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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
  1. Examples where the self-contained and competitive tests give different answers are in bold. For the inverted single block structure, a two-sided competitive null would be rejected whereas the one-sided competitive H A would be accepted. For the repeated block structure, H 0 will be rejected since a random sample of g genes from among all p genes will likely include some pairs with 0 covariance