Skip to content

S-LDSC

Stratified Linkage Disequilibrium Score Regression (S-LDSC)12 is an extension of Linkage Disequilibrium Score Regression (LDSC)3. S-LDSC can be used to generate hypotheses about the key cells and tissues underlying a phenotype.

High-level summary

Review of LDSC

Recall that LDSC assumes isotropic polygenicity: heritability is evenly distributed across the genome. Mathematically this means that in LDSC for all genetic variants \(i\) and \(j\):

\[ \begin{align} \mathbb{Var}(\beta_i)&=Q\\ \mathbb{Cov}(\beta_i,\beta_j) &= 0 & \text{ when }i\ne j \end{align} \]

where \(\beta_i\) is the true effect of variant \(i\) on the phenotype and \(Q\in\mathbb{R}\) is a constant.

Overview of S-LDSC

S-LSDC instead assumes that the genome is divided into regions, and isotropic polygenicity holds within each region. Mathematically, this means that S-LDSC for all genetic variants \(i\) and \(j\):

\[ \begin{align} \mathbb{Var}(\beta_i)&=\sum_{k} \tau_k I_{i \in C_k}\\ \mathbb{Cov}(\beta_i, \beta_j) &= 0 & \text{ when }i\ne j \end{align} \]

Where:

  • The \(\{C_k\}\) are sub of genetic variants. The \(C_k\) can describe functional chromosomal regions, like promoters, enhancers, or other regulatory regions. They can also reflect gene-tissue expression. For instance, a \(C_k\) could mark SNPs that are near genes that have been observed to be over-expressed in the liver according to a GTEx RNAseq dataset. The \(C_k\) may overlap.
  • The \(\{\tau_k\}\) are heritability weights. \(\tau_k\in\mathbb{R}\) measures the effect on the GWAS signal of a genetic variant belonging to category \(C_k\).

Procedure

Review of LDSC Procedure

Recall that LDSC runs a univariate regression based on the approximate equation:

\[ \mathbb{E} (\chi^2_i) \approx N\frac{h^2}{M}l_N+1, \]

where

  • \(\chi^2_i\) is the \(\chi^2\) statistic of the maginal GWAS regression for genetic variant \(i\).
  • \(M\) is the number of genetic variants in the GWAS.
  • \(N\) is the number of individuals in the GWAS.
  • \(l_i:=\sum_k r_{ik}^2\) is the linkage disequilibrium score of genetic variant \(i\), where \(r_{ik}\) denotes the correlation between variants \(i\) and \(k\).

This regression provides an estimate of \(h^2\), the trait's heritability.

S-LDSC Procedure

Analogously, S-LDSC runs a multivariate regression based on the approximate equation:

\[ \mathbb{E} (\chi^2_i) \approx N \sum_k \tau_k l_{i,k}+1, \]

where the notation is the same as above, with the addition that

  • \(l_{i,k}:=\sum_j r_{i,j}^2 I_{j\in C_k}\) is the linkage disequilibrium score of genetic variant \(i\) restricted to category \(k\).

This regression provides estimates of the heritability weights \(\{\tau_k\}\).

The output of S-LDSC

  • Using the \(\tau_k\) weights, we can estimate the overall heritability as \(h^2=\sum_i\sum_k \tau_k I_{i\in C_k}\).
  • We can evaluate the proportion of heritability due to category \(k'\) as \(\frac{\sum_i \tau_{k'} I_{i\in C_{k'}}}{\sum_i\sum_k \tau_k I_{i\in C_k}}\). We can compare this to the proportion of genetic variants in category \(k\), which is \(|C_k|/M\). This comparison gives us an estimate of the enrichment of heritability in the category.
  • We can run statistical tests to evaluate the evidence that \(\tau_k\ne 0\). If this evidence is strong, we can argue that category \(k\) reflects a meaningful grouping of genetic variants that differentially affect the phenotype.

  1. Hilary K Finucane, Yakir A Reshef, Verneri Anttila, Kamil Slowikowski, Alexander Gusev, Andrea Byrnes, Steven Gazal, Po-Ru Loh, Caleb Lareau, Noam Shoresh, and others. Heritability enrichment of specifically expressed genes identifies disease-relevant tissues and cell types. Nature Genetics, 50(4):621–629, 2018. URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC5896795/. ↩

  2. Hilary K Finucane, Brendan Bulik-Sullivan, Alexander Gusev, Gosia Trynka, Yakir Reshef, Po-Ru Loh, Verneri Anttila, Han Xu, Chongzhi Zang, Kyle Farh, and others. Partitioning heritability by functional annotation using genome-wide association summary statistics. Nature Genetics, 47(11):1228–1235, 2015. URL: https://www.nature.com/articles/ng.3404. ↩

  3. Brendan K Bulik-Sullivan, Po-Ru Loh, Hilary K Finucane, Stephan Ripke, Jian Yang, Nick Patterson, Mark J Daly, Alkes L Price, and Benjamin M Neale. LD Score regression distinguishes confounding from polygenicity in genome-wide association studies. Nature Genetics, 47(3):291–295, 2015. URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC4495769/. ↩