impute.zscore.RdImpute z-scores
impute.zscore(Z, V.t, observed)incomplete z-score matrix
t(V) matrix of the SVD result
SNP positions observed in Z
new z-score matrix
Assuming that \(\mathbf{y} = X_{0} \boldsymbol{\theta}_{0} + X_{1} \boldsymbol{\theta}_{1} \cdots\) and \(n^{-1/2}X = U D V^{\top}\), we have incomplete z-score \(\mathbf{z}_{0} \sim N(V_{0} D^2 V_{0}^{\top} , V_{0} D^2 V_{0}^{\top})\). By linear transformation, we have $$\mathbf{y}_{0} = U D V_{0}^{\top} \mathbf{\theta}_{0} \sim \mathcal{N}\!\left(U D^{-1} V_{0}^{\top}\mathbf{z}_{0}, I\right).$$ Using this we can derive imputed z-score: $$\tilde{\mathbf{z}} \approx VDU^{\top}\mathbf{y}_{0} \sim \mathcal{N}\!\left(V V_{0}^{\top}\mathbf{z}_{0}, VD^{2}V^{\top}\right).$$