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\begin{aligned} p o w e r = & {\begin{cases} P (Z \geq \frac{{S E}_{H_{0}} (\log ({W M W}_{o d d s}))}{S E (\log ({W M W}_{o d d s}))} z_{1 - α} - δ^{⋆} N^{\frac{1}{2}}), upper one-sided \\ P (Z \leq \frac{{S E}_{H_{0}} (\log ({W M W}_{o d d s}))}{S E (\log ({W M W}_{o d d s}))} z_{α} - δ^{⋆} N^{\frac{1}{2}}), lower one-sided \\ P (χ^{2} (1, (δ^{⋆})^{2} N) \geq {[\frac{{S E}_{H_{0}} (\log ({W M W}_{o d d s}))}{S E (\log ({W M W}_{o d d s}))}]}^{2} χ_{1 - α}^{2} (1)), two-sided \end{cases} \end{aligned}

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\begin{align*} \mr{power} = & \left\{ \begin{array}{l} P\left(Z \ge \frac{\mr{SE}_{H_0}(\log (\mr{WMW}_\mr {odds}))}{\mr{SE}(\log (\mr{WMW}_\mr {odds}))} z_{1-\alpha } - \delta ^\star N^\frac {1}{2} \right), \quad \mbox{upper one-sided} \\ P\left(Z \le \frac{\mr{SE}_{H_0}(\log (\mr{WMW}_\mr {odds}))}{\mr{SE}(\log (\mr{WMW}_\mr {odds}))} z_{\alpha } - \delta ^\star N^\frac {1}{2} \right), \quad \mbox{lower one-sided} \\ P\left(\chi ^2(1, (\delta ^\star )^2 N) \ge \left[ \frac{\mr{SE}_{H_0}(\log (\mr{WMW}_\mr {odds}))}{\mr{SE}(\log (\mr{WMW}_\mr {odds}))} \right]^2 \chi ^2_{1-\alpha }(1)\right), \quad \mbox{two-sided} \\ \end{array} \right. \\ \end{align*}