Biology Backgrounds
Last updated: 2021-04-21
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Knit directory: funcFinemapping/
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Methods for fine-mapping
TORUS:bayesian regression analysis
S-LDSC:LD score based linear regression
Standardized phenotypes are typically modeled using the linear model \(y=\sum\beta_jX_j+\epsilon\). The variance of explained by X is given by \[ \begin{align} Var(\sum\beta_jX_j) &=\sum Var(\beta_jX_j) \\ &= \sum \beta_j^2Var(X_j) \\ &=\sum \beta^2_{j} \end{align} \] assuming the predictors are independent. That is to say the genotype of the causal SNP at one site is independent from that of all other sites. Here \(\beta_j\) is the true effect size of the j-th SNP, assumed to be fixed. Thus, \(h^2\) can be written as \(\sum \beta_j^2\) as genotypes are standardidized.
In S-LDSC, the effect size of the j-th SNP \(\beta_j\) is modeled as a random variable with mean 0 and variance \(Var(\beta_j)\). Then \(\sum \beta_j^2=M*E[\beta_j^2]=M*Var(\beta_j)\). Thus, \[ \begin{align} \text{total heritability} &= \sum \beta_j^2 \\ &= M * Var(\beta_j) \\ \end{align} \] \(Var(\beta_j)\) can be estimated simply using total SNP heritability divided by M, number of SNPs across the whole genome. However, we know heritability is disproportionally distributed across genome. This motivates the modeling of \(Var(\beta_j)\) as the sum of the per SNP heritability for each functional category.
\[ \begin{align} Var(\beta_j) &= \sum_{c:j\in} \frac{h^2(c)}{M_c} \\ &= \sum_{c:j\in C_c}\tau_c\\ \end{align} \] ## Including Plots
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sessionInfo()R version 4.0.4 (2021-02-15)
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attached base packages:
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