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p o w e r = {\begin{cases} Φ (\frac{(\tilde{p_{2}} - {R R}_{0} \tilde{p_{1}}) (N w_{1} w_{2})^{\frac{1}{2}} - z_{1 - α} {[w_{2} {R R}_{0}^{2} \tilde{p_{1}} (1 - \tilde{p_{1}}) + w_{1} \tilde{p_{2}} (1 - \tilde{p_{2}})]}^{\frac{1}{2}}}{{[w_{2} {R R}_{0}^{2} p_{1} (1 - p_{1}) + w_{1} p_{2} (1 - p_{2})]}^{\frac{1}{2}}}), & upper one-sided \\ Φ (\frac{- (\tilde{p_{2}} - {R R}_{0} \tilde{p_{1}}) (N w_{1} w_{2})^{\frac{1}{2}} - z_{1 - α} {[w_{2} {R R}_{0}^{2} \tilde{p_{1}} (1 - \tilde{p_{1}}) + w_{1} \tilde{p_{2}} (1 - \tilde{p_{2}})]}^{\frac{1}{2}}}{{[w_{2} {R R}_{0}^{2} p_{1} (1 - p_{1}) + w_{1} p_{2} (1 - p_{2})]}^{\frac{1}{2}}}), & lower one-sided \\ Φ (\frac{(\tilde{p_{2}} - {R R}_{0} \tilde{p_{1}}) (N w_{1} w_{2})^{\frac{1}{2}} - z_{1 - \frac{α}{2}} {[w_{2} {R R}_{0}^{2} \tilde{p_{1}} (1 - \tilde{p_{1}}) + w_{1} \tilde{p_{2}} (1 - \tilde{p_{2}})]}^{\frac{1}{2}}}{{[w_{2} {R R}_{0}^{2} p_{1} (1 - p_{1}) + w_{1} p_{2} (1 - p_{2})]}^{\frac{1}{2}}}) + \\ Φ (\frac{- (\tilde{p_{2}} - {R R}_{0} \tilde{p_{1}}) (N w_{1} w_{2})^{\frac{1}{2}} - z_{1 - \frac{α}{2}} {[w_{2} {R R}_{0}^{2} \tilde{p_{1}} (1 - \tilde{p_{1}}) + w_{1} \tilde{p_{2}} (1 - \tilde{p_{2}})]}^{\frac{1}{2}}}{{[w_{2} {R R}_{0}^{2} p_{1} (1 - p_{1}) + w_{1} p_{2} (1 - p_{2})]}^{\frac{1}{2}}}), & two-sided \end{cases}

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\[ \mr{power} = \left\{ \begin{array}{ll} \Phi \left( \frac{(\tilde{p_2} - \mr{RR}_0 \tilde{p_1}) (N w_1 w_2)^\frac {1}{2} - z_{1-\alpha } \left[ w_2\mr{RR}^2_0\tilde{p_1}(1-\tilde{p_1}) + w_1\tilde{p_2}(1-\tilde{p_2}) \right]^\frac {1}{2}}{\left[ w_2 \mr{RR}^2_0 p_1 (1 - p_1) + w_1 p_2 (1 - p_2) \right]^\frac {1}{2}} \right), & \mbox{upper one-sided} \\ \Phi \left( \frac{-(\tilde{p_2} - \mr{RR}_0 \tilde{p_1}) (N w_1 w_2)^\frac {1}{2} - z_{1-\alpha } \left[ w_2\mr{RR}^2_0\tilde{p_1}(1-\tilde{p_1}) + w_1\tilde{p_2}(1-\tilde{p_2}) \right]^\frac {1}{2}}{\left[ w_2 \mr{RR}^2_0 p_1 (1 - p_1) + w_1 p_2 (1 - p_2) \right]^\frac {1}{2}} \right), & \mbox{lower one-sided} \\ \Phi \left( \frac{(\tilde{p_2} - \mr{RR}_0 \tilde{p_1}) (N w_1 w_2)^\frac {1}{2} - z_{1-\frac{\alpha }{2}} \left[ w_2\mr{RR}^2_0\tilde{p_1}(1-\tilde{p_1}) + w_1\tilde{p_2}(1-\tilde{p_2}) \right]^\frac {1}{2}}{\left[ w_2 \mr{RR}^2_0 p_1 (1 - p_1) + w_1 p_2 (1 - p_2) \right]^\frac {1}{2}} \right) + \\ \quad \Phi \left( \frac{-(\tilde{p_2} - \mr{RR}_0 \tilde{p_1}) (N w_1 w_2)^\frac {1}{2} - z_{1-\frac{\alpha }{2}} \left[ w_2\mr{RR}^2_0\tilde{p_1}(1-\tilde{p_1}) + w_1\tilde{p_2}(1-\tilde{p_2}) \right]^\frac {1}{2}}{\left[ w_2 \mr{RR}^2_0 p_1 (1 - p_1) + w_1 p_2 (1 - p_2) \right]^\frac {1}{2}} \right), & \mbox{two-sided} \\ \end{array} \right. \]