Table 1 Implemented methods in the metacp package and their corresponding formulas for combining p-values
From: metacp: a versatile software package for combining dependent or independent p-values
Method | Combined p-value |
|---|---|
Logitp | \(p_{c} = 1 - {\rm T}\left( {\left| { - \frac{{\mathop \sum \nolimits_{i = 1}^{k} \log \left( {\frac{{p_{i} }}{{1 - p_{i} }}} \right)}}{C}} \right|, 2k} \right)\) |
Meanp | \(p_{c} = 1 - \Phi \left( {\left( {0.5 - \overline{p}} \right)\sqrt {12k} } \right)\) |
Fisher | \(p_{c} = { }1 - F\left( { - 2\mathop \sum \limits_{i = 1}^{k} \ln p_{i} , 2k} \right)\) |
Lancaster | \(p_{c} = 1 - F\left( {\mathop \sum \limits_{i = 1}^{N} F_{{X^{2} }}^{ - 1} \left( {1 - p_{i} , n_{i} } \right), \mathop \sum \limits_{i = 1}^{k} n_{i } } \right)\) |
Stouffer | \(p_{c} { } = 1 - \Phi \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{k} z_{i} }}{\sqrt k }} \right)\) |
Weighted Stouffer | \(p_{c} { } = 1 - \Phi \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{k} w_{i} z_{i} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{k} w_{i}^{2} } }}} \right)\) |
Inverse chi2 | \(p_{c } = { }1 - F\left( {\mathop \sum \limits_{i = 1}^{k} { }F^{ - 1} \left( {1 - p_{i} ,{ }1} \right), k} \right)\) |
Binomial | \(p_{c} = \mathop \sum \limits_{x = r}^{k} \left( {\begin{array}{*{20}c} k \\ x \\ \end{array} } \right)a^{x} \left( {1 - a} \right)^{k - x}\) |
Bonferroni | \(p_{c} = min\left( {1, \min \left( {p_{1} , p_{2} , \ldots , p_{k} } \right) \times k} \right)\) |
Bonferroni Adjustments (CN, Li-Ji, Galwey, Gao and so on) | \(p_{c} = {\text{ min}}\left( {1, \min \left( {p_{1} , p_{2} , \ldots , p_{k} } \right) \times m} \right)\) |
MinP | \(p_{c} = 1 - \left( {1 - {\text{min}}\left( {p_{1} , p_{2} , \ldots , p_{k} } \right)} \right)^{k}\) |
CCT | \(p_{CCT} = P\left[ {C\left( {0, 1} \right) \ge t} \right]\) |
MCM | \(p_{MCM} = 2{\text{min}}\left( {p_{CCT} , p_{MinP} , 0.5} \right)\) |
CMC | \(p_{CMC} = CCT\left( {p_{CCT} , p_{MinP} } \right)\) |
HMP | \(p_{c} = k/\mathop \sum \limits_{i = 1}^{k} \left( {1/p_{i} } \right)\) |
EBM | \(p_{c} { } = 1 - F\left( { - \frac{{2\mathop \sum \nolimits_{i = 1}^{k} {\text{log}}p_{i} }}{c},2f} \right)\) |
Yang’s extension to EBM | \(p_{c} = 1 - \Gamma \left( { - 2\mathop \sum \limits_{i = 1}^{k} {\text{log}}p_{i} ,{ }v/2,{ }2\gamma } \right)\) |
Weighted Stouffer for Correlated Tests | \(p_{c} { } = 1 - \Phi \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{k} w_{i} z_{i} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{k} \mathop \sum \nolimits_{j = 1}^{k} w_{i} w_{j} r_{ij} } }}} \right)\) |
Stouffer for Correlated Tests (without weights) | \(p_{c} { } = 1 - \Phi \left( {\mathop \sum \limits_{i = 1}^{k} z_{i} /\sqrt {\left( {k + \mathop \sum \limits_{i = 1}^{k} \mathop \sum \limits_{j = 1}^{k} r_{ij} } \right)} } \right)\) |