Table 2 Formulation of enrichment score functions meanrank, ksmax and ksmean, defined for competitive testing

Meanrank

\(T_{meanrank} = \frac{1}{m_o}\sum \limits _{i\in S}\frac{(q+1)/2 - \rho _i}{q}\)

Ksmax

\(T_{ksmax} = \hbox {I}(A>|a|)A + \hbox {I}(A\le |a|)a,\)

\(A = \max \limits _{l \in S} ks(l|S), a = \min \limits _{l \in S} ks(l|S),\)

\(ks(l|S) = \frac{\sum \limits _{i\in S} |\gamma _i|^k I(\rho _i \le l)}{\sum \limits _{i\in S} |\gamma _i|^k} - \frac{\sum \limits _{i\not \in S} I(\rho _i \le l)}{q - m_o}\)

Ksmean

\(T_{ksmean} = \max \limits _{l \in S} ks(l|S) + \min \limits _{l \in S} ks(l|S),\)

\(ks(l|S) = \frac{\sum \limits _{i\in S} |\eta _i|^k I(\rho _i \le l)}{\sum \limits _{i\in S} |\eta _i|^k} - \frac{\sum \limits _{i\not \in S} I(\rho _i \le l)}{q - m_o}\)

Notation

\([\delta _i] \equiv \hbox {modt-statistics, } i\in \Omega = [1,\ldots ,q]\)

\([\rho _i] \equiv \hbox {rank for } [\delta _i] \hbox { in decreasing order}\)

\(S \equiv \hbox {Testing gene set, } S\subset \Omega , \,\,\, m_0 = |S|,\,\, C = \Omega \setminus S\)

\(\gamma _i = \delta _i - \bar{\delta },\,\,\, \eta _i = (2\rho _i+q+1)/2\)