ASCI 896 Statistical Genomics

Whole-genome regression - ridge regression 1

• \( \hat{\boldsymbol{\beta}}^{ridge} = (\mathbf{X'X} + \lambda \mathbf{I})^{-1} \mathbf{X^{'}y} \)
• \( \hat{\mathbf{y}}^{ridge} = \mathbf{X}\hat{\boldsymbol{\beta}}^{ridge} \)

The amount of shrinkage \[ \begin{align*} \hat{\beta}_1^{ridge} &= \frac{\sum^n_{i=1} x_{i1}(y_i - x_{i2}\hat{\beta_2} - \cdots - x_{im}\hat{\beta_m}) }{\sum^n_{i=1} x^2_{i1} + \lambda} \\ &= \frac{\sum^n_{i=1} x^2_{i1} }{\sum^n_{i=1} x^2_{i1} + \lambda} \times \hat{\beta}^{ols}_{1} \end{align*} \] where \[ \begin{align*} \hat{\beta}_1^{ols} &= \frac{\sum^n_{i=1} x_{i1}(y_i - x_{i2}\hat{\beta_2} - \cdots - x_{im}\hat{\beta_m}) }{\sum^n_{i=1} x^2_{i1} } \end{align*} \]