ジョン・ラチン（2020）『医薬データのための統計解析』 問題5.6 解答例

本稿は、ジョン・ラチン（2020）『医薬データのための統計解析』の「問題5.6」の自作解答例です。多項分布にもとづくマクネマー検定統計量の導出に関する問題です。

なお、閲覧にあたっては、以下の点にご注意ください。

問題5.6.1：対立仮説における標本比率の差の分散

確率変数の差の分散の公式 $V(X-Y)=V\left(X\right)+V\left(Y\right)-2\mathrm{Cov}(X,Y)$ より、 $$\begin{array}{rl}V(p_{12}-p_{21})& =V\left(p_{12}\right)+V\left(p_{21}\right)-2\mathrm{Cov}(p_{12},p_{21})\\ & =\frac{{\pi }_{12}(1-{\pi }_{12})}{N}+\frac{{\pi }_{21}(1-{\pi }_{21})}{N}-2(-\frac{{\pi }_{12}{\pi }_{11}}{N})\\ & =\frac{{\pi }_{12}-{\pi }_{12}^2+{\pi }_{21}-{\pi }_{21}^2+2{\pi }_{12}{\pi }_{11}}{N}\\ & =\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\end{array}$$ $◼$

問題5.6.2：帰無仮説における標本比率の差の分散

帰無仮説 $H_0:{\pi }_{12}={\pi }_{21}=\pi$ における標本比率の差の分散は、 $$\begin{array}{rl}V(p_{12}-p_{21})& =\frac{(\pi +\pi )-{(\pi -\pi )}^2}{N}\\ & =\frac{2\pi }{N}\end{array}$$ $◼$

問題5.6.3, 問題5.6.4：マクネマー検定統計量

標本比率の差の分散の一致推定量は、 $$\begin{array}{r}\hat{V}(p_{12}-p_{21})=\frac{p_{12}+p_{21}}{N}\end{array}$$ これを標準化した値は、 $$\begin{array}{r}\frac{\sqrt{N}(p_{12}-p_{21})}{\sqrt{p_{12}+p_{21}}}=Z_M\sim (0,1)\end{array}$$ また、 $$\begin{array}{rl}{Z}_M& =\frac{N}{N}\cdot \frac{\sqrt{N}(p_{12}-p_{21})}{\sqrt{p_{12}+p_{21}}}\\ & =\frac{N(p_{12}-p_{21})}{\sqrt{N(p_{12}+p_{21})}}\end{array}$$ $N{p}_{12}=f,N{p}_{21}=g$ より、 $$\begin{array}{r}{Z}_M=\frac{f-g}{\sqrt{f+g}}\end{array}$$ これを2乗した値は、 $$\begin{array}{c}\frac{{(f-g)}^2}{f+g}={\chi }_M^2\sim {\chi }^2\left(1\right)\end{array}$$ $◼$

問題5.6.5：対数条件付きオッズ比の漸近分散―四項分布にもとづく考え方

多項分布の周辺分布であることから、 $$\begin{array}{r}e\sim \mathrm{B}(N,{\pi }_{11})\\ f\sim \mathrm{B}(N,{\pi }_{12})\end{array}$$ 二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_{12}\sim N[{\pi }_{12},\frac{{\pi }_{12}(1-{\pi }_{12})}{N}]\\ p_{21}\sim N[{\pi }_{21},\frac{{\pi }_{21}(1-{\pi }_{21})}{N}]\end{array}$$ 標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_{12}\\ p_{21}\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_{12}\\ {\pi }_{21}\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_{12}(1-{\pi }_{12})}{N}& -\frac{{\pi }_{12}{\pi }_{21}}{N}\\ -\frac{{\pi }_{12}{\pi }_{21}}{N}& \frac{{\pi }_{21}(1-{\pi }_{21})}{N}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\log \frac{{\pi }_{12}}{{\pi }_{21}}\\ G\left(\mathit{p}\right)=\hat{\theta }=\log \frac{p_{12}}{p_{21}}\end{array}$$ と変数変換する。 多変量のデルタ法を用いて $G\left(\mathit{p}\right)$ を期待値 $E\left(\mathit{p}\right)=\mathit{\pi }$ まわりでテイラー展開すると、偏導関数ベクトルは、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_2}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\log \frac{{\pi }_{12}}{{\pi }_{21}}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_{12}(1-{\pi }_{12})}{N}& -\frac{{\pi }_{12}{\pi }_{21}}{N}\\ -\frac{{\pi }_{21}{\pi }_{12}}{N}& \frac{{\pi }_{21}(1-{\pi }_{21})}{N}\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{c}\frac{1}{N}\\ -\frac{1}{N}\end{array}\right]\\ & =\frac{1}{N{\pi }_{12}}+\frac{1}{N{\pi }_{21}}\end{array}$$ 母比率 ${\pi }_i$ を一致推定量である標本比率 $p_i$ で置き換えると、漸近分散の一致推定量は、 $$\begin{array}{r}\hat{V}\left[\hat{\theta }\right]=\frac{1}{N{p}_{12}}+\frac{1}{N{p}_{21}}=\frac{1}{f}+\frac{1}{g}\end{array}$$ $◼$

問題5.6.6：対数母集団平均相対リスクの漸近分散―四項分布にもとづく考え方

多項分布の周辺分布であることから、 $$\begin{array}{c}e\sim \mathrm{B}(N,{\pi }_{11})\\ f\sim \mathrm{B}(N,{\pi }_{12})\\ g\sim \mathrm{B}(N,{\pi }_{21})\\ e+f=n_{1\bullet }\sim \mathrm{B}(N,{\pi }_{1\bullet })\\ e+g=n_{\bullet 1}\sim \mathrm{B}(N,{\pi }_{\bullet 1})\end{array}$$ 二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_{11}\sim \mathrm{N}[{\pi }_{11},\frac{{\pi }_{11}(1-{\pi }_{11})}{N}]\\ p_{12}\sim \mathrm{N}[{\pi }_{12},\frac{{\pi }_{12}(1-{\pi }_{12})}{N}]\\ p_{21}\sim \mathrm{N}[{\pi }_{21},\frac{{\pi }_{21}(1-{\pi }_{21})}{N}]\\ p_{1\bullet }\sim \mathrm{N}[{\pi }_{1\bullet },\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}]\\ p_{\bullet 1}\sim \mathrm{N}[{\pi }_{\bullet 1},\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}]\end{array}$$ ここで、和の分散の公式より、 $$\begin{array}{rl}V(p_{12}-p_{21})& =V\left(p_{12}\right)+V\left(p_{21}\right)-2\mathrm{Cov}(p_{12},p_{21})\\ & =\frac{{\pi }_{12}(1-{\pi }_{12})}{N}+\frac{{\pi }_{21}(1-{\pi }_{21})}{N}+\frac{2{\pi }_{12}{\pi }_{21}}{N}\\ & =\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ \text{(1)}& V(p_{1\bullet }-p_{\bullet 1})& =V\left(p_{1\bullet }\right)+V\left(p_{\bullet 1}\right)-2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})\end{array}$$ いっぽう、 $$\begin{array}{rl}V(p_{1\bullet }-p_{\bullet 1})& =V[(p_{11}+p_{12})-(p_{11}+p_{21})]\\ \text{(2)}& & =V(p_{12}-p_{21})\end{array}$$ よって、式 $(1).(2)$ より、 $$\begin{array}{rl}\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}& =\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}+\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}-2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})\\ 2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})& =\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}+\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{{\pi }_{1\bullet }-{\pi }_{1\bullet }^2}{N}+\frac{{\pi }_{\bullet 1}-{\pi }_{\bullet 1}^2}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{({\pi }_{11}+{\pi }_{12})-{({\pi }_{11}+{\pi }_{12})}^2}{N}+\frac{({\pi }_{11}+{\pi }_{21})-{({\pi }_{11}+{\pi }_{21})}^2}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{({\pi }_{11}+{\pi }_{12})-({\pi }_{11}^2+2{\pi }_{11}{\pi }_{12}+{\pi }_{12}^2)}{N}+\frac{({\pi }_{11}+{\pi }_{21})-({\pi }_{11}^2+2{\pi }_{11}{\pi }_{21}+{\pi }_{21}^2)}{N}-\frac{({\pi }_{12}+{\pi }_{21})-({\pi }_{12}^2-2{\pi }_{12}{\pi }_{21}+{\pi }_{21}^2)}{N}& =\frac{2{\pi }_{11}-2\{{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21})+{\pi }_{12}{\pi }_{21}\}}{N}\\ & =\frac{2[{\pi }_{11}-{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21})-{\pi }_{12}{\pi }_{21}]}{N}\\ & =\frac{2[{\pi }_{11}\{1-({\pi }_{11}+{\pi }_{12}+{\pi }_{21})\}-{\pi }_{12}{\pi }_{21}]}{N}& =\frac{2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})}{N}\\ \mathrm{Cov}(p_{1\bullet },p_{\bullet 1})& =\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}\end{array}$$ 標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_{1\bullet }\\ p_{\bullet 1}\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_{1\bullet }\\ {\pi }_{\bullet 1}\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}& \frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}\\ \frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}& \frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\log \frac{{\pi }_{1\bullet }}{{\pi }_{\bullet 1}}\\ G\left(\mathit{p}\right)=\hat{\theta }=\log \frac{p_{1\bullet }}{p_{\bullet 1}}\end{array}$$ と変数変換する。 多変量のデルタ法を用いて $G\left(\mathit{p}\right)$ を期待値 $E\left(\mathit{p}\right)=\mathit{\pi }$ まわりでテイラー展開すると、偏導関数ベクトルは、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_{1\bullet }}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_{\bullet 1}}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_{1\bullet }}\\ -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\log \frac{{\pi }_{1\bullet }}{{\pi }_{\bullet 1}}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\frac{1}{N}\left[\begin{array}{cc}\frac{1}{{\pi }_{1\bullet }}& -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\left[\begin{array}{cc}{\pi }_{1\bullet }(1-{\pi }_{1\bullet })& {\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}\\ {\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}& {\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_{1\bullet }}\\ -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\\ & =\frac{1}{N}\left[\begin{array}{cc}\frac{1}{{\pi }_{1\bullet }}& -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\left[\begin{array}{c}1-{\pi }_{1\bullet }-\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{\bullet 1}}\\ \frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{1\bullet }}-(1-{\pi }_{\bullet 1})\end{array}\right]\\ & =\frac{1}{N}[\frac{1}{{\pi }_{1\bullet }}\{(1-{\pi }_{1\bullet })-\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{\bullet 1}}\}-\frac{1}{{\pi }_{\bullet 1}}\{\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{1\bullet }}-(1-{\pi }_{\bullet 1})\}]\\ & =\frac{1}{N}(\frac{1-{\pi }_{1\bullet }}{{\pi }_{1\bullet }}-\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{1\bullet }{\pi }_{\bullet 1}}-\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{1\bullet }{\pi }_{\bullet 1}}+\frac{1-{\pi }_{\bullet 1}}{{\pi }_{\bullet 1}})\\ & =\frac{{\pi }_{\bullet 1}(1-{\pi }_{1\bullet })-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})+{\pi }_{1\bullet }(1-{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{({\pi }_{\bullet 1}+{\pi }_{1\bullet }-2{\pi }_{1\bullet }{\pi }_{\bullet 1})-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{\bullet 1}+{\pi }_{1\bullet }-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{11}+{\pi }_{12}+{\pi }_{11}+{\pi }_{21}-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1}-{\pi }_{11})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+({\pi }_{11}+{\pi }_{12})({\pi }_{11}+{\pi }_{21})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}{\pi }_{22}+{\pi }_{11}^2+{\pi }_{11}({\pi }_{12}+{\pi }_{21})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21}+{\pi }_{22})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2({\pi }_{11}\cdot 1-{\pi }_{11})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\end{array}$$ 母比率 ${\pi }_i$ を一致推定量である標本比率 $p_i$ で置き換えると、漸近分散の一致推定量は、 $$\begin{array}{r}\hat{V}\left[\hat{\theta }\right]=\frac{p_{12}+p_{21}}{N{p}_{1\bullet }{p}_{\bullet 1}}=\frac{M}{N{n}_{1\bullet }{n}_{\bullet 1}}\end{array}$$ $◼$

参考文献

• ジョン・ラチン 著, 宮岡 悦良 監訳, 遠藤 輝, 黒沢 健, 下川 朝有, 寒水 孝司 訳. 医薬データのための統計解析. 共立出版, 2020, p.259
• ジョン・ラチン 著, 宮岡 悦良 監訳, 遠藤 輝, 黒沢 健, 下川 朝有, 寒水 孝司 訳. 医薬データのための統計解析. 共立出版, 2020, p.227-232

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