Algorithm 3.

aMeta

Inputs: The estimated marker-specific p-value (Pj) and the permuted (null) pooled effects (${\widehat{\mu }}_{{j,0}_1},...,{\widehat{\mu }}_{{j,0}_R}$) obtained based on the single-marker meta-analysis (Algorithm 1) for each marker (j = 1, …, Q).

Outputs: The p-value for aMeta.

1. Compute the null values of Pj (denoted as $P_{j_r}$) as $P_{j_r}$ = $[\sum _{r^{\text{'}}\ne r}I\left(\left|{\widehat{\mu }}_{A_{0,r^{\text{'}}}}\right|>\left|{\widehat{\mu }}_{A_{0,r}}\right|\right)+1]/(R+1)$, where r = 1, …, R and $r^{\text{'}}$ = 1, …, R, for each marker (j = 1, …, Q).

2. Compute the null statistic values of TaMeta (denoted as $T_{{aMeta}_r}$) as $T_{{aMeta}_r}=\underset{j\in \{1,\dots ,\mathit{Q}\}}{\mathit{min}}{P}_{j_r}$, for r = 1, …, R (Eq. 7).

3. Compute the observed statistic value (denoted as TaMeta) as $T_{aMeta}=\underset{j\in \{1,\dots ,\mathit{Q}\}}{\mathit{min}}{P}_j$ (Eq. 7).

4. Compute the p-value for aMeta as $[\sum _{r=1}^{R}I(T_{{aMeta}_r}<T_{aMeta})+1]/(R+1)$, where I(·) is an indicator function.