Model Description

The sum of single shared effect (SuShiE) extends the sum of single effect (SuSiE) [1] model by introducing a prior correlation estimator to account for the ancestral quantitative trait loci (QTL) effect size similarity. Specifically, for \(i^{\text{th}}\) of total \(k \in \mathbb{N}\) ancestries, we model the molecular data \(g_i \in \mathbb{R}^{n_i \times 1}\) for \(n_i \in \mathbb{N}\) individuals as a linear combination of standardized genotype matrix \(X_i \in \mathbb{R}^{n_i \times p}\) for \(p \in \mathbb{N}\) SNPs as

\begin{gather*} g_i = X_i \beta_i+\epsilon_i \\ \beta_i = \sum_{l=1}^{L}\beta_{i,l} \\ \beta_{i,l} = \gamma_l \cdot b_{i, l} \\ b_{l} = \begin{bmatrix} b_{1,l} \\ \vdots \\ b_{k,l} \end{bmatrix} \sim \mathcal{N}(0, C_l) \\ C_{i,i',l} = \begin{cases} \sigma_{i,b,l}^2 & \text{if } i = i' \\ \rho_{i,i',l} \cdot \sigma_{i,b,l} \cdot \sigma_{i',b,l} & \text{otherwise}\end{cases} \\ \gamma_l \sim \text{Multi}(1, \pi) \\ \epsilon_i \sim \mathcal{N}(0, \sigma^2_{i, e}I_{n_i}) \\ \end{gather*}

where \(\beta_i \in \mathbb{R}^{p \times1}\) is the shared QTL effects, \(\epsilon_i \in \mathbb{R}^{n_i \times 1}\) is the ancestry-specific effects and other environmental noises, \(L \in \mathbb{R}\) is the number of shared effects, for \(l^{\text{th}}\) single shared effect, \(b_{i,l} \in \mathbb{R}\) is a scaler representing effect size, \(C_l \in \mathbb{R}^{k \times k}\) is the prior covariance matrix with \(\sigma^2_{i,b}\) as variance and \(\rho\) as correlation, \(\gamma_l\) is an binary indicator vector specifying which single SNP is the QTL, \(\pi\) is the prior probability for each SNP to be QTL, and \(\sigma^2_e\) is the prior variance for noises.

SuShiE runs varitional inference to estimate the posterior distribution for \(\beta_l\) and \(\gamma_l\) for each \(l^{\text{th}}\) effect. We can quantify the probability of QTL for each SNP through Posterior Inclusion Probabilities (PIPs). If the posterior distribution of \(\gamma_l\) is \(\text{Multi}(1, \alpha_l)\), then for each SNP \(j\), we have:

\[\text{PIP}_j = 1 - \prod_{l=1}^L(1 - \alpha_{l, j})\]

For more details in math derivation and algorithm, stay tuned for our upcoming manuscript.

Reference

Last update: Oct 27, 2024