Table 1 Overview of gene-based test statistics considered

Sum of natural log-transformed p value (Sum (ln p))

\( {\sum}_{j=1}^m\ln \left({p}_j\right) \)

Sum of negative squared natural log-transformed p value (Sum (−(lnp)²))

\( {\sum}_{j=1}^m-{\left(\ln {p}_j\right)}^2 \)

Minimum p (Min p)

\( \underset{j\in \left\{1,\dots, m\right\}}{\mathit{\min}}\left({p}_j\right) \)

p value threshold 0.01 (pT 0.01)

\( {\sum}_{j=1}^m\left\{\begin{array}{cc}-{\left(\ln \left(\frac{p_j}{0.01}\right)\right)}^2& \mathrm{if}{p}_j\le 0.01\\ {}0& \mathrm{if}{p}_j>0.01\end{array}\right. \)

p value threshold 0.05 (pT 0.05)

\( {\sum}_{j=1}^m\left\{\begin{array}{cc}-{\left(\ln \left(\frac{p_j}{0.05}\right)\right)}^2& \mathrm{if}{p}_j\le 0.05\\ {}0& \mathrm{if}{p}_j>0.05\end{array}\right. \)

p value threshold 0.10 (pT 0.10)

\( {\sum}_{j=1}^m\left\{\begin{array}{cc}-{\left(\ln \left(\frac{p_j}{0.1}\right)\right)}^2& \mathrm{if}{p}_j\le 0.1\\ {}0& \mathrm{if}{p}_j>0.1\end{array}\right. \)