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