# Table 1 Covariance structure examples

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 