Method Tutorial

Shengtong Han

2025-09-08

MIRAGE is able to analyze both variant set and gene set, although its name focus on gene set. Mixture model is ultilized to model the risk uncertainty of either a variant or a gene and the risk probability depends on the annotations. We will start with a simple case-variant set.

Variant set (VS) analysis mirage_vs

Suppose there are $K$ variant groups and let’s focus on one variant group only. Variant $j$ in the group is modeled a mixture of risk variant and non-risk variant following Bernoulli distribution

$$P(Z_j=1)=\eta$$$Z_j$ is a binary variable indicating it’s a risk variant $Z_j=1$ and non-risk $Z_j=0$. All variants within the same group are assumed to be homogeneous sharing similar effect size and $\eta$ is the proportion of risk variants in the group. The posterior probability (PP) of being a risk variant is

$$P(Z_j=1|X_j, T_j)=\frac{P(Z_j=1, X_j, T_j)}{P(X_j, T_j)}=\frac{\eta BF_j}{\eta BF_j+1-\eta}$$ where $BF_j=\frac{P(X_j, T_j|Z_j=1)}{P(X_j, T_j|Z_j=0)}$, is the Bayes factor of variant $j$, $X_j, T_j$ are rare allele counts in cases and both cases and controls respectively.

Gene set analysis mirage

In a gene set, every gene is modeled as a mixture of risk gene and non-risk gene as

$$P(U_i=1)=\delta$$

gene $i$ is a risk gene when $U_i=1$ and non-risk gene $U_i=0$. $\delta$ is the proportion of risk genes in the gene set. If gene $i$ is a risk gene, its variant $(i,j)$ is from variant group $k$, then $$P(Z_{ij}=1)=\eta_k$$

$\eta_k$ is the proportion of risk variants in variant group $k$ where variants may be from multipe different genes. The posterior probability (PP) is

$$PP_i=\frac{\delta B_i}{\delta B_i+1-\delta}$$

$B_i$ is the Bayes factor of gene $i$. More details can be found in the reference.

References

A Bayesian method for rare variant analysis using functional annotations and its application to Autism: https://www.biorxiv.org/content/10.1101/828061v1