Table 1 Covariance structure examples
From: Unsupervised gene set testing based on random matrix theory
Name | Model | Self-contained | Competitive |
|---|---|---|---|
Identity | \( \boldsymbol {\Sigma } =\left [\begin {array}{cc} I & 0 \\ 0 & I \\ \end {array}\right ]\) | Accept H 0 | Accept H 0 |
Scaled identity | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} \alpha I & 0 \\ 0 & \alpha I \\ \end {array}\right ]\) | Reject H 0 | Accept H 0 |
Single block | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} \rho & 0 \\ 0 & I \\ \end {array}\right ]\) | Reject H 0 | Reject H 0 |
Multi-block | \(\boldsymbol {\Sigma } = \left [\begin {array}{cc} \left [\begin {array}{cc} \rho & 0 \\ 0 & \rho \\ \end {array}\right ] & 0 \\ 0 & I \\ \end {array}\right ]\) | Reject H 0 | Reject H 0 |
Anti-correlated multi-block | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} \left [\begin {array}{cc} \rho & -\rho \\ -\rho & \rho \\ \end {array}\right ] & 0 \\ 0 & I \\ \end {array}\right ] \) | Reject H 0 | Reject H 0 |
Inverted single block | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} I & 0 \\ 0 & \rho \\ \end {array}\right ] \) | Accept H 0 | ∗Accept H 0 |
Repeated single block | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} \rho & 0 \\ 0 & \rho \\ \end {array}\right ] \) | Reject H 0 | ∗Reject H 0 |
Compound symmetry | \( \boldsymbol {\Sigma } = \left [\begin {array}{cc} \rho & \rho \\ \rho & \rho \\ \end {array}\right ]\) | Reject H 0 | Accept H 0 |