regression slope to estimate true heritability and the intercept to filter out confounding from population stratification and cryptic relatedness.[1] Applied across 394,783 UK Biobank exomes, BHR revealed that while rare coding variants explain a modest $1.3\%$ of total trait heritability (compared to $13\%$ for common variants), rare-variant heritability is far more oligogenic—concentrated sharply in a small number of high-effect genes such as tumor suppressors.[1]

Mathematical Model & Statistical Formulation¶

BHR models the marginal association statistic ($\gamma_g$) for gene $g$ aggregated across $M_g$ rare coding variants as a random effect:[1]

$$\gamma_g = \sum_{j \in g} w_j \beta_j + \epsilon_g$$

where $w_j \in {0, 1}$ indicates variant carrying status and $\beta_j \sim \mathcal{N}(0, \sigma^2_b)$ represents the underlying effect size.[1]

The expectation of the gene-level marginal test statistic $\chi^2_g = N \hat{\gamma}_g^2$ is expressed as a linear regression on the gene burden score ($S_g$):[1]

$$\mathbb{E}[\chi^2_g] = N \cdot S_g \cdot h^2_{\text{burden}} + \alpha$$

Key Variables & Terms¶

Gene Burden Score ($S_g$): Quantifies the cumulative expected variance of rare minor alleles in gene $g$ under low LD assumptions:[1] $$S_g \approx \sum_{j \in g} 2 p_j (1 - p_j) w_j^2$$ where $p_j$ is the minor allele frequency (MAF) of variant $j$.
Regression Slope ($h^2_{\text{burden}}$): Directly measures the fraction of phenotypic variance explained by rare coding burden per unit of burden score across the gene set.[1]
Intercept ($\alpha$): Measures baseline test-statistic inflation caused by sample overlap, uncorrected population stratification, and cryptic relatedness.[1] Under null conditions ($\beta = 0$), $\alpha \to 1.0$ in unconfounded samples.[1]

BHR Regression Mechanics
  Gene Burden Test Statistic (χ²_g)
          │
          │         /  Slope = N * h²_burden (True Genetic Signal)
          │        /
          │       /
          │      /
          │     /│ Intercept = α (Confounding / Stratification)
          └────┴─────────────────────
               0       Gene Burden Score (S_g)

Stratification & Multi-Tier Functional Partitioning¶

BHR partitions $h^2_{\text{burden}}$ across minor allele frequency tiers and functional consequence classes to prevent attenuation from neutral variants:[1]

Allele Frequency Tiers:
Ultrarare: $MAF < 1.0 \times 10^{-5}$ (dominated by singletons/doubletons in UK Biobank).[1]
Rare: $1.0 \times 10^{-5} \le MAF < 1.0 \times 10^{-3}$.[1]
Functional Categories:
pLoF (Predicted Loss-of-Function): Stop-gained, essential splice site, frameshift insertions/deletions.[1]
Damaging Missense: Missense variants predicted damaging by PolyPhen-2, SIFT, or MPC ($MPC > 2$).[1]
Benign Missense: Missense variants predicted benign.[1]
Synonymous: Serves as a mandatory empirical negative control (expected $h^2_{\text{burden}} = 0$).[1]

Major Empirical Findings across 394,783 UK Biobank Exomes¶

Applying BHR to 22 quantitative and complex traits in the UK Biobank yielded fundamental insights into human genetic architecture:[1]

Total Heritability Contribution: Rare coding variants ($MAF < 10^{-3}$) explain an average $h^2_{\text{burden}} = 1.3\%$ (s.e. $0.03\%$) of phenotypic variance across complex traits, compared to common-variant $h^2_{\text{SNP}} \approx 13\%$ estimated via LDSC.[1]
Functional Severity Concentration: Over $90\%$ of total $h^2_{\text{burden}}$ is concentrated in pLoF and damaging missense variants.[1] Synonymous variants yielded $h^2_{\text{burden}} \approx 0.000$, validating BHR's resistance to false-positive inflation.[1]
Extreme Oligogenic Concentration vs Polygenicity:
In common-variant GWAS, heritability is polygenic, spread across thousands of loci due to "flattening" by natural selection.[1]
In rare coding variation, a median of 6 exome-wide significant genes per trait accounts for $19\%$ of total $h^2_{\text{burden}}$ (after correcting for Winner's Curse).[1]
Examples: APOB alone explains $39\%$ of LDL cholesterol burden heritability; GCK explains $15\%$ of diabetes burden heritability.[1]

Trait Category	Top Exome Gene	Gene $h^2_{\text{burden}}$ Share	Common Variant ($h^2_{\text{SNP}}$) Share	Architectural Interpretation
LDL Cholesterol	APOB	$39.2\%$	$< 1.5\%$	Massive rare coding effect concentrated in single transporter [1]
Type 2 Diabetes	GCK	$15.1\%$	$< 0.8\%$	Rare monogenic driver explaining large fraction of exome variance [1]
Height	Top 81 genes	$32.1\%$	$9.5\%$	Moderate concentration relative to common polygenic background [1]

Cancer Gene Census (CGC) & Tumor Suppressor Enrichment¶

BHR revealed a striking divergence between tumor suppressors and oncogenes across somatic/germline cancer traits:[1] - Tumor Suppressor Genes ($N = 172$): Account for $48\%$ (s.e. $10\%$) of total cancer burden heritability.[1] Germline loss-of-function in a single allele of a tumor suppressor requires only a somatic second hit for oncogenesis.[1] - Oncogenes ($N = 101$): Account for only $1\%$ (s.e. $2\%$) of cancer burden heritability.[1] Oncogenic gain-of-function mutations are extremely rare in germline pools due to embryonic lethality or strong negative selection.[1]

Method Comparison: BHR vs. LDSC vs. GCTA-LDMS¶

Feature / Dimension	Burden Heritability Regression (BHR)	LD Score Regression (LDSC)	GCTA-LDMS
Target Allele Frequency	Rare & Ultrarare ($MAF < 10^{-3}$)	Common ($MAF > 0.01$)	Full Spectrum ($MAF > 0.001$)
Data Requirement	Summary-level gene burden stats	Summary-level GWAS stats	Individual-level genotype arrays/WGS
LD Modeling	Assumes low LD among rare coding variants	Explicitly models LD structure via LD scores	Estimates GRM partitioned by MAF & LD
Confounding Filter	Regression intercept ($\alpha$)	Regression intercept ($\text{LDSC}_{\text{intercept}}$)	Principal Component adjustments
Primary Output	$h^2_{\text{burden}}$ & rare $r_g$	$h^2_{\text{SNP}}$ & common $r_g$	$h^2_{\text{total}}$

Applications & Downstream Extensions¶

Cross-Trait Genetic Correlations ($r_g$): BHR extends bivariate regression to compute rare-variant genetic correlations between phenotypes without individual-level data.[1]
Psychiatric & Complex Trait Application: Used by Tian et al. (2024) to calculate rare-variant $r_g$ across seven depression definitions, proving that rare PTV genetic correlations display phenotypic clustering distinct from common-variant correlations ($ARI = -0.167$).[2]
RICE Integration: Motivated the calibration of RICE (Rare-variant Integrated Polygenic Risk Scores) to combine rare coding carriers with common PRS.[1]

Citations¶

[1] Weiner, D. J., Nadig, A., Jagadeesh, K. A., Dey, K. K., Neale, B. M., Robinson, E. B., Karczewski, K. J., & O'Connor, L. J. (2023). Polygenic architecture of rare coding variation across 394,783 exomes. Nature, 614(7948), 492–499. Supports: BHR random-effects model, gene burden score definition, $h^2_{\text{burden}} = 1.3\%$, oligogenic concentration (6 genes = $19\%$), tumor suppressor enrichment ($48\%$ vs $1\%$), and summary-statistic estimation. Location: Main Text ("Estimation of burden heritability", "Burden heritability of 22 complex traits", "Concentration of burden heritability"); Methods ("Burden Heritability Regression model").

[2] Tian, R., Ge, T., Kweon, H., Rocha, D.B., Lam, M., Liu, J.Z., et al. (2024). Whole-exome sequencing in UK Biobank reveals rare genetic architecture for depression. Nature Communications, 15, 1755. Supports: application of BHR to calculate rare-variant genetic correlations ($r_g$) and phenotypic clustering discordance ($ARI = -0.167$). Location: Results ("Genetic relationships between depression definitions across frequency spectrum"); Figure 2.