マクネマー検定に関する優越性試験のサンプルサイズ設計の公式

対称性に関する仮説を検定するための検定統計量は、 \begin{align} \hat{\delta}={\hat{\pi}}_{12}-{\hat{\pi}}_{21} \end{align} 〔1〕標本リスク差の漸近分布
対立仮説の下で \begin{align} \hat{\delta}\xrightarrow[]{d}N \left[\pi_{12}-\pi_{21},\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\right] \end{align} 帰無仮説の下で \begin{align} \hat{\delta} \sim \mathrm{N} \left(0,\frac{\pi_{12}+\pi_{21}}{N}\right) \end{align} したがって、検定統計量の分散について、 \begin{gather} \frac{\phi_0^2}{N}=\frac{\pi_{12}+\pi_{21}}{N}\\ \phi_0^2=\pi_{12}+\pi_{21}\\ \phi_0=\sqrt{\pi_{12}+\pi_{21}} \end{gather} \begin{gather} \frac{\phi_1^2}{N}=\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\\ \phi_1^2= \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2\\ \phi_1=\sqrt{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2} \end{gather} これをサンプルサイズの設計公式に代入すると、 \begin{align} N= \left[\frac{Z_\alpha \cdot \sqrt{\pi_{12}+\pi_{21}}-Z_{1-\beta} \cdot \sqrt{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}}{ \left|\pi_{12}-\pi_{21}\right|}\right]^2 \end{align} 条件付きオッズ比が与えられた場合は、 \begin{gather} \varphi_c=\frac{\pi_{12}}{\pi_{21}}\Leftrightarrow\pi_{12}=\varphi_c\pi_{21} \end{gather} これを代入すると、 \begin{align} N&= \left[\frac{Z_\alpha \cdot \sqrt{\varphi_c\pi_{21}+\pi_{21}}-Z_{1-\beta} \cdot \sqrt{ \left(\varphi_c\pi_{21}+\pi_{21}\right)- \left(\varphi_c\pi_{21}-\pi_{21}\right)^2}}{ \left|\varphi_c\pi_{21}-\pi_{21}\right|}\right]^2\\ &= \left[\frac{Z_\alpha \cdot \sqrt{ \left(\varphi_c+1\right)\pi_{21}}-Z_{1-\beta} \cdot \sqrt{ \left(\varphi_c+1\right)\pi_{21}-\pi_{21}^2 \left(\varphi_c-1\right)^2}}{ \left| \left(\varphi_c-1\right)\pi_{21}\right|}\right]^2 \end{align} $\blacksquare$